Prove equation (11.7a), assuming that for all values of and .
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 and , and are angles of incidence, those for the deepest interface being critical angles, is the dip of the interface at the top of the layer. To get the traveltime from to , , we consider a plane wavefront that passes through at time 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 at time and at time where
The same wavefront will travel upward from to in time
Since is the critical angle,
Generalizing, we get for layers
But ; for layers, we get
all differences in dip being small, that is, . 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).
We are asked to prove that
where the differences in dip are all small, that is, . 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 and expand:
Since all differences in dip are small, we expand the right-hand side and set the products of the sines equal to zero. Thus,
Next we treat the right-hand cosine in the same way, writing it as . We now expand the factor and drop the sine term. Continuing in this way we eventually arrive at the result
We now take and the result is equation (11.7a).
Illustrating delay time.
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Proof of a generalized reciprocal method relation