Zoeppritz equation

The Zoeppritz equations describe seismic wave energy partitioning at an interface, for example the boundary between two different rocks. The equations relate the amplitude of incident P-waves to reflected and refracted P- and S-waves at a plane interface for a given angle of incidence.

P-wave incident on a planar interface

Mode conversion.

A planar P-wave hitting the boundary between two layers will produce both P and SV reflected transmitted waves. This is called mode conversion. The angles of the incident, reflected and transmitted rays are related by Snell's law as follows:

$p={\frac {\sin \theta _{1}}{V_{\mathrm {P1} }}}={\frac {\sin \theta _{2}}{V_{\mathrm {P2} }}}={\frac {\sin \phi _{1}}{V_{\mathrm {S1} }}}={\frac {\sin \phi _{1}}{V_{\mathrm {S2} }}}$,

where p is called the ray parameter.

Zoeppritz (1919) derived the particle motion amplitudes of the reflected and transmitted waves using the conservation of stress and displacement across the interface, which yields four equations with four unknowns:

need re-check the equation

${\begin{bmatrix}R_{\mathrm {P} }\\R_{\mathrm {S} }\\T_{\mathrm {P} }\\T_{\mathrm {S} }\\\end{bmatrix}}={\begin{bmatrix}-\sin \theta _{1}&-\cos \phi _{1}&\sin \theta _{2}&\cos \phi _{2}\\\cos \theta _{1}&-\sin \phi _{1}&\cos \theta _{2}&-\sin \phi _{2}\\\sin 2\theta _{1}&{\frac {V_{\mathrm {P1} }}{V_{\mathrm {S1} }}}\cos 2\phi _{1}&{\frac {\rho _{2}V_{\mathrm {S2} }^{2}V_{\mathrm {P1} }}{\rho _{1}V_{\mathrm {S1} }^{2}V_{\mathrm {P2} }}}\cos 2\phi _{1}&{\frac {\rho _{2}V_{\mathrm {S2} }V_{\mathrm {P1} }}{\rho _{1}V_{\mathrm {S1} }^{2}}}\cos 2\phi _{2}\\-\cos 2\phi _{1}&{\frac {V_{\mathrm {S1} }}{V_{\mathrm {P1} }}}\sin 2\phi _{1}&{\frac {\rho _{2}V_{P2}}{\rho _{1}V_{P1}}}\cos 2\phi _{2}&{\frac {\rho _{2}V_{S2}}{\rho _{1}V_{P1}}}\sin 2\phi _{2}\end{bmatrix}}^{-1}{\begin{bmatrix}\sin \theta _{1}\\\cos \theta _{1}\\\sin 2\theta _{1}\\\cos 2\phi _{1}\\\end{bmatrix}}$

R_P, R_S, T_P, and T_S, are the reflected P, reflected S, transmitted P, and transmitted S-wave amplitude coefficients, respectively. Inverting the matrix form of the Zoeppritz equations give the coefficients as a function of angle.

AVO and linear approximations

Main article: AVO

Although the Zoeppritz equations are exact, the equations do not lead to an intuitive understanding of the AVO process. Modeling can be routinely done with the Zoeppritz equations but most AVO methods for analyzing real time seismic data are based on linearized approximations to the Zoeppritz equations (e.g. Bortfeld, 1961, Richards and Frasier, 1976, and Aki and Richards, 1980), which include:

• Aki-Richards equation
• Shuey equation
• Fatti equation
• Bortfeld equation
• Hilterman approximation
• Verm & Hilterman approximation
• And lots more in here