Approximate sensitivities for the electromagnetic inverse problem

We present an approximate method for generating the Jacobian matrix of sensitivities required by linearized, iterative procedures for inverting electromagnetic measurements. The approximation is based on the adjoint-equation method in which the sensitivities are obtained by integrating, over each cell, the scalar product of an adjoint electric field (the adjoint Green's function) with the electric field produced by the forward modelling at the end of the preceding iteration. Instead of computing the adjoint field in the multidimensional conductivity model, we compute an approximate adjoint field, either in a homogeneous or layered half-space. Such an approximate adjoint field is significantly faster to compute than the true adjoint field. This leads to a considerable reduction in computation time over the exact sensitivities. The speed-up can be one or two orders of magnitude, with the relative difference increasing with the size of the problem. Sensitivities calculated using the approximate adjoint field appear to be good approximations to the exact sensitivities. This is verified by comparing true and approximate sensitivities for 2- and 3-D conductivity models, and for sources that are both finite and infinite in extent. The approximation is sufficiently accurate to allow an iterative inversion algorithm to converge to the desired result, and we illustrate this by inverting magnetotelluric data to recover a 2-D conductivity structure. Our approximate sensitivities should enable larger inverse problems to be solved than is currently feasible using exact sensitivities and present-day computing power.