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

### Background

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

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