P-wave anisotropy : Thomsen & Tsvankin Parameters

The weak anisotropy developed by Thomsen for the VTI case was extended to the Orthorhombic case by Tsanvskin ¹.

Thomsen Parameters in different symmetries.

Velocities in VTI

Reference velocities and equivalences: (cf. Thomsen 1986 ².) \[V_{P0} = V_{PV} = \sqrt{\frac{c_{33}}{\rho}},\quad V_{S0} = V_{SH1} = V_{SH2} = \sqrt{\frac{c_{55}}{\rho}},\]

Velocity as a function of incidence angle \(\theta\) \[\begin{aligned} V_P(\theta) = V_{P0}\bigg[1+\delta\sin^2 \theta \cos^2\theta + \epsilon \sin^4 \theta \bigg]\\ V_{SV}(\theta) = V_{S0}\bigg[ 1 +\frac{V_{P0}^ 2}{V_{S0}^2}(\epsilon - \delta) \sin^2 \theta \cos^2 \theta \bigg]\\ V_{SH}(\theta) = V_{S0}\bigg[ 1 + \gamma \sin^2\theta \bigg]\\ \end{aligned}\]

Velocities in HTI

Reference velocities and equivalences: (cf. Ruger 1997 ³.) \[V_{P0} = V_{PV} = V_{PH2} = \sqrt{\frac{c_{33}}{\rho}},\quad\]

Velocity as a function of incidence angle \(\theta\) \[\begin{aligned} V_P(\theta,\phi) = V_{P0}\bigg[1+\delta^{(V)} \sin^2 \theta \cos^2 \phi + (\epsilon^{(V)} - \delta^{(V)}) \sin^4 \theta\cos^4 \phi \bigg]\\ \end{aligned}\]

Velocities in Orthorhombic Medium

Reference velocities and equivalences: (cf. Ruger 1998 ⁴.) \[V_{P0} = V_{PV} = \sqrt{\frac{c_{33}}{\rho}},\quad\]

Velocity as a function of incidence angle \(\theta\) \[\begin{aligned} V_P(\theta,\phi) = V_{P0}\bigg[1+\delta(\psi)\sin^2 \theta \cos^2\theta + \epsilon(\phi) \sin^4 \theta \bigg]\\ \delta(\phi) = \delta^{(1)}\sin^2\phi + \delta^{(2)} \cos^2 \phi \\ \epsilon(\phi) = \epsilon^{(1)} \sin^4 \phi + \epsilon^{(2)} \cos^4 \phi + (2\epsilon^{(2)} + \delta^{(3)})\sin^2 \phi \cos^2 \phi \end{aligned}\]

References and Further Reading

1: Tsvankin 1997 ↩ Ilya Tsvankin. Anisotropic parameters and pwave velocity for orthorhombic media. GEOPHYSICS, 62(4) :1292–1309, 1997. doi :10.1190/1.1444231. URL

2: Thomsen 1986 ↩ Leon Thomsen. Weak elastic anisotropy. GEOPHYSICS, 51 (10) :1954–1966, 1986. doi : 10.1190/1.1442051. URL

3: Rüger 1997 ↩ Andreas Rüger. Pwave reflection coefficients for transversely isotropic models with vertical and horizontal axis of symmetry. GEOPHYSICS, 62(3) :713–722, 1997. doi : 10.1190/1.1444181. URL

4: Rüger 1998 ↩ Andreas Rüger. Variation of p-wave reflectivity with offset and azimuth in anisotropic media. GEOPHYSICS, 63(3) :935–947, 1998. doi :10.1190/1.1444405. URL