A Novel Immersed Boundary Approach for Irregular Topography with Acoustic Wave Equations

Irregular terrain has a pronounced effect on the propagation of seismic and acoustic wavefields but is not straightforwardly reconciled with structured finite-difference (FD) methods used to model such phenomena. Methods currently detailed in the literature are generally limited in scope application-wise or non-trivial to apply to real-world geometries. With this in mind, a general immersed boundary treatment capable of imposing a range of boundary conditions in a relatively equation-agnostic manner has been developed, alongside a framework implementing this approach, intending to complement emerging code-generation paradigms. The approach is distinguished by the use of N-dimensional Taylor-series extrapolants constrained by boundary conditions imposed at some suitably-distributed set of surface points. The extrapolation process is encapsulated in modified derivative stencils applied in the vicinity of the boundary, utilizing hyperspherical support regions. This method ensures boundary representation is consistent with the FD discretization: both must be considered in tandem. Furthermore, high-dimensional and vector boundary conditions can be applied without approximation prior to discretization. A consistent methodology can thus be applied across free and rigid surfaces with the first and second-order acoustic wave equation formulations. Application to both equations is demonstrated, and numerical examples based on analytic and real-world topography implementing free and rigid surfaces in 2D and 3D are presented.