Complex coefficient of reflection
 | |
| 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 3.6a
Using the expression $ \psi =Ae^{\mathrm {j} \omega \left(r/V-t\right)} $ to represent a plane wave incident on a plane interface, show that a complex coefficient of reflection,
$ {\begin{aligned}R=a+\mathrm {j} b,\quad a^{2}+b^{2}<1,\end{aligned}} $
R [defined by equation (3.6a) below] corresponds to a reduction in amplitude by the factor $ (a^{2}+b^{2})^{1/2} $ and an advance in phase by $ \tan ^{-1}\left(b/a\right) $.
Background
When a plane P-wave is incident perpendicularly on a plane interface, the tangential displacements and stresses vanish, so equations (3.2f,i) are not constraining and we are left with equations (3.2e,h). Moreover, $ \delta _{1}=0=\delta _{2} $, so the equations reduce to
$ {\begin{aligned}A_{1}+A_{2}&=A_{0},\\Z_{1}A_{1}-Z_{2}A_{2}&=-Z_{1}A_{0},\end{aligned}} $
$ Z_{1} $, $ Z_{2} $, being impedances (see problem 3.2). The solution of these equations is
$ {\begin{aligned}R&={\frac {A_{1}}{A_{0}}}={\frac {\rho _{2}\alpha _{2}-\rho _{1}\alpha _{1}}{\rho _{2}\alpha _{2}+\rho _{1}\alpha _{1}}}={\frac {Z_{2}-Z_{1}}{Z_{2}+Z_{1}}},\end{aligned}} $ ()
$ {\begin{aligned}T&={\frac {A_{2}}{A_{0}}}={\frac {2\rho _{1}\alpha _{1}}{\rho _{2}\alpha _{2}+\rho _{1}\alpha _{1}}}={\frac {2Z_{1}}{Z_{2}+Z_{1}}},\end{aligned}} $ ()
$ R $ and $ T $ being the coefficient of reflection and coefficient of transmission, respectively. Although equations (3.6a,b) hold only for normal incidence, the definitions $ R=A_{1}/A_{0} $ and $ T=A_{2}/A_{0} $ are valid for all angles of incidence. A negative value of $ R $ means that $ A_{1} $ in equation (3.6a) is opposite in sign to 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/":): A_{0} . Since 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/":): e^{\mathrm{j}\pi} =-1 , the minus sign is equivalent to adding 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/":): \pi to the phase of the waveform in part (a), that is, reversing the phase. Note that (except for phase reversal) 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 is independent of the direction of incidence on the interface; however, the magnitude 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/":): T depends upon this direction, and when necessary, we shall write 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/":): T\downarrow 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/":): T\uparrow to distinguish between the two values. Note the following relations:
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 + T\downarrow = 1,\quad T\uparrow + T\downarrow = 2,\quad T\uparrow T\downarrow = E_{T}, \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/":): E_{T} is the fraction of energy transmitted as defined in equation (3.7a).
Euler’s formulas (see Sheriff and Geldart, 1995, p. 564) express 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/":): \sin x 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/":): \cos x as
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} \sin x=\left(e^{\mathrm{j}x} -e^{-\mathrm{j}x} \right)/2\mathrm{j},\quad \cos x=\left(e^{\mathrm{j}x} +e^{-\mathrm{j}x} \right)/2. \end{align} ()
The hyperbolic sine and cosine are defined by the relations
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} \sinh x=\left(e^{x} -e^{-x} \right)/2,\quad \cosh x=\left(e^{x} +e^{-x} \right)/2. \end{align} ()
Solution
Writing 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/":): \psi 0=A_{0} e^{\mathrm{j}\omega \left(r/V-t\right)} = A_{0} e^{\mathrm{j}\left(\kappa r-\omega t\right)} , we have
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} \psi_{1} =R\psi_{0} = \left(a+\mathrm{j}b\right)A_{0} e^{\left(\kappa r-\omega t\right)}. \end{align}
But 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(a+\mathrm{j}b\right)=(a^{2} +b^{2} )^{1/2} e^{\mathrm{j}\phi} 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/":): \tan \phi = b/a (see Sheriff and Geldart, 1995, section 15.1.5), 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} \psi_{1} =(a^{2} +b^{2} )^{1/2} A_{0} e^{\mathrm{j}(kr-\omega t+\phi)}. \end{align}
Since 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/":): (a^{2} +b^{2} )^{1/2} <1 , the amplitude is reduced by this factor and the phase is advanced 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/":): \phi
Problem 3.6b
Show that an imaginary angle of refraction 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/":): \theta_{2} (see Figure 3.1b) in equations (3.2e,f,h,i) leads to a complex value 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/":): R and hence to phase shifts.
Solution
Let 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/":): \theta _{2} =\pi /2-\mathrm{j}\theta , 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/":): \theta is real. Then, using equations (3.6d,e), we have
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} \sin \theta_{2} =\cos \left(\mathrm{j}\theta \right)=\cosh \theta,\qquad \cos \theta_{2} = \sin \left(\mathrm{j} \theta \right) = \mathrm{j}\sinh \theta, \end{align}
hence some of the coefficients in equations (3.2e,f,h,i) are imaginary 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/":): R 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/":): T will in general be complex, so phase shifts will occur.
Problem 3.6c
Show that 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/":): R is negative, 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/":): A_{2} >A_{0} .
Solution
If 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 is negative, 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/":): Z_{1} >Z_{2} in equation (3.6a), 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/":): 2Z_{1} >\left(Z_{1} +Z_{2} \right) . Therefore, from equation (3.6b) we see 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/":): T>1 , and since 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/":): T=A_{2} /A_{0} , 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/":): A_{2} >A_{0} .
Continue reading
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| Reinforcement depth in marine recording | Reflection and transmission coefficients |
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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
- Reflection and transmission coefficients
- Amplitude/energy of reflections and multiples
- Reflection/transmission coefficients at small angles and magnitude
- Magnitude
- AVO versus AVA and effect of velocity gradient
- Variation of reflectivity with angle (AVA)
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