Fast multimodel finite-difference controlled-source electromagnetic simulations based on a Schur complement approach

We have developed an efficient numerical scheme for fast multimodel 3D electromagnetic simulations by applying a Schur complement approach to a frequency-domain finite-difference method. The scheme is based on direct solvers and developed with constrained inversion algorithms in view. Such algorithms normally need many forward modeling jobs with different resistivities for the target zone and/or background formation. We geometrically divide the computational domain into two subdomains: an anomalous subdomain, the resistivities of which were permitted to change, and a background subdomain, having fixed resistivities. The system matrix is partially factorized by precomputing a Schur complement to eliminate unknowns associated with the background subdomain. The Schur complement system is then solved to compute fields inside the anomalous subdomain. Finally, the background subdomain fields are computed using inexpensive local substitutions. For each successive simulation, only the relatively small Schur complement system has to be solved, which results in significant computational savings. We applied this approach to two moderately sized 3D problems in marine controlled-source electromagnetic modeling: (1) a deep-water model in which the resistivities of the seawater and the air layer were kept fixed and (2) a model in which focused inversion was performed in a scenario in which the resistivities of the background formation, the air layer, and the seawater were known. We found a significant reduction of the modeling time in inversion that depended on the relative sizes of the constrained and unconstrained volumes: the smaller the unconstrained volume, the larger the savings. Specifically, for a focused inversion of the Troll oil field in the North Sea, the gain amounted up to 80% of the total modeling time.