Fatti et al gave a formulation derived from the Aki-Richards equation, which does not account for the critical angle since it only computes with the incident angle (that is, it does not contain Snell's law):
where RPP is the P-wave reflectivity, θ is the incidence angle, I is the acoustic impedance for P and S waves (denoted by subscript), V is acoustic velocity, and ρ is bulk density. The deltas indicate that the difference at each interface is to be used, and given as a proportion of the average of the quantity across the interface (so that these coefficients are positive).
Many people read the equation as three terms: the first driven principally by acoustic P-wave impedance, the second by shear wave impedance, and the third by density.
Implementation in Modelr
Implementation in Python
Fatti's formula is implemented in bruges as
- Fatti et al (1994), Geophysics 59(9), p 1362