# Fatti equation

An approximate solution to the Zoeppritz equation, used in seismic analysis.

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): $R_\mathrm{PP}(\theta) = {(1+\tan^2\theta)\,\frac{\Delta I_\mathrm{P}}{2I_\mathrm{P}}}\ \ -\ \ {8\left(\frac{V_\mathrm{S}}{V_\mathrm{P}}\right)^2\sin^2\theta\,\frac{\Delta I_\mathrm{S}}{2I_\mathrm{S}}} \ \ - \ \ {\left[\frac{1}{2}\tan^2\theta - 2\left(\frac{V_\mathrm{S}}{V_\mathrm{P}}\right)^2\sin^2\theta\right]\frac{\Delta\rho}{\rho}}$

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 Python

Fatti's formula is implemented in bruges as bruges.reflection.fatti.