Elastic forward modeling by the fourier method

Elastic forward modeling is important both for the evaluation of conentional seismic reflection surveys, and for shear wave surveys. In the former, elastic effects are ignored and for different structures one constantly needs to evaluate the validity of this assumption. In shear surveys acoustic assumptions are no longer valid and full elastic modeling needs to be employed. In elastic forward modeling, it is important to reproduce correct amplitudes of events since claims for the justification of ignoring elastic effects are often based on their small amplitudes. On the other hand the elastic wave equation is considerably more complicated than the acoustic wave equation, and therefore elastic forward modeling is more challenging. Thus with elastic modeling there is more justification for using direct methods such as finite differences which are expensive but are capable of producing correct amplitudes. In this work we present an elastic forward modeling algorithim based on the Fourier method. The main advantage of the method is the high accuracy of the spatial derivative approximation. This allows for the separation ofP waves from shear waves through application of divergence and curl numerical operators, and hence to follow the generation of converted waves. The numerical algorithim also allows for modeling with materials with high Poisson ratios and thus to approximate wave propagation in heterogeneous structures which contain both solids and fluids in juxtaposition. In deriving the numerical algorithim, we depart from the approach of solving the vector wave equation for the displacements. Instead we derived a new set of equations for the stresses which does not include derivatives of the elastic constants. With this approach it becomes easier to affect the free surface boundary condition with the Fourier method and the algorithim appears more robust in modeling structures with severe velocity changes.