Finite Difference Elastic Anisotropic Wave Propagation
, by John T. Etgen

The heterogeneous anisotropic elastic wave equation can be solved on a discrete
grid using an explicit finite-difference technique. This solution of the wave equation is used for forward modeling as well as
for prestack migration of elastic wave fields. A three-dimensional version of the method can compute the elastic
wave field in a heterogeneous anisotropic medium due to a variety of sources.
For an axisymmetric-anisotropic 2-D medium, the equations that govern in-plane
pseudo-P and pseudo-Sv waves are uncoupled from the Sh wave equation. The P--Sv and S_h wave fields excited by a
line source can then be calculated on separate 2-D grids.
The finite-difference method uses spatial differentiation operators centered halfway between grid points,
and represents stresses and displacements on a staggered grid. For
common descriptions of an anisotropic solid, no interpolation of grid values is needed during the computations.
The wave fields in both solid and liquid media are computed using the same
equations; layer boundaries are represented by changes in elastic constants
throughout the computational grid without need for explicit
boundary conditions. Absorbing boundaries are used to reduce undesirable edge effects.
A free surface condition is also incorporated at the top of the model.
Elastic wave fields calculated using the 3-D and 2-D versions of the algorithm
agree with analytical solutions for simple Earth models. Wave fields in more
complicated models with varying degrees of anisotropy and heterogeneity
show expected behavior, although direct comparison with analytical
solutions is impossible. For 3-D calculations, an efficient ``out-of-core" algorithm computes
multiple time steps on each pass through the computational model to reduce the i/o cost of the method.
The algorithm is organized to deal with the data as a sequence of planes so the data does not have to be transposed
during the computations.