Efficient 3-D Inversion for Transient Electromagnetic Method Based on Rational Krylov Subspace

Conventional three-dimensional (3D) inversion algorithms for the transient electromagnetic method (TEM) require time-domain discretization in both forward and adjoint modeling. These algorithms start from the initial time and compute the electric field iteratively at each new time step, which involve repeatedly solving large-scale systems of linear equations and thus cause a substantial computational cost. In contrast, the Krylov subspace-based model order reduction technique computes the 3D time-domain electromagnetic response with just a one-time reduction process, allowing for the electric field at any time to be obtained by solving only a small-scale matrix exponential problem, significantly improving the efficiency of forward modeling. However, this approach has not yet been widely utilized in 3D TEM inversion. In this study, the rational Krylov subspace-based model order reduction technique is applied to both forward and adjoint modeling. The data misfit residuals at each observation time channel are treated as virtual sources for the adjoint problem, enabling the use of the rational Krylov subspace method to solve the corresponding adjoint field. The gradient is evaluated using results from both the forward and adjoint modeling, and the model is updated via the L-BFGS method. Throughout the gradient calculation, only two matrix factorizations and a limited number of backward substitutions for right-hand sides are required in total for all necessary forward and adjoint modeling, which substantially reduces the computation burden. The proposed algorithm is validated using synthetic data. Results show that the presented 3D TEM inversion algorithm using the rational Krylov subspace technique achieves satisfactory inversion accuracy while significantly improving computational efficiency compared to inversion methods that employ conventional time-stepping approaches. Therefore, this work can provide valuable insights for further application of rational Krylov subspace methods in TEM data inversion.