Elastic wave modelling in heterogeneous media with high velocity contrasts

In this paper a new numerical technique is proposed for modelling elastic wave propagation in heterogeneous media with high velocity contrasts, as for instance in the presence of weathered layers or soft marine sediments. The scheme allows a sharp jump in grid spacing across the interface with high velocity contrasts. That is, the number of nodes along the interface with high velocity contrasts may be different for two meshes beside the interface. Sharp jumps of grid spacing can occur along a curved interface. Therefore, the proposed scheme can use a fine mesh in regions with low velocities and a coarser mesh in regions with higher velocities by modelling the interface topography with a piecewise linear function, so both spatial and temporal oversampling is avoided. This leads to a distinct reduction in the storage requirements and in the computational cost. The scheme is developed by formulating the problem in terms of stresses (at nodes) in regions with low S-wave velocities and in terms of velocities (at nodes) in regions with higher velocities, and then defining an explicit boundary between them. The boundary conditions on this explicit boundary with a complex geometry are implemented by introducing an integral approach to the equations on the explicit boundary. The outcome of this is that no nodes need to be added through the interpolations of the field variables. Moreover, no smoothing of the material parameters is needed on the interface with high velocity contrasts. A stress-grid scheme is developed to solve the problem inside regions with low S-wave velocities, and the (velocity) grid method is used to solve the problem inside regions with higher velocities. The algorithm is a second-order approximation for an irregular mesh in comparison to conventional finite differences. Implementation of the free-surface conditions of a complex geometrical surface is obtained for the proposed stress-grid scheme when the surface topography is approximated to be piecewise linear. Thus, the new algorithm can model elastic wave propagation even when the regions of low S-wave velocities reach the surface. Numerical examples demonstrate the efficiency of the proposed numerical technique.