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.

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 () 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

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

and the RMS for a function over all time is

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

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)

Failed to parse (unknown function "\omegat"): {\displaystyle y_{RMS} = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omegat)]^2 dt}}

Thus, the RMS=√2/2 A, it’s 0.707 times the maximum amplitude.