Tensorial formulation of the wave equation for modelling curved interfaces

Many situations of practical interest involving seismic wave modelling require curved interfaces and free-surface topography to be taken into account. Collocation methods, for instance pseudospectral or finite-difference algorithms, are attractive approaches for modelling wave propagation through these complex realistic models, particularly in view of their ease of implementation. Nonetheless, these methods formulated in Cartesian coordinates are not well suited to such models because the sharp interfaces and free surface do not coincide with grid lines. This leads to a slow convergence rate, resulting in visible artefacts such as diffractions from staircase discretizations of interfaces and the free surface. Such problems can be overcome through the use of curved grids whose lines follow sharp interfaces and whose density increases in the vicinity of these interfaces. One approach is to solve the wave equation in Cartesian coordinates by using the chain rule to express the Cartesian partial derivatives in terms of derivatives computed in the new coordinate system. However, it is more natural to solve the tensorial form of the wave equation directly in the desired curvilinear coordinate system, making use of a transformation of a square grid onto the curved grid. The tensorial approach, which is independent of the coordinate system, requires the same number of derivatives to be computed as in the Cartesian case, whereas the chain rule approach requires 25 per cent more in 2-D and 50 per cent more in 3-D. While the tensorial approach is less computationally expensive than the chain rule method, it requires more memory. Numerical tests validate the tensorial approach by comparing the results with the analytical solution of the tilted Lamb problem. Other numerical experiments demonstrate the ability of the tensorial formulation to model wave propagation in the presence of free-surface topography. Mode conversions between Rayleigh and body waves are observed when bumps on the free surface are encountered.