2.5-D inversion of frequency domain CSEM data using a PDE constrained optimization technique

Abstract

We present a new algorithm for inverting frequency domain controlled source electromagnetic (CSEM) data using a partial differential equation (PDE) constrained optimization technique within a 2.5-D (three-dimensional source over a two-dimensional subsurface structure) approximation. A common approach applied to solve the CSEM inverse problem eliminates the field in the objective function using equality constraints, leaving an unconstrained optimization problem to be solved. In contrast, we propose to use a PDE constrained optimization. Using this methodology, fields remains as part fo the objective function and the PDE equations are added as constraints. Thus, allowing us to balance the accuracy of the iterations for solving the forward and the inverse problems. To solve the forward problem within the 2.5-D approach, we apply a Fourier transformation of the 3D Maxwell's equation onto the wavenumber domain on the parallel strike direction coordinate. Then we solve the resulting two coupled PDE equations numerically using the Finite Element (FE) method with unstructured grids for a set of strike parallel wavenumber values. We then finally, transform the solution fields back to the spatial domain. In order to introduce the PDE equations as objective function constraints in the inversion scheme, we formulate the linear system matrices obtained through the FE method to include the discretized equations for all the wavenumber values. We then use a direct solver (MUMPS) to solve the FE linear system for all the wavenumbers at the same time. As part of the PDE constrained optimization formulation, we introduce a Lagrangian functional and solve the optimization problem by finding a stationary point within it. For this purpose, in the algorithm presented in this work, we use the SNOPT package. SNOPT uses a sequential quadratic programming (SQP) method, using limited-memory quasi-Newton approximations to the Hessian of the Lagrangian