pylops.avo.avo.ps

pylops.avo.avo.ps(theta, vsvp, n=1)

PS reflection coefficient

Computes the coefficients for the PS 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 scalar)

Returns:

G1 : np.ndarray

first coefficient for VP \([n_{theta} \times n_{vsvp}]\)

G2 : np.ndarray

second coefficient for VS \([n_{theta} \times n_{vsvp}]\)

G3 : np.ndarray

third coefficient for density \([n_{theta} \times n_{vsvp}]\)

Notes

The approximation in [1] is used to compute the PS reflection coefficient as linear combination of contrasts in \(V_P\), \(V_S\), and \(\rho\). More specifically:

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

where \(G_2(\theta) = tan \theta / 2 [4 (V_S/V_P)^2 sin^2 \theta - 4(V_S/V_P) cos \theta cos \phi]\), \(G_3(\theta) = -tan \theta / 2 [1 - 2 (V_S/V_P)^2 sin^2 \theta + 2(V_S/V_P) cos \theta cos \phi]\), \(\frac{\Delta V_S}{\bar{V_S}} = 2 \frac{V_{S,2}-V_{S,1}}{V_{S,2}+V_{S,1}}\), and \(\frac{\Delta \rho}{\bar{\rho}} = 2 \frac{\rho_2-\rho_1}{\rho_2+\rho_1}\). Note that \(\theta\) is the P-incidence angle whilst \(\phi\) is the S-reflected angle which is computed using Snell’s law and the average \(VS/VP\) ratio.

[1]	Xu, Y., and Bancroft, J.C., “Joint AVO analysis of PP and PS seismic data”, CREWES Report, vol. 9. 1997.