, vsvp, n=1)[source]

PS reflection coefficient

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

theta : np.ndarray

Incident angles in degrees

vsvp : np.ndarray or float

\(V_S/V_P\) ratio

n : int, optional

Number of samples (if vsvp is a scalar)

G1 : np.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.

G2 : np.ndarray

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

G3 : np.ndarray

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


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:

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


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

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.