A Novel Restarted Rational Krylov Subspace Algorithm for 3-D Multifrequency CSEM Forward Modeling

Controlled-source electromagnetic (CSEM) method is a valuable technique used in geophysical prospecting. However, the efficiency of CSEM forward modeling is significantly limited by the number of frequencies. This article proposes a restarted rational Krylov (RK) subspace algorithm, which has significantly improved the computational efficiency of 3D multi-frequency CSEM forward modeling. Initially, a brief introduction is provided on the finite element forward modeling based on octree meshes, which can effectively and accurately discretize the subsurface and allow us to consider the details and complexity of the geological structures. Subsequently, the principle of using the rational Krylov subspace algorithm to solve the forward equations at multi-frequency is given. Then, based on error derivation, we present the correction formula for the approximate solution and the algorithm framework for the restarted rational Krylov subspace algorithm, which solves the problem that the calculation accuracy of the traditional rational Krylov subspace algorithm depends on the subspace dimension. Ultimately, a layered model is used to determine the optimal dimension of the restarted subspace and validate the accuracy and efficiency of the proposed algorithm. The Nihe iron deposit model is also employed to demonstrate the capability of the proposed algorithm in handling practical and complex models.