An efficient data-subspace inversion method for 2-D magnetotelluric data

There are currently three types of algorithms in use for regularized 2-D in version of magnetotelluric (MT) data. All seek to minimize some functional which penalizes data misfit and model structure. With the most straightforw ard approach (exemplified by OCCAM), the minimization is accomplished using some Variant on a linearized Gauss-Newton approach. A second approach is t o use a descent method [e.g., nonlinear conjugate gradients (NLCG)] to avoi d the expense of constructing large matrices (e.g., the sensitivity matrix) . Finally, approximate methods [e.g., rapid relaxation inversion (RRI)] hav e been developed which use cheaply computed approximations to the sensitivi ty matrix to search for a minimum of the penalty functional. Approximate ap proaches can be very fast, but in practice often fail to converge without s ignificant expert user intervention. On the other hand, the more straightfo rward methods can be prohibitively expensive to use for even moderate-size data sets. Here, we present a new and much more efficient variant on the OC CAM scheme. By expressing the solution as a linear combination of rows of t he sensitivity matrix smoothed by the model covariance (the "representers") , we transform the linearized inverse problem from the M-dimensional model space to the N-dimensional data space. This method is referred to as DASOCC , the data space OCCAM's inversion. Since generally N much less than M, thi s transformation by itself can result in significant computational saving. More importantly the data space formulation suggests a simple approximate m ethod for constructing the inverse solution. Since MT data an smooth and "r edundant," a subset of the representers is typically sufficient to form the model without significant loss of detail. Computations required for constr ucting sensitivities and the size of matrices to be inverted can be signifi cantly reduced by this approximation. We refer to this inversion as REBOCC, the reduced basis OCCAM's inversion. Numerical experiments on synthetic an d real data sets with REBOCC, DASOCC, NLCG, RRI, and OCCAM show that REBOCC is faster than both DASOCC and NLCG, which are comparable in speed. All of these methods are significantly faster than OCCAM, but are not competitive with RRI. However, even with a simple synthetic data set, we could not alw ays get RRI to converge to a reasonable solution. The basic idea behind REB OCC should be more broadly applicable, in particular to 3-D MT inversion.