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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