Generalised immersed boundary methods for finite-difference geophysical wavefield modelling

Abstract

Topography has considerable impact on seismic wavefields, producing complex patterns of reflections and diffractions. Careful representation is paramount in numerical models of such phenomena; omission or poor-quality implementation severely degrades simulation accuracy, carrying into any functionality built atop such models. Whilst structured finite-difference approaches are widely used in seismic simulation, they are difficult to reconcile with such geometries, producing surface representations which are artificially blocky, resulting in nonphysical behaviour.

At the large scales involved in seismic simulation, increasing model resolution becomes unfeasibly computationally expensive, necessitating efficient algorithms. Immersed boundary approaches embed curvilinear surfaces within regular computational grids, naturally accommodating irregular topography by suitably continuing fields beyond the interior domain, yielding accurate results with coarser discretisations than would be otherwise possible. By formulating a generalised immersed boundary, a consistent methodology can be applied across a wide range of boundary conditions and geometries, avoiding the need to develop a new numerical treatment for each particular problem.

Removing the need for interventions based on domain knowledge, it becomes possible to automatically devise immersed boundary treatments for a given discretisation and set of boundary conditions. A software framework, Schism, was designed for this purpose, creating separation of concerns between numerical model specification and underlying implementation by generating low-level boundary treatments from a high-level symbolic specification.

With this novel approach, topography implementation for a wide range of equations was explored, including several problems of seismological interest for which no published examples exist. Mountainous real-world topography could be accurately accommodated with relatively coarse discretisations, suppressing spurious scattering whilst capturing complex reflections and other free-surface effects generated by such irregular profiles. This approach offers a path towards implementation of complex geometries at scale across a diversity of geophysical applications.