METHODS FOR CALCULATING FRÉCHET DERIVATIVES AND SENSITIVITIES FOR THE NON‐LINEAR INVERSE PROBLEM: A COMPARATIVE STUDY¹

Abstract

MCGILLIVRAY and OLDENBURG, D.W. 1990. Methods for calculating Fréchet derivatives and sensitivities for the non-linear inverse problem: a comparative study. Geophysical Prospecting 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 modelled 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 adjointequation 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. 38,499-524.

Multiscale electrical impedance tomography

“…Addressing this problem by comparing the decrease of the misfit against the price paid by adding parameters in the model is time consuming since this needs to solve the refined model to apply the decision criteria. In order to save computer time we instead performed a prior sensitivity analysis [McGillivary and Oldenburg, 1990] to determine whether or not a given block refinement is significant.…”

Identification of subsurface structures using electromagnetic data and shape priors

“…S (a) is the sensitivity matrix dm/da, which is found using the direct (or gradient simulator) method, see, e.g., [47]. The computational cost in finding S is reduced due to the stiffness matrix A being common both in solving the forward problem and calculating the sensitivity; see Appendix B.…”

“…S is a N d × N a matrix, which is found using the direct (or gradient simulator) method at each iteration k, see, e.g., [47]. In the direct method we differentiate m directly with a.…”

Two-Dimensional Magnetotelluric Modelling of Ore Deposits: Improvements in Model Constraints by Inclusion of Borehole Measurements

“…In the subsequent inversion examples, all Jacobian matrices (sensitivities) were computed using the sensitivity equation method (McGillivray and Oldenburg 1990;Rodi and Mackie 2001;Kalscheuer et al 2008). For a homogeneous half-space model and a blocky model with large resistivity contrasts (not shown), the computed sensitivities were verified using the computationally more expensive but easily implemented perturbation approach (McGillivray and Oldenburg 1990).…”