# 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 a post-stack attribute that is often used, although its interpretation in thin-layered beds is not necessarily straightforward.

Definition

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.

Mathematical Expression

This attribute calculates the squared sum of the sample values in the specified time-gate divided by the number of samples in the gate.

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

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

Physical Description

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:

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

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).

At the point of maximum amplitude (A_w) during the pendulum's oscillation, all of the pendulum's energy is potential energy (PEs),

${\displaystyle E_{S}=PE_{S}={\tfrac {kx^{2}}{2}}}$

However, the x (displacement from equilibrium) is equal to the amplitude of the pendulum,

${\displaystyle E_{S}={\tfrac {kA_{S}^{2}}{2}}}$

Since the energy of the wave (Ew) corresponds to mechanical energy the pendulum has (E_s) and the point of maximum amplitude (Aw) corresponds to the farthest displacement from equilibrium the massive bob (As). Thus,

${\displaystyle E_{S}={\tfrac {kA_{W}^{2}}{2}}\rightarrow E_{W}\propto A_{W}}$

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.

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 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.

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.

References

1. A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.
2. Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002
3. Weisstein, Eric W. "Root-Mean-Square". MathWorld.