Including geological orientation information into geophysical inversions with unstructured tetrahedral meshes

SUMMARY Minimum-structure, or Occam’s style of, inversion introduces a regularization function into the underdetermined geophysical inverse problems to stabilize the inverse problem and mitigate its non-uniqueness. The regularization function is typically designed such that it can incorporate a priori information into the inversion framework, thus constructing models that have more plausible representations of the true Earth’s subsurface structure. One type of a priori information is geological orientation information such as strike, dip and tilt angles of the subsurface structure. This type of information can be incorporated into inverse problems through the roughness operators. Designing such roughness operators for inversion frameworks using unstructured tetrahedral meshes is not as straightforward as for inversion frameworks using structured meshes due to the arbitrary and complex geometry of unstructured meshes. Researchers have developed methods which allow us to incorporate geological orientation information into inversion frameworks with unstructured tetrahedral meshes. The majority of these methods consider each cell in a package with its neighbours, hence, the constructed models are not as sharp as desired if the regularization function is measured using an $\ell _1$-type measure instead of the $\ell _2$ norm. To address this issue, we propose a method that calculates the directional derivatives of physical property differences between two adjacent cells normalized by the distance between the cell centroids. This approach is able to both incorporate geological orientation information into the inversion framework and construct models with sharp boundaries for the scenarios in which the regularization term is quantified by an $\ell _1$-type measure. This method is an integral-based approach, therefore, the roughness operators are scaled appropriately by the cell volumes, which is an important characteristic for the inversions with unstructured meshes. To assess the performance and the capability of the proposed method, it was applied to 3-D synthetic gravity and magnetotelluric examples. The gravity example was also used to investigate the impact of applying the depth weighting function inside and outside the roughness operators for the scenarios that the model objective function is measured by an $\ell _1$ norm. The examples show that the proposed method is able to construct models with a reasonable representation of the strike and dip directions of the true subsurface model with sharper boundaries if the regularization function is quantified by an $\ell _1$-type measure. The examples also demonstrate the proposed method behaves numerically well, and has a fast convergence rate.

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