Reinforcement depth in marine recording

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 3.5a

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
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
• 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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