Proof of a generalized reciprocal method relation

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	11
Pages	415 - 468
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 11.7

Prove equation (11.7a), assuming that $\left(\xi _{i}-\xi _{j}\right)\approx 0$ for all values of $i$ and $j$.

Background

The generalized reciprocal method (GRM) can be used with beds of different dips provided all have the same strike. Figure 11.7a shows a series of such beds. Depths normal to the beds are denoted by $z_{Ai}$ and $z_{Bi}$, $\alpha _{i}$ and $\beta _{i}$ are angles of incidence, those for the deepest interface being critical angles, $\xi _{i}$ is the dip of the interface at the top of the $i^{\rm {th}}$ layer. To get the traveltime from $A$ to $B$, $t_{AB}$, we consider a plane wavefront $PQ$ that passes through $A$ at time $t=0$ in a direction such that it will be totally refracted at one of the interfaces, the third in the case of Figure 11.7a. The wavefront reaches $C$ at time $t_{AC}$ and $R$ at time $t_{AR}$ where

${\begin{aligned}t_{AC}=\left(z_{A1}\cos \alpha _{1}/V_{1}\right),\quad t_{AR}=\sum \limits _{i=1}^{3}z_{Ai}\cos \alpha _{i}/V_{i}.\end{aligned}}$

The same wavefront will travel upward from $T$ to $B$ in time

${\begin{aligned}t_{BT}=\sum \limits _{i=1}^{3}z_{Bi}\cos \beta _{i}/V_{i}.\end{aligned}}$

Since $\alpha _{3}$ is the critical angle,

${\begin{aligned}t_{AB}=\sum \limits _{i=1}^{3}(z_{Ai}+z_{Bi})/V_{i}+RV/V_{4}.\end{aligned}}$

Generalizing, we get for $n$ layers

${\begin{aligned}t_{AB}=\sum \limits _{i=1}^{n-1}(z_{Ai}+z_{Bi})/V_{i}+RV/V_{n}.\end{aligned}}$

But $RV=YJ=EJ\cos \left(\xi _{3}-\xi _{2}\right)=AB\cos \xi _{1}\cos \left(\xi _{2}-\xi _{1}\right)\cos \left(\xi _{3}-\xi _{2}\right)$; for $n$ layers, we get

${\begin{aligned}t_{AB}=\sum \limits _{i=1}^{n-1}(z_{Ai}+z_{Bi})/V_{i}+AB\left(S_{n}/V_{n}\right),\end{aligned}}$

where

${\begin{aligned}S_{n}=\cos \xi _{1}\cos \left(\xi _{2}-\xi _{1}\right)\ldots \cos \left(\xi _{n-1}-\xi _{n-2}\right)\approx \cos \xi _{n-1},\end{aligned}}$

(11.7a)

all differences in dip being small, that is, $\left(\xi _{i}-\xi _{j}\right)\approx 0$. We shall not carry the derivation of the GRM formulas beyond this point; those who are interested should consult Sheriff and Geldart, 1995, Section 11.3.3, or Palmer (1980).

Solution

We are asked to prove that

${\begin{aligned}\cos \xi _{1}\cos \left(\xi _{2}-\xi _{1}\right)\cos \left(\xi _{3}-\xi _{2}\right)\ldots \cos \left(\xi _{n-1}-\xi _{n-2}\right)\approx \cos \xi _{n-1},\end{aligned}}$

where the differences in dip are all small, that is, $\xi _{j}-\xi _{j-1}\approx 0$. We start with the single cosine on the right-hand side of the equation and try to express it as a product of cosines. We write it as $\cos \left(\xi _{n-1}-\xi _{m}\right)$ and expand:

${\begin{aligned}\cos \left(\xi _{n-1}-\xi _{m}\right)=\cos[\left(\xi _{n-1}-\xi _{n-2}\right)+(\xi _{n-2}-\xi _{m})].\end{aligned}}$

Since all differences in dip are small, we expand the right-hand side and set the products of the sines equal to zero. Thus,

${\begin{aligned}\cos \left(\xi _{n-1}-\xi _{m}\right)\approx \cos \left(\xi _{n-1}-\xi _{n-2}\right)\cos \left(\xi _{n-2}-\xi _{m}\right).\end{aligned}}$

Next we treat the right-hand cosine in the same way, writing it as $\cos[\left(\xi _{n-2}-\xi _{n-3}\right)+(\xi _{n-3}-\xi _{m})]$. We now expand the factor and drop the sine term. Continuing in this way we eventually arrive at the result

${\begin{aligned}\cos \left(\xi _{n-1}-\xi _{m}\right)&\approx \cos \left(\xi _{n-1}-\xi _{n-2}\right)\cos \left(\xi _{n-2}-\xi _{n-3}\right)\;\ldots \\&\qquad \times \cos \left(\xi _{2}-\xi _{1}\right)\cos \left(\xi _{1}-\xi _{m}\right).\end{aligned}}$

We now take $\xi _{m}=0$ and the result is equation (11.7a).

Figure 11.8a.  Illustrating delay time.

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Table of Contents (book)

Also in this chapter

• Salt lead time as a function of depth
• Effect of assumptions on refraction interpretation
• Effect of a hidden layer
• Proof of the ABC refraction equation
• Adachi’s method
• Refraction interpretation by stripping
• Proof of a generalized reciprocal method relation
• Delay time
• Barry’s delay-time refraction interpretation method
• Parallelism of half-intercept and delay-time curves
• Wyrobek’s refraction interpretation method
• Properties of a coincident-time curve
• Interpretation by the plus-minus method
• Comparison of refraction interpretation methods
• Feasibility of mapping a horizon using head waves
• Refraction blind spot
• Interpreting marine refraction data

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