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

• Published in: Earth and space science (Hoboken, N.J.) N.J.), 2024-09, Vol.11 (9), p.n/a
• Main Authors: Codd, Andrea, Gross, Lutz, Kerr, Janelle
• Format: Article
• Language: English
• Subjects: Algorithms; Approximation; Conductivity; Electric fields; finite element method; inversion; Magnetic fields; magnetoteulluric; Methods; Optimization; Partial differential equations
• ISSN: 2333-5084; 2333-5084
• DOI: 10.1029/2024EA003680

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. Plain Language Summary We have developed a new inversion scheme for 2D MT data. Like most inversion methods, the cost function is composed of data misfit and a measure of the smoothness of the model (regularization). Inversion solutions are found by taking the derivative of the cost function and setting it to zero to minimize the function. The minimizing equation is solved iteratively using a quasi‐Newton method. Different to most other approaches, our method uses a discretize last strategy. Only once the quasi‐Newton iteration is derived, will discretization occur using an irregular triangular finite element mesh. The mesh is fine at ground level, very fine near measurement stations and extremely coarse at