pylops.avo.avo.fatti

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

Three terms Fatti approximation.

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

Parameters:
theta : np.ndarray

Incident angles in degrees

vsvp : np.ndarray or float

VS/VP ratio

n : int, optional

number of samples (if vsvp is a scalare)

Returns:
G1 : np.ndarray

first coefficient of three terms Smith-Gidlow approximation \([n_{theta} \times n_{vsvp}]\)

G2 : np.ndarray

second coefficient of three terms Smith-Gidlow approximation \([n_{theta} \times n_{vsvp}]\)

G3 : np.ndarray

third coefficient of three terms Smith-Gidlow approximation \([n_{theta} \times n_{vsvp}]\)

Notes

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

\[R(\theta) = G_1(\theta) \frac{\Delta AI}{\bar{AI}} + G_2(\theta) \frac{\Delta SI}{\bar{SI}} + G_3(\theta) \frac{\Delta \rho}{\bar{\rho}}\]

where \(G_1(\theta) = 0.5 (1 + tan^2 \theta)\), \(G_2(\theta) = -4 (V_S/V_P)^2 sin^2 \theta\), \(G_3(\theta) = 0.5 (4 (V_S/V_P)^2 sin^2 \theta - tan^2 \theta)\), \(\frac{\Delta AI}{\bar{AI}} = 2 \frac{AI_2-AI_1}{AI_2+AI_1}\). \(\frac{\Delta SI}{\bar{SI}} = 2 \frac{SI_2-SI_1}{SI_2+SI_1}\). \(\frac{\Delta \rho}{\bar{\rho}} = 2 \frac{\rho_2-\rho_1}{\rho_2+\rho_1}\).