# RMS amplitude

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.

## Contents

### Definition

In statistics, the RMS of ${\displaystyle \{x_{1},x_{2},...,x_{n}\}}$ is equal to the square root of the sum of the squares of the values divided by n. [1]

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 ${\displaystyle {\tfrac {1}{\sqrt {2}}}}$times the peak amplitude.[2]

Figure 1 Graph of a sine wave's amplitude vs. time, showing RMS(ARMS), peak (APK), and peak-to-peak (APP) amplitude.

### Mathematical Expression

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. [3] It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2. In a set of n values ${\displaystyle \{x_{1},x_{2},...,x_{n}\},}$ the RMS is

${\displaystyle x_{RMS}={\sqrt {{\tfrac {1}{n}}\sum _{i=1}^{n}x_{i}^{2}}}}$

The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is

${\displaystyle f_{RMS}={\sqrt {{\tfrac {1}{T_{2}-T_{1}}}\int _{T_{1}}^{T_{2}}[f(t)]^{2}dt}}}$

and the RMS for a function over all time is

${\displaystyle f_{RMS}={\sqrt {{\tfrac {1}{T}}\int _{0}^{T}[f(t)]^{2}dt}}}$

For a sine wave

${\displaystyle y(t)=A\sin(2\pi ft+\varphi )=\sin(\omega t+\varphi )}$

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

${\displaystyle Y_{RMS}={\sqrt {{\tfrac {1}{T}}\int _{0}^{T}[A\sin(\omega t)]^{2}dt}}}$

Using the power reduction formula

${\displaystyle \sin ^{2}(\omega t)={\tfrac {1-\cos(2\omega t)}{2}}}$

The RMS of the sine wave becomes

${\displaystyle Y_{RMS}=A{\sqrt {{\tfrac {1}{T}}\int _{0}^{T}{\tfrac {1-\cos(2\omega t)}{2}}dt}}}$

According to the trigonometric Integral

${\displaystyle \int \cos(2\omega t)dt={\tfrac {1}{2\omega }}\sin(2\omega t)}$

The RMS can be converted to

${\displaystyle Y_{RMS}=A{\sqrt {{\tfrac {1}{T}}{\Big [}{\tfrac {t}{2}}-{\tfrac {\sin(2\omega t)}{4\omega }}{\Big ]}_{0}^{T}}}=A{\sqrt {{\tfrac {1}{2}}-{\tfrac {1}{4\omega T}}\sin(2\omega t){\Big |}_{0}^{T}}}}$

According to the periodicity of the sine function and for ${\displaystyle T}$ equal to one period,

${\displaystyle \sin(2\omega t)|_{0}^{T}=\sin(2\omega T)=0}$

Finally, it’s obtained that

${\displaystyle Y_{RMS}={\tfrac {A}{\sqrt {2}}}}$

Thus, the RMS is 0.707 times the peak amplitude

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. [4] 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

${\displaystyle A_{RMS}={\sqrt {\tfrac {\sum _{i=1}^{n}A_{i}^{2}*w_{i}}{\sum _{i=1}^{n}w_{i}}}}}$

Where A = amplitude, n = number of samples,  is the weight of each amplitude value.

### Physical Description

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.

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.

### Seismic Interpretation

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).[5] 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.

Figure 2 Seismic data without AGC (left) and the same data after AGC is applied (Onajite,2013)

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).

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)

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. Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view.[6] 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.[7] 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.

### References

[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.

[2] R.E.Shelf. Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002

[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.

[4] Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms" MasteringElectronicsDesign.com. Retrieved 21 January 2015

[5] Nanda, Niranjan C.Seismic Data Interpretation and Evaluation for Hydrocarbon Exploration and Production. Springer 2016

[6] Alistair R. Brown.Pitfalls in the study of seismic amplitude. interpretation 6: SL15-SL20. pp20-24,2018

[7] David H. Johnston, Methods and Applications in Reservoir Geophysics..SEG 2010