Difference between revisions of "RMS amplitude"

From SEG Wiki
Jump to: navigation, search
Line 1: Line 1:
<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 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 (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)
Line 12: Line 24:
 
               =\tfrac{A}{\sqrt{2}}                                                                                                (5)
 
               =\tfrac{A}{\sqrt{2}}                                                                                                (5)
  
</math><math>fRMS=\sqrt{\frac{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(2)]^2dt }</math><math>=\sqrt{\frac{1}{N}\sum_{i=1}^nx_i^2}</math>
+
</math><math>fRMS=\sqrt{\frac{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(2)]^2dt }</math><math>=\sqrt{\frac{1}{N}\sum_{i=1}^nx_i^2}</math><math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math>
=== 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>
 
 
 
In a set of n values {}, the RMS is
 
 
 
<math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math>
 
 
 
The RMS of the corresponding formula for a continuous waveform ''f(t)'' defined over the interval [T<sub>1</sub>, T<sub>2</sub>] 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. <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.
 
 
 
=== '''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.
 

Revision as of 10:10, 21 October 2019

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 {}, the RMS isFailed 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) }