Magnitude

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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 $\displaystyle \Delta \sigma /\sigma$ is not necessarily small when $\displaystyle \Delta \alpha /\alpha$ , $\displaystyle \Delta \beta /\beta$ , and $\displaystyle \Delta p/p$ are all small.

Solution

Equation (1,8) in Table 2.2a is

\displaystyle \begin{align} \beta /\alpha =\left(1-2\sigma \right)/2\left(1-\sigma \right). \end{align}

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

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

Differentiation gives

\displaystyle \begin{align} \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{align}

Thus,

\displaystyle \begin{align} \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{align}

Since $\displaystyle 0\le \sigma \le +0.5$ , the product of the two Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sigma -factors varies between $\displaystyle 0$ (when $\displaystyle \sigma = 0.5$ ) and $\displaystyle +\infty$ (when $\displaystyle 2\sigma =0$ ). Therefore, even though $\displaystyle \left(\frac{\Delta \beta }{\beta } -\frac{\Delta \alpha }{\alpha } \right)$ is small (being the difference between two small quantities), the right-hand side can be large.