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A numerical method has been developed to simulate the electromagnetic (EM) response of a 3-D earth to a dipole source at frequencies ranging from 100 Hz to 100 MHz. The problem is formulated in the frequency domain with a modified vector Helmholtz equation for the scattered electric fields. The differential equation is approximated on a staggered finite-difference grid, giving a sparse complex symmetric matrix equation. The system is solved by a preconditioned quasi-minimum–residual method.

Dirichlet boundary conditions are imposed at the edges of the mesh by setting the tangential electric fields equal to zero. At frequencies less than 1 MHz, grid stretching reduces reflections off the grid boundaries. At higher frequencies, absorbing boundary conditions are imposed by making the stretching parameters of the modified vector Helmholtz equation complex to introduce loss at the boundaries.

Iterative solution to the nonlinear 3-D EM inverse problem proceeds by linearized model updates using the method of conjugate gradients. Full wave equation modeling is employed to compute model sensitivities and predicted data in the frequency domain with the 3-D finite-difference algorithm. Both the forward and inverse solutions are implemented on a massively parallel computing platform which allows forward and inverse models with millions and ten of thousands of parameters, respectively.

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