# Magnitude

Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 3 47 - 77 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

## Problem 3.10

Using equation (1,8) in Table 2.2a, show that the fractional change $\Delta \sigma /\sigma$ is not necessarily small when $\Delta \alpha /\alpha$ , $\Delta \beta /\beta$ , and $\Delta p/p$ are all small.

### Solution

Equation (1,8) in Table 2.2a is

{\begin{aligned}\beta /\alpha =\left(1-2\sigma \right)/2\left(1-\sigma \right).\end{aligned}} Because $p$ does not enter into this equation, it has no effect upon $\Delta \sigma /\sigma$ . The fractions $\Delta \alpha /\alpha$ , $\Delta \beta /\beta$ , and $\Delta \sigma /\sigma$ are of the form $\Delta x/x$ which suggests that we use logs [since $\Delta \left(Inx\right)=\Delta x/x]$ . Taking logs of both sides of the above equation, we get

{\begin{aligned}\ln \beta -In\alpha =\ln \left(1-2\sigma \right)-In2-In\left(1-\sigma \right).\end{aligned}} .

Differentiation gives

{\begin{aligned}{\frac {\Delta \beta }{\beta }}-{\frac {\Delta \alpha }{\alpha }}={\frac {-2\Delta \sigma }{1-2\sigma }}+{\frac {\Delta \sigma }{1-\sigma }}=\left({\frac {\Delta \sigma }{\sigma }}\right){\frac {-1}{\left(1-2\sigma \right)\left({\frac {1}{\sigma }}-1\right)}}.\end{aligned}} Thus,

{\begin{aligned}\left|{\frac {\Delta \sigma }{\sigma }}\left|=\right|\left({\frac {\Delta \beta }{\beta }}-{\frac {\Delta \alpha }{\alpha }}\right)\left(2\sigma -1\right)\left(1-{\frac {1}{\sigma }}\right)\right|\end{aligned}} Since $0\leq \sigma \leq +0.5$ , the product of the two $\sigma$ -factors varies between $0$ (when $\sigma =0.5$ ) and $+\infty$ (when $2\sigma =0$ ). Therefore, even though $\left({\frac {\Delta \beta }{\beta }}-{\frac {\Delta \alpha }{\alpha }}\right)$ is small (being the difference between two small quantities), the right-hand side can be large.