Geophysical inversions on unstructured meshes using non-gradient based regularization

Geophysical inverse problems are commonly ill-posed and require proper regularization to gain acceptable solutions. Adopting constraints on the smoothness and/or specified structures of an inverted geophysical model requires the implementation of regularization terms when either structured or unstructured meshes are used. Spatial gradients of the model parameters along axial or arbitrary directions are thus required. However, the calculation of spatial gradients on unstructured meshes is not straightforward since the interfaces between adjacent model cells (i.e. common edges or common faces) are orientated individually and usually are not perpendicular to the axial directions. Depending on the algorithm used, an uneven performance of the regularization is observed on unstructured meshes. To enforce effective and robust regularization terms for geophysical inversions on the unstructured meshes, we propose algorithms for constructing the smoothness and structural similarity operators that take advantage of the inherent merits of the unstructured meshes. Following a detailed introduction of the general inversion formula that we adopted, the smoothness and reference model constraints on triangular and tetrahedral meshes are proposed based on the neighbouring relationships between different model cells within the meshes. Particularly, a quasi-cross-gradient formulation is derived for triangular meshes suitable for the joint inversion of different kinds of geophysical data. Compared to existing algorithms, the new smoothness operator presents an equal or better performance for constraining the model roughness. In addition, the operator exploits the preferred elongation directions of the underground structures by performing varied constraints in different directions. Furthermore, the other new operator could effectively measure structural information of the inverted model even if the algorithms have incorporated sophisticated constraints from other geophysical or geological data. Demonstrated with the applications on synthetic examples, the new algorithms provide advanced regularization techniques for conducting geophysical inversions using unstructured meshes.