## 1. Introduction

[2] The finite difference time domain (FDTD) method belongs to the most widely used numerical methods, employed to solve complex electromagnetic scattering problems. FDTD is designed to handle large problems, conformal to structured grids, using an explicit leapfrog time-stepping scheme [*Yee*, 1966]. The modeling of increasingly complex structures has created a need to improve the inherently high computational time requirements of the FDTD method. Engineering applications in particular require modeling very complex and electrically large structures. For such applications, various schemes for FDTD subgridding algorithms have been presented [*Zivanovic et al.*, 1991; *Prescott and Shuley*, 1992; *White et al.*, 1997, 2001]. These algorithms have in common that a local mesh refinement is carried out to allow for increased resolution in regions of interest. Simultaneously, a computationally more economic coarse grid is maintained for the rest of the model domain.

[3] In contrast to engineering problems, typically characterized by high-frequency field simulations, our solution is designed to treat transient electromagnetic (TEM) problems arising in geophysical exploration, such as mineral and hydrological reconnaissance, or general geological mapping. The frequency range of such applications allows us to neglect displacement currents; thus we solve the diffusive forms of Maxwell's equations in the quasi-static limit.

[4] One highly efficient three-dimensional (3-D) solver for modeling diffusive EM fields is based on the spectral Lanczos decomposition method (SLDM) [*Druskin and Knizhnerman*, 1994; *Knizhnerman*, 1995; *Druskin et al.*, 1998]. It has been widely applied and continuously improved for geophysical applications, such as for example 3-D EM induction modeling in boreholes [*Druskin et al.*, 1999]. Compared to FDTD, SLDM and its recent improvements [e.g., *Davydycheva et al.*, 2003] has the outstanding advantage of providing faster solutions. On the other hand, FDTD is more effective in the case of a large and spatially distributed number of receptors to be simulated, as is typical in large-scale EM inverse problems. It is also reliable for producing accurate results over large dynamic time ranges.

[5] In this study, we aim at improving one of the major drawbacks of FDTD, its large computational requirements when simulating responses at late times. Our FDTD method's capabilities and the implementation for parallel computers have been described in two recent works [*Commer and Newman*, 2004; *Newman and Commer*, 2005]. We simulate the TEM fields due to a step response signal, impressed by two different transmitter types. These are loops with an inductive coupling to the ground and galvanically coupled grounded wires. The employed explicit time-stepping algorithm is of second-order accuracy and allows for a variation of the electrical conductivity in the three dimensions of a Cartesian grid. A modified version of the DuFort-Frankel method [*DuFort and Frankel*, 1953] is used to achieve unconditional stability.

[6] To achieve a further speedup of the explicit time-stepping scheme, we use concepts known from classical multigrid (MG) schemes. Classical MG methods [e.g., *Brandt*, 1973; *Trottenberg et al.*, 2000] solve a partial differential equation of interest on a hierarchy of grids. These methods perform a certain number of (pre)smoothing iterations to the initial equation system. The change to a coarser grid level is done after a sufficient reduction of the solution's high-frequency errors. Applying this concept to our explicit time-stepping scheme translates to the frequency content of the EM field. While the decaying TEM field diffuses in the conductive Earth, which is a lossy medium, high frequencies are damped out first. Hence the spatial sampling rate can be decreased accordingly with progressing simulation time. After a certain simulation time on the initial grid, the electromagnetic field is transferred to a coarser mesh by a field restriction procedure. Similar to classical MG, where the optimal number of presmoothing steps is problem-dependent, the optimal restriction time needs to be individually evaluated for a given modeling problem.