Reflection/transmission coefficients at small angles and 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 3.9

Show that, when angles in the Zoeppritz equations (3.2e,f,h,i) are small (so that squares and products are negligible), equations (3.6a) and (3.6b) for reflection and transmission at normal incidence are still valid, and that the reflected and transmitted S-waves are given by

${\begin{aligned}\mathbf {{\frac {B_{1}}{A_{0}}}={\frac {2W_{2}q+4Z_{1}r}{\left(W_{1}+W_{2}\right)\left(Z_{1}+Z_{2}\right)}},\;{\frac {B_{2}}{A_{0}}}={\frac {2W_{1}q-4Z_{1}r}{\left(W_{1}+W_{2}\right)\left(Z_{1}+Z_{2}\right)}},} \end{aligned}}$

where $\mathbf {q=Z_{1}\theta _{2}-Z_{2}\theta _{1}}$, $\mathbf {r=W_{1}\delta _{1}}$ - $\mathbf {W_{2}\delta _{1}}$

Solution

When the angle of incidence $\theta$ is small, $\sin \theta \approx \theta$ and $\cos \theta \approx 1$ and the same is true for $\delta$. In this case Snell’s law and the Zoeppritz equations (3.2e,f,h,i) become

${\begin{aligned}\theta _{1}/\alpha _{1}\approx \delta _{1}/\beta _{1}\approx \theta _{2}/\alpha _{2}\approx \delta _{2}/\beta _{2},\;\beta _{i}/\alpha _{i}=\delta _{i}/\theta _{i};\\A_{1}-\;\delta _{1}B_{1}+\;A_{2}+\;\delta _{2}B_{2}=\;A_{0},\\\theta _{1}A_{1}+\;B_{1}-\;\theta _{2}A_{2}+\;B_{2}=\;-\theta _{1}A_{0},\\Z_{1}A_{1}-2\delta _{1}W_{1}B_{1}-\;Z_{2}A_{2}-2\delta _{2}W_{2}B_{2}=\;-Z_{1}A_{0},\\2\delta _{1}W_{1}A_{1}+\;W_{1}B_{1}+2\delta _{2}W_{2}A_{2}-\;W_{2}B_{2}=2\delta _{1}W_{1}A_{0}.\end{aligned}}$

In matrix notation, the Zoeppritz equations are now

${\begin{aligned}\left\|{\begin{array}{llll}1&-\delta _{1}&1&\delta _{2}\\\theta _{1}&1&-\theta _{2}&1\\Z_{1}&-2\delta _{1}W_{1}&-Z_{2}&-2\delta _{2}W_{2}\\2\delta _{1}W_{1}&W_{1}&2\delta _{2}W_{2}&-W_{2}\end{array}}\right\|\ \left\|{\begin{array}{l}A_{1}\\B_{1}\\A_{2}\\B_{2}\end{array}}\right\|=A_{0}\left\|{\begin{array}{l}1\\-\theta _{1}\\-Z_{1}\\2\delta _{1}W_{1}\end{array}}\right\|\end{aligned}}$

(3.9a)

To get the amplitude ratios $A_{i}/A_{0}$ and $B_{i}/A_{0}$, we solve this equation either by inverting the left-hand matrix [see Sheriff and Geldart, 1995, equation (15.20)] or by using Cramer’s rule (see Wylie, 1966, 453). Using the latter method, and neglecting squares and products of the angles, we first calculate the value of det($A$), the determinant of the $4\times 4$ matrix in equation (3.9a). We shall expand by elements in the first row [see Sheriff and Geldart, 1995, equation (15.2)]; when we do this we see that the 2nd and 4th $\left(3\times 3\right)\$$ determinants in the expansion are multiplied by $+\delta _{1}$ and $-\delta _{2}$, respectively, and since we are neglecting products and squares of angles, angles inside these two determinants have been replaced with zeros. The expansion about the first row becomes

${\begin{aligned}\mathrm {det} \left(A\right)=\left|{\begin{array}{ccc}{1}&{-\theta _{2}}&{1}\\{-2\delta _{1}W_{1}}&{-Z_{2}}&{-2\delta _{2}W_{2}}\\{W_{1}}&{2\delta _{2}W_{2}}&{-W_{2}}\end{array}}\right|+\delta _{1}\left|{\begin{array}{ccc}{0}&{0}&{1}\\{Z_{1}}&{-Z_{2}}&{0}\\{0}&{0}&{-W_{2}}\end{array}}\right|\\+\left|{\begin{array}{ccc}{\theta _{1}}&{1}&{1}\\{Z_{1}}&{-2\delta _{1}W_{1}}&{-2\delta _{2}W_{2}}\\{2\delta _{1}W_{1}}&{W_{1}}&{-W_{2}}\end{array}}\right|-\delta _{2}\left|{\begin{array}{ccc}{0}&{1}&{0}\\{Z_{1}}&{0}&{-Z_{2}}\\{0}&{W_{1}}&{0}\end{array}}\right|\\=\left(Z_{2}W_{2}+Z_{2}W_{1}\right)+\left(Z_{1}W_{1}+Z_{1}W_{2}\right)=\left(Z_{1}+Z_{2}\right)\left(W_{1}+W_{2}\right).\end{aligned}}$

Next we calculate the values of $\mathrm {det} \left(A_{i}\right)$ and $\mathrm {det} \left(B_{i}\right)\$$, $i=1$, 2, where $\mathrm {det} \left(A_{1}\right)$ is $\mathrm {det} \left(A\right)$ with column 1 replaced with the elements of the right-hand matrix in equation (3.9a), etc. (see Cramer’s rule in Sheriff and Geldart, 1995, problem 15.2j). Expanding $\mathrm {det} \left(A_{1}\right)$ about the first row and setting the angles in the 2nd and 4th $3\times 3$ determinant equal to zero as before, the expansion becomes

${\begin{aligned}\mathrm {det} \left(A\right)=\left|{\begin{array}{ccc}{1}&{-\theta _{2}}&{1}\\{-2\delta _{1}W_{1}}&{-Z_{2}}&{-2\delta _{2}W_{2}}\\{W_{1}}&{2\delta _{2}W_{2}}&{-W_{2}}\end{array}}\right|+\delta _{1}\left|{\begin{array}{ccc}{0}&{0}&{1}\\{Z_{1}}&{-Z_{2}}&{0}\\{0}&{0}&{-W_{2}}\end{array}}\right|\\+\left|{\begin{array}{ccc}{\theta _{1}}&{1}&{1}\\{Z_{1}}&{-2\delta _{1}W_{1}}&{-2\delta _{2}W_{2}}\\{2\delta _{1}W_{1}}&{W_{1}}&{-W_{2}}\end{array}}\right|-\delta _{2}\left|{\begin{array}{ccc}{0}&{1}&{0}\\{Z_{1}}&{0}&{-Z_{2}}\\{0}&{W_{1}}&{0}\end{array}}\right|\\=\left(Z_{2}W_{2}+Z_{2}W_{1}\right)+\left(Z_{1}W_{1}+Z_{1}W_{2}\right)=\left(Z_{1}+Z_{2}\right)\left(W_{1}+W_{2}\right).\end{aligned}}$

The second and fourth determinants are zero, so

${\begin{aligned}\mathrm {det} \left(A_{1}\right)=\left(Z_{2}W_{2}+Z_{2}W_{1}\right)-\left(Z_{1}W_{2}+Z_{1}W_{1}\right)=\left(W_{1}+W_{2}\right)\left(Z_{2}-Z_{1}\right).\end{aligned}}$

Dividing by $\mathrm {det} \left(A\right)$, we get

${\begin{aligned}R=A_{1}/A_{0}=\left(Z_{2}-Z_{1}\right)/\left(Z_{2}+Z_{1}\right),\end{aligned}}$

which is the same as equation (3.6a). Similarly, we find that

${\begin{aligned}T=A_{2}/A_{0}=2Z_{1}/\left(Z_{2}+Z_{1}\right),\\B_{1}/A_{0}=\left(2W_{2}q+4Z_{1}r\right)/\left(Z_{1}+Z_{2}\right)\left(W_{1}+W_{2}\right),\\B_{2}/A_{0}=\left(2W_{1}q-4Z_{1}r\right)/\left(Z_{1}+Z_{2}\right)\left(W_{1}+W_{2}\right),\end{aligned}}$

where $q=\left(Z_{1}\theta _{2}-Z_{2}\theta _{1}\right)$, $r=\left(W_{1}\delta _{1}-W_{2}\delta _{2}\right)$. Note that

${\begin{aligned}A_{1}+A_{2}=A_{0},\end{aligned}}$

${\begin{aligned}B_{1}+B_{2}=2qA_{0}/\left(Z_{1}+Z_{2}\right)={\frac {2A_{0}\left(Z_{1}\theta _{2}-Z_{2}\theta _{1}\right)}{\left(Z_{1}+Z_{2}\right)}}\\=0\qquad {\text{for normal incidence}}.\end{aligned}}$

Also, when $\rho _{1}=\rho _{2},$

${\begin{aligned}\theta _{2}=\left(\alpha _{2}/\alpha _{1}\right)\theta _{1}=\left(Z_{2}/Z_{1}\right)\theta _{1},\end{aligned}}$

so q $=0$ and $B_{1}=-B_{2}=4Z_{1}r/[\left(W_{1}+W_{2}\right)(Z_{1}+Z_{2})].$

Continue reading

Previous section	Next section
Amplitude/energy of reflections and multiples	Magnitude
Previous chapter	Next chapter
Theory of Seismic Waves	Geometry of seismic waves
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
• Magnitude
• AVO versus AVA and effect of velocity gradient
• Variation of reflectivity with angle (AVA)

External links

find literature about Reflection/transmission coefficients at small angles and magnitude