Zoeppritz equations

In seismology, the Zoeppritz equations describe how seismic waves are transmitted and reflected at media boundaries, which are boundaries between two different layers of earth.

The Zoeppritz equations, formulated by the German geophysicist Karl Bernhard Zoeppritz, relate the amplitudes of P-waves and S-waves at each side of an interface to the angle of incidence of the incoming wave.

For instance, a blast of dynamite will create a seismic wave (P-wave) that will travel through the earth, be reflected off of various layers, and return to the surface where it can be detected. The time it takes to return is related to the depth of the various layers. By detecting the seismic waves at various points on the surface, one can see how the reflections change with angle of incidence. One can then use this information along with the Zoeppritz equations to learn more about the density and velocity of each layer. This is helpful for instance in locating underground reservoirs and deposits.

Approximations

Zeoppritz's equations are complex to use so approximations such as Bortfeld's 1961, and Shuey's 1985 are often used. Shuey's approximation is:

$R(\theta )=R_{0}+[A_{0}R_{0}+{\frac {\triangle \sigma }{(1-\sigma )^{2}}}]sin^{2}\theta +{\frac {1}{2}}{\frac {\triangle V_{p}}{V_{p}}}(tan^{2}\theta -sin^{2}\theta )$

Where each term covers the reflection amplitude at greater angles. The first term $R_{0}$ gives the amplitude at normal incidence $(\theta =0)$, the second term characterizes $R(\theta )$ at intermediate angles, and the third term describes the approach to critical angles. Here $\sigma$ is poisson's ratio, $\theta$ is the angle of incidence, and $A_{0}$ is a slowly changing value proportional to ${\frac {1-2\theta }{1-\theta }}$. This approximation is accurate to about 60 degrees of the critical angle and assumes that the change in density and velocity across the boundary is much less than 1.

External links

• http://www.crewes.org/ResearchLinks/ExplorerPrograms/ZE/ZEcrewes.html