Forward modelling of gravity data on unstructured grids using an adaptive mimetic finite-difference method

One of the techniques to optimize the gradient-based inversion of geophysical gravity data is to deal with the large and dense Jacobian matrices implicitly. This is made possible by solving the normal systems iteratively and to use pseudo-forward problems to calculate the effect of the Jacobian matrices. For this, numerical forward solvers should be used instead of analytical solvers. In this study, we propose mimetic finite-difference schemes for the solution of the forward gravity problem on unstructured grids. While the mimetic finite-difference method shares many traits with the finite-element and finite-volume methods, it has the advantage of naturally accommodating grids with arbitrary polyhedral elements. We use cell-based and vertex-based mimetic finite-difference schemes to solve a diffusion problem in mixed form, and Poisson's problem, respectively. The cell-based scheme is solved for the gravitational potential and attraction at the centroids and the facets of the elements, respectively, and the vertex-based scheme is solved for the potential at the vertices of the elements. We compare these mimetic finite-difference schemes with cell-based and vertex-based finite-volume schemes and a vertex-based finite-element scheme, in terms of accuracy of the potential. We then utilize the versatility of the mimetic finite-difference method, in accommodating arbitrary elements, and develop an adaptive mesh refinement tool. The mesh adaptivity is performed by an iterative h-refinement where goal-oriented error estimates are used to mark the elements for refinement. The marked elements are decomposed into new tetrahedra by regular subdivision. Since arbitrary polyhedra are naturally permitted in the mimetic finite-difference method, the added nodes are not regarded as hanging nodes, and therefore, any modification of the scheme or extra refinement is avoided. This characteristic allows preserving the quality of the initial grid and simplifying the refinement procedure. We demonstrate the practicality of the presented mesh adaptivity for the forward modelling of gravity and gravity gradient data using realistic geological models. We show that the cell-based mimetic scheme is more efficient than the vertex-based scheme in terms of computational resources. We also demonstrate that in the presence of irregular interfaces in the input domain, the regular refinement approach, presented in this study, is more reliable than a standard mesh regeneration technique.