# 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
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 Aki-Richards approximation $$[n_\theta \times n_\text{vsvp}]$$

G2np.ndarray

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

G3np.ndarray

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

Notes

The three terms Aki-Richards approximation [1], [2], is used to compute the reflection coefficient as linear combination of contrasts in $$V_P$$, $$V_S$$, and $$\rho.$$ More specifically:

$R(\theta) = G_1(\theta) \frac{\Delta V_P}{\overline{V}_P} + G_2(\theta) \frac{\Delta V_S}{\overline{V}_S} + G_3(\theta) \frac{\Delta \rho}{\overline{\rho}}$

where

\begin{split}\begin{align} 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}{\overline{V}_P} &= 2 \frac{V_{P,2}-V_{P,1}}{V_{P,2}+V_{P,1}},\\ \frac{\Delta V_S}{\overline{V}_S} &= 2 \frac{V_{S,2}-V_{S,1}}{V_{S,2}+V_{S,1}}, \\ \frac{\Delta \rho}{\overline{\rho}} &= 2 \frac{\rho_2-\rho_1}{\rho_2+\rho_1}. \end{align}\end{split}
1

https://wiki.seg.org/wiki/AVO_equations

2

Aki, K., and Richards, P. G. (2002). Quantitative Seismology (2nd ed.). University Science Books.