Proof of the ABC refraction equation

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Problem 11.4

Prove the ABC refraction equation [equation (11.4a)].

Background

The ABC equation is often used to calculate the weathering thickness. Assuming reversed profiles as shown in Figure 11.4a and writing , for the traveltimes from the sources to a geophone at and for the traveltime from to , the ABC equation gives the depth as


(11.4a)

Solution

Assuming that , , and are coplanar and that elevation corrections have been made, we can write

Figure 11.4a.  The ABC method.

Thus,

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