Sensitivity analysis of a nonlinear inversion method for 3D electromagnetic imaging in anisotropic media

We present a detailed sensitivity analysis for a nonlinear electromagnetic inversion method which was introduced earlier by the authors. Whereas the earlier work was restricted to the 3D imaging of isotropic structures in the earth from cross-borehole electromagnetic data, the analysis presented here is focused on the imaging of anisotropic structures which often have to be taken into account in practical situations. The inversion scheme considered can be described as a single-step adjoint field scheme. It avoids calculating huge sensitivity matrices (which we call linearized residual operators) during the inversion and uses only the data corresponding to one source position at a time. Doing so, the action of the adjoint linearized residual operator on the corresponding (filtered) residual vector can be calculated very efficiently by just running one forward and one adjoint Maxwell problem on the most recent best guess for the parameters. The outcome of these two runs is combined to find a correction to the latest best guess. The anisotropic sensitivity functions have the property that they decompose the linearized residual operator as well as the corresponding adjoint linearized residual operator. Playing this dual role, they provide useful information about how sources and receivers should be arranged in a given experiment, and which structures in the earth can be expected to be resolved in the inversion from a given data set. In the paper, we present numerical examples of 3D anisotropic sensitivity functions for homogeneous as well as for inhomogeneous background parameter distributions, and discuss their dual role in the nonlinear adjoint field inversion scheme.