Reflection/transmission coefficients at small angles and magnitude
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| 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}} $ ()
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 $), the determinant of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 4\times 4 matrix in equation (3.9a). We shall expand by elements in the first row [see Sheriff and Geldart, 1995, equation (15.2)]; when we do this we see that the 2nd and 4th Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(3\times 3\right)$ determinants in the expansion are multiplied by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): +\delta_{1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): -\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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align}\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{align}
Next we calculate the values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{det}\left(A_{i} \right) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{det}\left(B_{i} \right)$ , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): i=1 , 2, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{det}\left(A_{1} \right) is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{det}\left(A_{1} \right) about the first row and setting the angles in the 2nd and 4th Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 3\times 3 determinant equal to zero as before, the expansion becomes
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align}\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{align}
The second and fourth determinants are zero, so
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \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{align}
Dividing by $ \mathrm {det} \left(A\right) $, we get
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} R=A_{1} /A_{0} =\left(Z_{2} -Z_{1} \right)/\left(Z_{2} +Z_{1} \right), \end{align}
which is the same as equation (3.6a). Similarly, we find that
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 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{align}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): q=\left(Z_{1} \theta _{2} -Z_{2} \theta _{1} \right) , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r=\left(W_{1} \delta_{1} -W_{2} \delta_{2} \right) . Note that
$ {\begin{aligned}A_{1}+A_{2}=A_{0},\end{aligned}} $
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 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{align}
Also, when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \rho_{1} =\rho_{2},
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \theta _{2} =\left(\alpha _{2} /\alpha _{1} \right)\theta _{1} =\left(Z_{2} /Z_{1} \right)\theta _{1}, \end{align}
so q Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): = 0 and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B_{1} = -B_{2} = 4Z_{1} r/[\left(W_{1} +W_{2} \right)(Z_{1} +Z_{2})].
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| Amplitude/energy of reflections and multiples | Magnitude |
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| Theory of Seismic Waves | Geometry of seismic waves |
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)
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