Methods for Calculating Frechet Derivatives and Sensitivities for the Non-Linear Inverse Problem: a Comparative Study

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A fundamental step in the solution of most non-linear inverse problems is to establish a relationship between changes in a proposed model and resulting changes in the forward mod- elled data. Once this relationship has been established, it becomes possible to refine an initial model to obtain an improved fit to the observed data. In a linearized analysis, the Fréchet derivative is the connecting link between changes in the model and changes in the data. In some simple cases an analytic expression for the Fréchet derivative may be derived. In this paper we present three techniques to accomplish this and illustrate them by computing the Fréchet derivative for the 1D resistivity problem. For more complicated problems, where it is not possible to obtain an expression for the Fréchet derivative, it is necessary to parameterize the model and solve numerically for the sensitivities - partial derivatives of the data with respect to model parameters. The standard perturbation method for computing first-order sensitivities is discussed and compared to the more efficient sensitivity-equation and adjoint- equation methods. Extensions to allow for the calculation of higher order, directional and objective function sensitivities are also presented. Finally, the application of these various techniques is illustrated for both the 1D and 2D resistivity problems.

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