Gradient and smoothness regularization operators for geophysical inversion on unstructured meshes

Abstract

The non-uniqueness of the underdetermined inverse problem requires that any available geological information be incorporated to constrain the results. Such information commonly comes in the form of a geological model comprising unstructured wireframe surfaces. Hence, we perform geophysical modelling on unstructured meshes, which provide the flexibility required to efficiently incorporate complicated geological information. Designing spatial matrix operators for unstructured meshes is a non-trivial task. Gradient operators are required for powerful inversion regularization schemes that allow for the incorporation of geological information. Other authors have developed simple regularization schemes for unstructured meshes but those approaches do not use true gradient operators and do not allow for the incorporation of structural information. In this paper we develop new methods for generating spatial gradient operators on unstructured meshes. Our approach is essentially to fit a linear trend in a small neighbourhood around each cell. This results in a small linear system of equations to solve for each cell. Solving for the linear trend parameters yields the required information to construct the stationary gradient operators. Care must be taken when setting up the linear systems to avoid potential numerical issues. We test and compare our methods against the rectilinear mesh equivalents using some simple illustrative 2-D synthetic examples. Our methods are then applied to more complicated 2-D and 3-D examples, including real earth scenarios. This work provides a new method for regularizing inversions on unstructured meshes while allowing for the incorporation of structural orientation information.