Zoeppritz equation

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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_2V_\mathrm{S2}^2 V_\mathrm{P1}}{\rho_1V_\mathrm{S1}^2 V_\mathrm{P2}}\cos2\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_1V_{P1}}\sin2\phi_2 \end{bmatrix}^{-1} \begin{bmatrix} \sin\theta_1 \\ \cos\theta_1 \\ \sin 2\theta_1 \\ \cos 2\phi_1\\ \end{bmatrix}

RP, RS, TP, and TS, 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.

AVOAmplitude vs Offset and linear approximations

Although the Zoeppritz equations are exact, the equations do not lead to an intuitive understanding of the AVOAmplitude vs Offset process. Modeling can be routinely done with the Zoeppritz equations but most AVOAmplitude vs Offset 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: