ANISOTROPIC WAVE-PROPAGATION THROUGH FINITE-DIFFERENCE GRIDS

An algorithm is presented to solve the elastic-wave equation by replac ing the partial differentials with finite differences. It enables wave propagation to be simulated in three dimensions through generally ani sotropic and heterogeneous models. The space derivatives are calculate d using discrete convolution sums, while the time derivatives are repl aced by a truncated Taylor expansion. A centered finite difference sch eme in cartesian coordinates is used for the space derivatives leading to staggered grids. The use of finite difference approximations to th e partial derivatives results in a frequency-dependent error in the gr oup and phase velocities of waves. For anisotropic media, the use of s taggered grids implies that some of the elements of the stress and str ain tensors must be interpolated to calculate the Hook sum. This inter polation induces an additional error in the wave properties. The overa ll error depends on the precision of the derivative and interpolation operators, the anisotropic symmetry system, its orientation and the de gree of anisotropy. The dispersion relation for the homogeneous case w as derived for the proposed scheme. Since we use a general description of convolution sums to describe the finite difference operators, the numerical wave properties can be calculated for any space operator and an arbitrary homogeneous elastic model. In particular, phase and grou p velocities of the three wave types can be determined in any directio n. We demonstrate that waves can be modeled accurately even through mo dels with strong anisotropy when the operators are properly designed.