Three-Dimensional Transient Electromagnetic Forward Modeling for Simulating Arbitrary Source Waveform and e, db/dt, b Responses Using Rational Krylov Subspace Method

The rational Krylov subspace methods can improve the computational speed compared to conventional time-stepping approaches for calculating 3-D transient electromagnetic (TEM) method forward modeling. However, the rational Krylov subspace method simulates only the step-off response. Because primary source waveforms have nonnegligible effects on the induced responses, it is crucial to model the response induced by any given source waveform. The electric field (e) and the time derivative of the magnetic induction (<inline-formula> <tex-math notation="LaTeX">$\mathrm {d} {\mathbf { b}} / \mathrm {d}t$ </tex-math></inline-formula>) are commonly measured TEM responses. Case studies also show the magnetic induction (<inline-formula> <tex-math notation="LaTeX">$\bf b$ </tex-math></inline-formula>) response measured by magnetometers has a good resolution for exploring conductive mineral deposits. Therefore, modern TEM forward modeling algorithms should be able to simulate different types of responses. We present a new algorithm for TEM modeling using the rational Krylov subspace method. The following improvements are implemented in our approach: 1) the algorithm can efficiently compute the e and <inline-formula> <tex-math notation="LaTeX">$\mathrm {d} {\mathbf { b}} / \mathrm {d}t$ </tex-math></inline-formula> responses, and especially the <inline-formula> <tex-math notation="LaTeX">$\bf b$ </tex-math></inline-formula> response, which was less considered in other 3-D TEM studies; 2) a convolution approach is employed that allows the Krylov subspace method to simulate the source waveform effects on all three types of responses; and 3) we present the approach for computing the initial condition of b in cases of using galvanic sources. This work extends the flexibility of existing 3-D TEM modeling algorithms. Numerical examples demonstrate that the new algorithm is accurate and computationally efficient.