Semblance

Problems in Exploration Seismology and their Solutions

Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
Chapter	9
Pages	295 - 366
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem

Semblance $S_{T}$ is the ratio of the energy of a stack of $N$ traces to $N$ times the sum of the energies of the $N$ component traces, all summed over some time interval. Show that $S_{T}=1$ when the traces are identical.

Background

While crosscorrelation is a quantitative measure of the similarity of two traces, semblance $S_{T}$ is a measure of the similarity of a number of traces. Assuming $N$ traces, we sum (stack) the traces at time $t$, square the sum to get the total energy, and sum the values over a time interval ${m}\Delta$. The equation for $S_{T}$ is

${\begin{aligned}S_{T}=\sum \limits _{t=t_{0}}^{t_{0}+m\Delta }\left(\sum \limits _{i=1}^{N}g_{ti}\right)^{2}/N\left[\sum \limits _{t=t_{0}}^{t_{0}+m\Delta }\sum \limits _{i=1}^{N}(g_{ti})^{2}\right].\end{aligned}}$

(9.10a)

Solution

The sum of $N$ traces is $(\sum {_{i=1}^{N}g_{ti}})$ and its energy is $(\sum {_{i=1}^{N}}g_{ti})^{2}$. If the traces are identical, the numerator of equation (9.10a) becomes $N^{2}[\sum {^{t_{0}+m\Delta }_{t=t_{0}}}g_{t}^{2}]$. The denominator becomes $N\sum {^{t_{0}+m\Delta }_{t=t_{0}}}(\sum {^{N}_{i=1}}g_{t}^{2})=N^{2}\sum {}_{t=t_{0}}^{t_{0}+m\Delta }g_{t}^{2}$. Since numerator and denominator are equal, $S_{T}=1$.

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Table of Contents (book)

Also in this chapter

• Fourier series
• Space-domain convolution and vibroseis acquisition
• Fourier transforms of the unit impulse and boxcar
• Extension of the sampling theorem
• Alias filters
• The convolutional model
• Water reverberation filter
• Calculating crosscorrelation and autocorrelation
• Digital calculations
• Semblance
• Convolution and correlation calculations
• Properties of minimum-phase wavelets
• Phase of composite wavelets
• Tuning and waveshape
• Making a wavelet minimum-phase
• Zero-phase filtering of a minimum-phase wavelet
• Deconvolution methods
• Calculation of inverse filters
• Inverse filter to remove ghosting and recursive filtering
• Ghosting as a notch filter
• Autocorrelation
• Wiener (least-squares) inverse filters
• Interpreting stacking velocity
• Effect of local high-velocity body
• Apparent-velocity filtering
• Complex-trace analysis
• Kirchhoff migration
• Using an upward-traveling coordinate system
• Finite-difference migration
• Effect of migration on fault interpretation
• Derivative and integral operators
• Effects of normal-moveout (NMO) removal
• Weighted least-squares

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