pylops.avo.avo.fatti#
- pylops.avo.avo.fatti(theta, vsvp, n=1)[source]#
Three terms Fatti approximation.
Computes the coefficients of the three terms Fatti approximation for a set of angles and a constant or variable VS/VP ratio.
- Parameters
- Returns
- G1
np.ndarray First coefficient of three terms Smith-Gidlow approximation \([n_{\theta} \times n_\text{vsvp}]\)
- G2
np.ndarray Second coefficient of three terms Smith-Gidlow approximation \([n_{\theta} \times n_\text{vsvp}]\)
- G3
np.ndarray Third coefficient of three terms Smith-Gidlow approximation \([n_{\theta} \times n_\text{vsvp}]\)
- G1
Notes
The three terms Fatti approximation [1], [2], is used to compute the reflection coefficient as linear combination of contrasts in \(\text{AI},\) \(\text{SI}\), and \(\rho.\) More specifically:
\[R(\theta) = G_1(\theta) \frac{\Delta \text{AI}}{\bar{\text{AI}}} + G_2(\theta) \frac{\Delta \text{SI}}{\overline{\text{SI}}} + G_3(\theta) \frac{\Delta \rho}{\overline{\rho}}\]where
\[\begin{split}\begin{align} G_1(\theta) &= 0.5 (1 + \tan^2 \theta),\\ G_2(\theta) &= -4 (V_S/V_P)^2 \sin^2 \theta,\\ G_3(\theta) &= 0.5 \left(4 (V_S/V_P)^2 \sin^2 \theta - \tan^2 \theta\right),\\ \frac{\Delta \text{AI}}{\overline{\text{AI}}} &= 2 \frac{\text{AI}_2-\text{AI}_1}{\text{AI}_2+\text{AI}_1},\\ \frac{\Delta \text{SI}}{\overline{\text{SI}}} &= 2 \frac{\text{SI}_2-\text{SI}_1}{\text{SI}_2+\text{SI}_1},\\ \frac{\Delta \rho}{\overline{\rho}} &= 2 \frac{\rho_2-\rho_1}{\rho_2+\rho_1}. \end{align}\end{split}\]