RMS amplitude

\\This page is currently being authored by a student at the University of Oklahoma. This page will be complete by December 1, 2018.

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.

Definition

In statistics, RMS is typical value of a number (n) of values of a quantity (x₁, x₂, x₃…) 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 ${\tfrac {\sqrt {2}}{2}}$times the peak amplitude.^[2]

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 {x_1,x_2,…,x_n}, the RMS is

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

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

and the RMS for a function over all time is

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

For a sine wave

$y(t)=Asin(2\pi ft+\varphi )=sin(\omega t+\varphi )$

where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)

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

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

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

$={\tfrac {A}{\sqrt {2}}}$

Thus, the RMS amplitude is 0.707 times the maximum amplitude.

A Figure cannot be displayed

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 (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.

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.

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.

Seismic Interpretation

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.

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.

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.

References

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

[2] Encyclopedic Dictionary of Applied Geophysics (Fourth Edition). SEG.2002

[3] Weisstein, Eric W. "Root-Mean-Square". MathWorld.