Large-scale Gauss-Newton inversion of transient controlled-source electromagnetic data using the model reduction approach

Abstract

The main cost of Gauss-Newton (GN) inversion of time-domain controlled-source electromagnetic (tCSEM) data is comprised of the multiple solutions of the forward problem and linear-algebraic operations using the Jacobian matrix. For large-scale 2.5D and 3D problems with multiple sources and receivers, this cost grows enormously for inversion algorithms using conventional finite-difference time-domain (FDTD) algorithms. To mitigate this problem, a fast 3D forward solver based on the rational Krylov subspace reduction (RKSR) algorithm using an optimal subspace selection was proposed by us earlier. Here, we apply the same approach to reduce the size of the time-domain Jacobian matrix. The reduced-order model (ROM) is obtained by projecting a discretized large-scale Maxwell system onto a rational Krylov subspace (RKS) with optimized poles. The RKS expansion replaces the time discretization for both the forward and inverse problems; however (for the same or better accuracy), its subspace dimension is much smaller than the number of the time steps of the conventional FDTD. The crucial new development of this work is the decomposition of the ROM's time-domain Jacobian matrix via a tensor product of time-dependent and space-dependent terms, thus effectively decoupling discretizations in the time and parameter spaces. All the above, together with the weighted L2-norm regularization technique for the GN inversion, allowed us to arrive at an efficient algorithm for inversion of tCSEM data. We illustrate our approach using synthetic 2.5D examples of hydrocarbon reservoirs in marine environment.