# Proof of the ABC refraction equation

Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 11 415 - 468 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

## 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 ${\displaystyle t_{AC}}$, ${\displaystyle t_{BC}}$ for the traveltimes from the sources to a geophone at ${\displaystyle C}$ and ${\displaystyle t_{AB}}$ for the traveltime from ${\displaystyle A}$ to ${\displaystyle B}$, the ABC equation gives the depth ${\displaystyle h_{C}}$ as

 {\displaystyle {\begin{aligned}h_{C}={\frac {1}{2}}\left(t_{AC}+t_{BC}-t_{AB}\right)\left[V_{1}V_{2}/\left(V_{2}^{2}-V_{1}^{2}\right)^{1/2}\right].\end{aligned}}} (11.4a)

### Solution

Assuming that ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$ are coplanar and that elevation corrections have been made, we can write

{\displaystyle {\begin{aligned}V_{1}\left(t_{AC}+t_{BC}-t_{AB}\right)&=V_{1}\left(t_{MC}+t_{NC}-t_{MN}\right)=2V_{1}t_{MC}-V_{1}t_{MN}\\&=2h_{C}/\cos \theta _{c}-MN\left(V_{1}/V_{2}\right)\\&=2h_{C}/\cos \theta _{c}-\left(2h_{C}\tan \theta _{c}\right)\sin \theta _{c}\\&=2h_{C}/\cos \theta _{c}-\left(2h_{C}\sin ^{2}\theta _{c}/\cos \theta _{c}\right)\\&=\left(2h_{C}/\cos \theta _{c}\right)(1-\sin ^{2}\theta _{c})=2h_{C}\cos \theta _{c}.\end{aligned}}}

Figure 11.4a.  The ABC method.

Thus,

{\displaystyle {\begin{aligned}h_{C}&={\frac {1}{2}}\left(t_{AC}+t_{BC}-t_{AB}\right)V_{1}/\cos \theta _{c}\\&={\frac {1}{2}}\left(t_{AC}+t_{BC}-t_{AB}\right)V_{1}/[1-(V_{1}/V_{2})^{2}]^{1/2}\\&={\frac {1}{2}}\left(t_{AC}+t_{BC}-t_{AB}\right)V_{1}V_{2}/[V_{2}^{2}-V_{1}^{2}]^{1/2}.\end{aligned}}}