Reinforcement depth in marine recording
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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 3.5a
For a source at a depth $ h $, show that the maximum amplitude of a downgoing incident wave and its reflection at the surface of the sea occurs at the depth $ \lambda /\left(4\cos \theta \right) $, where $ \theta $ is the angle of incidence, by expressing the pressure $ P $ in the form used in equations (3.1b,d) and applying appropriate boundary conditions.
Solution
Since the interface is liquid/vacuum, only two waves exist, the incident and reflected P-waves. Taking the z-axis positive downward, we take $ {\mathcal {P}} $ in the form
$ {\begin{aligned}{\mathcal {P}}=A_{0}e^{\mathrm {j} \omega p\left(x+z\cot \theta \right)}+A_{1}e^{\mathrm {j} \omega p\left(x-z\cot \theta \right)}.\end{aligned}} $
There is only one boundary condition, namely that $ {\mathcal {P}}=0 $ at $ z=0 $. This gives $ A_{1}=-A_{0} $ Using Euler’s formulas (see Sheriff and Geldart, 1995, problem 15.12a), we get
$ {\begin{aligned}{\mathcal {P}}&=A_{0}e^{\mathrm {j} \omega px}\left(e^{\mathrm {j} \omega pz\cot \theta }-e^{-\mathrm {j} \omega pz\cot \theta }\right)\\&=2\mathrm {j} A_{0}e^{\mathrm {j} \omega \left(px-t\right)}\sin \left(\omega pz\cot \theta \right)\end{aligned}} $
upon inserting the time factor. The amplitude of the combined incident and reflected waves is
$ {\begin{aligned}2A_{0}\sin[\left(\omega pz\cot \theta \right]=2A_{0}\sin \left[\left(\omega z/\alpha \right)\cos \theta \right].\end{aligned}} $
It is a maximum when $ \left(\omega z/\alpha \right)\cos \theta =\pi /2 $, that is, when
$ {\begin{aligned}z=\left(\pi /2\right)\alpha /\left(\omega \cos \theta \right)=\lambda /\left(4\cos \theta \right).\end{aligned}} $
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| Zoeppritz’s equations for incident SV- and SH-waves | Complex coefficient of reflection |
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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
- Complex coefficient of reflection
- 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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