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  • 1
    Electronic Resource
    Electronic Resource
    METHODS FOR CALCULATING FRÉCHET DERIVATIVES AND SENSITIVITIES FOR THE NON-LINEAR INVERSE PROBLEM: A COMPARATIVE STUDY (1990)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical prospecting 38 (1990), S. 0 
    Details
    ISSN: 1365-2478
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences , Physics
    Notes: 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 ID 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-2478.1990.tb01859.x
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  • 2
    Electronic Resource
    Electronic Resource
    DIRECT CURRENT ELECTRIC POTENTIAL FIELD ASSOCIATED WITH TWO SPHERICAL CONDUCTORS IN A WHOLE-SPACE (1989)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical prospecting 37 (1989), S. 0 
    Details
    ISSN: 1365-2478
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences , Physics
    Notes: Bispherical coordinates are used to derive an exact mathematical solution for the potential field generated by direct current electric conduction in an earth model consisting of two spherical inclusions in a uniform whole-space. The solution takes the form of a spherical harmonic expansion in bispherical coordinates; coefficients in the expansion are obtained by solving sets of linear equations. Rapid forward modelling of numerous interesting situations in d.c. resistivity prospecting is facilitated by the generality and computational efficiency inherent to this new solution. For example, the accuracy of image (or superposition) methods for calculating potential solutions can be quantified. Similarly, the ability of d.c. conduction methods to resolve two distinct bounded bodies in three-dimensional space can be examined by repeatedly calculating the secondary potential or apparent resistivity response of an earth model as a selected parameter is varied. Synthetic mise à la masse, crosshole, or areal potential data sets can be generated for subsequent use in inversion studies. Improvements in solution technique derived here also apply to a simpler model consisting of a single sphere buried in a half-space.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-2478.1989.tb02209.x
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  • 3
    Electronic Resource
    Electronic Resource
    Inversion of time-domain electromagnetic data for a horizontally layered Earth (1993)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 114 (1993), S. 0 
    Details
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: Time-domain electromagnetic (TEM) data are inverted to produce a conductivity model composed of horizontal layers of constant conductivity. The data can be values of the time decay of the vertical component of the magnetic field, or of its time derivative, measured at points either inside or outside a rectangular transmitter loop. Our inversion allows many more layers than there are data. This means that the constructed conductivity model not only fits the data to the required level, but also possesses particular characteristics. By suitable choice of the objective function to be minimized, our constructed model may have minimum structure in some well-defined sense and/or it may be close to some known background model. Our inversion algorithm works directly in the time domain. This requires fractionally more computing time than the alternative approach of transforming the data to the frequency domain before inversion. However, working in the time domain prevents distortion of the data and their associated measurement errors which may arise during the transformation. Also, the effects of the full transmitter current waveform can easily be incorporated by convolution in the time domain. Our inversion is applied to data from an environmental survey and the results are shown to compare favourably with a nearby well-log.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-246X.1993.tb06977.x
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  • 4
    Electronic Resource
    Electronic Resource
    Magnetotelluric appraisal using simulated annealing (1991)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 106 (1991), S. 0 
    Details
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: Conductivity models derived from magnetotelluric measurements can be appraised by constructing extremal models which minimize and maximize localized conductivity averages. These extremal models provide lower and upper bounds for the conductivity average over the region of interest. Previous applications of this method have constructed extremal models via (iterated) linearized inversion; however, it is difficult to verify that the computed bounds represent global (rather than local) extrema. In this paper, a method of constructing extremal models using simulated annealing optimization is developed. Simulated annealing requires no approximations and is renowned for its ability to avoid unfavourable local minima. The optimization procedure is flexible and general, and can be applied to construct models which extremize a linear or non-linear objective function in any inverse problem for which the corresponding forward solution exists. Appraisal via simulated annealing is demonstrated using synthetic data and field measurements, and the results are compared with those based on linearization. The comparisons suggest that the bounds calculated via linearization represent excellent approximations to the global extrema.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-246X.1991.tb03899.x
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  • 5
    Electronic Resource
    Electronic Resource
    The similitude equation in magnetotelluric inversion (1991)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 106 (1991), S. 0 
    Details
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: The similitude equation for electromagnetic induction represents an exact integral relationship between the conductivity model and field measurements, and has been suggested as a basis for the inversion of magnetotelluric data. In this note, inversion of the similitude equation is compared to linearized inversion and found to be inadequate in that it implicitly neglects first-order terms.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-246X.1991.tb03910.x
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  • 6
    Electronic Resource
    Electronic Resource
    Linear and non-linear appraisal using extremal models of bounded variation (1989)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 99 (1989), S. 0 
    Details
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: Model features may be appraised by computing upper and lower bounds for the average value of the model over a specified region. The bounds are computed by constructing extremal models which maximize and minimize this average. In order to compute the most meaningful bounds, it is important that the allowed models are geophysically realistic. In this paper, the appraisal analysis of Oldenburg (1983) is extended to incorporate a bound on the total variation of the extremal models. Restricting the variation discriminates against highly oscillatory models and, as a consequence, the difference between upper and lower bounds is often considerably reduced. The original presentation of the funnel function bound curves is extended to include the variation of the model as another dimension. The interpreter may make use of any knowledge or insight regarding the variation of the model to generate realistic extremal models and meaningful bounds.The appraisal analysis is extended to non-linear problems by altering the usual linearized equations so that a global norm of the model can be used in the objective function. The method is general, but is applied here specifically to compute bounds for localized conductivity averages of the Earth by inverting magnetotelluric measurements. The variation bound may be formulated in terms of conductivity or log conductivity. The appraisal is illustrated using synthetic data and field measurements from southeastern British Columbia, Canada.Bounding the total variation may be viewed as constraining the flatness of the model. This suggests a new method of calculating (piecewise-constant) l1 flattest models by minimizing the norm of the total variation. Unlike l2 flattest models which vary in a smooth, continuous manner, the l1 minimum-variation model is a least-structure model that resembles a layered earth with structural variations occurring at distinct depths.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-246X.1989.tb02034.x
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  • 7
    Electronic Resource
    Electronic Resource
    Generalized Subspace Methods For Large-Scale Inverse Problems (1993)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 114 (1993), S. 0 
    Details
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: Numerical 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-246X.1993.tb01462.x
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  • 8
    Electronic Resource
    Electronic Resource
    Approximate sensitivities for the electromagnetic inverse problem (1996)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 126 (1996), S. 0 
    Details
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-246X.1996.tb05282.x
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  • 9
    Electronic Resource
    Electronic Resource
    Calculation of sensitivities for the frequency-domain electromagnetic problem (1994)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 116 (1994), S. 0 
    Details
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-246X.1994.tb02121.x
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  • 10
    Electronic Resource
    Electronic Resource
    Applied geophysical inversion (1994)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 116 (1994), S. 0 
    Details
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: Using the 2-D DC-resistivity tomography experiment as an example, we examine some of the difficulties inherently associated with constructing a single maximally smooth model as a solution to a geophysical inverse problem. We argue that this conventional approach yields at best only a single model from a myriad of possible models and at worst produces a model which, although having minimum structure, frequently has little useful relation to the earth that gave rise to the observed data. In fact in applied geophysics it is usual to have significant prior information which is to be supplemented by further geophysical experiments. With this perspective we suggest an alternate approach to geophysical inverse problems which emphasizes the prior information and includes the data from the geopysical experiment as a supplementary constraint. To this end we take all available prior information and construct an inversion algorithm which, given an arbitrary starting model and the absence of any data, will produce a preconceived earth model and then introduce the observed data into the inversion to determine how the prior earth model is influenced by the supplementary geophysical data.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-246X.1994.tb02122.x
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