https://wiki.seg.org/api.php?action=feedcontributions&user=Zhai0629&feedformat=atomSEG Wiki - User contributions [en]2020-08-13T11:36:53ZUser contributionsMediaWiki 1.31.6https://wiki.seg.org/index.php?title=RMS_amplitude&diff=150628RMS amplitude2020-05-17T05:36:32Z<p>Zhai0629: </p>
<hr />
<div>The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[[A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.|[1]]]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is <math>\tfrac{1}{\sqrt{2}}</math>times the peak amplitude.<sup>[[R.E.Shelf. Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002|[2]]]<br />
[[File:Picture-new.png|thumb|Figure 1 Graph of a sine wave's amplitude vs. time, showing RMS(A<sub>RMS</sub>), peak (A<sub>PK</sub>), and peak-to-peak (A<sub>PP</sub>) amplitude.]]<br />
<br />
=== Mathematical Expression ===<br />
<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[[Kenney, J. F. and Keeping, E. S. "Root Mean Square." §4.15 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton|[3]]]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x1,x2,…,xn}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)，ω is the angular frequency, specifies how many oscillations occur in a unit time interval. The RMS of the sine wave can be calculated as follows <br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
Using the power reduction formula <br />
<br />
<math>sin^2 (\omega t) = \tfrac{1 -cos(2\omega t)}{2}</math><br />
<br />
<br />
The RMS of the sine wave becomes<br />
<br />
<math> Y_{RMS} = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
According to the trigonometric Integral<br />
<br />
<math>\int cos(2\omega t) dt = \tfrac{1}{2\omega} sin(2\omega t)</math><br />
<br />
The RMS can be converted to<br />
<br />
<math>Y_{RMS} = A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
According to the periodicity of the sine function<br />
<br />
<math>sin(2\omega t)|_0^T = sin (2 \tfrac{2\pi}{2} T) = 0</math> <br />
<br />
<br />
Finally, it’s obtained that <br />
<br />
<math>Y_{RMS} = \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS is 0.707 times the peak amplitude <br />
<br />
The RMS over all time of a periodic function is equal to the RMS of one period of the function, its value of a continuous function can be approximated by taking the RMS of a sample consisting of equally spaced observations, thus, the RMS value of various waveforms can also be determined without calculus. <sup>[[Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms"|[4]]]</sup> For instance, the expected value is used instead of the mean, the corresponding formula for a wave’s RMS amplitude defined over a time interval is <br />
<br />
<math>A_{RMS} = \sqrt{\tfrac{\sum_{i=1}^n A_i^2 * w_i }{\sum_{i=1}^n w_i}}</math> <br />
<br />
Where ''A'' = amplitude, ''n'' = number of samples, is the weight of each amplitude value. <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values. when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. <br />
<br />
RMS amplitude analyze the overall amplitude of a signal, conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefor, they are all positive, then the signal average is calculated, eventually followed by the square root operation. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
<br />
=== Seismic Interpretation ===<br />
In seismic processing, automatic gain control (AGC) method is often used in data processing to improve the visibility of seismic data in which attenuation or spherical divergence has caused amplitude decay, which is a system that controls the increase in the amplitude of an electrical signal from the original input to the amplified output, automatically (figure 2).<sup>[[Nanda, Niranjan C.Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production|[5]]]</sup> RMS AGC is used for amplitude equalization. It gives the processor a measure of the overall absolute amplitude in the window, both as positive and as negative values. During the processing, the data processor assume that the average absolute reflectivity varies little over a long time window, so a long-gate AGC can be applied to avoids destroying the lateral variation in amplitude, and ensure that the window include many reflectors so that the target event makes very little contribution to the average amplitude in the window. For instance, RMS AGC is often applied on structural imaging.<br />
[[File:2019-10-23 12-41-48.jpg|thumb|Figure 2 Seismic data without AGC (left) and the same data after AGC is applied (Onajite,2013)]]<br />
For seismic integration, RMS is a most commonly used post stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage.[5] This has splendid applications for various reconnaissance endeavors. The squaring of the amplitude values within the window gives the high amplitudes maximum opportunity to stand out above the background contamination. The RMS amplitude over a large flat or structured time window can be effectively used to identify bright spots at different levels within a formation (Figure 3).<br />
[[File:Picture3.jpg|thumb|Figure 3 Top Balder Formation amplitude map from multiple 3-D surveys covering 5500 sq km of the Faroe Basin west of Shetland in U.K. waters. The colors show RMS amplitude over a 50-ms window surrounding the Top Balder reflection, with red indicating high amplitude and blue indicating low amplitude (Brown,2004)]]<br />
RMS amplitude is popular type of windowed amplitude attributes, the window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view.<sup>[[Alistair R. Brown.Pitfalls in the study of seismic amplitude|[6]]]</sup> In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide.<sup>[[David H. Johnston, Methods and Applications in Reservoir Geophysics|[7]]]</sup> The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009. <br />
<br />
[2] R.E.Shelf. Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
<br />
[3] Kenney, J. F. and Keeping, E. S. "Root Mean Square." §4.15 in ''Mathematics of Statistics, Pt. 1, 3rd ed.'' Princeton, NJ: Van Nostrand, pp. 59-60, 1962.<br />
<br />
[4] Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms" MasteringElectronicsDesign.com. Retrieved 21 January 2015 <br />
<br />
[5] Nanda, Niranjan C.Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production. Springer 2016<br />
<br />
[6] Alistair R. Brown.Pitfalls in the study of seismic amplitude. i''nterpretation'' 6: SL15-SL20. pp20-24,2018<br />
<br />
[7] David H. Johnston, Methods and Applications in Reservoir Geophysics..SEG 2010<br />
<br />
=== '''External links''' ===<br />
[https://csegrecorder.com/articles/view/too-many-seismic-attributes Too many seismic attributes?] <br />
<br />
[https://liquidsdr.org/doc/agc/ Automatic Gain Control (agc)]<br />
<br />
[https://community.sw.siemens.com/s/article/root-mean-square-rms-and-overall-level Root Mean Square (RMS) and Overall Level]</div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=141965RMS amplitude2019-12-05T04:27:20Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[[A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.|[1]]]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is </sup><math>\tfrac{1}{\sqrt{2}}</math>times the peak amplitude.<sup>[[R.E.Shelf. Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002|[2]]]<br />
[[File:Picture-new.png|thumb|Figure 1 Graph of a sine wave's amplitude vs. time, showing RMS(A<sub>RMS</sub>), peak (A<sub>PK</sub>), and peak-to-peak (A<sub>PP</sub>) amplitude.]]<br />
<br />
=== Mathematical Expression ===<br />
<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[[Kenney, J. F. and Keeping, E. S. "Root Mean Square." §4.15 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton|[3]]]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x1,x2,…,xn}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)，ω is the angular frequency, specifies how many oscillations occur in a unit time interval. The RMS of the sine wave can be calculated as follows <br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
Using the power reduction formula <br />
<br />
<math>sin^2 (\omega t) = \tfrac{1 -cos(2\omega t)}{2}</math><br />
<br />
<br />
The RMS of the sine wave becomes<br />
<br />
<math> Y_{RMS} = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
According to the trigonometric Integral<br />
<br />
<math>\int cos(2\omega t) dt = \tfrac{1}{2\omega} sin(2\omega t)</math><br />
<br />
The RMS can be converted to<br />
<br />
<math>Y_{RMS} = A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
According to the periodicity of the sine function<br />
<br />
<math>sin(2\omega t)|_0^T = sin (2 \tfrac{2\pi}{2} T) = 0</math> <br />
<br />
<br />
Finally, it’s obtained that <br />
<br />
<math>Y_{RMS} = \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS is 0.707 times the peak amplitude <br />
<br />
The RMS over all time of a periodic function is equal to the RMS of one period of the function, its value of a continuous function can be approximated by taking the RMS of a sample consisting of equally spaced observations, thus, the RMS value of various waveforms can also be determined without calculus. <sup>[[Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms"|[4]]]</sup> For instance, the expected value is used instead of the mean, the corresponding formula for a wave’s RMS amplitude defined over a time interval is <br />
<br />
<math>A_{RMS} = \sqrt{\tfrac{\sum_{i=1}^n A_i^2 * w_i }{\sum_{i=1}^n w_i}}</math> <br />
<br />
Where ''A'' = amplitude, ''n'' = number of samples, is the weight of each amplitude value. <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values. when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. <br />
<br />
RMS amplitude analyze the overall amplitude of a signal, conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefor, they are all positive, then the signal average is calculated, eventually followed by the square root operation. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
<br />
=== Seismic Interpretation ===<br />
In seismic processing, automatic gain control (AGC) method is often used in data processing to improve the visibility of seismic data in which attenuation or spherical divergence has caused amplitude decay, which is a system that controls the increase in the amplitude of an electrical signal from the original input to the amplified output, automatically (figure 2).<sup>[[Nanda, Niranjan C.Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production|[5]]]</sup> RMS AGC is used for amplitude equalization. It gives the processor a measure of the overall absolute amplitude in the window, both as positive and as negative values. During the processing, the data processor assume that the average absolute reflectivity varies little over a long time window, so a long-gate AGC can be applied to avoids destroying the lateral variation in amplitude, and ensure that the window include many reflectors so that the target event makes very little contribution to the average amplitude in the window. For instance, RMS AGC is often applied on structural imaging.<br />
[[File:2019-10-23 12-41-48.jpg|thumb|Figure 2 Seismic data without AGC (left) and the same data after AGC is applied (Onajite,2013)]]<br />
For seismic integration, RMS is a most commonly used post stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage.[5] This has splendid applications for various reconnaissance endeavors. The squaring of the amplitude values within the window gives the high amplitudes maximum opportunity to stand out above the background contamination. The RMS amplitude over a large flat or structured time window can be effectively used to identify bright spots at different levels within a formation (Figure 3).<br />
[[File:Picture3.jpg|thumb|Figure 3 Top Balder Formation amplitude map from multiple 3-D surveys covering 5500 sq km of the Faroe Basin west of Shetland in U.K. waters. The colors show RMS amplitude over a 50-ms window surrounding the Top Balder reflection, with red indicating high amplitude and blue indicating low amplitude (Brown,2004)]]<br />
RMS amplitude is popular type of windowed amplitude attributes, the window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view.<sup>[[Alistair R. Brown.Pitfalls in the study of seismic amplitude|[6]]]</sup> In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide.<sup>[[David H. Johnston, Methods and Applications in Reservoir Geophysics|[7]]]</sup> The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009. <br />
<br />
[2] R.E.Shelf. Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
<br />
[3] Kenney, J. F. and Keeping, E. S. "Root Mean Square." §4.15 in ''Mathematics of Statistics, Pt. 1, 3rd ed.'' Princeton, NJ: Van Nostrand, pp. 59-60, 1962.<br />
<br />
[4] Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms" MasteringElectronicsDesign.com. Retrieved 21 January 2015 <br />
<br />
[5] Nanda, Niranjan C.Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production. Springer 2016<br />
<br />
[6] Alistair R. Brown.Pitfalls in the study of seismic amplitude. i''nterpretation'' 6: SL15-SL20. pp20-24,2018<br />
<br />
[7] David H. Johnston, Methods and Applications in Reservoir Geophysics..SEG 2010<br />
<br />
=== '''External links''' ===<br />
[https://csegrecorder.com/articles/view/too-many-seismic-attributes Too many seismic attributes?] <br />
<br />
[https://liquidsdr.org/doc/agc/ Automatic Gain Control (agc)]<br />
<br />
[https://community.sw.siemens.com/s/article/root-mean-square-rms-and-overall-level Root Mean Square (RMS) and Overall Level]</div>Zhai0629https://wiki.seg.org/index.php?title=Average_energy&diff=141964Average energy2019-12-05T04:19:07Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018.<br />
<br />
The stacked seismic data volume is commonly used for the interpretation of geologic structure and seismic attributes. The average energy of amplitude is one post-stack attribute that is often used in direct hydrocarbon indicators (DHIs), although its interpretation in thin-layered beds is not necessarily straightforward.<br />
<br />
=== Definition ===<br />
In physics, the wave is a periodic disturbance in a medium or in space. In a progressive wave, energy is transferred from one place to another by the vibrations.<sup>[[A Dictionary of Physics (Sixth Edition.)|[1]]]</sup> Like light waves, sound waves, or transverse oscillations of a string, these disturbances take energy to create and propagate, in order to move the constituent particles or change the electric/magnetic fields. The energy transported by a wave is directly proportional to the square of the amplitude of the wave.<br />
<br />
Seismic wave is an elastic disturbance that is propagated from point to point through a medium.<sup>[[RE Shelf. Encyclopedic Dictionary of Applied Geophysics (Forth Edition)|[2]]]</sup> The average energy attribute of the seismic wave is calculated by adding the square of each sample, then dividing by the number of samples in the window to yield the mean. <sup>[3]</sup><br />
<br />
=== Mathematical Expression ===<br />
Average energy calculates the squared sum of the sample values in the specified time window divided by the number of samples in the window.<br />
<br />
<math>Average Energy = \tfrac{\sum_{i=1}^nA_i^2}{n}</math><br />
<br />
Where A<sub>i</sub> is the amplitude of the sampling point in a given time window, n is the number of sampling points.<br />
<br />
=== '''Physical Description''' ===<br />
The total mechanical energy of the wave is the sum of its kinetic energy and potential energy<sup>[[Landau, L.D.; Lifshitz, E. M. Theory of Elasticity (3rd ed.). Oxford, England 1986|[3]]]</sup>. The energy associated with a traveling wave in a stretched string is conveniently expressed as the energy per wavelength. The energy of a small segment of the string can be expressed as the sum of the kinetic energy and elastic potential energy of the segment (Figure 1).<br />
[[File:AE-energy illustration.jpg|thumb|Figure 1 A sinusoidal wave in the string, each point of the string with mass Δm oscillates at the same frequency as the wave. The total energy of each point is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>).]]<br />
<br />
According to the kinetic energy (E<sub>K</sub>) formula <br />
<br />
<math>E_K = \tfrac{1}{2}mv^2</math><br />
<br />
Where m is the mass and v is the velocity of the body. The mass element oscillates perpendicular to the direction of the motion of the wave. Using the constant linear mass density, the kinetic energy of each mass element of the string with length Δx is<br />
<br />
<math>\vartriangle E_K = \tfrac{1}{2}(\mu\vartriangle x){v_y}^2</math><br />
<br />
Where μ is density of the particle, each mass element of the string has the mass element of the string oscillates with a velocity v<sub>y</sub>. When the mass element of the string approach zero <br />
<br />
<math>dE_K = \lim_{\vartriangle x \to \infty}\tfrac{1}{2}(\mu\vartriangle x){v_y}^2=\tfrac{1}{2}\mu{v_y}^2dx</math><br />
<br />
For a sinusoidal wave with simple harmonic motion, the position of each mass element may be modeled as<br />
<br />
<math>y(x,t)=Asin(nx-\omega t)</math><br />
<br />
<br />
Where ω is the angular frequency in time, ω =2πf. n is the wave number. The velocity of each mass element of the string oscillates is<br />
<br />
<math>v_y=\tfrac{\partial (x,t)}{\partial t} = -A\omega cos(nx-\omega t)</math><br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
<math>dE_K=\tfrac{1}{2}\mu \omega_2A^2cos^2(nx-\omega t)</math><br />
<br />
The wave can be consisting of multiple wavelengths. To standardize the energy, consider the kinetic energy associated with a wavelength of the wave. The kinetic energy can be integrated over the wavelength to find the energy associated with each wavelength of the wave<br />
<br />
<math>\int_{0}^{E_{K\lambda}} dE_K = \int_{0}^{\lambda} \tfrac{1}{2} \mu \omega^2A^2cos^2(nx-\omega t)dx</math><br />
<br />
The energy for a full wavelength can be found by integrating this expression at a given time, and it is most convenient to set t=0 for this integration. The energy for one wavelength along the string is<br />
<br />
<math>\int_{0}^{E_{K\lambda}} dE_K = \tfrac{1}{2}\mu \omega^2A^2\int_{0}^{\lambda} cos^2(nx)dx = \tfrac{1}{2}\mu \omega^2A^2 \int_{0}^{\lambda}\tfrac{1+sin(2nx)}{2}dx</math><br />
<br />
Thus, it’s obtained that<br />
<br />
<math>E_K=\tfrac{1}{2}\mu \omega^2A^2[\tfrac{1}{2}x+\tfrac{1}{4n}sin(2nx)]_0^\lambda = \tfrac{1}{4}\mu \omega^2A^2\lambda</math><br />
<br />
According to the potential energy formula <br />
<br />
<math>E_P=\tfrac{1}{2}kx^2</math><br />
<br />
The simple and harmonic motion with an angular frequency (ω) given by<br />
<br />
<math>\omega=\sqrt{\tfrac{k}{m}}</math><br />
<br />
Where k is the spring constant, m is the mass of the object, bring it into the formula<br />
<br />
<math>E_P=\tfrac{1}{2}m\omega^2x^2</math><br />
<br />
The differential form of the elastic potential energy is<br />
<br />
<math>dE_P=\tfrac{1}{2}\omega^2y^2\mu dx=\tfrac{1}{2}\mu \omega^2A^2sin^2(nx-\omega t)</math><br />
<br />
It’s obtained that<br />
<br />
<math>\int_{0}^{E_{P\lambda}} dE_P=\tfrac{1}{2}\mu \omega^2A^2\int_{0}^{\lambda} sin^2(nx-\omega t)dx=\tfrac{1}{4}\mu \omega^2A^2\lambda</math><br />
<br />
Finally, the total energy associated with a wavelength is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>)<br />
<br />
<math>E_P=E_K+E_P=\tfrac{1}{2}\mu \omega^2A^2\lambda</math><br />
<br />
This shows a wave is an energy transport phenomenon which transports energy along with a medium without transporting matter. The amount of energy transferred by a wave is related to the amplitude of the wave. A high energy wave is featured by a high amplitude; a low energy wave is featured by a low amplitude. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. Assuming that the frequency of the wave is unchanged, an increase in amplitude will result in an increase in energy and a decrease in amplitude will result in a decrease in energy. The energy of a wave is directly proportional to the square of the amplitude it contains: <br />
<br />
<math>E \varpropto A^2</math> <br />
<br />
The energy of the wave depends on both the amplitude and the frequency. If the energy of each wavelength is a discrete packet of energy, a high-frequency wave will deliver more of these packets per unit time than a low-frequency wave. The average rate of energy transfer in mechanical waves is proportional to both the square of the amplitude and the square of the frequency. If two mechanical waves have equal amplitudes, but one wave has a frequency equal to twice the frequency of the other, the higher-frequency wave will have a rate of energy transfer a factor of four times as great as the rate of energy transfer of the lower-frequency wave. A doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave. A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave. And a quadrupling of the amplitude of a wave is indicative of a 16-fold increase in the amount of energy transported by the wave. Thus, whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared (Figure 2). <br />
[[File:AE-AMP.jpg|thumb|Figure 2 The amplitude of a wave is related to the energy which it transport. Wave 1 with wavelength (λ), frequency (f), and amplitude (A); Wave 2 has the same wavelength and frequency as wave 1, but has two times the amplitude, has higher energy; Wave 3 with the shortest wavelength has the greatest number of wavelengths per unit time (the highest frequency). If two waves have the same frequency and speed, the one with the greater amplitude has the higher energy.]] <br />
<br />
Different materials also have differing degrees of elasticity. A more elastic medium will tend to offer less resistance to the force and allow a greater amplitude pulse to travel through it; being less rigid (and therefore more elastic), the same force causes a greater amplitude.<br />
=== '''Seismic Interpretation''' ===<br />
The average energy attribute of seismic waves is a measure of reflectivity in the specified time window. According to the above description, the higher the energy should have the higher the amplitude<sup>[[Landau, L.D.; Lifshitz, E. M. Theory of Elasticity (3rd ed.). Oxford, England 1986|[4]]]</sup>. This attribute enhances lateral variations within seismic events among others (Figure 3), therefore, it’s useful for seismic object detection, for instance, amplitude anomalies, chimney detection, etc. The response energy also characterizes acoustic rock properties and bed thickness.<br />
<br />
The average energy is often found to correlate strongly with liquid saturation (oil/water vs. gas) because those reservoir properties have a strong effect on both velocity and density, and energy of seismic reflections are generated at boundaries where the acoustic impedance (the product of velocity and density) changes.<sup>[[Eservoir Geophysics: Applications|[5]]]</sup> Thus, it’s commonly used in direct hydrocarbon indicators. In general, the values of average energy are not important, and often not cited, because it is the relative value of an attribute along a given horizon or interval that is important. [[File:AE.jpg|thumb|Figure 3 Average energy extraction in a 100ms window (25 samples) at a constant time (Barnes,2006). ]]<br />
<br />
=== '''References''' ===<br />
# A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
# RE Shelf. Encyclopedic Dictionary of Applied Geophysics (Forth Edition), SEG.2002<br />
# Landau, L.D.; Lifshitz, E. M. ''Theory of Elasticity'' (3rd ed.). Oxford, England 1986<br />
# Methods and Applications in Reservoir Geophysics.David H. Johnston, SEG 2010<br />
# Reservoir Geophysics: Applications (SEG Distinguished Instructor Series, No. 11). Willam L. Abriel SEG 2008<br />
<br />
=== '''External links''' ===<br />
# [https://www.physicsclassroom.com/class/waves/Lesson-2/Energy-Transport-and-the-Amplitude-of-a-Wave Properties of a Wave] <br />
# [https://csegrecorder.com/articles/view/too-many-seismic-attributes Too many seismic attributes?] <br />
# [https://www.geoinsights.com/tag/direct-hydrocarbon-indicators/ Direct hydrocarbon indicators]</div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=138576RMS amplitude2019-10-30T16:51:07Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[[A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.|[1]]]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is times the peak amplitude.<sup>[[R.E.Shelf. Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002|[2]]]</sup><math>\tfrac{1}{\sqrt{2}}</math><br />
[[File:Picture-new.png|thumb|Figure 1 Graph of a sine wave's amplitude vs. time, showing RMS(A<sub>RMS</sub>), peak (A<sub>PK</sub>), and peak-to-peak (A<sub>PP</sub>) amplitude.]]<br />
<br />
=== Mathematical Expression ===<br />
<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[[Kenney, J. F. and Keeping, E. S. "Root Mean Square." §4.15 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton|[3]]]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x_1,x_2,…,x_n}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)，ω is the angular frequency, specifies how many oscillations occur in a unit time interval. The RMS of the sine wave can be calculated as follows <br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
Using the power reduction formula <br />
<br />
<math>sin^2 (\omega t) = \tfrac{1 -cos(2\omega t)}{2}</math><br />
<br />
<br />
The RMS of the sine wave becomes<br />
<br />
<math> Y_{RMS} = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
According to the trigonometric Integral<br />
<br />
<math>\int cos(2\omega t) dt = \tfrac{1}{2\omega} sin(2\omega t)</math><br />
<br />
The RMS can be converted to<br />
<br />
<math>Y_{RMS} = A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
According to the periodicity of the sine function<br />
<br />
<math>sin(2\omega t)|_0^T = sin (2 \tfrac{2\pi}{2} T) = 0</math> <br />
<br />
<br />
Finally, it’s obtained that <br />
<br />
<math>Y_{RMS} = \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS is 0.707 times the peak amplitude <br />
<br />
The RMS over all time of a periodic function is equal to the RMS of one period of the function, its value of a continuous function can be approximated by taking the RMS of a sample consisting of equally spaced observations, thus, the RMS value of various waveforms can also be determined without calculus. <sup>[[Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms"|[4]]]</sup> For instance, the expected value is used instead of the mean, the corresponding formula for a wave’s RMS amplitude defined over a time interval is <br />
<br />
<math>A_{RMS} = \sqrt{\tfrac{\sum_{i=1}^n A_i^2 * w_i }{\sum_{i=1}^n w_i}}</math> <br />
<br />
Where ''A'' = amplitude, ''n'' = number of samples, is the weight of each amplitude value. <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values. when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. <br />
<br />
RMS amplitude analyze the overall amplitude of a signal, conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefor, they are all positive, then the signal average is calculated, eventually followed by the square root operation. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
<br />
=== Seismic Interpretation ===<br />
In seismic processing, automatic gain control (AGC) method is often used in data processing to improve the visibility of seismic data in which attenuation or spherical divergence has caused amplitude decay, which is a system that controls the increase in the amplitude of an electrical signal from the original input to the amplified output, automatically (figure 2).<sup>[[Nanda, Niranjan C.Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production|[5]]]</sup> RMS AGC is used for amplitude equalization. It gives the processor a measure of the overall absolute amplitude in the window, both as positive and as negative values. During the processing, the data processor assume that the average absolute reflectivity varies little over a long time window, so a long-gate AGC can be applied to avoids destroying the lateral variation in amplitude, and ensure that the window include many reflectors so that the target event makes very little contribution to the average amplitude in the window. For instance, RMS AGC is often applied on structural imaging.<br />
[[File:2019-10-23 12-41-48.jpg|thumb|Figure 2 Seismic data without AGC (left) and the same data after AGC is applied (Onajite,2013)]]<br />
For seismic integration, RMS is a most commonly used post stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage.[5] This has splendid applications for various reconnaissance endeavors. The squaring of the amplitude values within the window gives the high amplitudes maximum opportunity to stand out above the background contamination. The RMS amplitude over a large flat or structured time window can be effectively used to identify bright spots at different levels within a formation (Figure 3).<br />
[[File:Picture3.jpg|thumb|Figure 3 Top Balder Formation amplitude map from multiple 3-D surveys covering 5500 sq km of the Faroe Basin west of Shetland in U.K. waters. The colors show RMS amplitude over a 50-ms window surrounding the Top Balder reflection, with red indicating high amplitude and blue indicating low amplitude (Brown,2004)]]<br />
RMS amplitude is popular type of windowed amplitude attributes, the window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view.<sup>[[Alistair R. Brown.Pitfalls in the study of seismic amplitude|[6]]]</sup> In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide.<sup>[[David H. Johnston, Methods and Applications in Reservoir Geophysics|[7]]]</sup> The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009. <br />
<br />
[2] R.E.Shelf. Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
<br />
[3] Kenney, J. F. and Keeping, E. S. "Root Mean Square." §4.15 in ''Mathematics of Statistics, Pt. 1, 3rd ed.'' Princeton, NJ: Van Nostrand, pp. 59-60, 1962.<br />
<br />
[4] Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms" MasteringElectronicsDesign.com. Retrieved 21 January 2015 <br />
<br />
[5] Nanda, Niranjan C.Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production. Springer 2016<br />
<br />
[6] Alistair R. Brown.Pitfalls in the study of seismic amplitude. i''nterpretation'' 6: SL15-SL20. pp20-24,2018<br />
<br />
[7] David H. Johnston, Methods and Applications in Reservoir Geophysics..SEG 2010<br />
<br />
=== '''External links''' ===<br />
[https://csegrecorder.com/articles/view/too-many-seismic-attributes Too many seismic attributes?] <br />
<br />
[https://liquidsdr.org/doc/agc/ Automatic Gain Control (agc)]<br />
<br />
[https://community.sw.siemens.com/s/article/root-mean-square-rms-and-overall-level Root Mean Square (RMS) and Overall Level]</div>Zhai0629https://wiki.seg.org/index.php?title=Average_energy&diff=138575Average energy2019-10-30T16:35:28Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018.<br />
<br />
The stacked seismic data volume is commonly used for the interpretation of geologic structure and seismic attributes. The average energy of amplitude is one post-stack attribute that is often used in direct hydrocarbon indicators (DHIs), although its interpretation in thin-layered beds is not necessarily straightforward.<br />
<br />
=== Definition ===<br />
In physics, the wave is a periodic disturbance in a medium or in space. In a progressive wave, energy is transferred from one place to another by the vibrations.<sup>[[A Dictionary of Physics (Sixth Edition.)|[1]]]</sup> Like light waves, sound waves, or transverse oscillations of a string, these disturbances take energy to create and propagate, in order to move the constituent particles or change the electric/magnetic fields. The energy transported by a wave is directly proportional to the square of the amplitude of the wave.<br />
<br />
Seismic wave is an elastic disturbance that is propagated from point to point through a medium.<sup>[[RE Shelf. Encyclopedic Dictionary of Applied Geophysics (Forth Edition)|[2]]]</sup> The average energy attribute of the seismic wave is calculated by adding the square of each sample, then dividing by the number of samples in the window to yield the mean. <sup>[3]</sup><br />
<br />
=== Mathematical Expression ===<br />
This attribute calculates the squared sum of the sample values in the specified time window divided by the number of samples in the window.<br />
<br />
<math>Average Energy = \tfrac{\sum_{i=1}^nA_i^2}{n}</math><br />
<br />
Where A<sub>i</sub> is the amplitude of the sampling point in a given time window, n is the number of sampling points.<br />
<br />
=== '''Physical Description''' ===<br />
The total mechanical energy of the wave is the sum of its kinetic energy and potential energy<sup>[[Landau, L.D.; Lifshitz, E. M. Theory of Elasticity (3rd ed.). Oxford, England 1986|[3]]]</sup>. The energy associated with a traveling wave in a stretched string is conveniently expressed as the energy per wavelength. The energy of a small segment of the string can be expressed as the sum of the kinetic energy and elastic potential energy of the segment (Figure 1).<br />
[[File:AE-energy illustration.jpg|thumb|Figure 1 A sinusoidal wave in the string, each point of the string with mass Δm oscillates at the same frequency as the wave. The total energy of each point is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>).]]<br />
<br />
According to the kinetic energy (E<sub>K</sub>) formula <br />
<br />
<math>E_K = \tfrac{1}{2}mv^2</math><br />
<br />
Where m is the mass and v is the velocity of the body. The mass element oscillates perpendicular to the direction of the motion of the wave. Using the constant linear mass density, the kinetic energy of each mass element of the string with length Δx is<br />
<br />
<math>\vartriangle E_K = \tfrac{1}{2}(\mu\vartriangle x){v_y}^2</math><br />
<br />
Where μ is density of the particle, each mass element of the string has the mass element of the string oscillates with a velocity v<sub>y</sub>. When the mass element of the string approach zero <br />
<br />
<math>dE_K = \lim_{\vartriangle x \to \infty}\tfrac{1}{2}(\mu\vartriangle x){v_y}^2=\tfrac{1}{2}\mu{v_y}^2dx</math><br />
<br />
For a sinusoidal wave with simple harmonic motion, the position of each mass element may be modeled as<br />
<br />
<math>y(x,t)=Asin(nx-\omega t)</math><br />
<br />
<br />
Where ω is the angular frequency in time, ω =2πf. n is the wave number. The velocity of each mass element of the string oscillates is<br />
<br />
<math>v_y=\tfrac{\partial (x,t)}{\partial t} = -A\omega cos(nx-\omega t)</math><br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
<math>dE_K=\tfrac{1}{2}\mu \omega_2A^2cos^2(nx-\omega t)</math><br />
<br />
The wave can be consisting of multiple wavelengths. To standardize the energy, consider the kinetic energy associated with a wavelength of the wave. The kinetic energy can be integrated over the wavelength to find the energy associated with each wavelength of the wave<br />
<br />
<math>\int_{0}^{E_{K\lambda}} dE_K = \int_{0}^{\lambda} \tfrac{1}{2} \mu \omega^2A^2cos^2(nx-\omega t)dx</math><br />
<br />
The energy for a full wavelength can be found by integrating this expression at a given time, and it is most convenient to set t=0 for this integration. The energy for one wavelength along the string is<br />
<br />
<math>\int_{0}^{E_{K\lambda}} dE_K = \tfrac{1}{2}\mu \omega^2A^2\int_{0}^{\lambda} cos^2(nx)dx = \tfrac{1}{2}\mu \omega^2A^2 \int_{0}^{\lambda}\tfrac{1+sin(2nx)}{2}dx</math><br />
<br />
Thus, it’s obtained that<br />
<br />
<math>E_K=\tfrac{1}{2}\mu \omega^2A^2[\tfrac{1}{2}x+\tfrac{1}{4n}sin(2nx)]_0^\lambda = \tfrac{1}{4}\mu \omega^2A^2\lambda</math><br />
<br />
According to the potential energy formula <br />
<br />
<math>E_P=\tfrac{1}{2}kx^2</math><br />
<br />
The simple and harmonic motion with an angular frequency (ω) given by<br />
<br />
<math>\omega=\sqrt{\tfrac{k}{m}}</math><br />
<br />
Where k is the spring constant, m is the mass of the object, bring it into the formula<br />
<br />
<math>E_P=\tfrac{1}{2}m\omega^2x^2</math><br />
<br />
The differential form of the elastic potential energy is<br />
<br />
<math>dE_P=\tfrac{1}{2}\omega^2y^2\mu dx=\tfrac{1}{2}\mu \omega^2A^2sin^2(nx-\omega t)</math><br />
<br />
It’s obtained that<br />
<br />
<math>\int_{0}^{E_{P\lambda}} dE_P=\tfrac{1}{2}\mu \omega^2A^2\int_{0}^{\lambda} sin^2(nx-\omega t)dx=\tfrac{1}{4}\mu \omega^2A^2\lambda</math><br />
<br />
Finally, the total energy associated with a wavelength is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>)<br />
<br />
<math>E_P=E_K+E_P=\tfrac{1}{2}\mu \omega^2A^2\lambda</math><br />
<br />
This shows a wave is an energy transport phenomenon which transports energy along with a medium without transporting matter. The amount of energy transferred by a wave is related to the amplitude of the wave. A high energy wave is featured by a high amplitude; a low energy wave is featured by a low amplitude. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. Assuming that the frequency of the wave is unchanged, an increase in amplitude will result in an increase in energy and a decrease in amplitude will result in a decrease in energy. The energy of a wave is directly proportional to the square of the amplitude it contains: <br />
<br />
<math>E \varpropto A^2</math> <br />
<br />
The energy of the wave depends on both the amplitude and the frequency. If the energy of each wavelength is a discrete packet of energy, a high-frequency wave will deliver more of these packets per unit time than a low-frequency wave. The average rate of energy transfer in mechanical waves is proportional to both the square of the amplitude and the square of the frequency. If two mechanical waves have equal amplitudes, but one wave has a frequency equal to twice the frequency of the other, the higher-frequency wave will have a rate of energy transfer a factor of four times as great as the rate of energy transfer of the lower-frequency wave. A doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave. A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave. And a quadrupling of the amplitude of a wave is indicative of a 16-fold increase in the amount of energy transported by the wave. Thus, whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared (Figure 2). <br />
[[File:AE-AMP.jpg|thumb|Figure 2 The amplitude of a wave is related to the energy which it transport. Wave 1 with wavelength (λ), frequency (f), and amplitude (A); Wave 2 has the same wavelength and frequency as wave 1, but has two times the amplitude, has higher energy; Wave 3 with the shortest wavelength has the greatest number of wavelengths per unit time (the highest frequency). If two waves have the same frequency and speed, the one with the greater amplitude has the higher energy.]] <br />
<br />
Different materials also have differing degrees of elasticity. A more elastic medium will tend to offer less resistance to the force and allow a greater amplitude pulse to travel through it; being less rigid (and therefore more elastic), the same force causes a greater amplitude.<br />
=== '''Seismic Interpretation''' ===<br />
The average energy attribute of seismic waves is a measure of reflectivity in the specified time window. According to the above description, the higher the energy should have the higher the amplitude<sup>[[Landau, L.D.; Lifshitz, E. M. Theory of Elasticity (3rd ed.). Oxford, England 1986|[4]]]</sup>. This attribute enhances lateral variations within seismic events among others (Figure 3), therefore, it’s useful for seismic object detection, for instance, amplitude anomalies, chimney detection, etc. The response energy also characterizes acoustic rock properties and bed thickness.<br />
<br />
The average energy is often found to correlate strongly with liquid saturation (oil/water vs. gas) because those reservoir properties have a strong effect on both velocity and density, and energy of seismic reflections are generated at boundaries where the acoustic impedance (the product of velocity and density) changes.<sup>[[Eservoir Geophysics: Applications|[5]]]</sup> Thus, it’s commonly used in direct hydrocarbon indicators. In general, the values of average energy are not important, and often not cited, because it is the relative value of an attribute along a given horizon or interval that is important. [[File:AE.jpg|thumb|Figure 3 Average energy extraction in a 100ms window (25 samples) at a constant time (Barnes,2006). ]]<br />
<br />
=== '''References''' ===<br />
# A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
# RE Shelf. Encyclopedic Dictionary of Applied Geophysics (Forth Edition), SEG.2002<br />
# Landau, L.D.; Lifshitz, E. M. ''Theory of Elasticity'' (3rd ed.). Oxford, England 1986<br />
# Methods and Applications in Reservoir Geophysics.David H. Johnston, SEG 2010<br />
# Reservoir Geophysics: Applications (SEG Distinguished Instructor Series, No. 11). Willam L. Abriel SEG 2008<br />
<br />
=== '''External links''' ===<br />
# [https://www.physicsclassroom.com/class/waves/Lesson-2/Energy-Transport-and-the-Amplitude-of-a-Wave Properties of a Wave] <br />
# [https://csegrecorder.com/articles/view/too-many-seismic-attributes Too many seismic attributes?] <br />
# [https://www.geoinsights.com/tag/direct-hydrocarbon-indicators/ Direct hydrocarbon indicators]</div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=138574RMS amplitude2019-10-30T16:28:43Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[[A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.|[1]]]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is times the peak amplitude.<sup>[[R.E.Shelf. Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002|[2]]]</sup><math>\tfrac{1}{\sqrt{2}}</math><br />
[[File:Picture-new.png|thumb|Figure 1 Graph of a sine wave's amplitude vs. time, showing RMS(A<sub>RMS</sub>), peak (A<sub>PK</sub>), and peak-to-peak (A<sub>PP</sub>) amplitude.]]<br />
<br />
=== Mathematical Expression ===<br />
<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[[Kenney, J. F. and Keeping, E. S. "Root Mean Square." §4.15 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton|[3]]]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x_1,x_2,…,x_n}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)，ω is the angular frequency, specifies how many oscillations occur in a unit time interval. The RMS of the sine wave can be calculated as follows <br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
Using the power reduction formula <br />
<br />
<math>sin^2 (\omega t) = \tfrac{1 -cos(2\omega t)}{2}</math><br />
<br />
<br />
The RMS of the sine wave becomes<br />
<br />
<math> = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
According to the trigonometric Integral<br />
<br />
<math>\int cos(2\omega t) dt = \tfrac{1}{2\omega} sin(2\omega t)</math><br />
<br />
The RMS can be converted to<br />
<br />
<math>Y_{RMS} = A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
According to the periodicity of the sine function<br />
<br />
<math>sin(2\omega t)|_0^T = sin (2 \tfrac{2\pi}{2} T) = 0</math> <br />
<br />
<br />
Finally, it’s obtained that <br />
<br />
<math>Y_{RMS} = \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS is 0.707 times the peak amplitude <br />
<br />
The RMS over all time of a periodic function is equal to the RMS of one period of the function, its value of a continuous function can be approximated by taking the RMS of a sample consisting of equally spaced observations, thus, the RMS value of various waveforms can also be determined without calculus. <sup>[[Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms"|[4]]]</sup> For instance, the expected value is used instead of the mean, the corresponding formula for a wave’s RMS amplitude defined over a time interval is <br />
<br />
<math>A_{RMS} = \sqrt{\tfrac{\sum_{i=1}^n A_i^2 * w_i }{\sum_{i=1}^n w_i}}</math> <br />
<br />
Where ''A'' = amplitude, ''n'' = number of samples, is the weight of each amplitude value. <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values. when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. <br />
<br />
RMS amplitude analyze the overall amplitude of a signal, conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefor, they are all positive, then the signal average is calculated, eventually followed by the square root operation. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
<br />
=== Seismic Interpretation ===<br />
In seismic processing, automatic gain control (AGC) method is often used in data processing to improve the visibility of seismic data in which attenuation or spherical divergence has caused amplitude decay, which is a system that controls the increase in the amplitude of an electrical signal from the original input to the amplified output, automatically (figure 2).<sup>[[Nanda, Niranjan C.Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production|[5]]]</sup> RMS AGC is used for amplitude equalization. It gives the processor a measure of the overall absolute amplitude in the window, both as positive and as negative values. During the processing, the data processor assume that the average absolute reflectivity varies little over a long time window, so a long-gate AGC can be applied to avoids destroying the lateral variation in amplitude, and ensure that the window include many reflectors so that the target event makes very little contribution to the average amplitude in the window. For instance, RMS AGC is often applied on structural imaging.<br />
[[File:2019-10-23 12-41-48.jpg|thumb|Figure 2 Seismic data without AGC (left) and the same data after AGC is applied (Onajite,2013)]]<br />
For seismic integration, RMS is a most commonly used post stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage.[5] This has splendid applications for various reconnaissance endeavors. The squaring of the amplitude values within the window gives the high amplitudes maximum opportunity to stand out above the background contamination. The RMS amplitude over a large flat or structured time window can be effectively used to identify bright spots at different levels within a formation (Figure 3).<br />
[[File:Picture3.jpg|thumb|Figure 3 Top Balder Formation amplitude map from multiple 3-D surveys covering 5500 sq km of the Faroe Basin west of Shetland in U.K. waters. The colors show RMS amplitude over a 50-ms window surrounding the Top Balder reflection, with red indicating high amplitude and blue indicating low amplitude (Brown,2004)]]<br />
RMS amplitude is popular type of windowed amplitude attributes, the window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view.<sup>[[Alistair R. Brown.Pitfalls in the study of seismic amplitude|[6]]]</sup> In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide.<sup>[[David H. Johnston, Methods and Applications in Reservoir Geophysics|[7]]]</sup> The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009. <br />
<br />
[2] R.E.Shelf. Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
<br />
[3] Kenney, J. F. and Keeping, E. S. "Root Mean Square." §4.15 in ''Mathematics of Statistics, Pt. 1, 3rd ed.'' Princeton, NJ: Van Nostrand, pp. 59-60, 1962.<br />
<br />
[4] Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms" MasteringElectronicsDesign.com. Retrieved 21 January 2015 <br />
<br />
[5] Nanda, Niranjan C.Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production. Springer 2016<br />
<br />
[6] Alistair R. Brown.Pitfalls in the study of seismic amplitude. i''nterpretation'' 6: SL15-SL20. pp20-24,2018<br />
<br />
[7] David H. Johnston, Methods and Applications in Reservoir Geophysics..SEG 2010<br />
<br />
=== '''External links''' ===<br />
[https://csegrecorder.com/articles/view/too-many-seismic-attributes Too many seismic attributes?] <br />
<br />
[https://liquidsdr.org/doc/agc/ Automatic Gain Control (agc)]<br />
<br />
[https://community.sw.siemens.com/s/article/root-mean-square-rms-and-overall-level Root Mean Square (RMS) and Overall Level]</div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=138573RMS amplitude2019-10-30T16:20:29Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[1]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is times the peak amplitude.<sup>[2]</sup><math>\tfrac{1}{\sqrt{2}}</math><br />
[[File:Picture-new.png|thumb|Figure 1 Graph of a sine wave's amplitude vs. time, showing RMS(A<sub>RMS</sub>), peak (A<sub>PK</sub>), and peak-to-peak (A<sub>PP</sub>) amplitude.]]<br />
<br />
=== Mathematical Expression ===<br />
<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[3]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x_1,x_2,…,x_n}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)，ω is the angular frequency, specifies how many oscillations occur in a unit time interval. The RMS of the sine wave can be calculated as follows <br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
Using the power reduction formula <br />
<br />
<math>sin^2 (\omega t) = \tfrac{1 -cos(2\omega t)}{2}</math><br />
<br />
<br />
The RMS of the sine wave becomes<br />
<br />
<math> = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
According to the trigonometric Integral<br />
<br />
<math>\int cos(2\omega t) dt = \tfrac{1}{2\omega} sin(2\omega t)</math><br />
<br />
The RMS can be converted to<br />
<br />
<math>Y_{RMS} = A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
According to the periodicity of the sine function<br />
<br />
<math>sin(2\omega t)|_0^T = sin (2 \tfrac{2\pi}{2} T) = 0</math> <br />
<br />
<br />
Finally, it’s obtained that <br />
<br />
<math>Y_{RMS} = \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS is 0.707 times the peak amplitude <br />
<br />
The RMS over all time of a periodic function is equal to the RMS of one period of the function, its value of a continuous function can be approximated by taking the RMS of a sample consisting of equally spaced observations, thus, the RMS value of various waveforms can also be determined without calculus. <sup>[3]</sup> For instance, the expected value is used instead of the mean, the corresponding formula for a wave’s RMS amplitude defined over a time interval is <br />
<br />
<math>A_{RMS} = \sqrt{\tfrac{\sum_{i=1}^n A_i^2 * w_i }{\sum_{i=1}^n w_i}}</math> <br />
<br />
Where ''A'' = amplitude, ''n'' = number of samples, is the weight of each amplitude value. <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values. when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. <br />
<br />
RMS amplitude analyze the overall amplitude of a signal, conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefor, they are all positive, then the signal average is calculated, eventually followed by the square root operation. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
<br />
=== Seismic Interpretation ===<br />
In seismic processing, automatic gain control (AGC) method is often used in data processing to improve the visibility of seismic data in which attenuation or spherical divergence has caused amplitude decay, which is a system that controls the increase in the amplitude of an electrical signal from the original input to the amplified output, automatically (figure 2).[4] RMS AGC is used for amplitude equalization. It gives the processor a measure of the overall absolute amplitude in the window, both as positive and as negative values. During the processing, the data processor assume that the average absolute reflectivity varies little over a long time window, so a long-gate AGC can be applied to avoids destroying the lateral variation in amplitude, and ensure that the window include many reflectors so that the target event makes very little contribution to the average amplitude in the window. For instance, RMS AGC is often applied on structural imaging.<br />
[[File:2019-10-23 12-41-48.jpg|thumb|Figure 2 Seismic data without AGC (left) and the same data after AGC is applied (Onajite,2013)]]<br />
For seismic integration, RMS is a most commonly used post stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage.[5] This has splendid applications for various reconnaissance endeavors. The squaring of the amplitude values within the window gives the high amplitudes maximum opportunity to stand out above the background contamination. The RMS amplitude over a large flat or structured time window can be effectively used to identify bright spots at different levels within a formation (Figure3).<br />
[[File:Picture3.jpg|thumb|Figure 3 Top Balder Formation amplitude map from multiple 3-D surveys covering 5500 sq km of the Faroe Basin west of Shetland in U.K. waters. The colors show RMS amplitude over a 50-ms window surrounding the Top Balder reflection, with red indicating high amplitude and blue indicating low amplitude (Brown,2004)]]<br />
RMS amplitude is popular type of windowed amplitude attributes, the window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view. In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide. The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009. <br />
<br />
[2] R.E.Shelf. Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
<br />
[3] Kenney, J. F. and Keeping, E. S. "Root Mean Square." §4.15 in ''Mathematics of Statistics, Pt. 1, 3rd ed.'' Princeton, NJ: Van Nostrand, pp. 59-60, 1962.<br />
<br />
[4] Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms" MasteringElectronicsDesign.com. Retrieved 21 January 2015 <br />
<br />
[5] Nanda, Niranjan C.Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production. Springer 2016<br />
<br />
[6] Alistair R. Brown.Pitfalls in the study of seismic amplitude. i''nterpretation'' 6: SL15-SL20. pp20-24,2018<br />
<br />
[7] David H. Johnston, Methods and Applications in Reservoir Geophysics..SEG 2010<br />
<br />
=== '''External links''' ===<br />
[https://csegrecorder.com/articles/view/too-many-seismic-attributes Too many seismic attributes?] <br />
<br />
[https://liquidsdr.org/doc/agc/ Automatic Gain Control (agc)]<br />
<br />
[https://community.sw.siemens.com/s/article/root-mean-square-rms-and-overall-level Root Mean Square (RMS) and Overall Level]</div>Zhai0629https://wiki.seg.org/index.php?title=Average_energy&diff=138572Average energy2019-10-30T16:10:36Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018.<br />
<br />
The stacked seismic data volume is commonly used for the interpretation of geologic structure and seismic attributes. The average energy of amplitude is one post-stack attribute that is often used in direct hydrocarbon indicators (DHIs), although its interpretation in thin-layered beds is not necessarily straightforward.<br />
<br />
=== Definition ===<br />
In physics, the wave is a periodic disturbance in a medium or in space. In a progressive wave, energy is transferred from one place to another by the vibrations.<sup>[1]</sup> Like light waves, sound waves, or transverse oscillations of a string, these disturbances take energy to create and propagate, in order to move the constituent particles or change the electric/magnetic fields. The energy transported by a wave is directly proportional to the square of the amplitude of the wave.<br />
<br />
Seismic wave is an elastic disturbance that is propagated from point to point through a medium.<sup>[2]</sup> The average energy attribute of the seismic wave is calculated by adding the square of each sample, then dividing by the number of samples in the window to yield the mean. <sup>[3]</sup><br />
<br />
=== Mathematical Expression ===<br />
This attribute calculates the squared sum of the sample values in the specified time window divided by the number of samples in the window.<br />
<br />
<math>Average Energy = \tfrac{\sum_{i=1}^nA_i^2}{n}</math><br />
<br />
Where A<sub>i</sub> is the amplitude of the sampling point in a given time window, n is the number of sampling points.<br />
<br />
=== '''Physical Description''' ===<br />
The total mechanical energy of the wave is the sum of its kinetic energy and potential energy. The energy associated with a traveling wave in a stretched string is conveniently expressed as the energy per wavelength. The energy of a small segment of the string can be expressed as the sum of the kinetic energy and elastic potential energy of the segment (Figure 1).<br />
[[File:AE-energy illustration.jpg|thumb|Figure 1 A sinusoidal wave in the string, each point of the string with mass Δm oscillates at the same frequency as the wave. The total energy of each point is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>).]]<br />
<br />
According to the kinetic energy (E<sub>K</sub>) formula <br />
<br />
<math>E_K = \tfrac{1}{2}mv^2</math><br />
<br />
Where m is the mass and v is the velocity of the body. The mass element oscillates perpendicular to the direction of the motion of the wave. Using the constant linear mass density, the kinetic energy of each mass element of the string with length Δx is<br />
<br />
<math>\vartriangle E_K = \tfrac{1}{2}(\mu\vartriangle x){v_y}^2</math><br />
<br />
Where μ is density of the particle, each mass element of the string has the mass element of the string oscillates with a velocity v<sub>y</sub>. When the mass element of the string approach zero <br />
<br />
<math>dE_K = \lim_{\vartriangle x \to \infty}\tfrac{1}{2}(\mu\vartriangle x){v_y}^2=\tfrac{1}{2}\mu{v_y}^2dx</math><br />
<br />
For a sinusoidal wave with simple harmonic motion, the position of each mass element may be modeled as<br />
<br />
<math>y(x,t)=Asin(nx-\omega t)</math><br />
<br />
<br />
Where ω is the angular frequency in time, ω =2πf. n is the wave number. The velocity of each mass element of the string oscillates is<br />
<br />
<math>v_y=\tfrac{\partial (x,t)}{\partial t} = -A\omega cos(nx-\omega t)</math><br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
<math>dE_K=\tfrac{1}{2}\mu \omega_2A^2cos^2(nx-\omega t)</math><br />
<br />
The wave can be consisting of multiple wavelengths. To standardize the energy, consider the kinetic energy associated with a wavelength of the wave. The kinetic energy can be integrated over the wavelength to find the energy associated with each wavelength of the wave<br />
<br />
<math>\int_{0}^{E_{K\lambda}} dE_K = \int_{0}^{\lambda} \tfrac{1}{2} \mu \omega^2A^2cos^2(nx-\omega t)dx</math><br />
<br />
The energy for a full wavelength can be found by integrating this expression at a given time, and it is most convenient to set t=0 for this integration. The energy for one wavelength along the string is<br />
<br />
<math>\int_{0}^{E_{K\lambda}} dE_K = \tfrac{1}{2}\mu \omega^2A^2\int_{0}^{\lambda} cos^2(nx)dx = \tfrac{1}{2}\mu \omega^2A^2 \int_{0}^{\lambda}\tfrac{1+sin(2nx)}{2}dx</math><br />
<br />
Thus, it’s obtained that<br />
<br />
<math>E_K=\tfrac{1}{2}\mu \omega^2A^2[\tfrac{1}{2}x+\tfrac{1}{4n}sin(2nx)]_0^\lambda = \tfrac{1}{4}\mu \omega^2A^2\lambda</math><br />
<br />
According to the potential energy formula <br />
<br />
<math>E_P=\tfrac{1}{2}kx^2</math><br />
<br />
The simple and harmonic motion with an angular frequency (ω) given by<br />
<br />
<math>\omega=\sqrt{\tfrac{k}{m}}</math><br />
<br />
Where k is the spring constant, m is the mass of the object, bring it into the formula<br />
<br />
<math>E_P=\tfrac{1}{2}m\omega^2x^2</math><br />
<br />
The differential form of the elastic potential energy is<br />
<br />
<math>dE_P=\tfrac{1}{2}\omega^2y^2\mu dx=\tfrac{1}{2}\mu \omega^2A^2sin^2(nx-\omega t)</math><br />
<br />
It’s obtained that<br />
<br />
<math>\int_{0}^{E_{P\lambda}} dE_P=\tfrac{1}{2}\mu \omega^2A^2\int_{0}^{\lambda} sin^2(nx-\omega t)dx=\tfrac{1}{4}\mu \omega^2A^2\lambda</math><br />
<br />
Finally, the total energy associated with a wavelength is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>)<br />
<br />
<math>E_P=E_K+E_P=\tfrac{1}{2}\mu \omega^2A^2\lambda</math><br />
<br />
This shows a wave is an energy transport phenomenon which transports energy along with a medium without transporting matter. The amount of energy transferred by a wave is related to the amplitude of the wave. A high energy wave is featured by a high amplitude; a low energy wave is featured by a low amplitude. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. Assuming that the frequency of the wave is unchanged, an increase in amplitude will result in an increase in energy and a decrease in amplitude will result in a decrease in energy. The energy of a wave is directly proportional to the square of the amplitude it contains: <br />
<br />
<math>E \varpropto A^2</math> <br />
<br />
The energy of the wave depends on both the amplitude and the frequency. If the energy of each wavelength is a discrete packet of energy, a high-frequency wave will deliver more of these packets per unit time than a low-frequency wave. The average rate of energy transfer in mechanical waves is proportional to both the square of the amplitude and the square of the frequency. If two mechanical waves have equal amplitudes, but one wave has a frequency equal to twice the frequency of the other, the higher-frequency wave will have a rate of energy transfer a factor of four times as great as the rate of energy transfer of the lower-frequency wave. A doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave. A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave. And a quadrupling of the amplitude of a wave is indicative of a 16-fold increase in the amount of energy transported by the wave. Thus, whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared (Figure 2). <br />
[[File:AE-AMP.jpg|thumb|Figure 2 The amplitude of a wave is related to the energy which it transport. Wave 1 with wavelength (λ), frequency (f), and amplitude (A); Wave 2 has the same wavelength and frequency as wave 1, but has two times the amplitude, has higher energy; Wave 3 with the shortest wavelength has the greatest number of wavelengths per unit time (the highest frequency). If two waves have the same frequency and speed, the one with the greater amplitude has the higher energy.]] <br />
<br />
Different materials also have differing degrees of elasticity. A more elastic medium will tend to offer less resistance to the force and allow a greater amplitude pulse to travel through it; being less rigid (and therefore more elastic), the same force causes a greater amplitude.<br />
=== '''Seismic Interpretation''' ===<br />
The average energy attribute of seismic waves is a measure of reflectivity in the specified time window. According to the above description, the higher the energy should have the higher the amplitude. This attribute enhances lateral variations within seismic events among others (Figure 3), therefore, it’s useful for seismic object detection, for instance, amplitude anomalies, chimney detection, etc. The response energy also characterizes acoustic rock properties and bed thickness.<br />
<br />
The average energy is often found to correlate strongly with liquid saturation (oil/water vs. gas) because those reservoir properties have a strong effect on both velocity and density, and energy of seismic reflections are generated at boundaries where the acoustic impedance (the product of velocity and density) changes, thus, it’s commonly used in direct hydrocarbon indicators. In general, the values of average energy are not important, and often not cited, because it is the relative value of an attribute along a given horizon or interval that is important. [[File:AE.jpg|thumb|Figure 3 Average energy extraction in a 100ms window (25 samples) at a constant time (Barnes,2006). ]]<br />
<br />
=== '''References''' ===<br />
# A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
# Encyclopedic Dictionary of Applied Geophysics (Forth Edition).RE Shelf, SEG.2002<br />
# Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production. N.C. Nanda, SEG 2007<br />
# Methods and Applications in Reservoir Geophysics.David H. Johnston, SEG 2010<br />
# Reservoir Geophysics: Applications (SEG Distinguished Instructor Series, No. 11). Willam L. Abriel SEG 2008<br />
<br />
=== '''External links''' ===<br />
# [https://www.physicsclassroom.com/class/waves/Lesson-2/Energy-Transport-and-the-Amplitude-of-a-Wave Properties of a Wave] <br />
# [https://csegrecorder.com/articles/view/too-many-seismic-attributes Too many seismic attributes?] <br />
# [https://www.geoinsights.com/tag/direct-hydrocarbon-indicators/ Direct hydrocarbon indicators]</div>Zhai0629https://wiki.seg.org/index.php?title=Average_energy&diff=138571Average energy2019-10-30T16:05:33Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018.<br />
<br />
The stacked seismic data volume is commonly used for the interpretation of geologic structure and seismic attributes. The average energy of amplitude is one post-stack attribute that is often used in direct hydrocarbon indicators (DHIs), although its interpretation in thin-layered beds is not necessarily straightforward.<br />
<br />
=== Definition ===<br />
In physics, the wave is a periodic disturbance in a medium or in space. In a progressive wave, energy is transferred from one place to another by the vibrations.<sup>[1]</sup> Like light waves, sound waves, or transverse oscillations of a string, these disturbances take energy to create and propagate, in order to move the constituent particles or change the electric/magnetic fields. The energy transported by a wave is directly proportional to the square of the amplitude of the wave.<br />
<br />
Seismic wave is an elastic disturbance that is propagated from point to point through a medium.<sup>[2]</sup> The average energy attribute of the seismic wave is calculated by adding the square of each sample, then dividing by the number of samples in the window to yield the mean. <sup>[3]</sup><br />
<br />
=== Mathematical Expression ===<br />
This attribute calculates the squared sum of the sample values in the specified time window divided by the number of samples in the window.<br />
<br />
<math>Average Energy = \tfrac{\sum_{i=1}^nA_i^2}{n}</math><br />
<br />
Where A<sub>i</sub> is the amplitude of the sampling point in a given time window, n is the number of sampling points.<br />
<br />
=== '''Physical Description''' ===<br />
The total mechanical energy of the wave is the sum of its kinetic energy and potential energy. The energy associated with a traveling wave in a stretched string is conveniently expressed as the energy per wavelength. The energy of a small segment of the string can be expressed as the sum of the kinetic energy and elastic potential energy of the segment (Figure 1).<br />
[[File:AE-energy illustration.jpg|thumb|Figure 1 A sinusoidal wave in the string, each point of the string with mass Δm oscillates at the same frequency as the wave. The total energy of each point is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>).]]<br />
<br />
According to the kinetic energy (E<sub>K</sub>) formula <br />
<br />
<math>E_K = \tfrac{1}{2}mv^2</math><br />
<br />
Where m is the mass and v is the velocity of the body. The mass element oscillates perpendicular to the direction of the motion of the wave. Using the constant linear mass density, the kinetic energy of each mass element of the string with length Δx is<br />
<br />
<math>\vartriangle E_K = \tfrac{1}{2}(\mu\vartriangle x){v_y}^2</math><br />
<br />
Where μ is density of the particle, each mass element of the string has the mass element of the string oscillates with a velocity v<sub>y</sub>. When the mass element of the string approach zero <br />
<br />
<math>dE_K = \lim_{\vartriangle x \to \infty}\tfrac{1}{2}(\mu\vartriangle x){v_y}^2=\tfrac{1}{2}\mu{v_y}^2dx</math><br />
<br />
For a sinusoidal wave with simple harmonic motion, the position of each mass element may be modeled as<br />
<br />
<math>y(x,t)=Asin(nx-\omega t)</math><br />
<br />
<br />
Where ω is the angular frequency in time, ω =2πf. n is the wave number. The velocity of each mass element of the string oscillates is<br />
<br />
<math>v_y=\tfrac{\partial (x,t)}{\partial t} = -A\omega cos(nx-\omega t)</math><br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
<math>dE_K=\tfrac{1}{2}\mu \omega_2A^2cos^2(nx-\omega t)</math><br />
<br />
The wave can be consisting of multiple wavelengths. To standardize the energy, consider the kinetic energy associated with a wavelength of the wave. The kinetic energy can be integrated over the wavelength to find the energy associated with each wavelength of the wave<br />
<br />
<math>\int_{0}^{E_{K\lambda}} dE_K = \int_{0}^{\lambda} \tfrac{1}{2} \mu \omega^2A^2cos^2(nx-\omega t)dx</math><br />
<br />
The energy for a full wavelength can be found by integrating this expression at a given time, and it is most convenient to set t=0 for this integration. The energy for one wavelength along the string is<br />
<br />
<math>\int_{0}^{E_{K\lambda}} dE_K = \tfrac{1}{2}\mu \omega^2A^2\int_{0}^{\lambda} cos^2(nx)dx = \tfrac{1}{2}\mu \omega^2A^2 \int_{0}^{\lambda}\tfrac{1+sin(2nx)}{2}dx</math><br />
<br />
Thus, it’s obtained that<br />
<br />
<math>E_K=\tfrac{1}{2}\mu \omega^2A^2[\tfrac{1}{2}x+\tfrac{1}{4n}sin(2nx)]_0^\lambda = \tfrac{1}{4}\mu \omega^2A^2\lambda</math><br />
<br />
According to the potential energy formula <br />
<br />
<math>E_P=\tfrac{1}{2}kx^2</math><br />
<br />
The simple and harmonic motion with an angular frequency (ω) given by<br />
<br />
<math>\omega=\sqrt{\tfrac{k}{m}}</math><br />
<br />
Where k is the spring constant, m is the mass of the object, bring it into the formula<br />
<br />
<math>E_P=\tfrac{1}{2}m\omega^2x^2</math><br />
<br />
The differential form of the elastic potential energy is<br />
<br />
<math>dE_P=\tfrac{1}{2}\omega^2y^2\mu dx=\tfrac{1}{2}\mu \omega^2A^2sin^2(nx-\omega t)</math><br />
<br />
It’s obtained that<br />
<br />
<math>\int_{0}^{E_{P\lambda}} dE_P=\tfrac{1}{2}\mu \omega^2A^2\int_{0}^{\lambda} sin^2(nx-\omega t)dx=\tfrac{1}{4}\mu \omega^2A^2\lambda</math><br />
<br />
Finally, the total energy associated with a wavelength is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>)<br />
<br />
<math>E_P=E_K+E_P=\tfrac{1}{2}\mu \omega^2A^2\lambda</math><br />
<br />
This shows a wave is an energy transport phenomenon which transports energy along with a medium without transporting matter. The amount of energy transferred by a wave is related to the amplitude of the wave. A high energy wave is featured by a high amplitude; a low energy wave is featured by a low amplitude. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. Assuming that the frequency of the wave is unchanged, an increase in amplitude will result in an increase in energy and a decrease in amplitude will result in a decrease in energy. The energy of a wave is directly proportional to the square of the amplitude it contains: <br />
<br />
<math>E \varpropto A^2</math> <br />
<br />
The energy of the wave depends on both the amplitude and the frequency. If the energy of each wavelength is a discrete packet of energy, a high-frequency wave will deliver more of these packets per unit time than a low-frequency wave. The average rate of energy transfer in mechanical waves is proportional to both the square of the amplitude and the square of the frequency. If two mechanical waves have equal amplitudes, but one wave has a frequency equal to twice the frequency of the other, the higher-frequency wave will have a rate of energy transfer a factor of four times as great as the rate of energy transfer of the lower-frequency wave. A doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave. A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave. And a quadrupling of the amplitude of a wave is indicative of a 16-fold increase in the amount of energy transported by the wave. Thus, whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared (Figure 2). <br />
[[File:AE-AMP.jpg|thumb|Figure 2 The amplitude of a wave is related to the energy which it transport. Wave 1 with wavelength (λ), frequency (f), and amplitude (A); Wave 2 has the same wavelength and frequency as wave 1, but has two times the amplitude, has higher energy; Wave 3 with the shortest wavelength has the greatest number of wavelengths per unit time (the highest frequency). If two waves have the same frequency and speed, the one with the greater amplitude has the higher energy.]] <br />
<br />
Different materials also have differing degrees of elasticity. A more elastic medium will tend to offer less resistance to the force and allow a greater amplitude pulse to travel through it; being less rigid (and therefore more elastic), the same force causes a greater amplitude.<br />
=== '''Seismic Interpretation''' ===<br />
The average energy attribute of seismic waves is a measure of reflectivity in the specified time window. According to the above description, the higher the energy should have the higher the amplitude. This attribute enhances lateral variations within seismic events among others (Figure 3), therefore, it’s useful for seismic object detection, for instance, amplitude anomalies, chimney detection, etc. The response energy also characterizes acoustic rock properties and bed thickness.<br />
<br />
The average energy is often found to correlate strongly with liquid saturation (oil/water vs. gas) because those reservoir properties have a strong effect on both velocity and density, and energy of seismic reflections are generated at boundaries where the acoustic impedance (the product of velocity and density) changes, thus, it’s commonly used in direct hydrocarbon indicators. In general, the values of average energy are not important, and often not cited, because it is the relative value of an attribute along a given horizon or interval that is important. [[File:AE.jpg|thumb|Figure 3 Average energy extraction in a 100ms window (25 samples) at a constant time (Barnes,2006). ]]<br />
<br />
=== '''References''' ===<br />
# A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
# Encyclopedic Dictionary of Applied Geophysics (Forth Edition).RE Shelf, SEG.2002<br />
# Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production. N.C. Nanda, SEG 2007<br />
# Methods and Applications in Reservoir Geophysics.David H. Johnston, SEG 2010<br />
# Reservoir Geophysics: Applications (SEG Distinguished Instructor Series, No. 11). Willam L. Abriel SEG 2008<br />
<br />
=== '''External links''' ===<br />
# <nowiki>https://www.physicsclassroom.com/class/waves/Lesson-2/Energy-Transport-and-the-Amplitude-of-a-Wave</nowiki><br />
# <nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
# <nowiki>https://www.geoinsights.com/tag/direct-hydrocarbon-indicators/</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=138570RMS amplitude2019-10-30T15:52:58Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[1]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is times the peak amplitude.<sup>[2]</sup><math>\tfrac{1}{\sqrt{2}}</math><br />
[[File:Picture-new.png|thumb|Figure 1 Graph of a sine wave's amplitude vs. time, showing RMS(A<sub>RMS</sub>), peak (A<sub>PK</sub>), and peak-to-peak (A<sub>PP</sub>) amplitude.]]<br />
<br />
=== Mathematical Expression ===<br />
<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[3]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x_1,x_2,…,x_n}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)，ω is the angular frequency, specifies how many oscillations occur in a unit time interval. The RMS of the sine wave can be calculated as follows <br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
Using the power reduction formula <br />
<br />
<math>sin^2 (\omega t) = \tfrac{1 -cos(2\omega t)}{2}</math><br />
<br />
<br />
The RMS of the sine wave becomes<br />
<br />
<math> = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
According to the trigonometric Integral<br />
<br />
<math>\int cos(2\omega t) dt = \tfrac{1}{2\omega} sin(2\omega t)</math><br />
<br />
The RMS can be converted to<br />
<br />
<math>Y_{RMS} = A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
According to the periodicity of the sine function<br />
<br />
<math>sin(2\omega t)|_0^T = sin (2 \tfrac{2\pi}{2} T) = 0</math> <br />
<br />
<br />
Finally, it’s obtained that <br />
<br />
<math>Y_{RMS} = \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS is 0.707 times the peak amplitude <br />
<br />
The RMS over all time of a periodic function is equal to the RMS of one period of the function, its value of a continuous function can be approximated by taking the RMS of a sample consisting of equally spaced observations, thus, the RMS value of various waveforms can also be determined without calculus. <sup>[3]</sup> For instance, the expected value is used instead of the mean, the corresponding formula for a wave’s RMS amplitude defined over a time interval is <br />
<br />
<math>A_{RMS} = \sqrt{\tfrac{\sum_{i=1}^n A_i^2 * w_i }{\sum_{i=1}^n w_i}}</math> <br />
<br />
Where ''A'' = amplitude, ''n'' = number of samples, is the weight of each amplitude value. <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values. when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. <br />
<br />
RMS amplitude analyze the overall amplitude of a signal, conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefor, they are all positive, then the signal average is calculated, eventually followed by the square root operation. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
<br />
=== Seismic Interpretation ===<br />
In seismic processing, automatic gain control (AGC) method is often used in data processing to improve the visibility of seismic data in which attenuation or spherical divergence has caused amplitude decay, which is a system that controls the increase in the amplitude of an electrical signal from the original input to the amplified output, automatically (figure 2).[4] RMS AGC is used for amplitude equalization. It gives the processor a measure of the overall absolute amplitude in the window, both as positive and as negative values. During the processing, the data processor assume that the average absolute reflectivity varies little over a long time window, so a long-gate AGC can be applied to avoids destroying the lateral variation in amplitude, and ensure that the window include many reflectors so that the target event makes very little contribution to the average amplitude in the window. For instance, RMS AGC is often applied on structural imaging.<br />
[[File:2019-10-23 12-41-48.jpg|thumb|Figure 2 Seismic data without AGC (left) and the same data after AGC is applied (Onajite,2013)]]<br />
For seismic integration, RMS is a most commonly used post stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage.[5] This has splendid applications for various reconnaissance endeavors. The squaring of the amplitude values within the window gives the high amplitudes maximum opportunity to stand out above the background contamination. The RMS amplitude over a large flat or structured time window can be effectively used to identify bright spots at different levels within a formation (Figure3).<br />
[[File:Picture3.jpg|thumb|Figure 3 Top Balder Formation amplitude map from multiple 3-D surveys covering 5500 sq km of the Faroe Basin west of Shetland in U.K. waters. The colors show RMS amplitude over a 50-ms window surrounding the Top Balder reflection, with red indicating high amplitude and blue indicating low amplitude (Brown,2004)]]<br />
RMS amplitude is popular type of windowed amplitude attributes, the window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view. In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide. The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009. <br />
<br />
[2] Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
<br />
[3] Weisstein, Eric W. "Root-Mean-Square". MathWorld.<br />
<br />
[4] Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms" MasteringElectronicsDesign.com. Retrieved 21 January 2015 <br />
<br />
[5] Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
<br />
[6] Pitfalls in the study of seismic amplitude<br />
<br />
[7] SEG: Methods and Applications in Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
<nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
<br />
[[Rms amplitude AGC|https://wiki.seg.org/wiki/Rms_amplitude_AGC]]<br />
<br />
<nowiki>https://community.sw.siemens.com/s/article/root-mean-square-rms-and-overall-level</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=Average_energy&diff=138569Average energy2019-10-30T15:41:57Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018.<br />
<br />
The stacked seismic data volume is commonly used for the interpretation of geologic structure and seismic attributes. The average energy of amplitude is one post-stack attribute that is often used in direct hydrocarbon indicators (DHIs), although its interpretation in thin-layered beds is not necessarily straightforward.<br />
<br />
=== Definition ===<br />
In physics, the wave is a periodic disturbance in a medium or in space. In a progressive wave, energy is transferred from one place to another by the vibrations.<sup>[1]</sup> Like light waves, sound waves, or transverse oscillations of a string, these disturbances take energy to create and propagate, in order to move the constituent particles or change the electric/magnetic fields. The energy transported by a wave is directly proportional to the square of the amplitude of the wave.<br />
<br />
Seismic wave is an elastic disturbance that is propagated from point to point through a medium.<sup>[2]</sup> The average energy attribute of the seismic wave is calculated by adding the square of each sample, then dividing by the number of samples in the window to yield the mean. <sup>[3]</sup><br />
<br />
=== Mathematical Expression ===<br />
This attribute calculates the squared sum of the sample values in the specified time window divided by the number of samples in the window.<br />
<br />
<math>Average Energy = \tfrac{\sum_{i=1}^nA_i^2}{n}</math><br />
<br />
Where A<sub>i</sub> is the amplitude of the sampling point in a given time window, n is the number of sampling points.<br />
<br />
=== '''Physical Description''' ===<br />
The total mechanical energy of the wave is the sum of its kinetic energy and potential energy. The energy associated with a traveling wave in a stretched string is conveniently expressed as the energy per wavelength. The energy of a small segment of the string can be expressed as the sum of the kinetic energy and elastic potential energy of the segment (Figure 1).<br />
[[File:AE-energy illustration.jpg|thumb|Figure 1 A sinusoidal wave in the string, each point of the string with mass Δm oscillates at the same frequency as the wave. The total energy of each point is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>).]]<br />
<br />
According to the kinetic energy (E<sub>K</sub>) formula <br />
<br />
<math>E_K = \tfrac{1}{2}mv^2</math><br />
<br />
Where m is the mass and v is the velocity of the body. The mass element oscillates perpendicular to the direction of the motion of the wave. Using the constant linear mass density, the kinetic energy of each mass element of the string with length Δx is<br />
<br />
<math>\vartriangle E_K = \tfrac{1}{2}(\mu\vartriangle x){v_y}^2</math><br />
<br />
Where μ is density of the particle, each mass element of the string has the mass element of the string oscillates with a velocity v<sub>y</sub>. When the mass element of the string approach zero <br />
<br />
<math>dE_K = \lim_{\vartriangle x \to \infty}\tfrac{1}{2}(\mu\vartriangle x){v_y}^2=\tfrac{1}{2}\mu{v_y}^2dx</math><br />
<br />
For a sinusoidal wave with simple harmonic motion, the position of each mass element may be modeled as<br />
<br />
<math>y(x,t)=Asin(nx-\omega t)</math><br />
<br />
<br />
Where ω is the angular frequency in time, ω =2πf. n is the wave number. The velocity of each mass element of the string oscillates is<br />
<br />
<math>v_y=\tfrac{\partial (x,t)}{\partial t} = -A\omega cos(nx-\omega t)</math><br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
<math>dE_K=\tfrac{1}{2}\mu \omega_2A^2cos^2(nx-\omega t)</math><br />
<br />
The wave can be consisting of multiple wavelengths. To standardize the energy, consider the kinetic energy associated with a wavelength of the wave. The kinetic energy can be integrated over the wavelength to find the energy associated with each wavelength of the wave<br />
<br />
<math>\int_{0}^{E_{K\lambda}} dE_K = \int_{0}^{\lambda} \tfrac{1}{2} \mu \omega^2A^2cos^2(nx-\omega t)dx</math><br />
<br />
The energy for a full wavelength can be found by integrating this expression at a given time, and it is most convenient to set t=0 for this integration. The energy for one wavelength along the string is<br />
<br />
<math>\int_{0}^{E_{K\lambda}} dE_K = \tfrac{1}{2}\mu \omega^2A^2\int_{0}^{\lambda} cos^2(nx)dx = \tfrac{1}{2}\mu \omega^2A^2 \int_{0}^{\lambda}\tfrac{1+sin(2nx)}{2}dx</math><br />
<br />
Thus, it’s obtained that<br />
<br />
<math>E_K=\tfrac{1}{2}\mu \omega^2A^2[\tfrac{1}{2}x+\tfrac{1}{4n}sin(2nx)]_0^\lambda = \tfrac{1}{4}\mu \omega^2A^2\lambda</math><br />
<br />
According to the potential energy formula <br />
<br />
<math>E_P=\tfrac{1}{2}kx^2</math><br />
<br />
The simple and harmonic motion with an angular frequency (ω) given by<br />
<br />
<math>\omega=\sqrt{\tfrac{k}{m}}</math><br />
<br />
Where k is the spring constant, m is the mass of the object, bring it into the formula<br />
<br />
<math>E_P=\tfrac{1}{2}m\omega^2x^2</math><br />
<br />
The differential form of the elastic potential energy is<br />
<br />
<math>dE_P=\tfrac{1}{2}\omega^2y^2\mu dx=\tfrac{1}{2}\mu \omega^2A^2sin^2(nx-\omega t)</math><br />
<br />
It’s obtained that<br />
<br />
<math>\int_{0}^{E_{P\lambda}} dE_P=\tfrac{1}{2}\mu \omega^2A^2\int_{0}^{\lambda} sin^2(nx-\omega t)dx=\tfrac{1}{4}\mu \omega^2A^2\lambda</math><br />
<br />
Finally, the total energy associated with a wavelength is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>)<br />
<br />
<math>E_P=E_K+E_P=\tfrac{1}{2}\mu \omega^2A^2\lambda</math><br />
<br />
This shows a wave is an energy transport phenomenon which transports energy along with a medium without transporting matter. The amount of energy transferred by a wave is related to the amplitude of the wave. A high energy wave is featured by a high amplitude; a low energy wave is featured by a low amplitude. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. Assuming that the frequency of the wave is unchanged, an increase in amplitude will result in an increase in energy and a decrease in amplitude will result in a decrease in energy. The energy of a wave is directly proportional to the square of the amplitude it contains: <br />
<br />
<math>E \varpropto A^2</math> <br />
<br />
The energy of the wave depends on both the amplitude and the frequency. If the energy of each wavelength is a discrete packet of energy, a high-frequency wave will deliver more of these packets per unit time than a low-frequency wave. The average rate of energy transfer in mechanical waves is proportional to both the square of the amplitude and the square of the frequency. If two mechanical waves have equal amplitudes, but one wave has a frequency equal to twice the frequency of the other, the higher-frequency wave will have a rate of energy transfer a factor of four times as great as the rate of energy transfer of the lower-frequency wave. A doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave. A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave. And a quadrupling of the amplitude of a wave is indicative of a 16-fold increase in the amount of energy transported by the wave. Thus, whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared (Figure 2). <br />
[[File:AE-AMP.jpg|thumb|Figure 2 The amplitude of a wave is related to the energy which it transport. Wave 1 with wavelength (λ), frequency (f), and amplitude (A); Wave 2 has the same wavelength and frequency as wave 1, but has two times the amplitude, has higher energy; Wave 3 with the shortest wavelength has the greatest number of wavelengths per unit time (the highest frequency). If two waves have the same frequency and speed, the one with the greater amplitude has the higher energy.]] <br />
<br />
Different materials also have differing degrees of elasticity. A more elastic medium will tend to offer less resistance to the force and allow a greater amplitude pulse to travel through it; being less rigid (and therefore more elastic), the same force causes a greater amplitude.<br />
=== '''Seismic Interpretation''' ===<br />
The average energy attribute of seismic waves is a measure of reflectivity in the specified time window. According to the above description, the higher the energy should have the higher the amplitude. This attribute enhances lateral variations within seismic events among others (Figure 3), therefore, it’s useful for seismic object detection, for instance, amplitude anomalies, chimney detection, etc. The response energy also characterizes acoustic rock properties and bed thickness.<br />
<br />
The average energy is often found to correlate strongly with liquid saturation (oil/water vs. gas) because those reservoir properties have a strong effect on both velocity and density, and energy of seismic reflections are generated at boundaries where the acoustic impedance (the product of velocity and density) changes, thus, it’s commonly used in direct hydrocarbon indicators. In general, the values of average energy are not important, and often not cited, because it is the relative value of an attribute along a given horizon or interval that is important. [[File:AE.jpg|thumb|Figure 3 Average energy extraction in a 100ms window (25 samples) at a constant time (Barnes,2006). ]]<br />
<br />
=== '''References''' ===<br />
# A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
# Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
# Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
# SEG: Methods and Applications in Reservoir Geophysics<br />
# Seismic Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
# <nowiki>https://www.physicsclassroom.com/class/waves/Lesson-2/Energy-Transport-and-the-Amplitude-of-a-Wave</nowiki><br />
# <nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
# <nowiki>https://www.geoinsights.com/tag/direct-hydrocarbon-indicators/</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=Average_energy&diff=138435Average energy2019-10-26T09:06:21Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018.<br />
<br />
The stacked seismic data volume is commonly used for the interpretation of geologic structure and seismic attributes. The average energy of amplitude is one post-stack attribute that is often used in direct hydrocarbon indicators (DHIs), although its interpretation in thin-layered beds is not necessarily straightforward.<br />
<br />
=== Definition ===<br />
In physics, the wave is a periodic disturbance in a medium or in space. In a progressive wave, energy is transferred from one place to another by the vibrations.<sup>[1]</sup> Like light waves, sound waves, or transverse oscillations of a string, these disturbances take energy to create and propagate, in order to move the constituent particles or change the electric/magnetic fields. The energy transported by a wave is directly proportional to the square of the amplitude of the wave.<br />
<br />
Seismic wave is an elastic disturbance that is propagated from point to point through a medium.<sup>[2]</sup> The average energy attribute of the seismic wave is calculated by adding the square of each sample, then dividing by the number of samples in the window to yield the mean. <sup>[3]</sup><br />
<br />
=== Mathematical Expression ===<br />
This attribute calculates the squared sum of the sample values in the specified time window divided by the number of samples in the window.<br />
<br />
<math>Average Energy = \tfrac{\sum_{i=1}^nA_i^2}{n}</math><br />
<br />
Where A<sub>i</sub> is the amplitude of the sampling point in a given time window, n is the number of sampling points.<br />
<br />
=== '''Physical Description''' ===<br />
The total mechanical energy of the wave is the sum of its kinetic energy and potential energy. The energy associated with a traveling wave in a stretched string is conveniently expressed as the energy per wavelength. The energy of a small segment of the string can be expressed as the sum of the kinetic energy and elastic potential energy of the segment (Figure 1).<br />
[[File:AE-energy illustration.jpg|thumb|Figure 1 A sinusoidal wave in the string, each point of the string with mass Δm oscillates at the same frequency as the wave. The total energy of each point is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>).]]<br />
<br />
According to the kinetic energy (E<sub>K</sub>) formula <br />
<br />
<math>E_K = \tfrac{1}{2}mv^2</math><br />
<br />
Where m is the mass and v is the velocity of the body. The mass element oscillates perpendicular to the direction of the motion of the wave. Using the constant linear mass density, the kinetic energy of each mass element of the string with length Δx is<br />
<br />
<math>\vartriangle E_K = \tfrac{1}{2}(\mu\vartriangle x){v_y}^2</math><br />
<br />
Where μ is density of the particle, each mass element of the string has the mass element of the string oscillates with a velocity v<sub>y</sub>. When the mass element of the string approach zero <br />
<br />
<math>dE_K = \lim_{\vartriangle x \to \infty}\tfrac{1}{2}(\mu\vartriangle x){v_y}^2=\tfrac{1}{2}\mu{v_y}^2dx</math><br />
<br />
For a sinusoidal wave with simple harmonic motion, the position of each mass element may be modeled as<br />
<br />
<math>y(x,t)=Asin(nx-\omega t)</math><br />
<br />
<br />
Where ω is the angular frequency in time, ω =2πf. n is the wave number. The velocity of each mass element of the string oscillates is<br />
<br />
<math>v_y=\tfrac{\partial (x,t)}{\partial t} = -A\omega cos(nx-\omega t)</math><br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
<math>dE_K=\tfrac{1}{2}\mu \omega_2A^2cos^2(nx-\omega t)</math><br />
<br />
The wave can be consisting of multiple wavelengths. To standardize the energy, consider the kinetic energy associated with a wavelength of the wave. The kinetic energy can be integrated over the wavelength to find the energy associated with each wavelength of the wave<br />
<br />
<math>\int_{0}^{E_{K\lambda}} dE_K = \int_{0}^{\lambda} \tfrac{1}{2} \mu \omega^2A^2cos^2(nx-\omega t)dx</math><br />
<br />
The energy for a full wavelength can be found by integrating this expression at a given time, and it is most convenient to set t=0 for this integration. The energy for one wavelength along the string is<br />
<br />
<math>\int_{0}^{E_{K\lambda}} dE_K = \tfrac{1}{2}\mu \omega^2A^2\int_{0}^{\lambda} cos^2(nx)dx = \tfrac{1}{2}\mu \omega^2A^2 \int_{0}^{\lambda}\tfrac{1+sin(2nx)}{2}dx</math><br />
<br />
Thus, it’s obtained that<br />
<br />
<math>E_K=\tfrac{1}{2}\mu \omega^2A^2[\tfrac{1}{2}x+\tfrac{1}{4n}sin(2nx)]_0^\lambda = \tfrac{1}{4}\mu \omega^2A^2\lambda</math><br />
<br />
According to the potential energy formula <br />
<br />
<math>E_P=\tfrac{1}{2}kx^2</math><br />
<br />
The simple and harmonic motion with an angular frequency (ω) given by<br />
<br />
<math>\omega=\sqrt{\tfrac{k}{m}}</math><br />
<br />
Where k is the spring constant, m is the mass of the object, bring it into the formula<br />
<br />
<math>E_P=\tfrac{1}{2}m\omega^2x^2</math><br />
<br />
The differential form of the elastic potential energy is<br />
<br />
<math>dE_P=\tfrac{1}{2}\omega^2y^2\mu dx=\tfrac{1}{2}\mu \omega^2A^2sin^2(nx-\omega t)</math><br />
<br />
It’s obtained that<br />
<br />
<math>\int_{0}^{E_{P\lambda}} dE_P=\tfrac{1}{2}\mu \omega^2A^2\int_{0}^{\lambda} sin^2(nx-\omega t)dx=\tfrac{1}{4}\mu \omega^2A^2\lambda</math><br />
<br />
Finally, the total energy associated with a wavelength is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>)<br />
<br />
<math>E_P=E_K+E_P=\tfrac{1}{2}\mu \omega^2A^2\lambda</math><br />
<br />
This shows a wave is an energy transport phenomenon which transports energy along with a medium without transporting matter. The amount of energy transferred by a wave is related to the amplitude of the wave. A high energy wave is featured by a high amplitude; a low energy wave is featured by a low amplitude. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. Assuming that the frequency of the wave is unchanged, an increase in amplitude will result in an increase in energy and a decrease in amplitude will result in a decrease in energy. The energy of a wave is directly proportional to the square of the amplitude it contains: <br />
<br />
<math>E \varpropto A^2</math> <br />
<br />
The energy of the wave depends on both the amplitude and the frequency. If the energy of each wavelength is a discrete packet of energy, a high-frequency wave will deliver more of these packets per unit time than a low-frequency wave. The average rate of energy transfer in mechanical waves is proportional to both the square of the amplitude and the square of the frequency. If two mechanical waves have equal amplitudes, but one wave has a frequency equal to twice the frequency of the other, the higher-frequency wave will have a rate of energy transfer a factor of four times as great as the rate of energy transfer of the lower-frequency wave. A doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave. A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave. And a quadrupling of the amplitude of a wave is indicative of a 16-fold increase in the amount of energy transported by the wave. Thus, whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared (Figure 2). <br />
[[File:AE-AMP.jpg|thumb|Figure 2 The amplitude of a wave is related to the energy which it transport. Wave 1 with wavelength (λ), frequency (f), and amplitude (A); Wave 2 has the same wavelength and frequency as wave 1, but has two times the amplitude, has higher energy; Wave 3 with the shortest wavelength has the greatest number of wavelengths per unit time (the highest frequency). If two waves have the same frequency and speed, the one with the greater amplitude has the higher energy.]] <br />
<br />
Different materials also have differing degrees of elasticity. A more elastic medium will tend to offer less resistance to the force and allow a greater amplitude pulse to travel through it; being less rigid (and therefore more elastic), the same force causes a greater amplitude.<br />
=== '''Seismic Interpretation''' ===<br />
The average energy attribute of seismic waves is a measure of reflectivity in the specified time window. According to the above description, the higher the energy should have the higher the amplitude. This attribute enhances lateral variations within seismic events among others (Figure 3), therefore, it’s useful for seismic object detection, for instance, amplitude anomalies, chimney detection, etc. The response energy also characterizes acoustic rock properties and bed thickness.<br />
<br />
The average energy is often found to correlate strongly with liquid saturation (oil/water vs. gas) because those reservoir properties have a strong effect on both velocity and density, and energy of seismic reflections are generated at boundaries where the acoustic impedance (the product of velocity and density) changes, thus, it’s commonly used in direct hydrocarbon indicators. In general, the values of average energy are not important, and often not cited, because it is the relative value of an attribute along a given horizon or interval that is important. [[File:AE.jpg|thumb|Figure 3 Average energy extraction in a 100ms window (25 samples) at a constant time (Barnes,2006). ]]<br />
<br />
=== '''References''' ===<br />
# A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
# Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
# Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
# SEG: Methods and Applications in Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
# <nowiki>https://www.physicsclassroom.com/class/waves/Lesson-2/Energy-Transport-and-the-Amplitude-of-a-Wave</nowiki><br />
# <nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
# <nowiki>https://www.geoinsights.com/tag/direct-hydrocarbon-indicators/</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=File:AE-AMP.jpg&diff=138434File:AE-AMP.jpg2019-10-26T09:05:51Z<p>Zhai0629: </p>
<hr />
<div>AE-AMP</div>Zhai0629https://wiki.seg.org/index.php?title=File:AE-energy_illustration.jpg&diff=138433File:AE-energy illustration.jpg2019-10-26T09:04:13Z<p>Zhai0629: </p>
<hr />
<div>AE-energy illustration</div>Zhai0629https://wiki.seg.org/index.php?title=Average_energy&diff=138432Average energy2019-10-26T09:00:33Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018.<br />
<br />
The stacked seismic data volume is commonly used for the interpretation of geologic structure and seismic attributes. The average energy of amplitude is one post-stack attribute that is often used in direct hydrocarbon indicators (DHIs), although its interpretation in thin-layered beds is not necessarily straightforward.<br />
<br />
=== Definition ===<br />
In physics, the wave is a periodic disturbance in a medium or in space. In a progressive wave, energy is transferred from one place to another by the vibrations.<sup>[1]</sup> Like light waves, sound waves, or transverse oscillations of a string, these disturbances take energy to create and propagate, in order to move the constituent particles or change the electric/magnetic fields. The energy transported by a wave is directly proportional to the square of the amplitude of the wave.<br />
<br />
Seismic wave is an elastic disturbance that is propagated from point to point through a medium.<sup>[2]</sup> The average energy attribute of the seismic wave is calculated by adding the square of each sample, then dividing by the number of samples in the window to yield the mean. <sup>[3]</sup><br />
<br />
=== Mathematical Expression ===<br />
This attribute calculates the squared sum of the sample values in the specified time window divided by the number of samples in the window.<br />
<br />
<math>Average Energy = \tfrac{\sum_{i=1}^nA_i^2}{n}</math><br />
<br />
Where A<sub>i</sub> is the amplitude of the sampling point in a given time window, n is the number of sampling points.<br />
<br />
=== '''Physical Description''' ===<br />
The total mechanical energy of the wave is the sum of its kinetic energy and potential energy. The energy associated with a traveling wave in a stretched string is conveniently expressed as the energy per wavelength. The energy of a small segment of the string can be expressed as the sum of the kinetic energy and elastic potential energy of the segment (Figure 1).<br />
<br />
According to the kinetic energy (E<sub>K</sub>) formula <br />
<br />
<math>E_K = \tfrac{1}{2}mv^2</math><br />
<br />
Where m is the mass and v is the velocity of the body. The mass element oscillates perpendicular to the direction of the motion of the wave. Using the constant linear mass density, the kinetic energy of each mass element of the string with length Δx is<br />
<br />
<math>\vartriangle E_K = \tfrac{1}{2}(\mu\vartriangle x){v_y}^2</math><br />
<br />
Where μ is density of the particle, each mass element of the string has the mass element of the string oscillates with a velocity v<sub>y</sub>. When the mass element of the string approach zero <br />
<br />
<math>dE_K = \lim_{\vartriangle x \to \infty}\tfrac{1}{2}(\mu\vartriangle x){v_y}^2=\tfrac{1}{2}\mu{v_y}^2dx</math><br />
<br />
For a sinusoidal wave with simple harmonic motion, the position of each mass element may be modeled as<br />
<br />
<math>y(x,t)=Asin(nx-\omega t)</math><br />
<br />
<br />
Where ω is the angular frequency in time, ω =2πf. n is the wave number. The velocity of each mass element of the string oscillates is<br />
<br />
<math>v_y=\tfrac{\partial (x,t)}{\partial t} = -A\omega cos(nx-\omega t)</math><br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
<math>dE_K=\tfrac{1}{2}\mu \omega_2A^2cos^2(nx-\omega t)</math><br />
<br />
The wave can be consisting of multiple wavelengths. To standardize the energy, consider the kinetic energy associated with a wavelength of the wave. The kinetic energy can be integrated over the wavelength to find the energy associated with each wavelength of the wave<br />
<br />
<math>\int_{0}^{E_{K\lambda}} dE_K = \int_{0}^{\lambda} \tfrac{1}{2} \mu \omega^2A^2cos^2(nx-\omega t)dx</math><br />
<br />
The energy for a full wavelength can be found by integrating this expression at a given time, and it is most convenient to set t=0 for this integration. The energy for one wavelength along the string is<br />
<br />
<math>\int_{0}^{E_{K\lambda}} dE_K = \tfrac{1}{2}\mu \omega^2A^2\int_{0}^{\lambda} cos^2(nx)dx = \tfrac{1}{2}\mu \omega^2A^2 \int_{0}^{\lambda}\tfrac{1+sin(2nx)}{2}dx</math><br />
<br />
Thus, it’s obtained that<br />
<br />
<math>E_K=\tfrac{1}{2}\mu \omega^2A^2[\tfrac{1}{2}x+\tfrac{1}{4n}sin(2nx)]_0^\lambda = \tfrac{1}{4}\mu \omega^2A^2\lambda</math><br />
<br />
According to the potential energy formula <br />
<br />
<math>E_P=\tfrac{1}{2}kx^2</math><br />
<br />
The simple and harmonic motion with an angular frequency (ω) given by<br />
<br />
<math>\omega=\sqrt{\tfrac{k}{m}}</math><br />
<br />
Where k is the spring constant, m is the mass of the object, bring it into the formula<br />
<br />
<math>E_P=\tfrac{1}{2}m\omega^2x^2</math><br />
<br />
The differential form of the elastic potential energy is<br />
<br />
<math>dE_P=\tfrac{1}{2}\omega^2y^2\mu dx=\tfrac{1}{2}\mu \omega^2A^2sin^2(nx-\omega t)</math><br />
<br />
It’s obtained that<br />
<br />
<math>\int_{0}^{E_{P\lambda}} dE_P=\tfrac{1}{2}\mu \omega^2A^2\int_{0}^{\lambda} sin^2(nx-\omega t)dx=\tfrac{1}{4}\mu \omega^2A^2\lambda</math><br />
<br />
Finally, the total energy associated with a wavelength is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>)<br />
<br />
<math>E_P=E_K+E_P=\tfrac{1}{2}\mu \omega^2A^2\lambda</math><br />
<br />
This shows a wave is an energy transport phenomenon which transports energy along with a medium without transporting matter. The amount of energy transferred by a wave is related to the amplitude of the wave. A high energy wave is featured by a high amplitude; a low energy wave is featured by a low amplitude. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. Assuming that the frequency of the wave is unchanged, an increase in amplitude will result in an increase in energy and a decrease in amplitude will result in a decrease in energy. The energy of a wave is directly proportional to the square of the amplitude it contains: <br />
<br />
<math>E \varpropto A^2</math> <br />
<br />
The energy of the wave depends on both the amplitude and the frequency. If the energy of each wavelength is a discrete packet of energy, a high-frequency wave will deliver more of these packets per unit time than a low-frequency wave. The average rate of energy transfer in mechanical waves is proportional to both the square of the amplitude and the square of the frequency. If two mechanical waves have equal amplitudes, but one wave has a frequency equal to twice the frequency of the other, the higher-frequency wave will have a rate of energy transfer a factor of four times as great as the rate of energy transfer of the lower-frequency wave. A doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave. A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave. And a quadrupling of the amplitude of a wave is indicative of a 16-fold increase in the amount of energy transported by the wave. Thus, whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared (Figure 2). <br />
<br />
Different materials also have differing degrees of elasticity. A more elastic medium will tend to offer less resistance to the force and allow a greater amplitude pulse to travel through it; being less rigid (and therefore more elastic), the same force causes a greater amplitude.<br />
=== '''Seismic Interpretation''' ===<br />
The average energy attribute of seismic waves is a measure of reflectivity in the specified time window. According to the above description, the higher the energy should have the higher the amplitude. This attribute enhances lateral variations within seismic events among others (Figure 3), therefore, it’s useful for seismic object detection, for instance, amplitude anomalies, chimney detection, etc. The response energy also characterizes acoustic rock properties and bed thickness.<br />
<br />
The average energy is often found to correlate strongly with liquid saturation (oil/water vs. gas) because those reservoir properties have a strong effect on both velocity and density, and energy of seismic reflections are generated at boundaries where the acoustic impedance (the product of velocity and density) changes, thus, it’s commonly used in direct hydrocarbon indicators. In general, the values of average energy are not important, and often not cited, because it is the relative value of an attribute along a given horizon or interval that is important. [[File:AE.jpg|thumb|Figure 3 Average energy extraction in a 100ms window (25 samples) at a constant time (Barnes,2006). ]]<br />
<br />
=== '''References''' ===<br />
# A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
# Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
# Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
# SEG: Methods and Applications in Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
# <nowiki>https://www.physicsclassroom.com/class/waves/Lesson-2/Energy-Transport-and-the-Amplitude-of-a-Wave</nowiki><br />
# <nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
# <nowiki>https://www.geoinsights.com/tag/direct-hydrocarbon-indicators/</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=Average_energy&diff=138431Average energy2019-10-26T08:59:28Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018.<br />
<br />
The stacked seismic data volume is commonly used for the interpretation of geologic structure and seismic attributes. The average energy of amplitude is one post-stack attribute that is often used in direct hydrocarbon indicators (DHIs), although its interpretation in thin-layered beds is not necessarily straightforward.<br />
<br />
=== Definition ===<br />
In physics, the wave is a periodic disturbance in a medium or in space. In a progressive wave, energy is transferred from one place to another by the vibrations.<sup>[1]</sup> Like light waves, sound waves, or transverse oscillations of a string, these disturbances take energy to create and propagate, in order to move the constituent particles or change the electric/magnetic fields. The energy transported by a wave is directly proportional to the square of the amplitude of the wave.<br />
<br />
Seismic wave is an elastic disturbance that is propagated from point to point through a medium.<sup>[2]</sup> The average energy attribute of the seismic wave is calculated by adding the square of each sample, then dividing by the number of samples in the window to yield the mean. <sup>[3]</sup><br />
<br />
=== Mathematical Expression ===<br />
This attribute calculates the squared sum of the sample values in the specified time window divided by the number of samples in the window.<br />
<br />
<math>Average Energy = \tfrac{\sum_{i=1}^nA_i^2}{n}</math><br />
<br />
Where A<sub>i</sub> is the amplitude of the sampling point in a given time window, n is the number of sampling points.<br />
<br />
=== '''Physical Description''' ===<br />
The total mechanical energy of the wave is the sum of its kinetic energy and potential energy. The energy associated with a traveling wave in a stretched string is conveniently expressed as the energy per wavelength. The energy of a small segment of the string can be expressed as the sum of the kinetic energy and elastic potential energy of the segment (Figure 1).<br />
<br />
According to the kinetic energy (E<sub>K</sub>) formula <br />
<br />
<math>E_K = \tfrac{1}{2}mv^2</math><br />
<br />
Where m is the mass and v is the velocity of the body. The mass element oscillates perpendicular to the direction of the motion of the wave. Using the constant linear mass density, the kinetic energy of each mass element of the string with length Δx is<br />
<br />
<math>\vartriangle E_K = \tfrac{1}{2}(\mu\vartriangle x){v_y}^2</math><br />
<br />
Where μ is density of the particle, each mass element of the string has the mass element of the string oscillates with a velocity v<sub>y</sub>. When the mass element of the string approach zero <br />
<br />
<math>dE_K = \lim_{\vartriangle x \to \infty}\tfrac{1}{2}(\mu\vartriangle x){v_y}^2=\tfrac{1}{2}\mu{v_y}^2dx</math><br />
<br />
For a sinusoidal wave with simple harmonic motion, the position of each mass element may be modeled as<br />
<br />
<math>y(x,t)=Asin(nx-\omega t)</math><br />
<br />
<math>v_y=\tfrac{\partial (x,t)}{\partial t} = -A\omega cos(nx-\omega t)</math><br />
<br />
Where ω is the angular frequency in time, ω =2πf. n is the wave number. The velocity of each mass element of the string oscillates is<br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
<math>dE_K=\tfrac{1}{2}\mu \omega_2A^2cos^2(nx-\omega t)</math><br />
<br />
The wave can be consisting of multiple wavelengths. To standardize the energy, consider the kinetic energy associated with a wavelength of the wave. The kinetic energy can be integrated over the wavelength to find the energy associated with each wavelength of the wave<br />
<br />
<math>\int_{0}^{E_{K\lambda}} dE_K = \int_{0}^{\lambda} \tfrac{1}{2} \mu \omega^2A^2cos^2(nx-\omega t)dx</math><br />
<br />
The energy for a full wavelength can be found by integrating this expression at a given time, and it is most convenient to set t=0 for this integration. The energy for one wavelength along the string is<br />
<br />
<math>\int_{0}^{E_{K\lambda}} dE_K = \tfrac{1}{2}\mu \omega^2A^2\int_{0}^{\lambda} cos^2(nx)dx = \tfrac{1}{2}\mu \omega^2A^2 \int_{0}^{\lambda}\tfrac{1+sin(2nx)}{2}dx</math><br />
<br />
Thus, it’s obtained that<br />
<br />
<math>E_K=\tfrac{1}{2}\mu \omega^2A^2[\tfrac{1}{2}x+\tfrac{1}{4n}sin(2nx)]_0^\lambda = \tfrac{1}{4}\mu \omega^2A^2\lambda</math><br />
<br />
According to the potential energy formula <br />
<br />
<math>E_P=\tfrac{1}{2}kx^2</math><br />
<br />
The simple and harmonic motion with an angular frequency (ω) given by<br />
<br />
<math>\omega=\sqrt{\tfrac{k}{m}}</math><br />
<br />
Where k is the spring constant, m is the mass of the object, bring it into the formula<br />
<br />
<math>E_P=\tfrac{1}{2}m\omega^2x^2</math><br />
<br />
The differential form of the elastic potential energy is<br />
<br />
<math>dE_P=\tfrac{1}{2}\omega^2y^2\mu dx=\tfrac{1}{2}\mu \omega^2A^2sin^2(nx-\omega t)</math><br />
<br />
It’s obtained that<br />
<br />
<math>\int_{0}^{E_{P\lambda}} dE_P=\tfrac{1}{2}\mu \omega^2A^2\int_{0}^{\lambda} sin^2(nx-\omega t)dx=\tfrac{1}{4}\mu \omega^2A^2\lambda</math><br />
<br />
Finally, the total energy associated with a wavelength is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>)<br />
<br />
<math>E_P=E_K+E_P=\tfrac{1}{2}\mu \omega^2A^2\lambda</math><br />
<br />
This shows a wave is an energy transport phenomenon which transports energy along with a medium without transporting matter. The amount of energy transferred by a wave is related to the amplitude of the wave. A high energy wave is featured by a high amplitude; a low energy wave is featured by a low amplitude. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. Assuming that the frequency of the wave is unchanged, an increase in amplitude will result in an increase in energy and a decrease in amplitude will result in a decrease in energy. The energy of a wave is directly proportional to the square of the amplitude it contains: <br />
<br />
<math>E \varpropto A^2</math> <br />
<br />
The energy of the wave depends on both the amplitude and the frequency. If the energy of each wavelength is a discrete packet of energy, a high-frequency wave will deliver more of these packets per unit time than a low-frequency wave. The average rate of energy transfer in mechanical waves is proportional to both the square of the amplitude and the square of the frequency. If two mechanical waves have equal amplitudes, but one wave has a frequency equal to twice the frequency of the other, the higher-frequency wave will have a rate of energy transfer a factor of four times as great as the rate of energy transfer of the lower-frequency wave. A doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave. A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave. And a quadrupling of the amplitude of a wave is indicative of a 16-fold increase in the amount of energy transported by the wave. Thus, whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared (Figure 2). <br />
<br />
Different materials also have differing degrees of elasticity. A more elastic medium will tend to offer less resistance to the force and allow a greater amplitude pulse to travel through it; being less rigid (and therefore more elastic), the same force causes a greater amplitude.<br />
=== '''Seismic Interpretation''' ===<br />
The average energy attribute of seismic waves is a measure of reflectivity in the specified time window. According to the above description, the higher the energy should have the higher the amplitude. This attribute enhances lateral variations within seismic events among others (Figure 3), therefore, it’s useful for seismic object detection, for instance, amplitude anomalies, chimney detection, etc. The response energy also characterizes acoustic rock properties and bed thickness.<br />
<br />
The average energy is often found to correlate strongly with liquid saturation (oil/water vs. gas) because those reservoir properties have a strong effect on both velocity and density, and energy of seismic reflections are generated at boundaries where the acoustic impedance (the product of velocity and density) changes, thus, it’s commonly used in direct hydrocarbon indicators. In general, the values of average energy are not important, and often not cited, because it is the relative value of an attribute along a given horizon or interval that is important. [[File:AE.jpg|thumb|Figure 3 Average energy extraction in a 100ms window (25 samples) at a constant time (Barnes,2006). ]]<br />
<br />
=== '''References''' ===<br />
# A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
# Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
# Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
# SEG: Methods and Applications in Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
# <nowiki>https://www.physicsclassroom.com/class/waves/Lesson-2/Energy-Transport-and-the-Amplitude-of-a-Wave</nowiki><br />
# <nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
# <nowiki>https://www.geoinsights.com/tag/direct-hydrocarbon-indicators/</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=Average_energy&diff=138430Average energy2019-10-26T08:57:03Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018.<br />
<br />
The stacked seismic data volume is commonly used for the interpretation of geologic structure and seismic attributes. The average energy of amplitude is one post-stack attribute that is often used in direct hydrocarbon indicators (DHIs), although its interpretation in thin-layered beds is not necessarily straightforward.<br />
<br />
=== Definition ===<br />
In physics, the wave is a periodic disturbance in a medium or in space. In a progressive wave, energy is transferred from one place to another by the vibrations.<sup>[1]</sup> Like light waves, sound waves, or transverse oscillations of a string, these disturbances take energy to create and propagate, in order to move the constituent particles or change the electric/magnetic fields. The energy transported by a wave is directly proportional to the square of the amplitude of the wave.<br />
<br />
Seismic wave is an elastic disturbance that is propagated from point to point through a medium.<sup>[2]</sup> The average energy attribute of the seismic wave is calculated by adding the square of each sample, then dividing by the number of samples in the window to yield the mean. <sup>[3]</sup><br />
<br />
=== Mathematical Expression ===<br />
This attribute calculates the squared sum of the sample values in the specified time window divided by the number of samples in the window.<br />
<br />
<math>Average Energy = \tfrac{\sum_{i=1}^nA_i^2}{n}</math><br />
<br />
Where A<sub>i</sub> is the amplitude of the sampling point in a given time window, n is the number of sampling points.<br />
<br />
=== '''Physical Description''' ===<br />
The total mechanical energy of the wave is the sum of its kinetic energy and potential energy. The energy associated with a traveling wave in a stretched string is conveniently expressed as the energy per wavelength. The energy of a small segment of the string can be expressed as the sum of the kinetic energy and elastic potential energy of the segment (Figure 1).<br />
<br />
According to the kinetic energy (E<sub>K</sub>) formula <br />
<br />
<math>E_K = \tfrac{1}{2}mv^2</math><br />
<br />
Where m is the mass and v is the velocity of the body. The mass element oscillates perpendicular to the direction of the motion of the wave. Using the constant linear mass density, the kinetic energy of each mass element of the string with length Δx is<br />
<br />
<math>\vartriangle E_K = \tfrac{1}{2}(\mu\vartriangle x){v_y}^2</math><br />
<br />
Where μ is density of the particle, each mass element of the string has the mass element of the string oscillates with a velocity v<sub>y</sub>. When the mass element of the string approach zero <br />
<br />
<math>dE_K = \lim_{\vartriangle x \to \infty}\tfrac{1}{2}(\mu\vartriangle x){v_y}^2=\tfrac{1}{2}\mu{v_y}^2dx</math><br />
<br />
For a sinusoidal wave with simple harmonic motion, the position of each mass element may be modeled as<br />
<br />
<math>y(x,t)=Asin(nx-\omega t)</math><br />
<br />
<math>v_y=\tfrac{\partial (x,t)}{\partial t} = -A\omega cos(nx-\omega t)</math><br />
<br />
Where ω is the angular frequency in time, ω =2πf. n is the wave number. The velocity of each mass element of the string oscillates is<br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
<math>dE_K=\tfrac{1}{2}\mu \omega_2A^2cos^2(nx-\omega t)</math><br />
<br />
The wave can be consisting of multiple wavelengths. To standardize the energy, consider the kinetic energy associated with a wavelength of the wave. The kinetic energy can be integrated over the wavelength to find the energy associated with each wavelength of the wave<br />
<br />
<math>\int_{0}^{E_{K\lambda}} dE_K = \int_{0}^{\lambda} \tfrac{1}{2} \mu \omega^2A^2cos^2(nx-\omega t)dx</math><br />
<br />
The energy for a full wavelength can be found by integrating this expression at a given time, and it is most convenient to set t=0 for this integration. The energy for one wavelength along the string is<br />
<br />
<math>\int_{0}^{E_{K\lambda}} dE_K = \tfrac{1}{2}\mu \omega^2A^2\int_{0}^{\lambda} cos^2(nx)dx = \tfrac{1}{2}\mu \omega^2A^2 \int_{0}^{\lambda}\tfrac{1+sin(2nx)}{2}dx</math><br />
<br />
Thus, it’s obtained that<br />
<br />
<math>E_K=\tfrac{1}{2}\mu \omega^2A^2[\tfrac{1}{2}x+\tfrac{1}{4n}sin(2nx)]_0^\lambda = \tfrac{1}{4}\mu \omega^2A^2\lambda</math><br />
<br />
According to the potential energy formula <br />
<br />
<math>E_P=\tfrac{1}{2}kx^2</math><br />
<br />
The simple and harmonic motion with an angular frequency (ω) given by<br />
<br />
<math>\omega=\sqrt{\tfrac{k}{m}}</math><br />
<br />
Where k is the spring constant, m is the mass of the object, bring it into the formula<br />
<br />
<math>E_P=\tfrac{1}{2}m\omega^2x^2</math><br />
<br />
The differential form of the elastic potential energy is<br />
<br />
<math>dE_P=\tfrac{1}{2}\omega^2y^2\mu dx=\tfrac{1}{2}\mu \omega^2A^2sin^2(nx-\omega t)</math><br />
<br />
It’s obtained that<br />
<br />
<math>\int_{0}^{E_{P\lambda}} dE_P=\tfrac{1}{2}\mu \omega^2A^2\int_{0}^{\lambda} sin^2(nx-\omega t)dx=\tfrac{1}{4}\mu \omega^2A^2\lambda</math><br />
<br />
Finally, the total energy associated with a wavelength is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>)<br />
<br />
This shows a wave is an energy transport phenomenon which transports energy along with a medium without transporting matter. The amount of energy transferred by a wave is related to the amplitude of the wave. A high energy wave is featured by a high amplitude; a low energy wave is featured by a low amplitude. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. Assuming that the frequency of the wave is unchanged, an increase in amplitude will result in an increase in energy and a decrease in amplitude will result in a decrease in energy. The energy of a wave is directly proportional to the square of the amplitude it contains: <br />
<br />
The energy of the wave depends on both the amplitude and the frequency. If the energy of each wavelength is a discrete packet of energy, a high-frequency wave will deliver more of these packets per unit time than a low-frequency wave. The average rate of energy transfer in mechanical waves is proportional to both the square of the amplitude and the square of the frequency. If two mechanical waves have equal amplitudes, but one wave has a frequency equal to twice the frequency of the other, the higher-frequency wave will have a rate of energy transfer a factor of four times as great as the rate of energy transfer of the lower-frequency wave. A doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave. A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave. And a quadrupling of the amplitude of a wave is indicative of a 16-fold increase in the amount of energy transported by the wave. Thus, whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared (Figure 2). <br />
<br />
Different materials also have differing degrees of elasticity. A more elastic medium will tend to offer less resistance to the force and allow a greater amplitude pulse to travel through it; being less rigid (and therefore more elastic), the same force causes a greater amplitude.<br />
=== '''Seismic Interpretation''' ===<br />
The average energy attribute of seismic waves is a measure of reflectivity in the specified time window. According to the above description, the higher the energy should have the higher the amplitude. This attribute enhances lateral variations within seismic events among others (Figure 3), therefore, it’s useful for seismic object detection, for instance, amplitude anomalies, chimney detection, etc. The response energy also characterizes acoustic rock properties and bed thickness.<br />
<br />
The average energy is often found to correlate strongly with liquid saturation (oil/water vs. gas) because those reservoir properties have a strong effect on both velocity and density, and energy of seismic reflections are generated at boundaries where the acoustic impedance (the product of velocity and density) changes, thus, it’s commonly used in direct hydrocarbon indicators. In general, the values of average energy are not important, and often not cited, because it is the relative value of an attribute along a given horizon or interval that is important. [[File:AE.jpg|thumb|Figure 3 Average energy extraction in a 100ms window (25 samples) at a constant time (Barnes,2006). ]]<br />
<br />
=== '''References''' ===<br />
# A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
# Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
# Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
# SEG: Methods and Applications in Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
# <nowiki>https://www.physicsclassroom.com/class/waves/Lesson-2/Energy-Transport-and-the-Amplitude-of-a-Wave</nowiki><br />
# <nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
# <nowiki>https://www.geoinsights.com/tag/direct-hydrocarbon-indicators/</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=Average_energy&diff=138429Average energy2019-10-26T08:44:44Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018.<br />
<br />
The stacked seismic data volume is commonly used for the interpretation of geologic structure and seismic attributes. The average energy of amplitude is one post-stack attribute that is often used in direct hydrocarbon indicators (DHIs), although its interpretation in thin-layered beds is not necessarily straightforward.<br />
<br />
=== Definition ===<br />
In physics, the wave is a periodic disturbance in a medium or in space. In a progressive wave, energy is transferred from one place to another by the vibrations.<sup>[1]</sup> Like light waves, sound waves, or transverse oscillations of a string, these disturbances take energy to create and propagate, in order to move the constituent particles or change the electric/magnetic fields. The energy transported by a wave is directly proportional to the square of the amplitude of the wave.<br />
<br />
Seismic wave is an elastic disturbance that is propagated from point to point through a medium.<sup>[2]</sup> The average energy attribute of the seismic wave is calculated by adding the square of each sample, then dividing by the number of samples in the window to yield the mean. <sup>[3]</sup><br />
<br />
=== Mathematical Expression ===<br />
This attribute calculates the squared sum of the sample values in the specified time window divided by the number of samples in the window.<br />
<br />
<math>Average Energy = \tfrac{\sum_{i=1}^nA_i^2}{n}</math><br />
<br />
Where A<sub>i</sub> is the amplitude of the sampling point in a given time window, n is the number of sampling points.<br />
<br />
=== '''Physical Description''' ===<br />
The total mechanical energy of the wave is the sum of its kinetic energy and potential energy. The energy associated with a traveling wave in a stretched string is conveniently expressed as the energy per wavelength. The energy of a small segment of the string can be expressed as the sum of the kinetic energy and elastic potential energy of the segment (Figure 1).<br />
<br />
According to the kinetic energy (E<sub>K</sub>) formula <br />
<br />
<math>E_K = \tfrac{1}{2}mv^2</math><br />
<br />
Where m is the mass and v is the velocity of the body. The mass element oscillates perpendicular to the direction of the motion of the wave. Using the constant linear mass density, the kinetic energy of each mass element of the string with length Δx is<br />
<br />
<math>\vartriangle E_K = \tfrac{1}{2}(\mu\vartriangle x){v_y}^2</math><br />
<br />
Where μ is density of the particle, each mass element of the string has the mass element of the string oscillates with a velocity v<sub>y</sub>. When the mass element of the string approach zero <br />
<br />
<math>dE_K = \lim_{\vartriangle x \to \infty}\tfrac{1}{2}(\mu\vartriangle x){v_y}^2=\tfrac{1}{2}\mu{v_y}^2dx</math><br />
<br />
For a sinusoidal wave with simple harmonic motion, the position of each mass element may be modeled as<br />
<br />
<math>v_y=\tfrac{\partial (x,t)}{\partial t} = -A\omega cos(nx-\omega t)</math><br />
<br />
Where ω is the angular frequency in time, ω =2πf. n is the wave number. The velocity of each mass element of the string oscillates is<br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
<math>dE_K=\tfrac{1}{2}\mu \omega_2A^2cos^2(nx-\omega t)</math><br />
<br />
The wave can be consisting of multiple wavelengths. To standardize the energy, consider the kinetic energy associated with a wavelength of the wave. The kinetic energy can be integrated over the wavelength to find the energy associated with each wavelength of the wave<br />
<br />
<math>\int_{0}^{E_{K\lambda}} dE_K = \int_{0}^{\lambda} \tfrac{1}{2} \mu \omega^2A^2cos^2(nx-\omega t)dx</math><br />
<br />
The energy for a full wavelength can be found by integrating this expression at a given time, and it is most convenient to set t=0 for this integration. The energy for one wavelength along the string is<br />
<br />
<math>\int_{0}^{E_{K\lambda}} dE_K = \tfrac{1}{2}\mu \omega^2A^2\int_{0}^{\lambda} cos^2(nx)dx = \tfrac{1}{2}\mu \omega^2A^2 \int_{0}^{\lambda}\tfrac{1+sin(2nx)}{2}dx</math><br />
<br />
Thus, it’s obtained that<br />
<br />
According to the potential energy formula <br />
<br />
The simple and harmonic motion with an angular frequency (ω) given by<br />
<br />
Where k is the spring constant, m is the mass of the object, bring it into the formula<br />
<br />
The differential form of the elastic potential energy is<br />
<br />
It’s obtained that<br />
<br />
Finally, the total energy associated with a wavelength is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>)<br />
<br />
This shows a wave is an energy transport phenomenon which transports energy along with a medium without transporting matter. The amount of energy transferred by a wave is related to the amplitude of the wave. A high energy wave is featured by a high amplitude; a low energy wave is featured by a low amplitude. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. Assuming that the frequency of the wave is unchanged, an increase in amplitude will result in an increase in energy and a decrease in amplitude will result in a decrease in energy. The energy of a wave is directly proportional to the square of the amplitude it contains: <br />
<br />
The energy of the wave depends on both the amplitude and the frequency. If the energy of each wavelength is a discrete packet of energy, a high-frequency wave will deliver more of these packets per unit time than a low-frequency wave. The average rate of energy transfer in mechanical waves is proportional to both the square of the amplitude and the square of the frequency. If two mechanical waves have equal amplitudes, but one wave has a frequency equal to twice the frequency of the other, the higher-frequency wave will have a rate of energy transfer a factor of four times as great as the rate of energy transfer of the lower-frequency wave. A doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave. A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave. And a quadrupling of the amplitude of a wave is indicative of a 16-fold increase in the amount of energy transported by the wave. Thus, whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared (Figure 2). <br />
<br />
Different materials also have differing degrees of elasticity. A more elastic medium will tend to offer less resistance to the force and allow a greater amplitude pulse to travel through it; being less rigid (and therefore more elastic), the same force causes a greater amplitude.<br />
=== '''Seismic Interpretation''' ===<br />
The average energy attribute of seismic waves is a measure of reflectivity in the specified time window. According to the above description, the higher the energy should have the higher the amplitude. This attribute enhances lateral variations within seismic events among others (Figure 3), therefore, it’s useful for seismic object detection, for instance, amplitude anomalies, chimney detection, etc. The response energy also characterizes acoustic rock properties and bed thickness.<br />
<br />
The average energy is often found to correlate strongly with liquid saturation (oil/water vs. gas) because those reservoir properties have a strong effect on both velocity and density, and energy of seismic reflections are generated at boundaries where the acoustic impedance (the product of velocity and density) changes, thus, it’s commonly used in direct hydrocarbon indicators. In general, the values of average energy are not important, and often not cited, because it is the relative value of an attribute along a given horizon or interval that is important. [[File:AE.jpg|thumb|Figure 3 Average energy extraction in a 100ms window (25 samples) at a constant time (Barnes,2006). ]]<br />
<br />
=== '''References''' ===<br />
# A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
# Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
# Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
# SEG: Methods and Applications in Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
# <nowiki>https://www.physicsclassroom.com/class/waves/Lesson-2/Energy-Transport-and-the-Amplitude-of-a-Wave</nowiki><br />
# <nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
# <nowiki>https://www.geoinsights.com/tag/direct-hydrocarbon-indicators/</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=Average_energy&diff=138428Average energy2019-10-26T08:33:05Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018.<br />
<br />
The stacked seismic data volume is commonly used for the interpretation of geologic structure and seismic attributes. The average energy of amplitude is one post-stack attribute that is often used in direct hydrocarbon indicators (DHIs), although its interpretation in thin-layered beds is not necessarily straightforward.<br />
<br />
=== Definition ===<br />
In physics, the wave is a periodic disturbance in a medium or in space. In a progressive wave, energy is transferred from one place to another by the vibrations.<sup>[1]</sup> Like light waves, sound waves, or transverse oscillations of a string, these disturbances take energy to create and propagate, in order to move the constituent particles or change the electric/magnetic fields. The energy transported by a wave is directly proportional to the square of the amplitude of the wave.<br />
<br />
Seismic wave is an elastic disturbance that is propagated from point to point through a medium.<sup>[2]</sup> The average energy attribute of the seismic wave is calculated by adding the square of each sample, then dividing by the number of samples in the window to yield the mean. <sup>[3]</sup><br />
<br />
=== Mathematical Expression ===<br />
This attribute calculates the squared sum of the sample values in the specified time window divided by the number of samples in the window.<br />
<br />
<math>Average Energy = \tfrac{\sum_{i=1}^nA_i^2}{n}</math><br />
<br />
Where A<sub>i</sub> is the amplitude of the sampling point in a given time window, n is the number of sampling points.<br />
<br />
=== '''Physical Description''' ===<br />
The total mechanical energy of the wave is the sum of its kinetic energy and potential energy. The energy associated with a traveling wave in a stretched string is conveniently expressed as the energy per wavelength. The energy of a small segment of the string can be expressed as the sum of the kinetic energy and elastic potential energy of the segment (Figure 1).<br />
<br />
According to the kinetic energy (E<sub>K</sub>) formula <br />
<br />
<math>E_K = \tfrac{1}{2}mv^2</math><br />
<br />
Where m is the mass and v is the velocity of the body. The mass element oscillates perpendicular to the direction of the motion of the wave. Using the constant linear mass density, the kinetic energy of each mass element of the string with length Δx is<br />
<br />
<math>\vartriangle E_K = \tfrac{1}{2}(\mu\vartriangle x){v_y}^2</math><br />
<br />
Where μ is density of the particle, each mass element of the string has the mass element of the string oscillates with a velocity v<sub>y</sub>. When the mass element of the string approach zero <br />
<br />
<math>dE_K = \lim_{\vartriangle x \to \infty}\tfrac{1}{2}(\mu\vartriangle x){v_y}^2=\tfrac{1}{2}\mu{v_y}^2dx</math><br />
<br />
For a sinusoidal wave with simple harmonic motion, the position of each mass element may be modeled as<br />
<br />
Where ω is the angular frequency in time, ω =2πf. n is the wave number. The velocity of each mass element of the string oscillates is<br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
The energy for a full wavelength can be found by integrating this expression at a given time, and it is most convenient to set t=0 for this integration. The energy for one wavelength along the string is<br />
<br />
Thus, it’s obtained that<br />
<br />
According to the potential energy formula <br />
<br />
The simple and harmonic motion with an angular frequency (ω) given by<br />
<br />
Where k is the spring constant, m is the mass of the object, bring it into the formula<br />
<br />
The differential form of the elastic potential energy is<br />
<br />
It’s obtained that<br />
<br />
Finally, the total energy associated with a wavelength is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>)<br />
<br />
This shows a wave is an energy transport phenomenon which transports energy along with a medium without transporting matter. The amount of energy transferred by a wave is related to the amplitude of the wave. A high energy wave is featured by a high amplitude; a low energy wave is featured by a low amplitude. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. Assuming that the frequency of the wave is unchanged, an increase in amplitude will result in an increase in energy and a decrease in amplitude will result in a decrease in energy. The energy of a wave is directly proportional to the square of the amplitude it contains: <br />
<br />
The energy of the wave depends on both the amplitude and the frequency. If the energy of each wavelength is a discrete packet of energy, a high-frequency wave will deliver more of these packets per unit time than a low-frequency wave. The average rate of energy transfer in mechanical waves is proportional to both the square of the amplitude and the square of the frequency. If two mechanical waves have equal amplitudes, but one wave has a frequency equal to twice the frequency of the other, the higher-frequency wave will have a rate of energy transfer a factor of four times as great as the rate of energy transfer of the lower-frequency wave. A doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave. A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave. And a quadrupling of the amplitude of a wave is indicative of a 16-fold increase in the amount of energy transported by the wave. Thus, whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared (Figure 2). <br />
<br />
Different materials also have differing degrees of elasticity. A more elastic medium will tend to offer less resistance to the force and allow a greater amplitude pulse to travel through it; being less rigid (and therefore more elastic), the same force causes a greater amplitude.<br />
=== '''Seismic Interpretation''' ===<br />
The average energy attribute of seismic waves is a measure of reflectivity in the specified time window. According to the above description, the higher the energy should have the higher the amplitude. This attribute enhances lateral variations within seismic events among others (Figure 3), therefore, it’s useful for seismic object detection, for instance, amplitude anomalies, chimney detection, etc. The response energy also characterizes acoustic rock properties and bed thickness.<br />
<br />
The average energy is often found to correlate strongly with liquid saturation (oil/water vs. gas) because those reservoir properties have a strong effect on both velocity and density, and energy of seismic reflections are generated at boundaries where the acoustic impedance (the product of velocity and density) changes, thus, it’s commonly used in direct hydrocarbon indicators. In general, the values of average energy are not important, and often not cited, because it is the relative value of an attribute along a given horizon or interval that is important. [[File:AE.jpg|thumb|Figure 3 Average energy extraction in a 100ms window (25 samples) at a constant time (Barnes,2006). ]]<br />
<br />
=== '''References''' ===<br />
# A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
# Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
# Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
# SEG: Methods and Applications in Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
# <nowiki>https://www.physicsclassroom.com/class/waves/Lesson-2/Energy-Transport-and-the-Amplitude-of-a-Wave</nowiki><br />
# <nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
# <nowiki>https://www.geoinsights.com/tag/direct-hydrocarbon-indicators/</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=Average_energy&diff=138427Average energy2019-10-26T08:29:11Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018.<br />
<br />
The stacked seismic data volume is commonly used for the interpretation of geologic structure and seismic attributes. The average energy of amplitude is one post-stack attribute that is often used in direct hydrocarbon indicators (DHIs), although its interpretation in thin-layered beds is not necessarily straightforward.<br />
<br />
=== Definition ===<br />
In physics, the wave is a periodic disturbance in a medium or in space. In a progressive wave, energy is transferred from one place to another by the vibrations.<sup>[1]</sup> Like light waves, sound waves, or transverse oscillations of a string, these disturbances take energy to create and propagate, in order to move the constituent particles or change the electric/magnetic fields. The energy transported by a wave is directly proportional to the square of the amplitude of the wave.<br />
<br />
Seismic wave is an elastic disturbance that is propagated from point to point through a medium.<sup>[2]</sup> The average energy attribute of the seismic wave is calculated by adding the square of each sample, then dividing by the number of samples in the window to yield the mean. <sup>[3]</sup><br />
<br />
=== Mathematical Expression ===<br />
This attribute calculates the squared sum of the sample values in the specified time window divided by the number of samples in the window.<br />
<br />
<math>Average Energy = \tfrac{\sum_{i=1}^nA_i^2}{n}</math><br />
<br />
Where A<sub>i</sub> is the amplitude of the sampling point in a given time window, n is the number of sampling points.<br />
<br />
=== '''Physical Description''' ===<br />
The total mechanical energy of the wave is the sum of its kinetic energy and potential energy. The energy associated with a traveling wave in a stretched string is conveniently expressed as the energy per wavelength. The energy of a small segment of the string can be expressed as the sum of the kinetic energy and elastic potential energy of the segment (Figure 1).<br />
<br />
According to the kinetic energy (E<sub>K</sub>) formula <br />
<br />
<math>E_K = \tfrac{1}{2}mv^2</math><br />
<br />
Where m is the mass and v is the velocity of the body. The mass element oscillates perpendicular to the direction of the motion of the wave. Using the constant linear mass density, the kinetic energy of each mass element of the string with length Δx is<br />
<br />
<math>\vartriangle E_K = \tfrac{1}{2}(\mu\vartriangle x){v_y}^2</math><br />
<br />
Where μ is density of the particle, each mass element of the string has the mass element of the string oscillates with a velocity v<sub>y</sub>. When the mass element of the string approach zero <br />
<br />
For a sinusoidal wave with simple harmonic motion, the position of each mass element may be modeled as<br />
<br />
Where ω is the angular frequency in time, ω =2πf. n is the wave number. The velocity of each mass element of the string oscillates is<br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
The energy for a full wavelength can be found by integrating this expression at a given time, and it is most convenient to set t=0 for this integration. The energy for one wavelength along the string is<br />
<br />
Thus, it’s obtained that<br />
<br />
According to the potential energy formula <br />
<br />
The simple and harmonic motion with an angular frequency (ω) given by<br />
<br />
Where k is the spring constant, m is the mass of the object, bring it into the formula<br />
<br />
The differential form of the elastic potential energy is<br />
<br />
It’s obtained that<br />
<br />
Finally, the total energy associated with a wavelength is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>)<br />
<br />
This shows a wave is an energy transport phenomenon which transports energy along with a medium without transporting matter. The amount of energy transferred by a wave is related to the amplitude of the wave. A high energy wave is featured by a high amplitude; a low energy wave is featured by a low amplitude. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. Assuming that the frequency of the wave is unchanged, an increase in amplitude will result in an increase in energy and a decrease in amplitude will result in a decrease in energy. The energy of a wave is directly proportional to the square of the amplitude it contains: <br />
<br />
The energy of the wave depends on both the amplitude and the frequency. If the energy of each wavelength is a discrete packet of energy, a high-frequency wave will deliver more of these packets per unit time than a low-frequency wave. The average rate of energy transfer in mechanical waves is proportional to both the square of the amplitude and the square of the frequency. If two mechanical waves have equal amplitudes, but one wave has a frequency equal to twice the frequency of the other, the higher-frequency wave will have a rate of energy transfer a factor of four times as great as the rate of energy transfer of the lower-frequency wave. A doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave. A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave. And a quadrupling of the amplitude of a wave is indicative of a 16-fold increase in the amount of energy transported by the wave. Thus, whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared (Figure 2). <br />
<br />
Different materials also have differing degrees of elasticity. A more elastic medium will tend to offer less resistance to the force and allow a greater amplitude pulse to travel through it; being less rigid (and therefore more elastic), the same force causes a greater amplitude.<br />
=== '''Seismic Interpretation''' ===<br />
The average energy attribute of seismic waves is a measure of reflectivity in the specified time window. According to the above description, the higher the energy should have the higher the amplitude. This attribute enhances lateral variations within seismic events among others (Figure 3), therefore, it’s useful for seismic object detection, for instance, amplitude anomalies, chimney detection, etc. The response energy also characterizes acoustic rock properties and bed thickness.<br />
<br />
The average energy is often found to correlate strongly with liquid saturation (oil/water vs. gas) because those reservoir properties have a strong effect on both velocity and density, and energy of seismic reflections are generated at boundaries where the acoustic impedance (the product of velocity and density) changes, thus, it’s commonly used in direct hydrocarbon indicators. In general, the values of average energy are not important, and often not cited, because it is the relative value of an attribute along a given horizon or interval that is important. [[File:AE.jpg|thumb|Figure 3 Average energy extraction in a 100ms window (25 samples) at a constant time (Barnes,2006). ]]<br />
<br />
=== '''References''' ===<br />
# A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
# Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
# Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
# SEG: Methods and Applications in Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
# <nowiki>https://www.physicsclassroom.com/class/waves/Lesson-2/Energy-Transport-and-the-Amplitude-of-a-Wave</nowiki><br />
# <nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
# <nowiki>https://www.geoinsights.com/tag/direct-hydrocarbon-indicators/</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=Average_energy&diff=138426Average energy2019-10-26T08:27:06Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018.<br />
<br />
The stacked seismic data volume is commonly used for the interpretation of geologic structure and seismic attributes. The average energy of amplitude is one post-stack attribute that is often used in direct hydrocarbon indicators (DHIs), although its interpretation in thin-layered beds is not necessarily straightforward.<br />
<br />
=== Definition ===<br />
In physics, the wave is a periodic disturbance in a medium or in space. In a progressive wave, energy is transferred from one place to another by the vibrations.<sup>[1]</sup> Like light waves, sound waves, or transverse oscillations of a string, these disturbances take energy to create and propagate, in order to move the constituent particles or change the electric/magnetic fields. The energy transported by a wave is directly proportional to the square of the amplitude of the wave.<br />
<br />
Seismic wave is an elastic disturbance that is propagated from point to point through a medium.<sup>[2]</sup> The average energy attribute of the seismic wave is calculated by adding the square of each sample, then dividing by the number of samples in the window to yield the mean. <sup>[3]</sup><br />
<br />
=== Mathematical Expression ===<br />
This attribute calculates the squared sum of the sample values in the specified time window divided by the number of samples in the window.<br />
<br />
<math>Average Energy = \tfrac{\sum_{i=1}^nA_i^2}{n}</math><br />
<br />
Where A<sub>i</sub> is the amplitude of the sampling point in a given time window, n is the number of sampling points.<br />
<br />
=== '''Physical Description''' ===<br />
The total mechanical energy of the wave is the sum of its kinetic energy and potential energy. The energy associated with a traveling wave in a stretched string is conveniently expressed as the energy per wavelength. The energy of a small segment of the string can be expressed as the sum of the kinetic energy and elastic potential energy of the segment (Figure 1).<br />
<br />
According to the kinetic energy (E<sub>K</sub>) formula <br />
<br />
<math>E_K = \tfrac{1}{2}mv^2</math><math>Average Energy = \tfrac{\sum_{i=1}^n A_i^2}{n} </math>Where m is the mass and v is the velocity of the body. The mass element oscillates perpendicular to the direction of the motion of the wave. Using the constant linear mass density, the kinetic energy of each mass element of the string with length Δx is<br />
<br />
<math>\vartriangleE_K = \tfrac{1}{2}(\mu\vartrianglex){v_y}^2</math><br />
<br />
Where μ is density of the particle, each mass element of the string has the mass element of the string oscillates with a velocity v<sub>y</sub>. When the mass element of the string approach zero <br />
<br />
For a sinusoidal wave with simple harmonic motion, the position of each mass element may be modeled as<br />
<br />
Where ω is the angular frequency in time, ω =2πf. n is the wave number. The velocity of each mass element of the string oscillates is<br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
The energy for a full wavelength can be found by integrating this expression at a given time, and it is most convenient to set t=0 for this integration. The energy for one wavelength along the string is<br />
<br />
Thus, it’s obtained that<br />
<br />
According to the potential energy formula <br />
<br />
The simple and harmonic motion with an angular frequency (ω) given by<br />
<br />
Where k is the spring constant, m is the mass of the object, bring it into the formula<br />
<br />
The differential form of the elastic potential energy is<br />
<br />
It’s obtained that<br />
<br />
Finally, the total energy associated with a wavelength is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>)<br />
<br />
This shows a wave is an energy transport phenomenon which transports energy along with a medium without transporting matter. The amount of energy transferred by a wave is related to the amplitude of the wave. A high energy wave is featured by a high amplitude; a low energy wave is featured by a low amplitude. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. Assuming that the frequency of the wave is unchanged, an increase in amplitude will result in an increase in energy and a decrease in amplitude will result in a decrease in energy. The energy of a wave is directly proportional to the square of the amplitude it contains: <br />
<br />
The energy of the wave depends on both the amplitude and the frequency. If the energy of each wavelength is a discrete packet of energy, a high-frequency wave will deliver more of these packets per unit time than a low-frequency wave. The average rate of energy transfer in mechanical waves is proportional to both the square of the amplitude and the square of the frequency. If two mechanical waves have equal amplitudes, but one wave has a frequency equal to twice the frequency of the other, the higher-frequency wave will have a rate of energy transfer a factor of four times as great as the rate of energy transfer of the lower-frequency wave. A doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave. A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave. And a quadrupling of the amplitude of a wave is indicative of a 16-fold increase in the amount of energy transported by the wave. Thus, whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared (Figure 2). <br />
<br />
Different materials also have differing degrees of elasticity. A more elastic medium will tend to offer less resistance to the force and allow a greater amplitude pulse to travel through it; being less rigid (and therefore more elastic), the same force causes a greater amplitude.<br />
=== '''Seismic Interpretation''' ===<br />
The average energy attribute of seismic waves is a measure of reflectivity in the specified time window. According to the above description, the higher the energy should have the higher the amplitude. This attribute enhances lateral variations within seismic events among others (Figure 3), therefore, it’s useful for seismic object detection, for instance, amplitude anomalies, chimney detection, etc. The response energy also characterizes acoustic rock properties and bed thickness.<br />
<br />
The average energy is often found to correlate strongly with liquid saturation (oil/water vs. gas) because those reservoir properties have a strong effect on both velocity and density, and energy of seismic reflections are generated at boundaries where the acoustic impedance (the product of velocity and density) changes, thus, it’s commonly used in direct hydrocarbon indicators. In general, the values of average energy are not important, and often not cited, because it is the relative value of an attribute along a given horizon or interval that is important. [[File:AE.jpg|thumb|Figure 3 Average energy extraction in a 100ms window (25 samples) at a constant time (Barnes,2006). ]]<br />
<br />
=== '''References''' ===<br />
# A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
# Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
# Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
# SEG: Methods and Applications in Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
# <nowiki>https://www.physicsclassroom.com/class/waves/Lesson-2/Energy-Transport-and-the-Amplitude-of-a-Wave</nowiki><br />
# <nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
# <nowiki>https://www.geoinsights.com/tag/direct-hydrocarbon-indicators/</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=Average_energy&diff=138425Average energy2019-10-26T08:21:21Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018.<br />
<br />
The stacked seismic data volume is commonly used for the interpretation of geologic structure and seismic attributes. The average energy of amplitude is one post-stack attribute that is often used in direct hydrocarbon indicators (DHIs), although its interpretation in thin-layered beds is not necessarily straightforward.<br />
<br />
=== Definition ===<br />
In physics, the wave is a periodic disturbance in a medium or in space. In a progressive wave, energy is transferred from one place to another by the vibrations.<sup>[1]</sup> Like light waves, sound waves, or transverse oscillations of a string, these disturbances take energy to create and propagate, in order to move the constituent particles or change the electric/magnetic fields. The energy transported by a wave is directly proportional to the square of the amplitude of the wave.<br />
<br />
Seismic wave is an elastic disturbance that is propagated from point to point through a medium.<sup>[2]</sup> The average energy attribute of the seismic wave is calculated by adding the square of each sample, then dividing by the number of samples in the window to yield the mean. <sup>[3]</sup><br />
<br />
=== Mathematical Expression ===<br />
This attribute calculates the squared sum of the sample values in the specified time window divided by the number of samples in the window.<br />
<br />
Where A<sub>i</sub> is the amplitude of the sampling point in a given time window, n is the number of sampling points.<br />
<br />
=== '''Physical Description''' ===<br />
The total mechanical energy of the wave is the sum of its kinetic energy and potential energy. The energy associated with a traveling wave in a stretched string is conveniently expressed as the energy per wavelength. The energy of a small segment of the string can be expressed as the sum of the kinetic energy and elastic potential energy of the segment (Figure 1).<math>Average Energy = \tfrac{\sum_{i=1}^n A_i^2}{n} </math>According to the kinetic energy (E<sub>K</sub>) formula <br />
<br />
Where m is the mass and v is the velocity of the body. The mass element oscillates perpendicular to the direction of the motion of the wave. Using the constant linear mass density, the kinetic energy of each mass element of the string with length Δx is<br />
<br />
Where μ is density of the particle, each mass element of the string has the mass element of the string oscillates with a velocity v<sub>y</sub>. When the mass element of the string approach zero <br />
<br />
For a sinusoidal wave with simple harmonic motion, the position of each mass element may be modeled as<br />
<br />
Where ω is the angular frequency in time, ω =2πf. n is the wave number. The velocity of each mass element of the string oscillates is<br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
The energy for a full wavelength can be found by integrating this expression at a given time, and it is most convenient to set t=0 for this integration. The energy for one wavelength along the string is<br />
<br />
Thus, it’s obtained that<br />
<br />
According to the potential energy formula <br />
<br />
The simple and harmonic motion with an angular frequency (ω) given by<br />
<br />
Where k is the spring constant, m is the mass of the object, bring it into the formula<br />
<br />
The differential form of the elastic potential energy is<br />
<br />
It’s obtained that<br />
<br />
Finally, the total energy associated with a wavelength is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>)<br />
<br />
This shows a wave is an energy transport phenomenon which transports energy along with a medium without transporting matter. The amount of energy transferred by a wave is related to the amplitude of the wave. A high energy wave is featured by a high amplitude; a low energy wave is featured by a low amplitude. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. Assuming that the frequency of the wave is unchanged, an increase in amplitude will result in an increase in energy and a decrease in amplitude will result in a decrease in energy. The energy of a wave is directly proportional to the square of the amplitude it contains: <br />
<br />
The energy of the wave depends on both the amplitude and the frequency. If the energy of each wavelength is a discrete packet of energy, a high-frequency wave will deliver more of these packets per unit time than a low-frequency wave. The average rate of energy transfer in mechanical waves is proportional to both the square of the amplitude and the square of the frequency. If two mechanical waves have equal amplitudes, but one wave has a frequency equal to twice the frequency of the other, the higher-frequency wave will have a rate of energy transfer a factor of four times as great as the rate of energy transfer of the lower-frequency wave. A doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave. A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave. And a quadrupling of the amplitude of a wave is indicative of a 16-fold increase in the amount of energy transported by the wave. Thus, whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared (Figure 2). <br />
<br />
Different materials also have differing degrees of elasticity. A more elastic medium will tend to offer less resistance to the force and allow a greater amplitude pulse to travel through it; being less rigid (and therefore more elastic), the same force causes a greater amplitude.<br />
=== '''Seismic Interpretation''' ===<br />
The average energy attribute of seismic waves is a measure of reflectivity in the specified time window. According to the above description, the higher the energy should have the higher the amplitude. This attribute enhances lateral variations within seismic events among others (Figure 3), therefore, it’s useful for seismic object detection, for instance, amplitude anomalies, chimney detection, etc. The response energy also characterizes acoustic rock properties and bed thickness.<br />
<br />
The average energy is often found to correlate strongly with liquid saturation (oil/water vs. gas) because those reservoir properties have a strong effect on both velocity and density, and energy of seismic reflections are generated at boundaries where the acoustic impedance (the product of velocity and density) changes, thus, it’s commonly used in direct hydrocarbon indicators. In general, the values of average energy are not important, and often not cited, because it is the relative value of an attribute along a given horizon or interval that is important. [[File:AE.jpg|thumb|Figure 3 Average energy extraction in a 100ms window (25 samples) at a constant time (Barnes,2006). ]]<br />
<br />
=== '''References''' ===<br />
# A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
# Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
# Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
# SEG: Methods and Applications in Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
# <nowiki>https://www.physicsclassroom.com/class/waves/Lesson-2/Energy-Transport-and-the-Amplitude-of-a-Wave</nowiki><br />
# <nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
# <nowiki>https://www.geoinsights.com/tag/direct-hydrocarbon-indicators/</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=Average_energy&diff=138424Average energy2019-10-26T08:20:31Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018.<br />
<br />
The stacked seismic data volume is commonly used for the interpretation of geologic structure and seismic attributes. The average energy of amplitude is one post-stack attribute that is often used in direct hydrocarbon indicators (DHIs), although its interpretation in thin-layered beds is not necessarily straightforward.<br />
<br />
=== Definition ===<br />
In physics, the wave is a periodic disturbance in a medium or in space. In a progressive wave, energy is transferred from one place to another by the vibrations.<sup>[1]</sup> Like light waves, sound waves, or transverse oscillations of a string, these disturbances take energy to create and propagate, in order to move the constituent particles or change the electric/magnetic fields. The energy transported by a wave is directly proportional to the square of the amplitude of the wave.<br />
<br />
Seismic wave is an elastic disturbance that is propagated from point to point through a medium.<sup>[2]</sup> The average energy attribute of the seismic wave is calculated by adding the square of each sample, then dividing by the number of samples in the window to yield the mean. <sup>[3]</sup><br />
<br />
=== Mathematical Expression ===<br />
This attribute calculates the squared sum of the sample values in the specified time window divided by the number of samples in the window.<br />
<br />
Where A<sub>i</sub> is the amplitude of the sampling point in a given time window, n is the number of sampling points.<br />
<br />
=== '''Physical Description''' ===<br />
The total mechanical energy of the wave is the sum of its kinetic energy and potential energy. The energy associated with a traveling wave in a stretched string is conveniently expressed as the energy per wavelength. The energy of a small segment of the string can be expressed as the sum of the kinetic energy and elastic potential energy of the segment (Figure 1).<math>Average Energy = \tfrac{\sum_{i=1}^n A_i^2}{n} </math>According to the kinetic energy (E<sub>K</sub>) formula <br />
<br />
Where m is the mass and v is the velocity of the body. The mass element oscillates perpendicular to the direction of the motion of the wave. Using the constant linear mass density, the kinetic energy of each mass element of the string with length Δx is<br />
<br />
Where μ is density of the particle, each mass element of the string has the mass element of the string oscillates with a velocity v<sub>y</sub>. When the mass element of the string approach zero <br />
<br />
For a sinusoidal wave with simple harmonic motion, the position of each mass element may be modeled as<br />
<br />
Where ω is the angular frequency in time, ω =2πf. n is the wave number. The velocity of each mass element of the string oscillates is<br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
The energy for a full wavelength can be found by integrating this expression at a given time, and it is most convenient to set t=0 for this integration. The energy for one wavelength along the string is<br />
<br />
Thus, it’s obtained that<br />
<br />
According to the potential energy formula <br />
<br />
The simple and harmonic motion with an angular frequency (ω) given by<br />
<br />
Where k is the spring constant, m is the mass of the object, bring it into the formula<br />
<br />
The differential form of the elastic potential energy is<br />
<br />
It’s obtained that<br />
<br />
Finally, the total energy associated with a wavelength is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>)<br />
<br />
This shows a wave is an energy transport phenomenon which transports energy along with a medium without transporting matter. The amount of energy transferred by a wave is related to the amplitude of the wave. A high energy wave is featured by a high amplitude; a low energy wave is featured by a low amplitude. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. Assuming that the frequency of the wave is unchanged, an increase in amplitude will result in an increase in energy and a decrease in amplitude will result in a decrease in energy. The energy of a wave is directly proportional to the square of the amplitude it contains: <br />
<br />
The energy of the wave depends on both the amplitude and the frequency. If the energy of each wavelength is a discrete packet of energy, a high-frequency wave will deliver more of these packets per unit time than a low-frequency wave. The average rate of energy transfer in mechanical waves is proportional to both the square of the amplitude and the square of the frequency. If two mechanical waves have equal amplitudes, but one wave has a frequency equal to twice the frequency of the other, the higher-frequency wave will have a rate of energy transfer a factor of four times as great as the rate of energy transfer of the lower-frequency wave. A doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave. A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave. And a quadrupling of the amplitude of a wave is indicative of a 16-fold increase in the amount of energy transported by the wave. Thus, whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared (Figure 2). <br />
<br />
Different materials also have differing degrees of elasticity. A more elastic medium will tend to offer less resistance to the force and allow a greater amplitude pulse to travel through it; being less rigid (and therefore more elastic), the same force causes a greater amplitude.<math>E \propto A^2</math><math>E_S = PE_S = \tfrac{kx^2}{2}</math><math>E_S = \tfrac{kA_S^2}{2} </math><math>E_S = \tfrac{kA_W^2}{2} \rightarrow E_W \propto A_W</math><br />
=== '''Seismic Interpretation''' ===<br />
The average energy attribute of seismic waves is a measure of reflectivity in the specified time window. According to the above description, the higher the energy should have the higher the amplitude. This attribute enhances lateral variations within seismic events among others (Figure 3), therefore, it’s useful for seismic object detection, for instance, amplitude anomalies, chimney detection, etc. The response energy also characterizes acoustic rock properties and bed thickness.<br />
<br />
The average energy is often found to correlate strongly with liquid saturation (oil/water vs. gas) because those reservoir properties have a strong effect on both velocity and density, and energy of seismic reflections are generated at boundaries where the acoustic impedance (the product of velocity and density) changes, thus, it’s commonly used in direct hydrocarbon indicators. In general, the values of average energy are not important, and often not cited, because it is the relative value of an attribute along a given horizon or interval that is important. [[File:AE.jpg|thumb|Figure 3 Average energy extraction in a 100ms window (25 samples) at a constant time (Barnes,2006). ]]<br />
<br />
=== '''References''' ===<br />
# A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
# Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
# Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
# SEG: Methods and Applications in Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
# <nowiki>https://www.physicsclassroom.com/class/waves/Lesson-2/Energy-Transport-and-the-Amplitude-of-a-Wave</nowiki><br />
# <nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
# <nowiki>https://www.geoinsights.com/tag/direct-hydrocarbon-indicators/</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=Average_energy&diff=138423Average energy2019-10-26T08:18:16Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018.<br />
<br />
The stacked seismic data volume is commonly used for the interpretation of geologic structure and seismic attributes. The average energy of amplitude is one post-stack attribute that is often used in direct hydrocarbon indicators (DHIs), although its interpretation in thin-layered beds is not necessarily straightforward.<math>Average Energy = \tfrac{\sum_{i=1}^n A_i^2}{n} </math><br />
The total mechanical energy of the wave is the sum of its kinetic energy and potential energy. The energy associated with a traveling wave in a stretched string is conveniently expressed as the energy per wavelength. The energy of a small segment of the string can be expressed as the sum of the kinetic energy and elastic potential energy of the segment (Figure 1).<br />
<br />
According to the kinetic energy (E<sub>K</sub>) formula <br />
<br />
Where m is the mass and v is the velocity of the body. The mass element oscillates perpendicular to the direction of the motion of the wave. Using the constant linear mass density, the kinetic energy of each mass element of the string with length Δx is<br />
<br />
Where μ is density of the particle, each mass element of the string has the mass element of the string oscillates with a velocity v<sub>y</sub>. When the mass element of the string approach zero <br />
<br />
For a sinusoidal wave with simple harmonic motion, the position of each mass element may be modeled as<br />
<br />
Where ω is the angular frequency in time, ω =2πf. n is the wave number. The velocity of each mass element of the string oscillates is<br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
The energy for a full wavelength can be found by integrating this expression at a given time, and it is most convenient to set t=0 for this integration. The energy for one wavelength along the string is<br />
<br />
Thus, it’s obtained that<br />
<br />
According to the potential energy formula <br />
<br />
The simple and harmonic motion with an angular frequency (ω) given by<br />
<br />
Where k is the spring constant, m is the mass of the object, bring it into the formula<br />
<br />
The differential form of the elastic potential energy is<br />
<br />
It’s obtained that<br />
<br />
Finally, the total energy associated with a wavelength is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>)<br />
<br />
This shows a wave is an energy transport phenomenon which transports energy along with a medium without transporting matter. The amount of energy transferred by a wave is related to the amplitude of the wave. A high energy wave is featured by a high amplitude; a low energy wave is featured by a low amplitude. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. Assuming that the frequency of the wave is unchanged, an increase in amplitude will result in an increase in energy and a decrease in amplitude will result in a decrease in energy. The energy of a wave is directly proportional to the square of the amplitude it contains: <br />
<br />
The energy of the wave depends on both the amplitude and the frequency. If the energy of each wavelength is a discrete packet of energy, a high-frequency wave will deliver more of these packets per unit time than a low-frequency wave. The average rate of energy transfer in mechanical waves is proportional to both the square of the amplitude and the square of the frequency. If two mechanical waves have equal amplitudes, but one wave has a frequency equal to twice the frequency of the other, the higher-frequency wave will have a rate of energy transfer a factor of four times as great as the rate of energy transfer of the lower-frequency wave. A doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave. A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave. And a quadrupling of the amplitude of a wave is indicative of a 16-fold increase in the amount of energy transported by the wave. Thus, whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared (Figure 2). <br />
<br />
Different materials also have differing degrees of elasticity. A more elastic medium will tend to offer less resistance to the force and allow a greater amplitude pulse to travel through it; being less rigid (and therefore more elastic), the same force causes a greater amplitude.<math>E \propto A^2</math><math>E_S = PE_S = \tfrac{kx^2}{2}</math><math>E_S = \tfrac{kA_S^2}{2} </math><math>E_S = \tfrac{kA_W^2}{2} \rightarrow E_W \propto A_W</math><br />
=== '''Seismic Interpretation''' ===<br />
The average energy attribute of seismic waves is a measure of reflectivity in the specified time window. According to the above description, the higher the energy should have the higher the amplitude. This attribute enhances lateral variations within seismic events among others (Figure 3), therefore, it’s useful for seismic object detection, for instance, amplitude anomalies, chimney detection, etc. The response energy also characterizes acoustic rock properties and bed thickness.<br />
<br />
The average energy is often found to correlate strongly with liquid saturation (oil/water vs. gas) because those reservoir properties have a strong effect on both velocity and density, and energy of seismic reflections are generated at boundaries where the acoustic impedance (the product of velocity and density) changes, thus, it’s commonly used in direct hydrocarbon indicators. In general, the values of average energy are not important, and often not cited, because it is the relative value of an attribute along a given horizon or interval that is important. [[File:AE.jpg|thumb|Figure 3 Average energy extraction in a 100ms window (25 samples) at a constant time (Barnes,2006). ]]<br />
<br />
=== '''References''' ===<br />
# A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
# Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
# Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
# SEG: Methods and Applications in Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
# <nowiki>https://www.physicsclassroom.com/class/waves/Lesson-2/Energy-Transport-and-the-Amplitude-of-a-Wave</nowiki><br />
# <nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
# <nowiki>https://www.geoinsights.com/tag/direct-hydrocarbon-indicators/</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=File:AE-1.jpg&diff=138422File:AE-1.jpg2019-10-26T08:11:17Z<p>Zhai0629: </p>
<hr />
<div>AE-1</div>Zhai0629https://wiki.seg.org/index.php?title=Average_energy&diff=138421Average energy2019-10-26T08:09:08Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018.<br />
<br />
The stacked seismic data volume is commonly used for the interpretation of geologic structure and seismic attributes. The average energy of amplitude is one post-stack attribute that is often used in direct hydrocarbon indicators (DHIs), although its interpretation in thin-layered beds is not necessarily straightforward.<br />
<br />
=== '''Definition''' ===<br />
In physics, wave is a periodic disturbance in a medium or in space. In a progressive wave, energy is transferred from one place to another by the vibrations.<sup>[1]</sup> Like light waves, sound waves, or transverse oscillations of a string, these disturbances take energy to create and propagate, in order to move the constituent particles or change the electric/magnetic fields. The energy transported by a wave is directly proportional to the square of the amplitude of the wave.<br />
<br />
Seismic wave is an elastic disturbance that is propagated from point to point through a medium.<sup>[2]</sup> The average energy attribute of seismic wave is calculated by adding the square of each sample, then dividing by the number of samples in the window to yield the mean. <sup>[3]</sup>[[File:Amplitude Energy.jpg|thumb|Figure 1(a) Wavelength (λ), frequency (ν, labeled in Hz), and amplitude are indicated on this drawing of a wave. (b) The wave with the shortest wavelength has the greatest number of wavelengths per unit time (i.e., the highest frequency). If two waves have the same frequency and speed, the one with the greater amplitude has the higher energy.]]<br />
<br />
=== '''Mathematical Expression''' ===<br />
This attribute calculates the squared sum of the sample values in the specified time window divided by the number of samples in the window.<br />
<br />
<math>Average Energy = \tfrac{\sum_{i=1}^n A_i^2}{n} </math><br />
<br />
Where A<sub>i</sub> is the amplitude of the sampling point in a given time window, n is the number of sampling points.<br />
<br />
=== '''Physical Description''' ===<br />
The total mechanical energy of the wave is the sum of its kinetic energy and potential energy. The energy associated with a traveling wave in a stretched string is conveniently expressed as the energy per wavelength. The energy of a small segment of the string can be expressed as the sum of the kinetic energy and elastic potential energy of the segment (Figure 1).<br />
<br />
According to the kinetic energy (E<sub>K</sub>) formula <br />
<br />
Where m is the mass and v is the velocity of the body. The mass element oscillates perpendicular to the direction of the motion of the wave. Using the constant linear mass density, the kinetic energy of each mass element of the string with length Δx is<br />
<br />
Where μ is density of the particle, each mass element of the string has the mass element of the string oscillates with a velocity v<sub>y</sub>. When the mass element of the string approach zero <br />
<br />
For a sinusoidal wave with simple harmonic motion, the position of each mass element may be modeled as<br />
<br />
Where ω is the angular frequency in time, ω =2πf. n is the wave number. The velocity of each mass element of the string oscillates is<br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes<br />
<br />
The energy for a full wavelength can be found by integrating this expression at a given time, and it is most convenient to set t=0 for this integration. The energy for one wavelength along the string is<br />
<br />
Thus, it’s obtained that<br />
<br />
According to the potential energy formula <br />
<br />
The simple and harmonic motion with an angular frequency (ω) given by<br />
<br />
Where k is the spring constant, m is the mass of the object, bring it into the formula<br />
<br />
The differential form of the elastic potential energy is<br />
<br />
It’s obtained that<br />
<br />
Finally, the total energy associated with a wavelength is the sum of the kinetic energy (E<sub>K</sub>) and the potential energy (E<sub>P</sub>)<br />
<br />
This shows a wave is an energy transport phenomenon which transports energy along with a medium without transporting matter. The amount of energy transferred by a wave is related to the amplitude of the wave. A high energy wave is featured by a high amplitude; a low energy wave is featured by a low amplitude. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. Assuming that the frequency of the wave is unchanged, an increase in amplitude will result in an increase in energy and a decrease in amplitude will result in a decrease in energy. The energy of a wave is directly proportional to the square of the amplitude it contains: <br />
<br />
The energy of the wave depends on both the amplitude and the frequency. If the energy of each wavelength is a discrete packet of energy, a high-frequency wave will deliver more of these packets per unit time than a low-frequency wave. The average rate of energy transfer in mechanical waves is proportional to both the square of the amplitude and the square of the frequency. If two mechanical waves have equal amplitudes, but one wave has a frequency equal to twice the frequency of the other, the higher-frequency wave will have a rate of energy transfer a factor of four times as great as the rate of energy transfer of the lower-frequency wave. A doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave. A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave. And a quadrupling of the amplitude of a wave is indicative of a 16-fold increase in the amount of energy transported by the wave. Thus, whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared (Figure 2). <br />
<br />
Different materials also have differing degrees of elasticity. A more elastic medium will tend to offer less resistance to the force and allow a greater amplitude pulse to travel through it; being less rigid (and therefore more elastic), the same force causes a greater amplitude.<math>E \propto A^2</math><math>E_S = PE_S = \tfrac{kx^2}{2}</math><math>E_S = \tfrac{kA_S^2}{2} </math><math>E_S = \tfrac{kA_W^2}{2} \rightarrow E_W \propto A_W</math><br />
=== '''Seismic Interpretation''' ===<br />
The average energy attribute of seismic waves is a measure of reflectivity in the specified time window. According to the above description, the higher the energy should have the higher the amplitude. This attribute enhances lateral variations within seismic events among others (Figure 3), therefore, it’s useful for seismic object detection, for instance, amplitude anomalies, the chimney detection, etc. The response energy also characterizes acoustic rock properties and bed thickness.<br />
<br />
The average energy is often found to correlate strongly with liquid saturation (oil/water vs. gas) because those reservoir properties have a strong effect on both velocity and density, and energy of seismic reflections are generated at boundaries where the acoustic impedance (the product of velocity and density) changes, thus, it’s commonly used in direct hydrocarbon indicators. In general, the values of average energy are not important, and often not cited, because it is the relative value of an attribute along a given horizon or interval that is important. [[File:AE.jpg|thumb|Figure 3 Average energy extraction in a 100ms window (25 samples) at a constant time (Barnes,2006). ]]<br />
<br />
=== '''References''' ===<br />
# A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
# Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
# Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
# SEG: Methods and Applications in Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
# <nowiki>https://www.physicsclassroom.com/class/waves/Lesson-2/Energy-Transport-and-the-Amplitude-of-a-Wave</nowiki><br />
# <nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
# <nowiki>https://www.geoinsights.com/tag/direct-hydrocarbon-indicators/</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=138211RMS amplitude2019-10-23T22:53:44Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[1]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is times the peak amplitude.<sup>[2]</sup><math>\tfrac{1}{\sqrt{2}}</math><br />
[[File:Picture-new.png|thumb|Figure 1 Graph of a sine wave's amplitude vs. time, showing RMS(A<sub>RMS</sub>), peak (A<sub>PK</sub>), and peak-to-peak (A<sub>PP</sub>) amplitude.]]<br />
<br />
=== Mathematical Expression ===<br />
<u>I give the detailed derivation process in this step, if you feel cumbersome, please let me know, I can delete some.</u><br />
<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[3]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x_1,x_2,…,x_n}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)，ω is the angular frequency, specifies how many oscillations occur in a unit time interval. The RMS of the sine wave can be calculated as follows <br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
Using the power reduction formula <br />
<br />
<math>sin^2 (\omega t) = \tfrac{1 -cos(2\omega t)}{2}</math><br />
<br />
<br />
The RMS of the sine wave becomes<br />
<br />
<math> = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
According to the trigonometric Integral<br />
<br />
<math>\int cos(2\omega t) dt = \tfrac{1}{2\omega} sin(2\omega t)</math><br />
<br />
The RMS can be converted to<br />
<br />
<math>Y_{RMS} = A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
According to the periodicity of the sine function<br />
<br />
<math>sin(2\omega t)|_0^T = sin (2 \tfrac{2\pi}{2} T) = 0</math> <br />
<br />
<br />
Finally, it’s obtained that <br />
<br />
<math>Y_{RMS} = \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS is 0.707 times the peak amplitude <br />
<br />
The RMS over all time of a periodic function is equal to the RMS of one period of the function, its value of a continuous function can be approximated by taking the RMS of a sample consisting of equally spaced observations, thus, the RMS value of various waveforms can also be determined without calculus. <sup>[3]</sup> For instance, the expected value is used instead of the mean, the corresponding formula for a wave’s RMS amplitude defined over a time interval is <br />
<br />
<math>A_{RMS} = \sqrt{\tfrac{\sum_{i=1}^n A_i^2 * w_i }{\sum_{i=1}^n w_i}}</math> <br />
<br />
Where ''A'' = amplitude, ''n'' = number of samples, is the weight of each amplitude value. <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values. when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. <br />
<br />
RMS amplitude analyze the overall amplitude of a signal, conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefor, they are all positive, then the signal average is calculated, eventually followed by the square root operation. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
<br />
=== Seismic Interpretation ===<br />
In seismic processing, automatic gain control (AGC) method is often used in data processing to improve the visibility of seismic data in which attenuation or spherical divergence has caused amplitude decay, which is a system that controls the increase in the amplitude of an electrical signal from the original input to the amplified output, automatically (figure 2).[4] RMS AGC is used for amplitude equalization. It gives the processor a measure of the overall absolute amplitude in the window, both as positive and as negative values. During the processing, the data processor assume that the average absolute reflectivity varies little over a long time window, so a long-gate AGC can be applied to avoids destroying the lateral variation in amplitude, and ensure that the window include many reflectors so that the target event makes very little contribution to the average amplitude in the window. For instance, RMS AGC is often applied on structural imaging.<br />
[[File:2019-10-23 12-41-48.jpg|thumb|Figure 2 Seismic data without AGC (left) and the same data after AGC is applied (Onajite,2013)]]<br />
For seismic integration, RMS is a most commonly used post stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage.[5] This has splendid applications for various reconnaissance endeavors. The squaring of the amplitude values within the window gives the high amplitudes maximum opportunity to stand out above the background contamination. The RMS amplitude over a large flat or structured time window can be effectively used to identify bright spots at different levels within a formation (Figure3).<br />
[[File:Picture3.jpg|thumb|Figure 3 Top Balder Formation amplitude map from multiple 3-D surveys covering 5500 sq km of the Faroe Basin west of Shetland in U.K. waters. The colors show RMS amplitude over a 50-ms window surrounding the Top Balder reflection, with red indicating high amplitude and blue indicating low amplitude (Brown,2004)]]<br />
RMS amplitude is popular type of windowed amplitude attributes, the window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view. In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide. The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009. <br />
<br />
[2] Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
<br />
[3] Weisstein, Eric W. "Root-Mean-Square". MathWorld.<br />
<br />
[4] Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms" MasteringElectronicsDesign.com. Retrieved 21 January 2015 <br />
<br />
[5] Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
<br />
[6] Pitfalls in the study of seismic amplitude<br />
<br />
[7] SEG: Methods and Applications in Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
<nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
<br />
[[Rms amplitude AGC|https://wiki.seg.org/wiki/Rms_amplitude_AGC]]<br />
<br />
<nowiki>https://community.sw.siemens.com/s/article/root-mean-square-rms-and-overall-level</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=Average_energy&diff=138194Average energy2019-10-23T20:16:49Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018.<br />
<br />
The stacked seismic data volume is commonly used for the interpretation of geologic structure and seismic attributes. the average energy of amplitude is a post-stack attribute that is often used, although its interpretation in thin-layered beds is not necessarily straightforward.<br />
<br />
=== '''Definition''' ===<br />
This attribute is calculated by adding the square of each sample, then dividing by the number of samples in the window to yield the mean.<br />
[[File:Amplitude Energy.jpg|thumb|Figure 1(a) Wavelength (λ), frequency (ν, labeled in Hz), and amplitude are indicated on this drawing of a wave. (b) The wave with the shortest wavelength has the greatest number of wavelengths per unit time (i.e., the highest frequency). If two waves have the same frequency and speed, the one with the greater amplitude has the higher energy.]]<br />
<br />
=== '''Mathematical Expression''' ===<br />
This attribute calculates the squared sum of the sample values in the specified time-gate divided by the number of samples in the gate.<br />
<br />
<math>Average Energy = \tfrac{\sum_{i=1}^n A_i^2}{n} </math><br />
<br />
Where Ai is the amplitude of the sampling point in a given time window, n is the number of sampling points.<br />
<br />
=== '''Physical Description''' ===<br />
A wave is an energy transport phenomenon which transports energy along with a medium without transporting matter. The amount of energy transferred by a wave is related to the amplitude of the wave. A high energy wave is featured by a high amplitude; a low energy wave is featured by a low amplitude. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. <br />
<br />
Assuming that the frequency of the wave is unchanged, an increase in amplitude will result in an increase in energy and a decrease in amplitude will result in a decrease in energy. The energy of a wave is directly proportional to the square of the amplitude it contains: <br />
<br />
<math>E \propto A^2</math><br />
<br />
A most common example of a simple harmonic wave - the pendulum. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period (T). The period depends on the length of the pendulum and also to a slight degree on the amplitude, the width of the pendulum's swing. The amplitude (Aw) of the wave corresponds to the farthest displacement from equilibrium the massive bob can go (As) and the energy of the wave (Ew) corresponds to the amount of mechanical energy the massive bob has (Es).<br />
<br />
At the point of maximum amplitude (Aw) during the pendulum's oscillation, all of the pendulum's energy is potential energy (PEs), <br />
<br />
<math>E_S = PE_S = \tfrac{kx^2}{2}</math><br />
<br />
However, the x (displacement from equilibrium) is equal to the amplitude of the pendulum, <br />
<br />
<math>E_S = \tfrac{kA_S^2}{2} </math><br />
<br />
Since the energy of the wave (Ew) corresponds to mechanical energy the pendulum has (Es) and the point of maximum amplitude (Aw) corresponds to the farthest displacement from equilibrium the massive bob (As). Thus,<br />
<br />
<math>E_S = \tfrac{kA_W^2}{2} \rightarrow E_W \propto A_W</math><br />
<br />
This means that a doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave. A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave. And a quadrupling of the amplitude of a wave is indicative of a 16-fold increase in the amount of energy transported by the wave. Thus, whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared. <br />
<br />
Different materials also have differing degrees of elasticity. A more elastic medium will tend to offer less resistance to the force and allow a greater amplitude pulse to travel through it; being less rigid (and therefore more elastic), the same force causes a greater amplitude.<br />
<br />
=== '''Seismic Interpretation''' ===<br />
The average energy is a measure of reflectivity in the specified time-gate. The higher the energy, the higher the amplitude. This attribute enhances, among others, lateral variations within seismic events and is, therefore, useful for seismic object detection, for instance the chimney detection. The response energy also characterizes acoustic rock properties and bed thickness.<br />
[[File:AE.jpg|thumb|Figure 2 Average energy result in a 100ms window (25 samples) at a constant time (Arthur Barnes,2006). ]]<br />
<br />
The average energy is often found to correlate strongly with liquid saturation (oil/water vs. gas) because those reservoir properties have a strong effect on both velocity and density, and energy of seismic reflections are generated at boundaries where the acoustic impedance (the product of velocity and density) changes. In general, the values of average energy are not important, and often not cited, because it is the relative value of an attribute along a given horizon or interval that is important.<br />
<br />
=== '''References''' ===<br />
# A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
# Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
# Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
# SEG: Methods and Applications in Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
# <nowiki>https://www.physicsclassroom.com/class/waves/Lesson-2/Energy-Transport-and-the-Amplitude-of-a-Wave</nowiki><br />
# <nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
# <nowiki>https://www.geoinsights.com/tag/direct-hydrocarbon-indicators/</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=Average_energy&diff=138193Average energy2019-10-23T20:15:32Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018.<br />
<br />
The stacked seismic data volume is commonly used for the interpretation of geologic structure and seismic attributes. the average energy of amplitude is a post-stack attribute that is often used, although its interpretation in thin-layered beds is not necessarily straightforward.<br />
<br />
=== '''Definition''' ===<br />
This attribute is calculated by adding the square of each sample, then dividing by the number of samples in the window to yield the mean.<br />
[[File:Amplitude Energy.jpg|thumb|Figure 1(a) Wavelength (λ), frequency (ν, labeled in Hz), and amplitude are indicated on this drawing of a wave. (b) The wave with the shortest wavelength has the greatest number of wavelengths per unit time (i.e., the highest frequency). If two waves have the same frequency and speed, the one with the greater amplitude has the higher energy.]]<br />
<br />
=== '''Mathematical Expression''' ===<br />
This attribute calculates the squared sum of the sample values in the specified time-gate divided by the number of samples in the gate.<br />
<br />
<math>Average Energy = \tfrac{\sum_{i=1}^n A_i^2}{n} </math><br />
<br />
Where Ai is the amplitude of the sampling point in a given time window, n is the number of sampling points.<br />
<br />
=== '''Physical Description''' ===<br />
A wave is an energy transport phenomenon which transports energy along with a medium without transporting matter. The amount of energy transferred by a wave is related to the amplitude of the wave. A high energy wave is featured by a high amplitude; a low energy wave is featured by a low amplitude. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. <br />
<br />
Assuming that the frequency of the wave is unchanged, an increase in amplitude will result in an increase in energy and a decrease in amplitude will result in a decrease in energy. The energy of a wave is directly proportional to the square of the amplitude it contains: <br />
<br />
<math>E \propto A^2</math><br />
<br />
A most common example of a simple harmonic wave - the pendulum. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period (T). The period depends on the length of the pendulum and also to a slight degree on the amplitude, the width of the pendulum's swing. The amplitude (Aw) of the wave corresponds to the farthest displacement from equilibrium the massive bob can go (As) and the energy of the wave (Ew) corresponds to the amount of mechanical energy the massive bob has (Es).<br />
<br />
At the point of maximum amplitude (Aw) during the pendulum's oscillation, all of the pendulum's energy is potential energy (PEs), <br />
<br />
<math>E_S = PE_S = \tfrac{kx^2}{2}</math><br />
<br />
However, the x (displacement from equilibrium) is equal to the amplitude of the pendulum, <br />
<br />
<math>E_S = \tfrac{kA_S^2}{2} </math><br />
<br />
Since the energy of the wave (Ew) corresponds to mechanical energy the pendulum has (Es) and the point of maximum amplitude (Aw) corresponds to the farthest displacement from equilibrium the massive bob (As). Thus,<br />
<br />
<math>E_S = \tfrac{kA_W^2}{2} \rightarrow E_W \propto A_W</math><br />
<br />
This means that a doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave. A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave. And a quadrupling of the amplitude of a wave is indicative of a 16-fold increase in the amount of energy transported by the wave. Thus, whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared. <br />
<br />
Different materials also have differing degrees of elasticity. A more elastic medium will tend to offer less resistance to the force and allow a greater amplitude pulse to travel through it; being less rigid (and therefore more elastic), the same force causes a greater amplitude.<br />
<br />
=== '''Seismic Interpretation''' ===<br />
The average energy is a measure of reflectivity in the specified time-gate. The higher the energy, the higher the amplitude. This attribute enhances, among others, lateral variations within seismic events and is, therefore, useful for seismic object detection, for instance the chimney detection. The response energy also characterizes acoustic rock properties and bed thickness.<br />
[[File:AE.jpg|thumb|Figure 2 Average energy result in a 100ms window (25 samples) at a constant time (Arthur Barnes,2006). ]]<br />
<br />
The average energy is often found to correlate strongly with liquid saturation (oil/water vs. gas) because those reservoir properties have a strong effect on both velocity and density, and energy of seismic reflections are generated at boundaries where the acoustic impedance (the product of velocity and density) changes. In general, the values of average energy are not important, and often not cited, because it is the relative value of an attribute along a given horizon or interval that is important.<br />
<br />
=== '''References''' ===<br />
# A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
# Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
# Weisstein, Eric W. "Root-Mean-Square". MathWorld.</div>Zhai0629https://wiki.seg.org/index.php?title=File:AE.jpg&diff=138192File:AE.jpg2019-10-23T20:14:30Z<p>Zhai0629: </p>
<hr />
<div>ae</div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=138188RMS amplitude2019-10-23T19:24:03Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[1]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is times the peak amplitude.<sup>[2]</sup><math>\tfrac{1}{\sqrt{2}}</math><br />
[[File:Picture-new.png|thumb|Figure 1 Graph of a sine wave's amplitude vs. time, showing RMS(A<sub>RMS</sub>), peak (A<sub>PK</sub>), and peak-to-peak (A<sub>PP</sub>) amplitude.]]<br />
<br />
=== Mathematical Expression ===<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[3]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x_1,x_2,…,x_n}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)，ω is the angular frequency, specifies how many oscillations occur in a unit time interval. The RMS of the sine wave can be calculated as follows <br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
Using the power reduction formula <br />
<br />
<math>sin^2 (\omega t) = \tfrac{1 -cos(2\omega t)}{2}</math><br />
<br />
<br />
The RMS of the sine wave becomes<br />
<br />
<math> = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
According to the trigonometric Integral<br />
<br />
<math>\int cos(2\omega t) dt = \tfrac{1}{2\omega} sin(2\omega t)</math><br />
<br />
The RMS can be converted to<br />
<br />
<math>Y_{RMS} = A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
According to the periodicity of the sine function<br />
<br />
<math>sin(2\omega t)|_0^T = sin (2 \tfrac{2\pi}{2} T) = 0</math> <br />
<br />
<br />
Finally, it’s obtained that <br />
<br />
<math>Y_{RMS} = \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS is 0.707 times the peak amplitude <br />
<br />
The RMS over all time of a periodic function is equal to the RMS of one period of the function, its value of a continuous function can be approximated by taking the RMS of a sample consisting of equally spaced observations, thus, the RMS value of various waveforms can also be determined without calculus. <sup>[3]</sup> For instance, the expected value is used instead of the mean, the corresponding formula for a wave’s RMS amplitude defined over a time interval is <br />
<br />
<math>A_{RMS} = \sqrt{\tfrac{\sum_{i=1}^n A_i^2 * w_i }{\sum_{i=1}^n w_i}}</math> <br />
<br />
Where ''A'' = amplitude, ''n'' = number of samples, is the weight of each amplitude value. <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values. when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. <br />
<br />
RMS amplitude analyze the overall amplitude of a signal, conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefor, they are all positive, then the signal average is calculated, eventually followed by the square root operation. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
<br />
=== Seismic Interpretation ===<br />
In seismic processing, automatic gain control (AGC) method is often used in data processing to improve the visibility of seismic data in which attenuation or spherical divergence has caused amplitude decay, which is a system that controls the increase in the amplitude of an electrical signal from the original input to the amplified output, automatically (figure 2).[4] RMS AGC is used for amplitude equalization. It gives the processor a measure of the overall absolute amplitude in the window, both as positive and as negative values. During the processing, the data processor assume that the average absolute reflectivity varies little over a long time window, so a long-gate AGC can be applied to avoids destroying the lateral variation in amplitude, and ensure that the window include many reflectors so that the target event makes very little contribution to the average amplitude in the window. For instance, RMS AGC is often applied on structural imaging.<br />
[[File:2019-10-23 12-41-48.jpg|thumb|Figure 2 Seismic data without AGC (left) and the same data after AGC is applied (Onajite,2013)]]<br />
For seismic integration, RMS is a most commonly used post stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage.[5] This has splendid applications for various reconnaissance endeavors. The squaring of the amplitude values within the window gives the high amplitudes maximum opportunity to stand out above the background contamination. The RMS amplitude over a large flat or structured time window can be effectively used to identify bright spots at different levels within a formation (Figure3).<br />
[[File:Picture3.jpg|thumb|Figure 3 Top Balder Formation amplitude map from multiple 3-D surveys covering 5500 sq km of the Faroe Basin west of Shetland in U.K. waters. The colors show RMS amplitude over a 50-ms window surrounding the Top Balder reflection, with red indicating high amplitude and blue indicating low amplitude (Brown,2004)]]<br />
RMS amplitude is popular type of windowed amplitude attributes, the window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view. In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide. The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009. <br />
<br />
[2] Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
<br />
[3] Weisstein, Eric W. "Root-Mean-Square". MathWorld.<br />
<br />
[4] Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms" MasteringElectronicsDesign.com. Retrieved 21 January 2015 <br />
<br />
[5] Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
<br />
[6] Pitfalls in the study of seismic amplitude<br />
<br />
[7] SEG: Methods and Applications in Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
<nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
<br />
[[Rms amplitude AGC|https://wiki.seg.org/wiki/Rms_amplitude_AGC]]<br />
<br />
<nowiki>https://community.sw.siemens.com/s/article/root-mean-square-rms-and-overall-level</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=File:Picture-new.png&diff=138187File:Picture-new.png2019-10-23T19:23:44Z<p>Zhai0629: </p>
<hr />
<div>Picture-new</div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=138186RMS amplitude2019-10-23T19:21:10Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[1]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is <math>\tfrac{1}{\sqrt{2}}</math> times the peak amplitude.<sup>[2]</sup><br />
<br />
Figure cannot be insert<br />
<br />
=== Mathematical Expression ===<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[3]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x_1,x_2,…,x_n}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)，ω is the angular frequency, specifies how many oscillations occur in a unit time interval. The RMS of the sine wave can be calculated as follows <br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
Using the power reduction formula <br />
<br />
<math>sin^2 (\omega t) = \tfrac{1 -cos(2\omega t)}{2}</math><br />
<br />
<br />
The RMS of the sine wave becomes<br />
<br />
<math> = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
According to the trigonometric Integral<br />
<br />
<math>\int cos(2\omega t) dt = \tfrac{1}{2\omega} sin(2\omega t)</math><br />
<br />
The RMS can be converted to<br />
<br />
<math>Y_{RMS} = A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
According to the periodicity of the sine function<br />
<br />
<math>sin(2\omega t)|_0^T = sin (2 \tfrac{2\pi}{2} T) = 0</math> <br />
<br />
<br />
Finally, it’s obtained that <br />
<br />
<math>Y_{RMS} = \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS is 0.707 times the peak amplitude <br />
<br />
The RMS over all time of a periodic function is equal to the RMS of one period of the function, its value of a continuous function can be approximated by taking the RMS of a sample consisting of equally spaced observations, thus, the RMS value of various waveforms can also be determined without calculus. <sup>[3]</sup> For instance, the expected value is used instead of the mean, the corresponding formula for a wave’s RMS amplitude defined over a time interval is <br />
<br />
<math>A_{RMS} = \sqrt{\tfrac{\sum_{i=1}^n A_i^2 * w_i }{\sum_{i=1}^n w_i}}</math> <br />
<br />
Where ''A'' = amplitude, ''n'' = number of samples, is the weight of each amplitude value. <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values. when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. <br />
<br />
RMS amplitude analyze the overall amplitude of a signal, conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefor, they are all positive, then the signal average is calculated, eventually followed by the square root operation. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
<br />
=== Seismic Interpretation ===<br />
In seismic processing, automatic gain control (AGC) method is often used in data processing to improve the visibility of seismic data in which attenuation or spherical divergence has caused amplitude decay, which is a system that controls the increase in the amplitude of an electrical signal from the original input to the amplified output, automatically (figure 2).[4] RMS AGC is used for amplitude equalization. It gives the processor a measure of the overall absolute amplitude in the window, both as positive and as negative values. During the processing, the data processor assume that the average absolute reflectivity varies little over a long time window, so a long-gate AGC can be applied to avoids destroying the lateral variation in amplitude, and ensure that the window include many reflectors so that the target event makes very little contribution to the average amplitude in the window. For instance, RMS AGC is often applied on structural imaging.<br />
[[File:2019-10-23 12-41-48.jpg|thumb|Figure 2 Seismic data without AGC (left) and the same data after AGC is applied (Onajite,2013)]]<br />
For seismic integration, RMS is a most commonly used post stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage.[5] This has splendid applications for various reconnaissance endeavors. The squaring of the amplitude values within the window gives the high amplitudes maximum opportunity to stand out above the background contamination. The RMS amplitude over a large flat or structured time window can be effectively used to identify bright spots at different levels within a formation (Figure3).<br />
[[File:Picture3.jpg|thumb|Figure 3 Top Balder Formation amplitude map from multiple 3-D surveys covering 5500 sq km of the Faroe Basin west of Shetland in U.K. waters. The colors show RMS amplitude over a 50-ms window surrounding the Top Balder reflection, with red indicating high amplitude and blue indicating low amplitude (Brown,2004)]]<br />
RMS amplitude is popular type of windowed amplitude attributes, the window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view. In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide. The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009. <br />
<br />
[2] Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
<br />
[3] Weisstein, Eric W. "Root-Mean-Square". MathWorld.<br />
<br />
[4] Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms" MasteringElectronicsDesign.com. Retrieved 21 January 2015 <br />
<br />
[5] Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
<br />
[6] Pitfalls in the study of seismic amplitude<br />
<br />
[7] SEG: Methods and Applications in Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
<nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
<br />
[[Rms amplitude AGC|https://wiki.seg.org/wiki/Rms_amplitude_AGC]]<br />
<br />
<nowiki>https://community.sw.siemens.com/s/article/root-mean-square-rms-and-overall-level</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=138185RMS amplitude2019-10-23T19:18:45Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[1]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is times the peak amplitude.<sup>[2]</sup><br />
<br />
<math>\tfrac{1}{\sqrt{2}}</math><br />
=== Mathematical Expression ===<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[3]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x_1,x_2,…,x_n}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)，ω is the angular frequency, specifies how many oscillations occur in a unit time interval. The RMS of the sine wave can be calculated as follows <br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
Using the power reduction formula <br />
<br />
<math>sin^2 (\omega t) = \tfrac{1 -cos(2\omega t)}{2}</math><br />
<br />
<br />
The RMS of the sine wave becomes<br />
<br />
<math> = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
According to the trigonometric Integral<br />
<br />
<math>\int cos(2\omega t) dt = \tfrac{1}{2\omega} sin(2\omega t)</math><br />
<br />
The RMS can be converted to<br />
<br />
<math>Y_{RMS} = A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
According to the periodicity of the sine function<br />
<br />
<math>sin(2\omega t)|_0^T = sin (2 \tfrac{2\pi}{2} T) = 0</math> <br />
<br />
<br />
Finally, it’s obtained that <br />
<br />
<math>Y_{RMS} = \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS is 0.707 times the peak amplitude <br />
<br />
The RMS over all time of a periodic function is equal to the RMS of one period of the function, its value of a continuous function can be approximated by taking the RMS of a sample consisting of equally spaced observations, thus, the RMS value of various waveforms can also be determined without calculus. <sup>[3]</sup> For instance, the expected value is used instead of the mean, the corresponding formula for a wave’s RMS amplitude defined over a time interval is <br />
<br />
<math>A_{RMS} = \sqrt{\tfrac{\sum_{i=1}^n A_i^2 * w_i }{\sum_{i=1}^n w_i}}</math> <br />
<br />
Where ''A'' = amplitude, ''n'' = number of samples, is the weight of each amplitude value. <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values. when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. <br />
<br />
RMS amplitude analyze the overall amplitude of a signal, conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefor, they are all positive, then the signal average is calculated, eventually followed by the square root operation. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
<br />
=== Seismic Interpretation ===<br />
In seismic processing, automatic gain control (AGC) method is often used in data processing to improve the visibility of seismic data in which attenuation or spherical divergence has caused amplitude decay, which is a system that controls the increase in the amplitude of an electrical signal from the original input to the amplified output, automatically (figure 2).[4] RMS AGC is used for amplitude equalization. It gives the processor a measure of the overall absolute amplitude in the window, both as positive and as negative values. During the processing, the data processor assume that the average absolute reflectivity varies little over a long time window, so a long-gate AGC can be applied to avoids destroying the lateral variation in amplitude, and ensure that the window include many reflectors so that the target event makes very little contribution to the average amplitude in the window. For instance, RMS AGC is often applied on structural imaging.<br />
[[File:2019-10-23 12-41-48.jpg|thumb|Figure 2 Seismic data without AGC (left) and the same data after AGC is applied (Onajite,2013)]]<br />
For seismic integration, RMS is a most commonly used post stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage.[5] This has splendid applications for various reconnaissance endeavors. The squaring of the amplitude values within the window gives the high amplitudes maximum opportunity to stand out above the background contamination. The RMS amplitude over a large flat or structured time window can be effectively used to identify bright spots at different levels within a formation (Figure3).<br />
[[File:Picture3.jpg|thumb|Figure 3 Top Balder Formation amplitude map from multiple 3-D surveys covering 5500 sq km of the Faroe Basin west of Shetland in U.K. waters. The colors show RMS amplitude over a 50-ms window surrounding the Top Balder reflection, with red indicating high amplitude and blue indicating low amplitude (Brown,2004)]]<br />
RMS amplitude is popular type of windowed amplitude attributes, the window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view. In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide. The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009. <br />
<br />
[2] Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
<br />
[3] Weisstein, Eric W. "Root-Mean-Square". MathWorld.<br />
<br />
[4] Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms" MasteringElectronicsDesign.com. Retrieved 21 January 2015 <br />
<br />
[5] Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
<br />
[6] Pitfalls in the study of seismic amplitude<br />
<br />
[7] SEG: Methods and Applications in Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
<nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
<br />
[[Rms amplitude AGC|https://wiki.seg.org/wiki/Rms_amplitude_AGC]]<br />
<br />
<nowiki>https://community.sw.siemens.com/s/article/root-mean-square-rms-and-overall-level</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=File:Picture1-1.jpg&diff=138184File:Picture1-1.jpg2019-10-23T19:15:43Z<p>Zhai0629: </p>
<hr />
<div>Picture1</div>Zhai0629https://wiki.seg.org/index.php?title=File:Picture3.jpg&diff=138183File:Picture3.jpg2019-10-23T19:13:34Z<p>Zhai0629: </p>
<hr />
<div>Picture3</div>Zhai0629https://wiki.seg.org/index.php?title=File:2019-10-23_12-41-48.jpg&diff=138181File:2019-10-23 12-41-48.jpg2019-10-23T19:12:20Z<p>Zhai0629: </p>
<hr />
<div>FIG2</div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=138180RMS amplitude2019-10-23T19:05:41Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[1]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is times the peak amplitude.<sup>[2]</sup><math>\tfrac{1}{\sqrt{2}}</math><br />
<br />
'''''Figure 1 Graph of a sine wave's amplitude vs. time, showing RMS, peak (PK), and peak-to-peak (PP) amplitude.'''''<br />
<br />
=== Mathematical Expression ===<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[3]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x_1,x_2,…,x_n}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)，ω is the angular frequency, specifies how many oscillations occur in a unit time interval. The RMS of the sine wave can be calculated as follows <br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
Using the power reduction formula <br />
<br />
<math>sin^2 (\omega t) = \tfrac{1 -cos(2\omega t)}{2}</math><br />
<br />
<br />
The RMS of the sine wave becomes<br />
<br />
<math> = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
According to the trigonometric Integral<br />
<br />
<math>\int cos(2\omega t) dt = \tfrac{1}{2\omega} sin(2\omega t)</math><br />
<br />
The RMS can be converted to<br />
<br />
<math>Y_{RMS} = A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
According to the periodicity of the sine function<br />
<br />
<math>sin(2\omega t)|_0^T = sin (2 \tfrac{2\pi}{2} T) = 0</math> <br />
<br />
<br />
Finally, it’s obtained that <br />
<br />
<math>Y_{RMS} = \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS is 0.707 times the peak amplitude <br />
<br />
The RMS over all time of a periodic function is equal to the RMS of one period of the function, its value of a continuous function can be approximated by taking the RMS of a sample consisting of equally spaced observations, thus, the RMS value of various waveforms can also be determined without calculus. <sup>[3]</sup> For instance, the expected value is used instead of the mean, the corresponding formula for a wave’s RMS amplitude defined over a time interval is <br />
<br />
<math>A_{RMS} = \sqrt{\tfrac{\sum_{i=1}^n A_i^2 * w_i }{\sum_{i=1}^n w_i}}</math> <br />
<br />
Where ''A'' = amplitude, ''n'' = number of samples, is the weight of each amplitude value. <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values. when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. <br />
<br />
RMS amplitude analyze the overall amplitude of a signal, conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefor, they are all positive, then the signal average is calculated, eventually followed by the square root operation. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
<br />
=== Seismic Interpretation ===<br />
In seismic processing, automatic gain control (AGC) method is often used in data processing to improve the visibility of seismic data in which attenuation or spherical divergence has caused amplitude decay, which is a system that controls the increase in the amplitude of an electrical signal from the original input to the amplified output, automatically (figure 2).[4] RMS AGC is used for amplitude equalization. It gives the processor a measure of the overall absolute amplitude in the window, both as positive and as negative values. During the processing, the data processor assume that the average absolute reflectivity varies little over a long time window, so a long-gate AGC can be applied to avoids destroying the lateral variation in amplitude, and ensure that the window include many reflectors so that the target event makes very little contribution to the average amplitude in the window. For instance, RMS AGC is often applied on structural imaging.<br />
<br />
'''''Figure 2 Seismic data without AGC (left) and the same data after AGC is applied (Onajite,2013)'''''<br />
<br />
For seismic integration, RMS is a most commonly used post stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage.[5] This has splendid applications for various reconnaissance endeavors. The squaring of the amplitude values within the window gives the high amplitudes maximum opportunity to stand out above the background contamination. The RMS amplitude over a large flat or structured time window can be effectively used to identify bright spots at different levels within a formation (Figure3).<br />
<br />
'''''Figure 3 Top Balder Formation amplitude map from multiple 3-D surveys covering 5500 sq km of the Faroe Basin west of Shetland in U.K. waters. The colors show RMS amplitude over a 50-ms window surrounding the Top Balder reflection, with red indicating high amplitude and blue indicating low amplitude (Brown,2004)'''''<br />
<br />
<br />
RMS amplitude is popular type of windowed amplitude attributes, the window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view. In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide. The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009. <br />
<br />
[2] Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
<br />
[3] Weisstein, Eric W. "Root-Mean-Square". MathWorld.<br />
<br />
[4] Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms" MasteringElectronicsDesign.com. Retrieved 21 January 2015 <br />
<br />
[5] Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
<br />
[6] Pitfalls in the study of seismic amplitude<br />
<br />
[7] SEG: Methods and Applications in Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
<nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
<br />
[[Rms amplitude AGC|https://wiki.seg.org/wiki/Rms_amplitude_AGC]]<br />
<br />
<nowiki>https://community.sw.siemens.com/s/article/root-mean-square-rms-and-overall-level</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=138179RMS amplitude2019-10-23T19:04:27Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[1]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is times the peak amplitude.<sup>[2]</sup><math>\tfrac{1}{\sqrt{2}}</math><br />
<br />
'''''Figure 1 Graph of a sine wave's amplitude vs. time, showing RMS, peak (PK), and peak-to-peak (PP) amplitude.'''''<br />
<br />
=== Mathematical Expression ===<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[3]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x_1,x_2,…,x_n}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)，ω is the angular frequency, specifies how many oscillations occur in a unit time interval. The RMS of the sine wave can be calculated as follows <br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
Using the power reduction formula <br />
<br />
<math>sin^2 (\omega t) = \tfrac{1 -cos(2\omega t)}{2}</math><br />
<br />
<br />
The RMS of the sine wave becomes<br />
<br />
<math> = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
According to the trigonometric Integral<br />
<br />
<math>\int cos(2\omega t) dt = \tfrac{1}{2\omega} sin(2\omega t)</math><br />
<br />
The RMS can be converted to<br />
<br />
<math>Y_{RMS} = A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
According to the periodicity of the sine function<br />
<br />
<math>sin(2\omega t)|_0^T = sin (2 \tfrac{2\pi}{2} T) = 0</math> <br />
<br />
<br />
Finally, it’s obtained that <br />
<br />
<math>Y_{RMS} = \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS is 0.707 times the peak amplitude <br />
<br />
The RMS over all time of a periodic function is equal to the RMS of one period of the function, its value of a continuous function can be approximated by taking the RMS of a sample consisting of equally spaced observations, thus, the RMS value of various waveforms can also be determined without calculus. <sup>[3]</sup> For instance, the expected value is used instead of the mean, the corresponding formula for a wave’s RMS amplitude defined over a time interval is <br />
<br />
<math>A_{RMS} = \sqrt{\tfrac{\sum_{i=1}^n a_i^2 * w_i }{\sum_{i=1}^n w_i}}</math> <br />
<br />
Where ''A'' = amplitude, ''n'' = number of samples, is the weight of each amplitude value. <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values. when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. <br />
<br />
RMS amplitude analyze the overall amplitude of a signal, conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefor, they are all positive, then the signal average is calculated, eventually followed by the square root operation. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
<br />
=== Seismic Interpretation ===<br />
In seismic processing, automatic gain control (AGC) method is often used in data processing to improve the visibility of seismic data in which attenuation or spherical divergence has caused amplitude decay, which is a system that controls the increase in the amplitude of an electrical signal from the original input to the amplified output, automatically (figure 2).[4] RMS AGC is used for amplitude equalization. It gives the processor a measure of the overall absolute amplitude in the window, both as positive and as negative values. During the processing, the data processor assume that the average absolute reflectivity varies little over a long time window, so a long-gate AGC can be applied to avoids destroying the lateral variation in amplitude, and ensure that the window include many reflectors so that the target event makes very little contribution to the average amplitude in the window. For instance, RMS AGC is often applied on structural imaging.<br />
<br />
'''''Figure 2 Seismic data without AGC (left) and the same data after AGC is applied (Onajite,2013)'''''<br />
<br />
For seismic integration, RMS is a most commonly used post stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage.[5] This has splendid applications for various reconnaissance endeavors. The squaring of the amplitude values within the window gives the high amplitudes maximum opportunity to stand out above the background contamination. The RMS amplitude over a large flat or structured time window can be effectively used to identify bright spots at different levels within a formation (Figure3).<br />
<br />
'''''Figure 3 Top Balder Formation amplitude map from multiple 3-D surveys covering 5500 sq km of the Faroe Basin west of Shetland in U.K. waters. The colors show RMS amplitude over a 50-ms window surrounding the Top Balder reflection, with red indicating high amplitude and blue indicating low amplitude (Brown,2004)'''''<br />
<br />
<br />
RMS amplitude is popular type of windowed amplitude attributes, the window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view. In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide. The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009. <br />
<br />
[2] Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
<br />
[3] Weisstein, Eric W. "Root-Mean-Square". MathWorld.<br />
<br />
[4] Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms" MasteringElectronicsDesign.com. Retrieved 21 January 2015 <br />
<br />
[5] Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
<br />
[6] Pitfalls in the study of seismic amplitude<br />
<br />
[7] SEG: Methods and Applications in Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
<nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
<br />
[[Rms amplitude AGC|https://wiki.seg.org/wiki/Rms_amplitude_AGC]]<br />
<br />
<nowiki>https://community.sw.siemens.com/s/article/root-mean-square-rms-and-overall-level</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=138178RMS amplitude2019-10-23T19:00:39Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[1]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is times the peak amplitude.<sup>[2]</sup><math>\tfrac{1}{\sqrt{2}}</math><br />
<br />
'''''Figure 1 Graph of a sine wave's amplitude vs. time, showing RMS, peak (PK), and peak-to-peak (PP) amplitude.'''''<br />
<br />
=== Mathematical Expression ===<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[3]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x_1,x_2,…,x_n}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)，ω is the angular frequency, specifies how many oscillations occur in a unit time interval. The RMS of the sine wave can be calculated as follows <br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
Using the power reduction formula <br />
<br />
<math>sin^2 (\omega t) = \tfrac{1 -cos(2\omega t)}{2}</math><br />
<br />
<br />
The RMS of the sine wave becomes<br />
<br />
<math> = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
According to the trigonometric Integral<br />
<br />
<math>\int cos(2\omega t) dt = \tfrac{1}{2\omega} sin(2\omega t)</math><br />
<br />
The RMS can be converted to<br />
<br />
<math>Y_{RMS} = A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
According to the periodicity of the sine function<br />
<br />
<math>sin(2\omega t)|_0^T = sin (2 \tfrac{2\pi}{2} T) = 0</math> <br />
<br />
<br />
Finally, it’s obtained that <br />
<br />
<math>Y_{RMS} = \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS is 0.707 times the peak amplitude <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values. when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. <br />
<br />
RMS amplitude analyze the overall amplitude of a signal, conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefor, they are all positive, then the signal average is calculated, eventually followed by the square root operation. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
<br />
=== Seismic Interpretation ===<br />
In seismic processing, automatic gain control (AGC) method is often used in data processing to improve the visibility of seismic data in which attenuation or spherical divergence has caused amplitude decay, which is a system that controls the increase in the amplitude of an electrical signal from the original input to the amplified output, automatically (figure 2).[4] RMS AGC is used for amplitude equalization. It gives the processor a measure of the overall absolute amplitude in the window, both as positive and as negative values. During the processing, the data processor assume that the average absolute reflectivity varies little over a long time window, so a long-gate AGC can be applied to avoids destroying the lateral variation in amplitude, and ensure that the window include many reflectors so that the target event makes very little contribution to the average amplitude in the window. For instance, RMS AGC is often applied on structural imaging.<br />
<br />
'''''Figure 2 Seismic data without AGC (left) and the same data after AGC is applied (Onajite,2013)'''''<br />
<br />
For seismic integration, RMS is a most commonly used post stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage.[5] This has splendid applications for various reconnaissance endeavors. The squaring of the amplitude values within the window gives the high amplitudes maximum opportunity to stand out above the background contamination. The RMS amplitude over a large flat or structured time window can be effectively used to identify bright spots at different levels within a formation (Figure3).<br />
<br />
'''''Figure 3 Top Balder Formation amplitude map from multiple 3-D surveys covering 5500 sq km of the Faroe Basin west of Shetland in U.K. waters. The colors show RMS amplitude over a 50-ms window surrounding the Top Balder reflection, with red indicating high amplitude and blue indicating low amplitude (Brown,2004)'''''<br />
<br />
<br />
RMS amplitude is popular type of windowed amplitude attributes, the window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view. In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide. The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009. <br />
<br />
[2] Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
<br />
[3] Weisstein, Eric W. "Root-Mean-Square". MathWorld.<br />
<br />
[4] Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms" MasteringElectronicsDesign.com. Retrieved 21 January 2015 <br />
<br />
[5] Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
<br />
[6] Pitfalls in the study of seismic amplitude<br />
<br />
[7] SEG: Methods and Applications in Reservoir Geophysics<br />
<br />
=== '''External links''' ===<br />
<nowiki>https://csegrecorder.com/articles/view/too-many-seismic-attributes</nowiki><br />
<br />
[[Rms amplitude AGC|https://wiki.seg.org/wiki/Rms_amplitude_AGC]]<br />
<br />
<nowiki>https://community.sw.siemens.com/s/article/root-mean-square-rms-and-overall-level</nowiki></div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=138177RMS amplitude2019-10-23T18:56:47Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[1]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is <math>\tfrac{1}{\sqrt{2}}</math>times the peak amplitude.<sup>[2]</sup><br />
<br />
=== Mathematical Expression ===<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[3]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x_1,x_2,…,x_n}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)，ω is the angular frequency, specifies how many oscillations occur in a unit time interval. The RMS of the sine wave can be calculated as follows <br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
Using the power reduction formula <br />
<br />
<math>sin^2 (\omega t) = \tfrac{1 -cos(2\omega t)}{2}</math><br />
<br />
<br />
The RMS of the sine wave becomes<br />
<br />
<math> = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
According to the trigonometric Integral<br />
<br />
<math>\int cos(2\omega t) dt = \tfrac{1}{2\omega} sin(2\omega t)</math><br />
<br />
The RMS can be converted to<br />
<br />
<math>Y_{RMS} = A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
According to the periodicity of the sine function<br />
<br />
<math>sin(2\omega t)|_0^T = sin (2 \tfrac{2\pi}{2} T) = 0</math> <br />
<br />
<br />
Finally, it’s obtained that <br />
<br />
<math>Y_{RMS} = \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS is 0.707 times the peak amplitude<br />
<br />
'''A Figure cannot be displayed''' <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values. when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. <br />
<br />
RMS amplitude analyze the overall amplitude of a signal, conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefor, they are all positive, then the signal average is calculated, eventually followed by the square root operation. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
<br />
=== Seismic Interpretation ===<br />
In seismic processing, automatic gain control (AGC) method is often used in data processing to improve the visibility of seismic data in which attenuation or spherical divergence has caused amplitude decay, which is a system that controls the increase in the amplitude of an electrical signal from the original input to the amplified output, automatically (figure 2).[4] RMS AGC is used for amplitude equalization. It gives the processor a measure of the overall absolute amplitude in the window, both as positive and as negative values. During the processing, the data processor assume that the average absolute reflectivity varies little over a long time window, so a long-gate AGC can be applied to avoids destroying the lateral variation in amplitude, and ensure that the window include many reflectors so that the target event makes very little contribution to the average amplitude in the window. For instance, RMS AGC is often applied on structural imaging.<br />
<br />
Figure 2 Seismic data without AGC (left) and the same data after AGC is applied (Onajite,2013)<br />
<br />
For seismic integration, RMS is a most commonly used post stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage.[5] This has splendid applications for various reconnaissance endeavors. The squaring of the amplitude values within the window gives the high amplitudes maximum opportunity to stand out above the background contamination. The RMS amplitude over a large flat or structured time window can be effectively used to identify bright spots at different levels within a formation (Figure3).<br />
<br />
Figure 3 Top Balder Formation amplitude map from multiple 3-D surveys covering 5500 sq km of the Faroe Basin west of Shetland in U.K. waters. The colors show RMS amplitude over a 50-ms window surrounding the Top Balder reflection, with red indicating high amplitude and blue indicating low amplitude (Brown,2004)<br />
<br />
<br />
RMS amplitude is popular type of windowed amplitude attributes, the window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view. In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide. The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
<br />
<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
[2] Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
[3] Weisstein, Eric W. "Root-Mean-Square". MathWorld.<br />
[4] Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms". <br />
MasteringElectronicsDesign.com. Retrieved 21 January 2015<br />
[5] Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
[6] Pitfalls in the study of seismic amplitude<br />
[7] SEG: Methods and Applications in Reservoir Geophysics</div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=138175RMS amplitude2019-10-23T18:53:56Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[1]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is <math>\tfrac{1}{\sqrt{2}}</math>times the peak amplitude.<sup>[2]</sup><br />
<br />
=== Mathematical Expression ===<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[3]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x_1,x_2,…,x_n}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)，ω is the angular frequency, specifies how many oscillations occur in a unit time interval. The RMS of the sine wave can be calculated as follows <br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
Using the power reduction formula <br />
<br />
<math>sin^2 (\omega t) = \tfrac{1 -cos(2\omega t)}{2}</math><br />
<br />
<br />
The RMS of the sine wave becomes<br />
<br />
<math> = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
According to the trigonometric Integral<br />
<br />
<math>\int cos(2\omega t) dt = \tfrac{1}{2\omega} sin(2\omega t)</math><br />
<br />
The RMS can be converted to<br />
<br />
<math>Y_{RMS} = A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
According to the periodicity of the sine function<br />
<br />
<br />
Finally, it’s obtained that <br />
<br />
<math>Y_{RMS} = \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS is 0.707 times the peak amplitude<br />
<br />
'''A Figure cannot be displayed''' <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values. when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. <br />
<br />
RMS amplitude analyze the overall amplitude of a signal, conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefor, they are all positive, then the signal average is calculated, eventually followed by the square root operation. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
<br />
=== Seismic Interpretation ===<br />
In seismic processing, automatic gain control (AGC) method is often used in data processing to improve the visibility of seismic data in which attenuation or spherical divergence has caused amplitude decay, which is a system that controls the increase in the amplitude of an electrical signal from the original input to the amplified output, automatically (figure 2).[4] RMS AGC is used for amplitude equalization. It gives the processor a measure of the overall absolute amplitude in the window, both as positive and as negative values. During the processing, the data processor assume that the average absolute reflectivity varies little over a long time window, so a long-gate AGC can be applied to avoids destroying the lateral variation in amplitude, and ensure that the window include many reflectors so that the target event makes very little contribution to the average amplitude in the window. For instance, RMS AGC is often applied on structural imaging.<br />
<br />
Figure 2 Seismic data without AGC (left) and the same data after AGC is applied (Onajite,2013)<br />
<br />
For seismic integration, RMS is a most commonly used post stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage.[5] This has splendid applications for various reconnaissance endeavors. The squaring of the amplitude values within the window gives the high amplitudes maximum opportunity to stand out above the background contamination. The RMS amplitude over a large flat or structured time window can be effectively used to identify bright spots at different levels within a formation (Figure3).<br />
<br />
Figure 3 Top Balder Formation amplitude map from multiple 3-D surveys covering 5500 sq km of the Faroe Basin west of Shetland in U.K. waters. The colors show RMS amplitude over a 50-ms window surrounding the Top Balder reflection, with red indicating high amplitude and blue indicating low amplitude (Brown,2004)<br />
<br />
<br />
RMS amplitude is popular type of windowed amplitude attributes, the window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view. In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide. The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
<br />
<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
[2] Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
[3] Weisstein, Eric W. "Root-Mean-Square". MathWorld.<br />
[4] Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms". <br />
MasteringElectronicsDesign.com. Retrieved 21 January 2015<br />
[5] Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
[6] Pitfalls in the study of seismic amplitude<br />
[7] SEG: Methods and Applications in Reservoir Geophysics</div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=138173RMS amplitude2019-10-23T18:51:00Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[1]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is <math>\tfrac{1}{\sqrt{2}}</math>times the peak amplitude.<sup>[2]</sup><br />
<br />
=== Mathematical Expression ===<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[3]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x_1,x_2,…,x_n}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)，ω is the angular frequency, specifies how many oscillations occur in a unit time interval. The RMS of the sine wave can be calculated as follows <br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
Using the power reduction formula <br />
<br />
<math>sin^2 (\omega t) = \tfrac{1 -cos(2\omega t)}{2}</math><br />
<br />
<br />
The RMS of the sine wave becomes<br />
<br />
<math> = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
According to the trigonometric Integral<br />
<br />
<math>\int cos(2\omega t) dt = \tfrac{1}{2\omega} sin(2\omega t)</math><br />
<br />
<math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math>The RMS can be converted to<br />
Finally, it’s obtained that <br />
<br />
<math>= \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS is 0.707 times the peak amplitude<br />
<br />
'''A Figure cannot be displayed''' <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values. when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. <br />
<br />
RMS amplitude analyze the overall amplitude of a signal, conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefor, they are all positive, then the signal average is calculated, eventually followed by the square root operation. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
<br />
=== Seismic Interpretation ===<br />
In seismic processing, automatic gain control (AGC) method is often used in data processing to improve the visibility of seismic data in which attenuation or spherical divergence has caused amplitude decay, which is a system that controls the increase in the amplitude of an electrical signal from the original input to the amplified output, automatically (figure 2).[4] RMS AGC is used for amplitude equalization. It gives the processor a measure of the overall absolute amplitude in the window, both as positive and as negative values. During the processing, the data processor assume that the average absolute reflectivity varies little over a long time window, so a long-gate AGC can be applied to avoids destroying the lateral variation in amplitude, and ensure that the window include many reflectors so that the target event makes very little contribution to the average amplitude in the window. For instance, RMS AGC is often applied on structural imaging.<br />
<br />
Figure 2 Seismic data without AGC (left) and the same data after AGC is applied (Onajite,2013)<br />
<br />
For seismic integration, RMS is a most commonly used post stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage.[5] This has splendid applications for various reconnaissance endeavors. The squaring of the amplitude values within the window gives the high amplitudes maximum opportunity to stand out above the background contamination. The RMS amplitude over a large flat or structured time window can be effectively used to identify bright spots at different levels within a formation (Figure3).<br />
<br />
Figure 3 Top Balder Formation amplitude map from multiple 3-D surveys covering 5500 sq km of the Faroe Basin west of Shetland in U.K. waters. The colors show RMS amplitude over a 50-ms window surrounding the Top Balder reflection, with red indicating high amplitude and blue indicating low amplitude (Brown,2004)<br />
<br />
<br />
RMS amplitude is popular type of windowed amplitude attributes, the window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view. In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide. The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
<br />
<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
[2] Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
[3] Weisstein, Eric W. "Root-Mean-Square". MathWorld.<br />
[4] Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms". <br />
MasteringElectronicsDesign.com. Retrieved 21 January 2015<br />
[5] Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
[6] Pitfalls in the study of seismic amplitude<br />
[7] SEG: Methods and Applications in Reservoir Geophysics</div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=138172RMS amplitude2019-10-23T18:46:45Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[1]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is <math>\tfrac{1}{\sqrt{2}}</math>times the peak amplitude.<sup>[2]</sup><br />
<br />
=== Mathematical Expression ===<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[3]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x_1,x_2,…,x_n}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)，ω is the angular frequency, specifies how many oscillations occur in a unit time interval. The RMS of the sine wave can be calculated as follows <br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
Using the power reduction formula <br />
<br />
<br />
The RMS of the sine wave becomes<br />
<br />
<br />
<br />
<math> = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
According to the trigonometric Integral<br />
<br />
<math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
The RMS can be converted to<br />
Finally, it’s obtained that <br />
<br />
<math>= \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS is 0.707 times the peak amplitude<br />
<br />
'''A Figure cannot be displayed''' <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values. when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. <br />
<br />
RMS amplitude analyze the overall amplitude of a signal, conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefor, they are all positive, then the signal average is calculated, eventually followed by the square root operation. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
<br />
=== Seismic Interpretation ===<br />
In seismic processing, automatic gain control (AGC) method is often used in data processing to improve the visibility of seismic data in which attenuation or spherical divergence has caused amplitude decay, which is a system that controls the increase in the amplitude of an electrical signal from the original input to the amplified output, automatically (figure 2).[4] RMS AGC is used for amplitude equalization. It gives the processor a measure of the overall absolute amplitude in the window, both as positive and as negative values. During the processing, the data processor assume that the average absolute reflectivity varies little over a long time window, so a long-gate AGC can be applied to avoids destroying the lateral variation in amplitude, and ensure that the window include many reflectors so that the target event makes very little contribution to the average amplitude in the window. For instance, RMS AGC is often applied on structural imaging.<br />
<br />
Figure 2 Seismic data without AGC (left) and the same data after AGC is applied (Onajite,2013)<br />
<br />
For seismic integration, RMS is a most commonly used post stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage.[5] This has splendid applications for various reconnaissance endeavors. The squaring of the amplitude values within the window gives the high amplitudes maximum opportunity to stand out above the background contamination. The RMS amplitude over a large flat or structured time window can be effectively used to identify bright spots at different levels within a formation (Figure3).<br />
<br />
Figure 3 Top Balder Formation amplitude map from multiple 3-D surveys covering 5500 sq km of the Faroe Basin west of Shetland in U.K. waters. The colors show RMS amplitude over a 50-ms window surrounding the Top Balder reflection, with red indicating high amplitude and blue indicating low amplitude (Brown,2004)<br />
<br />
<br />
RMS amplitude is popular type of windowed amplitude attributes, the window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view. In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide. The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
<br />
<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
[2] Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002<br />
[3] Weisstein, Eric W. "Root-Mean-Square". MathWorld.<br />
[4] Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms". <br />
MasteringElectronicsDesign.com. Retrieved 21 January 2015<br />
[5] Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production<br />
[6] Pitfalls in the study of seismic amplitude<br />
[7] SEG: Methods and Applications in Reservoir Geophysics</div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=138171RMS amplitude2019-10-23T18:42:13Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[1]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is <math>\tfrac{1}{\sqrt{2}}</math>times the peak amplitude.<sup>[2]</sup><br />
<br />
=== Mathematical Expression ===<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[3]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x_1,x_2,…,x_n}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)，ω is the angular frequency, specifies how many oscillations occur in a unit time interval. The RMS of the sine wave can be calculated as follows <br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
<math> = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
<math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
<math>= \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS amplitude is 0.707 times the maximum amplitude.<br />
<br />
'''A Figure cannot be displayed''' <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values (Figure 1). when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
An analysis used for the overall amplitude of a signal is called RMS amplitude. Conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefore, they are all positive, then the signal average is calculated, eventually followed by the square root operation. <br />
<br />
RMS mainly used in the context of sine waves. It can be considered as an alternative way of specifying how big a sine wave is, but with the advantage of allowing direct comparison with a non-oscillating source of energy. <br />
<br />
=== Seismic Interpretation ===<br />
For seismic, RMS is a most commonly used post-stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage. <br />
<br />
The window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires a careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view. In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide. The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
<br />
[2] Encyclopedic Dictionary of Applied Geophysics (Fourth Edition). SEG.2002<br />
<br />
[3] Weisstein, Eric W. "Root-Mean-Square". MathWorld.</div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=138170RMS amplitude2019-10-23T18:39:51Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[1]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is <math>\tfrac{1}{\sqrt{2}}</math>times the peak amplitude.<sup>[2]</sup><br />
<br />
=== Mathematical Expression ===<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[3]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x_1,x_2,…,x_n}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)<br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
<math> = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
<math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
<math>= \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS amplitude is 0.707 times the maximum amplitude.<br />
<br />
'''A Figure cannot be displayed''' <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values (Figure 1). when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
An analysis used for the overall amplitude of a signal is called RMS amplitude. Conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefore, they are all positive, then the signal average is calculated, eventually followed by the square root operation. <br />
<br />
RMS mainly used in the context of sine waves. It can be considered as an alternative way of specifying how big a sine wave is, but with the advantage of allowing direct comparison with a non-oscillating source of energy. <br />
<br />
=== Seismic Interpretation ===<br />
For seismic, RMS is a most commonly used post-stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage. <br />
<br />
The window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires a careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view. In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide. The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
<br />
[2] Encyclopedic Dictionary of Applied Geophysics (Fourth Edition). SEG.2002<br />
<br />
[3] Weisstein, Eric W. "Root-Mean-Square". MathWorld.</div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=138128RMS amplitude2019-10-23T01:11:25Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[1]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is <math>\tfrac{\sqrt{2}}{2}</math>times the peak amplitude.<sup>[2]</sup><br />
<br />
=== Mathematical Expression ===<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[3]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x_1,x_2,…,x_n}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)<br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
<math> = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
<math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
<math>= \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS amplitude is 0.707 times the maximum amplitude.<br />
<br />
'''A Figure cannot be displayed''' <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values (Figure 1). when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
An analysis used for the overall amplitude of a signal is called RMS amplitude. Conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefore, they are all positive, then the signal average is calculated, eventually followed by the square root operation. <br />
<br />
RMS mainly used in the context of sine waves. It can be considered as an alternative way of specifying how big a sine wave is, but with the advantage of allowing direct comparison with a non-oscillating source of energy. <br />
<br />
=== Seismic Interpretation ===<br />
For seismic, RMS is a most commonly used post-stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage. <br />
<br />
The window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires a careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view. In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide. The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
<br />
[2] Encyclopedic Dictionary of Applied Geophysics (Fourth Edition). SEG.2002<br />
<br />
[3] Weisstein, Eric W. "Root-Mean-Square". MathWorld.</div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=138127RMS amplitude2019-10-23T01:10:24Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[1]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is <math>\tfrac{\sqrt{2}}{2}</math>times the peak amplitude.<sup>[2]</sup><br />
<br />
=== Mathematical Expression ===<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[3]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x_1,x_2,…,x_n}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2} dt}</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)<br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
<math> = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
<math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
<math>= \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS amplitude is 0.707 times the maximum amplitude.<br />
<br />
'''A Figure cannot be displayed''' <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values (Figure 1). when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
An analysis used for the overall amplitude of a signal is called RMS amplitude. Conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefore, they are all positive, then the signal average is calculated, eventually followed by the square root operation. <br />
<br />
RMS mainly used in the context of sine waves. It can be considered as an alternative way of specifying how big a sine wave is, but with the advantage of allowing direct comparison with a non-oscillating source of energy. <br />
<br />
=== Seismic Interpretation ===<br />
For seismic, RMS is a most commonly used post-stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage. <br />
<br />
The window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires a careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view. In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide. The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
<br />
[2] Encyclopedic Dictionary of Applied Geophysics (Fourth Edition). SEG.2002<br />
<br />
[3] Weisstein, Eric W. "Root-Mean-Square". MathWorld.</div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=138126RMS amplitude2019-10-23T01:07:58Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[1]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is <math>\tfrac{\sqrt{2}}{2}</math>times the peak amplitude.<sup>[2]</sup><br />
<br />
=== Mathematical Expression ===<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[3]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x_1,x_2,…,x_n}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T \int_{0}^{T} [f(t)]^2} dt}</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)<br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
<math> = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
<math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
<math>= \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS amplitude is 0.707 times the maximum amplitude.<br />
<br />
'''A Figure cannot be displayed''' <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values (Figure 1). when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
An analysis used for the overall amplitude of a signal is called RMS amplitude. Conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefore, they are all positive, then the signal average is calculated, eventually followed by the square root operation. <br />
<br />
RMS mainly used in the context of sine waves. It can be considered as an alternative way of specifying how big a sine wave is, but with the advantage of allowing direct comparison with a non-oscillating source of energy. <br />
<br />
=== Seismic Interpretation ===<br />
For seismic, RMS is a most commonly used post-stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage. <br />
<br />
The window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires a careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
<br />
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view. In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide. The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
<br />
=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
<br />
[2] Encyclopedic Dictionary of Applied Geophysics (Fourth Edition). SEG.2002<br />
<br />
[3] Weisstein, Eric W. "Root-Mean-Square". MathWorld.</div>Zhai0629https://wiki.seg.org/index.php?title=RMS_amplitude&diff=138125RMS amplitude2019-10-23T01:07:06Z<p>Zhai0629: </p>
<hr />
<div>\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018. <br />
<br />
The root mean square amplitude (RMS) is a commonly used technique to display amplitude values in a specified window of stack data. With RMS amplitude, hydrocarbon indicators can be mapped directly by measure reflectivity in a zone of interest. <br />
<br />
=== Definition ===<br />
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[1]</sup><br />
<br />
In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is <math>\tfrac{\sqrt{2}}{2}</math>times the peak amplitude.<sup>[2]</sup><br />
<br />
=== Mathematical Expression ===<br />
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous-time waveform. <sup>[3]</sup> It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.<br />
In a set of n values {x_1,x_2,…,x_n}, the RMS is<br />
<br />
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math><br />
<br />
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T_2 - T_1} \int_{T_1}^{T_2} [f(t)]_2 dt }</math><br />
<br />
and the RMS for a function over all time is<br />
<br />
<math>f_{RMS} = \sqrt{\tfrac{1}{T \int_{0}^{T} [f(t)]^2} dt}</math><br />
<br />
For a sine wave<br />
<br />
<math>y(t) = Asin(2\pi ft + \varphi) = sin(\omega t + \varphi)</math><br />
<br />
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)<br />
<br />
<math>Y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omega t)]^2 dt }</math><br />
<br />
<math> = A \sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omega t)}{2} dt }</math><br />
<br />
<math>= A \sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omega t)}{4\omega }]_0^T }</math><br />
<br />
<math>= \tfrac{A}{\sqrt{2}}</math><br />
<br />
Thus, the RMS amplitude is 0.707 times the maximum amplitude.<br />
<br />
'''A Figure cannot be displayed''' <br />
<br />
=== Physical Description ===<br />
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values (Figure 1). when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.<br />
<br />
An analysis used for the overall amplitude of a signal is called RMS amplitude. Conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefore, they are all positive, then the signal average is calculated, eventually followed by the square root operation. <br />
<br />
RMS mainly used in the context of sine waves. It can be considered as an alternative way of specifying how big a sine wave is, but with the advantage of allowing direct comparison with a non-oscillating source of energy. <br />
<br />
=== Seismic Interpretation ===<br />
For seismic, RMS is a most commonly used post-stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage. <br />
<br />
The window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires a careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.<br />
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Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view. In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide. The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.<br />
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=== References ===<br />
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.<br />
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[2] Encyclopedic Dictionary of Applied Geophysics (Fourth Edition). SEG.2002<br />
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[3] Weisstein, Eric W. "Root-Mean-Square". MathWorld.</div>Zhai0629