We have developed an approximation to the Hessian for inversion of 3D controlled source electromagnetic data. Our approach can considerably reduce the numerical complexity in terms of the number of forward solutions as well as the size and complexity of the calculations required to compute the update direction from the Gauss-Newton equation. The approach makes use of “supershots,” in which several source positions are combined for simultaneous-source simulations. The resulting Hessian can be described as a low-rank approximation to the Gauss-Newton Hessian. The structure of the approximate Hessian facilitates a matrix-free direct solver for the Gauss-Newton equation, and the reduced memory complexity allows the use of a large number of unknowns. We studied the crosstalk introduced in the approximation, and we determined how the dissipative nature of marine electromagnetic field propagation reduces the impact of this noise. Inversion results from recent field data demonstrated the numerical and practical feasibility of the approach.
Fast Multipole Method as a Matrix-Free Hierarchical Low-Rank Approximation