Zoeppritz equations

In geophysics and reflection seismology, the Zoeppritz equations are a set of equations that describe the partitioning of seismic wave energy at an interface, typically a boundary between two different layers of rock. They are named after their author, the German geophysicist Karl Bernhard Zoeppritz, who died before they were published in 1919.

The equations are important in geophysics because they relate the amplitude of P-wave, incident upon a plane interface, and the amplitude of reflected and refracted P- and S-waves to the angle of incidence. They are the basis for investigating the factors affecting the amplitude of a returning seismic wave when the angle of incidence is altered — also known as amplitude versus offset analysis — which is a helpful technique in the detection of petroleum reservoirs.

The Zoeppritz equations were not the first to describe the amplitudes of reflected and refracted waves at a plane interface. Cargill Gilston Knott used an approach in terms of potentials almost 20 years earlier, in 1899, to derive Knott's equations. Both approaches are valid, but Zoeppritz's approach is more easily understood.

The Zoeppritz equations consist of four equations with four unknowns

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

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