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  • 1990-1994  (23)
  • 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
    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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  • 3
    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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  • 4
    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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  • 5
    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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  • 6
    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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  • 7
    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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  • 8
    Electronic Resource
    Electronic Resource
    Reply to “Comment on ‘Magnetotelluric appraisal using simulated annealing’” by S. Constable (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
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-246X.1991.tb03901.x
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  • 9
    Electronic Resource
    Electronic Resource
    Inversion of geophysical data using an approximate inverse mapping (1991)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 105 (1991), S. 0 
    Details
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: Realistic geologic features are 3-D and inverse techniques which rely upon linearization and computation of a sensitivity matrix to show how a change in the model affects a particular datum, can require prohibitive amounts of computation. Even 104 data collected over an earth parametrized into 102× 102× 102 elements has a sensitivity matrix which is 104× 106. The generation of that matrix requires the solution of many 3-D forward problems and its solution is also computationally intensive. In this paper we formulate a general technique for solving large-scale inverse problems which does not involve full linearization and which can obviate the need to solve a large system of equations. The method uses accurate forward modelling to compute responses, but only uses an approximate inverse mapping to map data back to model space. The approximate inverse mapping is chosen with emphasis on the physics of the problem and on computational expediency. There are two ways to implement the AIM (Approximate Inverse Mapping) inversion. At any iteration step, AIM-MS applies the approximate inverse mapping to forward modelled data and also applies the same mapping to the observations; the model perturbation is taken as the difference between the resulting functions. In AIM-DS, an alteration to the data is sought, such that the approximate inverse mapping applied to the altered data yields a model which adequately satisfies the observations. The approximate mapping inversion is illustrated with a simple parametric inverse problem and with the inversion of magnetotelluric (MT) data to recover a 1-D conductivity model. To illustrate the technique in a realistically complicated problem we invert MT data acquired from a line of stations over a 2-D conductivity structure. TE and TM mode data are inverted individually and as determinant averages. As a final example we invert 900 data, with and without noise, to recover a model that is parametrized by 1500 cells of unknown conductivity. The inversion is found to be computationally efficient and robust.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-246X.1991.tb06717.x
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  • 10
    Electronic Resource
    Electronic Resource
    The pole-pole 3-D Dc-resistivity inverse problem: a conjugategradient approach (1994)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 119 (1994), S. 0 
    Details
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: The pole-pole 3-D DC-resistivity inverse problem is solved by converting the inverse problem into an objective-function optimization problem, using the adjoint equation to compute the gradient of the objective function, and using a conjugategraient minimization. Two examples of the application of the resulting inversion algorithm are given. First, a large synthetic data set is inverted, and second, the inversion algorithm is used to invert E-SCAN field data of relevance to mineral exploration.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-246X.1994.tb00921.x
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