Difference between revisions of "Average energy"

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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.

Definition

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.[1] 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.

Seismic wave is an elastic disturbance that is propagated from point to point through a medium.[2] 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. [3]

Mathematical Expression

Average energy calculates the squared sum of the sample values in the specified time window divided by the number of samples in the window.

${\displaystyle AverageEnergy={\tfrac {\sum _{i=1}^{n}A_{i}^{2}}{n}}}$

Where Ai is the amplitude of the sampling point in a given time window, n is the number of sampling points.

Physical Description

The total mechanical energy of the wave is the sum of its kinetic energy and potential energy[3].  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).

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 (EK) and the potential energy (EP).

According to the kinetic energy (EK) formula

${\displaystyle E_{K}={\tfrac {1}{2}}mv^{2}}$

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

${\displaystyle \vartriangle E_{K}={\tfrac {1}{2}}(\mu \vartriangle x){v_{y}}^{2}}$

Where μ is density of the particle, each mass element of the string has the mass element of the string oscillates with a velocity vy. When the mass element of the string approach zero

${\displaystyle dE_{K}=\lim _{\vartriangle x\to \infty }{\tfrac {1}{2}}(\mu \vartriangle x){v_{y}}^{2}={\tfrac {1}{2}}\mu {v_{y}}^{2}dx}$

For a sinusoidal wave with simple harmonic motion, the position of each mass element may be modeled as

${\displaystyle y(x,t)=Asin(nx-\omega t)}$

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

${\displaystyle v_{y}={\tfrac {\partial (x,t)}{\partial t}}=-A\omega cos(nx-\omega t)}$

Bring this velocity expression into the formula, the kinetic energy of each mass element of the string becomes

${\displaystyle dE_{K}={\tfrac {1}{2}}\mu \omega _{2}A^{2}cos^{2}(nx-\omega t)}$

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

${\displaystyle \int _{0}^{E_{K\lambda }}dE_{K}=\int _{0}^{\lambda }{\tfrac {1}{2}}\mu \omega ^{2}A^{2}cos^{2}(nx-\omega t)dx}$

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

${\displaystyle \int _{0}^{E_{K\lambda }}dE_{K}={\tfrac {1}{2}}\mu \omega ^{2}A^{2}\int _{0}^{\lambda }cos^{2}(nx)dx={\tfrac {1}{2}}\mu \omega ^{2}A^{2}\int _{0}^{\lambda }{\tfrac {1+sin(2nx)}{2}}dx}$

Thus, it’s obtained that

${\displaystyle E_{K}={\tfrac {1}{2}}\mu \omega ^{2}A^{2}[{\tfrac {1}{2}}x+{\tfrac {1}{4n}}sin(2nx)]_{0}^{\lambda }={\tfrac {1}{4}}\mu \omega ^{2}A^{2}\lambda }$

According to the potential energy formula

${\displaystyle E_{P}={\tfrac {1}{2}}kx^{2}}$

The simple and harmonic motion with an angular frequency (ω) given by

${\displaystyle \omega ={\sqrt {\tfrac {k}{m}}}}$

Where k is the spring constant, m is the mass of the object, bring it into the formula

${\displaystyle E_{P}={\tfrac {1}{2}}m\omega ^{2}x^{2}}$

The differential form of the elastic potential energy is

${\displaystyle dE_{P}={\tfrac {1}{2}}\omega ^{2}y^{2}\mu dx={\tfrac {1}{2}}\mu \omega ^{2}A^{2}sin^{2}(nx-\omega t)}$

It’s obtained that

${\displaystyle \int _{0}^{E_{P\lambda }}dE_{P}={\tfrac {1}{2}}\mu \omega ^{2}A^{2}\int _{0}^{\lambda }sin^{2}(nx-\omega t)dx={\tfrac {1}{4}}\mu \omega ^{2}A^{2}\lambda }$

Finally, the total energy associated with a wavelength is the sum of the kinetic energy (EK) and the potential energy (EP)

${\displaystyle E_{P}=E_{K}+E_{P}={\tfrac {1}{2}}\mu \omega ^{2}A^{2}\lambda }$

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:

${\displaystyle E\varpropto A^{2}}$

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).

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.

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.

Seismic Interpretation

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[4]. 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.

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.[5] 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.

Figure 3 Average energy extraction in a 100ms window (25 samples) at a constant time (Barnes,2006).

References

1. A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.
2. RE Shelf. Encyclopedic Dictionary of Applied Geophysics (Forth Edition), SEG.2002
3. Landau, L.D.; Lifshitz, E. M. Theory of Elasticity (3rd ed.). Oxford, England 1986
4. Methods and Applications in Reservoir Geophysics.David H. Johnston, SEG 2010
5. Reservoir Geophysics: Applications (SEG Distinguished Instructor Series, No. 11). Willam L. Abriel SEG 2008