3D minimum-structure inversion of controlled-source EM data using unstructured grids

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In this study, a minimum-structure inversion procedure for controlled-source electromagnetic data that parameterizes the Earth model in terms of an unstructured tetrahedral grid is presented. To minimize the objective function defining the inversion problem Gauss–Newton iterations are used. Iterative solvers are employed for the solution of the system of equations at each Gauss–Newton step. These solvers do not need the matrix to be explicitly formed to solve the linear system, and hence, these kinds of solvers ensure memory efficiency of the developed algorithm. For the forward-modelling problem, a potentials formulation, in which the electric field is expressed in terms of vector and scalar potentials and the relevant partial differential equations are the Helmholtz and conservation of charge equations, is discretized by the finite-element method. This decomposition results in a well conditioned matrix equation compared to the E-field equation and gives a more natural description of electromagnetic phenomena. The linear system of equations for the forward-modelling problem is solved by a direct solver. The advantage of this type of solver is that the factorization that is generated, which takes up the bulk of the solution time for this system, can be reused when solving for multiple right-hand sides. This is important for the efficient calculation of the matrix–vector products in the above-mentioned Gauss–Newton systems. Two synthetic examples are given to illustrate the capabilities and efficiency of the presented procedure. In the first example, inversion of a dataset generated over a geometric shaped conductive body for a flat Earth’s surface is presented. In the second example, a realistic scenario is given. This example shows that topography and a realistic ore body can be naturally and effectively handled by unstructured grids.