Earth As An Unstructured Mesh and Its Recovery from Seismic Waveform Data

Abstract:

We consider multi-scale representations of Earth's interior from the
point of view of their possible recovery from multi- and
high-frequency seismic waveform data. These representations are
intrinsically connected to (geologic, tectonic) structures, that is,
geometric parametrizations of Earth's interior. Indeed, we address the
construction and recovery of such parametrizations using local
iterative methods with appropriately designed data misfits and
guaranteed convergence. The geometric parametrizations contain
interior boundaries (defining, for example, faults, salt bodies,
tectonic blocks, slabs) which can, in principle, be obtained from
successive segmentation. We make use of unstructured meshes.

For the adaptation and recovery of an unstructured mesh we introduce
an energy functional which is derived from the Hausdorff distance. Via
an augmented Lagrangian method, we incorporate the mentioned data
misfit. The recovery is constrained by shape optimization of the
interior boundaries, and is reminiscent of Hausdorff warping. We use
elastic deformation via finite elements as a regularization while
following a two-step procedure. The first step is an update determined
by the energy functional; in the second step, we modify the outcome of
the first step where necessary to ensure that the new mesh is
regular. This modification entails an array of techniques including
topology correction involving interior boundary contacting and
breakup, edge warping and edge removal. We implement this as a
feed-back mechanism from volume to interior boundary meshes
optimization. We invoke and apply a criterion of mesh quality control
for coarsening, and for dynamical local multi-scale refinement. We
present a novel (fluid-solid) numerical framework based on the
Discontinuous Galerkin method.