3D time-domain airborne electromagnetic forward modeling using the rational Krylov method

Traditional Three Dimensional (3D) time domain airborne Electromagnetic (EM) modeling utilizes an implicit time discretization technique to calculate the EM responses in the time domain, which requires several matrix decompositions and hundreds of forward and backward substitutions. This reduces largely the modeling efficiency. To speed up the 3D forward modeling for time domain Airborne Electromagnetic (AEM), we propose to solve the diffusion equation for the electric field using the rational Krylov method. The spatial discretization is completed by the unstructured tetrahedral grids, where lowest order Nedelec first kind vector basis functions are adopted to approximate the electric field inside each element. Then we derive the electric field solutions directly in terms of the product of a matrix exponential function with a vector. The rational Krylov method is used to solve the matrix function, where the orthogonal basis vectors in the rational Krylov subspace are constructed using the rational Arnoldi algorithm. Finally, the electric field is calculated for any desired time by the rational Arnoldi approximation, which avoids explicit or implicit time stepping. In addition, an exponential weight is incorporated into the rational Arnoldi approximation errors when optimizing the shift parameters, resulting in small approximation errors at late times. This further reduces the size of the rational Krylov subspace and improves the computational efficiency. We verify the correctness and accuracy of the proposed rational Krylov method by comparing the forward modeling results for a homogeneous half-space model with the semi-analytic solution. Furthermore, we prove the effectiveness of the presented rational Krylov method by comparing our solutions with time-domain finite-element and finite-volume solutions for a 3D abnormal model using global and local grids. The simulations of AEM responses for two typical 3D abnormal models demonstrate that the rational Krylov method can speed up the time-domain AEM forward modeling for as up to 7 times while keeping high modeling accuracy.