# Difference between revisions of "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.${\sqrt {2}}/2$ $x_{R}MS={\sqrt {{\tfrac {1}{N}}\sum _{i=1}^{n}x_{i}^{2}}}$ $f_{R}MS={\sqrt {{\tfrac {1}{T_{2}-T_{1}}}\int _{T_{1}}^{T_{2}}[f(t)]^{2}dt}}$ $f_{R}MS={\sqrt {{\tfrac {1}{T}}\int _{0}^{T}[f(t)]^{2}dt}}$ $\displaystyle y(t) = Asin (2\pift+\varphi) = Asin(\omegat+\varphi)$ $\displaystyle Y_RMS = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omegat)]^2 dt}$ $\displaystyle = A\sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omegat)}{2} dt}$ $\displaystyle = A\sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omegat)}{4\omega}]_0^T}$ $={\tfrac {A}{\sqrt {2}}}$ $={\sqrt {{\frac {1}{N}}\sum _{i=1}^{n}x_{i}^{2}}}$ $\displaystyle x_RMS=\sqrt{\tfrac{1}{n}\sum_{i=1}^nx_i^2} (1) The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is f_{RMS}=\sqrt{\tfrac{T_2}{T_1}\textstyle \int_{T_1}^{T_2} \displaystyle [f(t)]^2}dt (2) and the RMS for a function over all time is f_{RMS}=\sqrt{\tfrac{1}{T}\int_{0}^{T} [f(t)]^2}dt (3) For a sine wave y(t)=A sin⁡(2πft+φ)=Asin(ωt+φ) (4) where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero) Y_{RMS}=\sqrt{\tfrac{1}{T}\int_{0}^{T} [Asin(\omegat)]^2dt} =A\sqrt{\tfrac{1}{T}\int\limits_{0}^{T} \tfrac{1-cos(2\omegat)}{2}dt} =A\sqrt{\tfrac{1}{T}[\tfrac{T}{2}-\tfrac{sin(2\omegat)}{4\omega}]_0^T} =\tfrac{A}{\sqrt{2}} (5)$ $fRMS={\sqrt {{\frac {1}{T_{2}-T_{1}}}\int _{T_{1}}^{T_{2}}[f(t)]^{2}}}dt$ $f_{R}MS={\sqrt {{\tfrac {1}{T}}\int _{0}^{T}[f(2)]^{2}dt}}$ $={\sqrt {{\frac {1}{N}}\sum _{i=1}^{n}x_{i}^{2}}}$ 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. 

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.

In a set of n values {}, 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

### 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.  It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.