pylops.avo.avo.ps¶

pylops.avo.avo.ps(theta, 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.

Parameters: 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 $$[n_{\theta} \times n_\text{vsvp}]$$

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.
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