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

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 for VP \([n_{\theta} \times n_\text{vsvp}]\). Since the PS reflection at zero angle is zero, this value is not used and is only available to ensure function signature compatibility with other linearization routines.

G2np.ndarray

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

G3np.ndarray

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

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:

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

where

\[\begin{split}\begin{align} G_2(\theta) &= \tan \frac{\theta}{2} \left\{4 (V_S/V_P)^2 \sin^2 \theta - 4(V_S/V_P) \cos \theta \cos \phi \right\},\\ G_3(\theta) &= -\tan \frac{\theta}{2} \left\{1 - 2 (V_S/V_P)^2 \sin^2 \theta + 2(V_S/V_P) \cos \theta \cos \phi\right\},\\ \frac{\Delta V_S}{\overline{V_S}} &= 2 \frac{V_{S,2}-V_{S,1}}{V_{S,2}+V_{S,1}},\\ \frac{\Delta \rho}{\overline{\rho}} &= 2 \frac{\rho_2-\rho_1}{\rho_2+\rho_1}. \end{align}\end{split}\]

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 \(V_S/V_P\) ratio.

1

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