Finite difference methods for orthorhombic media: perfectly reflecting and absorbing boundaries

P.F. Daley

ABSTRACT

An approach for the numerical solution of the forward problem for elastic wave propagation in a plane layered anisotropic (orthorhombic) elastic media is revisited. The introduction of an absorbing boundary at the model bottom is considered in this report. These boundary conditions are similar to those derived in Clayton and Engquist (1977). The stiffness coefficients (in Voigt notation), C ij and the density, ρ , may vary arbitrarily with depth. The method discussed here employs finite Fourier transforms to temporarily remove the x and y coordinates resulting in a coupled system of three finite difference equations in the 3 Cartesian coordinate particle displacements in terms of depth ( z ) and time ( t ). The return to the ( x,y,z,t ) domain is done using a double inverse summation over the two horizontal wave numbers ( k x ,k y ). The absorbing boundary conditions are only considered for the model bottom as there are alternate methods for the free surface and side boundaries. The full elastic equations are not used at the model bottom, but rather their scalar approximations. This may appear highly suspect, but reasonable results have been obtained for less complex media types and it was thought that it should at least be investigated for this case.

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