Difference between revisions of "RMS amplitude"

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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.
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<math>\sqrt{2}/2</math><math>x_RMS = \sqrt{\tfrac{1}{N} \sum_{i=1}^n x_i^2}                        </math><math>f_RMS = \sqrt{\tfrac{1}{T_2-T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><math>f_RMS = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><math>y(t) = Asin (2\pift+\varphi) = Asin(\omegat+\varphi)</math><math>Y_RMS = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omegat)]^2 dt}</math><math>      = A\sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omegat)}{2} dt}</math><math>      = A\sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omegat)}{4\omega}]_0^T}</math><math>      = \tfrac{A}{\sqrt{2}}</math><math>=\sqrt{\frac{1}{N}\sum_{i=1}^nx_i^2}</math><math>x_RMS=\sqrt{\tfrac{1}{n}\sum_{i=1}^nx_i^2}                                                                    (1)
 
 
Definition
 
 
 
In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. '''<sup>[1]</sup>'''
 
 
 
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.'''<sup>[2]</sup>'''
 
 
 
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. '''<sup>[3]</sup>''' 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 {}, the RMS is<math>\sqrt{2}/2</math><math>x_RMS = \sqrt{\tfrac{1}{N} \sum_{i=1}^n x_i^2}                        </math><math>f_RMS = \sqrt{\tfrac{1}{T_2-T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><math>f_RMS = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><math>y(t) = Asin (2\pift+\varphi) = Asin(\omegat+\varphi)</math><math>Y_RMS = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omegat)]^2 dt}</math><math>      = A\sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omegat)}{2} dt}</math><math>      = A\sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omegat)}{4\omega}]_0^T}</math><math>      = \tfrac{A}{\sqrt{2}}</math><math>=\sqrt{\frac{1}{N}\sum_{i=1}^nx_i^2}</math><math>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
 
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)
 
f_{RMS}=\sqrt{\tfrac{T_2}{T_1}\textstyle \int_{T_1}^{T_2} \displaystyle [f(t)]^2}dt            (2)

Revision as of 10:11, 21 October 2019

Failed to parse (unknown function "\pift"): {\displaystyle y(t) = Asin (2\pift+\varphi) = Asin(\omegat+\varphi)}
Failed to parse (unknown function "\omegat"): {\displaystyle Y_RMS = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omegat)]^2 dt}}
Failed to parse (unknown function "\omegat"): {\displaystyle       = A\sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omegat)}{2} dt}}
Failed to parse (unknown function "\omegat"): {\displaystyle       = A\sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omegat)}{4\omega}]_0^T}}
Failed to parse (syntax error): {\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)  }