Hybrid conjugate gradient-Occam algorithms for inversion of multifrequency and multitransmitter EM data

Abstract

We describe novel hybrid algorithms for inversion of electromagnetic geophysical data, combining the computational and storage efficiency of a conjugate gradient approach with an Occam scheme for regularization and step-length control. The basic algorithm is based on the observation that iterative solution of the symmetric (Gauss-Newton) normal equations with conjugate gradients effectively generates a sequence of sensitivities for different linear combinations of the data, allowing construction of the Jacobian for a projection of the original full data space. The Occam scheme can then be applied to this projected problem, with the tradeoff parameter chosen by assessing fit to the full data set. For EM geophysical problems with multiple transmitters (either multiple frequencies or source geometries) an extension of the basic hybrid algorithm is possible. In this case multiple forward and adjoint solutions (one each for each transmitter) are required for each step in the iterative normal equation solver, and each corresponds to the sensitivity for a separate linear combination of data. From the perspective of the hybrid approach, with conjugate gradients generating an approximation to the full Jacobian, it is advantageous to save all of the component sensitivities, and use these to solve the projected problem in a larger subspace. We illustrate the algorithms on a simple problem, 2-D magnetotelluric inversion, using synthetic data. Both the basic and modified hybrid schemes produce essentially the same result as an Occam inversion based on a full calculation of the Jacobian, and the modified scheme requires significantly fewer steps (relative to the basic hybrid scheme) to converge to an adequate solution to the normal equations. The algorithms are expected to be useful primarily for 3-D inverse problems for which the computational burden is heavily dominated by solution to the forward and adjoint problems.