2D Magnetotelluric Inversion Using Linear Finite Element Methods and a Discretize‐Last Strategy With First and Second–Order Anisotropic Regularization

Abstract We present a new inversion scheme for 2D magnetotelluric data. In contrast to established approaches, it is based on a mesh‐free formulation of the Quasi‐Newton Broyden–Fletcher–Goldfarb–Shanno (BFGS) iteration which uses the cost function gradient to implicitly construct approximations of the Hessian inverse to update the unknown conductivity. We introduce conventional first–order regularization as well as second–order regularization where inversions based on the latter are more appropriate for sparse data and can be read as maximum likelihood estimation of the unknown conductivity. We apply first–order finite element method (FEM) discretizations of the inversion scheme, forward and adjoint problems, where the latter is required for the construction of the cost function gradients. We allow for unstructured first–order triangular meshes supporting an enhanced ground level resolution including topographical features and coarsening at the far field leading to significant reduction in computational costs from using structured mesh. Formulating the inversion iteration in continuous form prior to discretization eliminates bias due to local refinements in the mesh and gives way for computationally efficient sparse matrix techniques in the implementation. A keystone in the new scheme is the multi‐grid approximation of the Hessian of the regularizations to construct efficient preconditioning for the inversion iteration. The method is applied to the Commeni4 benchmark and two field data sets. Tests show that for both first and second–order regularization an anisotropic approach is important to address the vast differences in horizontal and vertical spatial scale which in conventional approaches is implicitly introduced through the elongated shape of grid cells.