A simple approximation to the P-wave reflection coefficient and its implication in the inversion of amplitude variation with offset data

I derive an approximate formula for the plane P-wave reflection coefficient as a function of ray-parameter. The approximation shows that the behavior of the P-wave reflection coefficient at nonnormal angles of incidence is mainly controlled by two parameters: (1) R f , the fluid-fluid reflection coefficient (i.e., the reflection coefficient when the S-wave velocities in both media are set to zero) and (2) delta mu /rho , the ratio of the contrast in shear moduli to the average bulk density. I also show that the other formulas for the P-wave reflection coefficient given by R. Bortfeld and R. Shuey can be approximately derived from my formula. I give numerical examples to demonstrate the accuracy of the formula. Using a least-squares inversion of theoretical values for the reflection coefficients, I demonstrate that, in a linear inversion of amplitude versus offset data, delta mu /rho is better estimated than the contrast in Poisson's ratio, delta sigma . Finally, comparing the exact reflection coefficient with Shuey's approximation for typical shale and Class 1 gas-sand models, I show that Shuey's approximation under-estimates the value of delta sigma for such reflections.