Boundary element solutions for broad-band 3-D geo-electromagnetic problems accelerated by an adaptive multilevel fast multipole method

We have developed a generalized and stable surface integral formula for 3-D uniform inducing field and plane wave electromagnetic induction problems, which works reliably over a wide frequency range. Vector surface electric currents and magnetic currents, scalar surface electric charges and magnetic charges are treated as the variables. This surface integral formula is successfully applied to compute the electromagnetic responses of 3-D topography to low frequency magnetotelluric and high frequency radio-magnetotelluric fields. The standard boundary element method which is used to solve this surface integral formula quickly exceeds the memory capacity of modern computers for problems involving hundreds of thousands of unknowns. To make the surface integral formulation applicable and capable of dealing with large-scale 3-D geo-electromagnetic problems, we have developed a matrix-free adaptive multilevel fast multipole boundary element solver. By means of the fast multipole approach, the time-complexity of solving the final system of linear equations is reduced to O(m log m) and the memory cost is reduced to O(m), where m is the number of unknowns. The analytical solutions for a half-space model were used to verify our numerical solutions over the frequency range 0.001–300kHz. In addition, our numerical solution shows excellent agreement with a published numerical solution for an edge-based finite-element method on a trapezoidal hill model at a frequency of 2Hz. Then, a high frequency simulation for a similar trapezoidal hill model was used to study the effects of displacement currents in the radio-magnetotelluric frequency range. Finally, the newly developed algorithm was applied to study the effect of moderate topography and to evaluate the applicability of a 2-D RMT inversion code that assumes a flat air–Earth interface, on RMT field data collected at Smørgrav, southern Norway. This paper constitutes the first part of a hybrid boundary element-finite element approach to compute the electromagnetic field inside structures involving complex 3-D conductivity and permittivity distributions.