Proof of a generalized reciprocal method relation

ADVERTISEMENT
From SEG Wiki
Jump to: navigation, search

Problem 11.7

Prove equation (11.7a), assuming that for all values of and .

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 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

where


(11.7a)

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).

Solution

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).

Figure 11.8a.  Illustrating delay time.

Continue reading

Previous section Next section
Refraction interpretation by stripping Delay time
Previous chapter Next chapter
Geologic interpretation of reflection data 3D methods

Table of Contents (book)

Also in this chapter

External links

find literature about
Proof of a generalized reciprocal method relation
SEG button search.png Datapages button.png GeoScienceWorld button.png OnePetro button.png Schlumberger button.png Google button.png AGI button.png