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## Appendix: Zoeppritz

In the current project, I simply model the wave propagation in a porous rock. In the future, I want to invert recorded seismograms to estimate the subsurface properties. In general, I could attempt to find the inverse operator to the wave propagation simulator. Traditionally, it was useful to approximate seismic inversion by two separate estimations of the subsurface velocity and the reflectivity. The pore fluids effect on velocities we explicitly derived in the previous sections. Their effect on amplitudes is not explicit in the equations above.

Studies of amplitude-versus-offset formulate the amplitude behavior of seismic reflections explicitly. The derivation is based on the Zoeppritz equations for reflected and transmitted amplitudes at concrete interfaces. Zoeppritz requires the continuity of displacement and stress (acceleration) at the interface and derives a matrix equation for the amplitude of waves at any impedance angle. The matrix coefficients depend on P-, S-wave velocity, and density, which in turn are dependent on lithology, porosity, pore fluid content, and temperature. Bortfeld , and Aki 1980 have published scalar approximations to the matrix equation usually based on the assumption of small incident angles.

In sophisticated AVO studies, the amplitude changes with offset (or more correctly with angle) are translated into hydrocarbon indicators. Considering the success of this approach, it seems attractive to use a similar formulation in time-lapse processing. To integrate the time-lapse experiments with petroleum engineering the AVO inversion needs to resolve physical parameters, such as saturation, rather than a generic hydrocarbon indicator.

Since Zoeppritz equations are based on discrete interfaces, the AVO approach comes natural for models that are parameterized as objects. To apply an AVO analysis to a subsurface model parameterized by pixels requires the explicit or implicit estimation of the local reflector slope and the specular reflection.

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Stanford Exploration Project
3/9/1999