Magnitude

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	3
Pages	47 - 77
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	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.

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Reflection/transmission coefficients at small angles and magnitude	AVO versus AVA and effect of velocity gradient
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Table of Contents (book)

Also in this chapter

• General form of Snell’s law
• Reflection/refraction at a solid/solid interface and displacement of a free surface
• Reflection/refraction at a liquid/solid interface
• Zoeppritz’s equations for incident SV- and SH-waves
• Reinforcement depth in marine recording
• Complex coefficient of reflection
• Reflection and transmission coefficients
• Amplitude/energy of reflections and multiples
• Reflection/transmission coefficients at small angles and magnitude
• AVO versus AVA and effect of velocity gradient
• Variation of reflectivity with angle (AVA)

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