Dictionary:Zoeppritz equations

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English • ‎español • ‎中文 Diagram showing the mode conversions that occur when a P-wave reflects off an interface at non-normal incidence

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 30 years earlier, in 1899, to derive Knott's equations. Both approaches are valid and Zoeppritz's approach is more easily understood.

Equations

There are 4 equations with 4 unknowns and although they can be solved, they do not give an intuitive understanding for how the reflection amplitudes vary with the rock properties involved (density, velocity etc.). Several attempts have been made to develop approximations to the Zoeppritz Equations, such as Bortfeld’s (1961) and Aki & Richards’ (1980), but the most successful of these is the Shuey's, which assumes Poisson's ratio to be the elastic property most directly related to the angular dependence of the reflection coefficient.

Shuey Equation

The 3-term Shuey Equation can be written a number of ways, the following is a common form:

$R(\theta )=R(0)+G\sin ^{2}\theta +F(\tan ^{2}\theta -\sin ^{2}\theta )$ where

$R(0)={\frac {1}{2}}\left({\frac {\Delta V_{\mathrm {P} }}{V_{\mathrm {P} }}}+{\frac {\Delta \rho }{\rho }}\right)$ and

$G={\frac {1}{2}}{\frac {\Delta V_{\mathrm {P} }}{V_{\mathrm {P} }}}-2{\frac {V_{\mathrm {S} }^{2}}{V_{\mathrm {P} }^{2}}}\left({\frac {\Delta \rho }{\rho }}+2{\frac {\Delta V_{\mathrm {S} }}{V_{\mathrm {S} }}}\right)$ ; $F={\frac {1}{2}}{\frac {\Delta V_{\mathrm {P} }}{V_{\mathrm {P} }}}$ where ${\theta }$ =angle of incidence; ${V_{p}}$ = P-wave velocity in medium; ${{\Delta }V_{p}}$ = P-wave velocity contrast across interface;${V_{s}}$ = S-wave velocity in medium; ${{\Delta }V_{s}}$ = S-wave velocity contrast across interface; ${\rho }$ = density in medium; ${{\Delta }{\rho }}$ = density contrast across interface;

A proposed better approximation of Zoeppritz equations:

$R(\theta )=R(0)-A\sin ^{2}\theta$ and

$A=2{\frac {V_{\mathrm {S} }^{2}}{V_{\mathrm {P} }^{2}}}\left({\frac {\Delta \rho }{\rho }}+2{\frac {\Delta V_{\mathrm {S} }}{V_{\mathrm {S} }}}\right)$ In the Shuey Equation, R(0) is the reflection coefficient at normal incidence and is controlled by the contrast in acoustic impedances. G, often referred to as the AVO gradient, describes the variation of reflection amplitudes at intermediate offsets and the third term, F, describes the behaviour at large angles/far offsets that are close to the critical angle. This equation can be further simplified by assuming that the angle of incidence is less than 30 degrees (i.e. the offset is relatively small), so the third term will tend to zero. This is the case in most seismic surveys and gives the “Shuey Approximation”:

$R(\theta )=R(0)+G\sin ^{2}\theta$ From Sheriff's Dictionary

(zō' pritz) Equations that express the partition of energy when a plane wave impinges on an acoustic-impedance contrast. In the general case for an interface between two solids when the incident angle is not zero, four waves are generated: reflected P-wave and S-wave and transmitted P-wave and S-wave. The partition of energy among these is found from four boundary conditions which require continuity of normal and tangential displacement and stress. Using the symbols given in Figure S-12, Snell's law states

${\frac {\sin \theta _{\mathrm {P1} }}{V_{\mathrm {P1} }}}={\frac {\sin \theta _{\mathrm {S1} }}{V_{\mathrm {S1} }}}={\frac {\sin \theta _{\mathrm {P2} }}{V_{\mathrm {P2} }}}={\frac {\sin \theta _{\mathrm {S2} }}{V_{\mathrm {S2} }}}$ and this defines all angles. For an incident plane P-wave of unity amplitude, the continuity conditions yield the four Zoeppritz equations shown here:

${\begin{bmatrix}\cos \theta _{\mathrm {P1} }&-\sin \theta _{\mathrm {S1} }&\cos \theta _{\mathrm {P2} }&\sin \theta _{\mathrm {S2} }\\\sin \theta _{\mathrm {P1} }&\cos \theta _{\mathrm {S1} }&-\sin \theta _{\mathrm {P2} }&\cos \theta _{\mathrm {S2} }\\Z_{1}\cos 2\theta _{\mathrm {S1} }&-W_{1}\sin 2\theta _{\mathrm {S1} }&-Z_{2}\cos 2\theta _{\mathrm {S2} }&-W_{2}\sin 2\theta _{\mathrm {S2} }\\{\tfrac {V_{\mathrm {S1} }}{V_{\mathrm {P1} }}}W_{1}\sin 2\theta _{\mathrm {P1} }&W_{1}\cos 2\theta _{\mathrm {S1} }&{\tfrac {V_{\mathrm {S2} }}{V_{\mathrm {P2} }}}W_{2}\sin 2\theta _{\mathrm {P2} }&-W_{2}\cos 2\theta _{\mathrm {S2} }\\\end{bmatrix}}{\begin{bmatrix}R_{\mathrm {P} }\\R_{\mathrm {S} }\\T_{\mathrm {P} }\\T_{\mathrm {S} }\\\end{bmatrix}}={\begin{bmatrix}\cos \theta _{\mathrm {P1} }\\-\sin \theta _{\mathrm {P1} }\\-Z_{1}\cos 2\theta _{\mathrm {S1} }\\{\tfrac {V_{\mathrm {S1} }}{V_{\mathrm {P1} }}}W_{1}\sin 2\theta _{\mathrm {P1} }\\\end{bmatrix}}$ where Zi = ρiVPi, Wi = ρiVSi, and RP, RS, TP, and TS are respectively the amplitudes of the reflected P- and S-waves and the transmitted P- and S-waves. However, their derivation does not consider head waves and hence they do not yield reliable values at and beyond the critical angle.

Figure Z-1 also shows the variation of energy with angle for several sets of parameters. Beyond the critical angles for P- and S-waves, the respective refracted waves vanish. The increase in reflection energy near the critical angle is sometimes referred to as the wide-angle phenomenon and is sometimes exploited in seismic surveying. The same relationships in terms of potentials are called Knott’s equations. Because no provision was made in the equation’s derivation for the head waves, these equations do not give head-wave amplitudes or correct values beyond the critical angle.