A comprehensive study of gradient and smoothness regularization operators on unstructured tetrahedral meshes

The regularization function in minimum-structure, or “Occam’s”, style of inversion can stabilize the underdetermined inverse problem and generate models with certain characteristics. The regularization term measures the amount of structure added to the model using the spatial gradient operators. The method commonly employed for calculating the gradient operators on unstructured tetrahedral meshes calculates the physical property differences across the cell faces of two adjacent cells. However, this method is not able to incorporate orientation information of the geological structures such as strike, dip, and tilt angles into the inversion. Providing this information for the inversion leads to constructing geophysical models that have a sensible representation of the true Earth models, especially when geophysical data with limited depth resolution such as gravity and magnetics data are inverted. Designing spatial gradient operators that allow one to incorporate this geological orientation information into the inversion on unstructured tetrahedral meshes is not as straightforward as for structured meshes due to the geometrical complexity of the unstructured tetrahedral meshes.

A few methods have been proposed for calculating the gradient operators for unstructured tetrahedral meshes that allow one to incorporate orientation information into the inversion framework and obtain more sensible geophysical models. Most of these methods consider a cell along with its nearest neighbours as a package and commonly use an l2 norm for the measure of the regularization term. These methods work well, however, the constructed models using these regularization methods are not as sharp as expected if an l1-type measure of roughness is used instead of an l2 norm.

In this study, the method that calculates the spatial gradient operators across the cell faces between two adjacent cells is extended such that structural orientation information can be incorporated into the inversion. The synthetic gravity examples demonstrate that this method allows models with desired strike and dip directions to be built successfully. Also, the constructed models using this proposed method have sharper boundaries compared to the constructed models that consider each cell as a package with its neighbours for the scenarios in which an l1-norm measure is employed in the regularization term.

Keywords: Gradient operators, inversion, structural orientation information, unstructured tetrahedral meshes.