pylops.avo.avo.akirichards

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

Three terms Aki-Richards approximation.

Computes the coefficients of the of three terms Aki-Richards 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 Aki-Richards approximation \([n_{theta} \times n_{vsvp}]\)

G2 : np.ndarray

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

G3 : np.ndarray

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

Notes

The three terms Aki-Richards approximation is used to compute the reflection coefficient as linear combination of contrasts in \(V_P\), \(V_S\), and \(\rho\). More specifically:

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

where \(G_1(\theta) = \frac{1}{2 cos^2 \theta}\), \(G_2(\theta) = -4 (V_S/V_P)^2 sin^2 \theta\), \(G_3(\theta) = 0.5 - 2 (V_S/V_P)^2 sin^2 \theta\), \(\frac{\Delta V_P}{\bar{V_P}} = 2 \frac{V_{P,2}-V_{P,1}}{V_{P,2}+V_{P,1}}\), \(\frac{\Delta V_S}{\bar{V_S}} = 2 \frac{V_{S,2}-V_{S,1}}{V_{S,2}+V_{S,1}}\), and \(\frac{\Delta \rho}{\bar{\rho}} = 2 \frac{\rho_2-\rho_1}{\rho_2+\rho_1}\).