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.

S U M M A R YNumerical efficiency and efficacy of subspace methods for solving large-scale geophysical inverse problems are investigated. The primary advantage of subspace techniques over traditional Gauss-Newton algorithms lies in the need to invert only a matrix equal to the dimension of the subspace. The efficacy of the method lies in a judicious choice of basis vectors. Vectors associated with gradients of the data misfit or gradients of the model component of the objective function are of great utility, but substantial improvement in convergence rates can be obtained by using basis vectors associated with gradients of a segmented objective function. To quantify these benefits we invert data acquired in a synthetic dc resistivity experiment. 420 electric potentials obtained at the surface of a 2-D earth are inverted to recover estimates of the electrical conductivity of 1296 cells. The number of basis vectors range from two to 95 and convergence rates, model norms and final models are compared. In an effort to reduce the computations we investigate the possibility of using only linear information in the data-misfit objective function. This is shown to be effective at early iterations and is computationally efficient since it obviates the need to calculate curvature information in the data misfit and because it can also be implemented without a line search. The effects of using gradient vectors versus steepest descent vectors in the inversion are examined. Accordingly we introduce two methods by which approximate descent vectors can be fabricated from gradient vectors. They show that even simple preconditioning of gradient vectors can dramatically improve convergence rates provided that all vectors are preconditioned in the same manner.

SUMMARY A simple derivation is presented for the computation of sensitivities needed to solve parametric inverse problems in electromagnetic induction. It is shown that sensitivities for any component of an electromagnetic field can be obtained by solving two boundary‐value problems which are identical except for the specification of the source terms and (possibly) prescribed boundary conditions. The electric fields from these primal and auxiliary problems are multiplied and integrated to produce a numerical value for the sensitivity. Although the final formulae derived here are equivalent to those developed through the use of formal adjoint or Green's functions approaches, our work does not require explicit derivation of the adjoint operator and boundary conditions and does not formally invoke reciprocity.

Surface deformation measurements have been used for years in oilfields to monitor production, waterflooding, waste injection, steam flooding, and Cyclic Steam Stimulation (CSS). They have been proven to be a very effective way to monitor the field operations and save money for operators wishing to avoid unwanted surface breeches, casing failures and excessive subsidence due to production. This paper demonstrates that more information can be extracted from surface deformation measurements by inverting the surface deformation for the volumetric deformation at the reservoir level, so the aerial distribution of volumetric deformation can be identified. First, a poroelastic model is presented to calculate the deformation due to the volumetric change in the reservoir. Then, a linear geophysical model is formulated to invert for the reservoir volumetric deformation from the measured surface deformation (or tilt). Constraints are added into the procedure as necessary to better resolve the inversion problem. After each inversion, the theoretical surface deformation (displacement, tilt, reservoir compaction and volumetric strain) can be calculated from the inverted volumetric deformation distribution which best fits the measured deformation data (or tilt) at the surface. The technique of mapping fluid flow using surface deformation was applied to real data from a cyclic steam injection project. Introduction Through the decades, many oil companies and individual researchers have studied reservoir compaction and its associated surface subsidence. Two techniques are used: forward modeling for prediction and direct measurements (or monitoring). The forward modeling includes numerical analysis using Finite Element Method and analytical or semi-analytical analysis. The most common monitoring techniques used in oil and gas fields are:Optical instrument leveling surveys or Global Positioning System (GPS) surveys[1]. These are conducted continuously or periodically to determine changes in position of monuments across the field.Interferometric Synthetic Aperture Radar (InSAR) [1]. This enables mapping of surface displacement along the satellite line of sight over large areas.Tiltmeter-based surface deformation monitoring[2,3]. High precision tiltmeters are placed near the earth's surface to measure the displacement gradient (tilt) induced by field operations such as fluid injection and production. Each technique has advantages and disadvantages, and in some cases two or even all three can be used in combination to get the necessary combination of precision, spatial coverage and temporal resolution. In the case history shown here, only tiltmeter data is used and the inversion process calculates and compares measured and theoretical tilt, but only minor changes are needed to perform the same calculations with displacement.

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