# Semblance

Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 9 295 - 366 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 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, $S_{T}=1$ .