# pylops.avo.avo.fatti#

pylops.avo.avo.fatti(theta, vsvp, n=1)[source]#

Three terms Fatti approximation.

Computes the coefficients of the three terms Fatti approximation for a set of angles and a constant or variable VS/VP ratio.

Parameters
thetanp.ndarray

Incident angles in degrees

vsvpnp.ndarray or float

$$V_S/V_P$$ ratio

nint, optional

Number of samples (if vsvp is a scalar)

Returns
G1np.ndarray

First coefficient of three terms Smith-Gidlow approximation $$[n_{\theta} \times n_\text{vsvp}]$$

G2np.ndarray

Second coefficient of three terms Smith-Gidlow approximation $$[n_{\theta} \times n_\text{vsvp}]$$

G3np.ndarray

Third coefficient of three terms Smith-Gidlow approximation $$[n_{\theta} \times n_\text{vsvp}]$$

Notes

The three terms Fatti approximation , , is used to compute the reflection coefficient as linear combination of contrasts in $$\text{AI},$$ $$\text{SI}$$, and $$\rho.$$ More specifically:

$R(\theta) = G_1(\theta) \frac{\Delta \text{AI}}{\bar{\text{AI}}} + G_2(\theta) \frac{\Delta \text{SI}}{\overline{\text{SI}}} + G_3(\theta) \frac{\Delta \rho}{\overline{\rho}}$

where

\begin{split}\begin{align} G_1(\theta) &= 0.5 (1 + \tan^2 \theta),\\ G_2(\theta) &= -4 (V_S/V_P)^2 \sin^2 \theta,\\ G_3(\theta) &= 0.5 \left(4 (V_S/V_P)^2 \sin^2 \theta - \tan^2 \theta\right),\\ \frac{\Delta \text{AI}}{\overline{\text{AI}}} &= 2 \frac{\text{AI}_2-\text{AI}_1}{\text{AI}_2+\text{AI}_1},\\ \frac{\Delta \text{SI}}{\overline{\text{SI}}} &= 2 \frac{\text{SI}_2-\text{SI}_1}{\text{SI}_2+\text{SI}_1},\\ \frac{\Delta \rho}{\overline{\rho}} &= 2 \frac{\rho_2-\rho_1}{\rho_2+\rho_1}. \end{align}\end{split}
1

https://www.subsurfwiki.org/wiki/Fatti_equation

2

Jan L. Fatti, George C. Smith, Peter J. Vail, Peter J. Strauss, and Philip R. Levitt, (1994), “Detection of gas in sandstone reservoirs using AVO analysis: A 3-D seismic case history using the Geostack technique,” Geophysics 59: 1362-1376.