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1

Electronic Resource

Inversion of geophysical data using an approximate inverse mapping (1991)

Oldenburg, D. W. ; Ellis, R. G.

Oxford, UK : Blackwell Publishing Ltd

Geophysical journal international 105 (1991), S. 0

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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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2

Electronic Resource

Generalized Subspace Methods For Large-Scale Inverse Problems (1993)

Oldenburg, D. W. ; Mcgillivray, P. R. ; Ellis, R. G.

Oxford, UK : Blackwell Publishing Ltd

Geophysical journal international 114 (1993), S. 0

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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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Applied geophysical inversion (1994)

Ellis, R. G. ; Oldenburg, D. W.

Oxford, UK : Blackwell Publishing Ltd

Geophysical journal international 116 (1994), S. 0

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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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4

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The pole-pole 3-D Dc-resistivity inverse problem: a conjugategradient approach (1994)

Ellis, R. G. ; Oldenburg, D. W.

Oxford, UK : Blackwell Publishing Ltd

Geophysical journal international 119 (1994), S. 0

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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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Calculation of sensitivities for the frequency-domain electromagnetic problem (1994)

McGillivray, P. R. ; Oldenburg, D. W. ; Ellis, R. G. ; [et al.]

Oxford, UK : Blackwell Publishing Ltd

Geophysical journal international 116 (1994), S. 0

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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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Applied geophysical inversion (1994)

Ellis, R. G. ; Oldenburg, D. W.

Oxford University Press

In: Geophysical Journal International. 1994; 116(1): 5-11. Published 1994 Jan 01. doi: 10.1111/j.1365-246x.1994.tb02122.x.

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Publication Date: 1994-01-01

Description: SummaryUsing 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.

Print ISSN: 0956-540X

Electronic ISSN: 1365-246X

Topics: Geosciences