The character of non-uniqueness in the conductivity modelling problem for the earth

Anderssen, R. S.

doi:10.1007/bf00880211

Cited by 17 publications

(10 citation statements)

References 22 publications

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“…Nevertheless, uniform random search methods still found applications. In addition to the regional and global travel time studies, other applications of MCI have included electromagnetic induction [ Anderssen , 1970], Rayleigh wave attenuation [ Mills and Fitch , 1977; Mills and Hales , 1978], regional magnetotelluric studies [ Hermance and Grillot , 1974; Jones and Hutton , 1979], estimation of mantle viscosities [ Ricard et al ., 1989], and estimation of plate rotation vectors [ Jestin et al ., 1994].…”

Section: A Brief History Of Monte Carlo Inversion In Geophysicsmentioning

confidence: 99%

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Monte Carlo Methods in Geophysical Inverse Problems

Sambridge

Mosegaard

2002

Reviews of Geophysics

665

395

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[1] Monte Carlo inversion techniques were first used by Earth scientists more than 30 years ago. Since that time they have been applied to a wide range of problems, from the inversion of free oscillation data for whole Earth seismic structure to studies at the meter-scale lengths encountered in exploration seismology. This paper traces the development and application of Monte Carlo methods for inverse problems in the Earth sciences and in particular geophysics. The major developments in theory and application are traced from the earliest work of the Russian school and the pioneering studies in the west by Press [1968] to modern importance sampling and ensemble inference methods. The paper is divided into two parts. The first is a literature review, and the second is a summary of Monte Carlo techniques that are currently popular in geophysics. These include simulated annealing, genetic algorithms, and other importance sampling approaches. The objective is to act as both an introduction for newcomers to the field and a comprehensive reference source for researchers already familiar with Monte Carlo inversion. It is our hope that the paper will serve as a timely summary of an expanding and versatile methodology and also encourage applications to new areas of the Earth sciences. INDEX TERMS:3260

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Section: A Brief History Of Monte Carlo Inversion In Geophysicsmentioning

confidence: 99%

“…[10] An approach developed by Anderssen andSenata [1971, 1972] went some way to answering these criticisms. They developed a statistical procedure for estimating the reliability of a given set of nonuniqueness bounds.…”

Section: Beginnings Of Monte Carlo Inversionmentioning

confidence: 99%

Monte Carlo Methods in Geophysical Inverse Problems

Sambridge

Mosegaard

2002

Reviews of Geophysics

665

395

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“…In geomagnetic induction studies, the Hedgehog inversion procedure does not appear to have been employed to date, but Monte-Carlo inversion has been applied by Anderssen (1970), to global electromagnetic induction data, and by Hermance & Grillot (1974), to regional MT data.…”

Section: Introductionmentioning

confidence: 99%

A multi-station magnetotelluric study in southern Scotland -- II. Monte-Carlo inversion of the data and its geophysical and tectonic implications

Jones

Hutton

1979

Geophysical Journal International

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A Monte-Carlo inversion procedure is developed and applied to magnetotelluric data from six locations, two of which are in the Midland Valley of Scotland, three in the Southern Uplands, and one in northern England. The method is described in full in respect of one of the six locations to illustrate both the importance of satisfying the phase as well as the amplitude data and the effect of model acceptance level. The electrical resistivity profiles resulting from application of the method indicate that; (a) there is a conducting zone under the Midland Valley at a depth no greater than 12 km, (b) the crust under the Southerri Uplands is mainly resistive, (c) there is a conductor at a depth greater than 24 km in this region, and (d) under northern England there is probably a very highly conducting region very close to the surface. A brief discussion of the possible geophysical and tectonic significance of these models follows. Monte-Carlo inversionAs stated by Jackson (1973), the complete solution to an inverse problem consists of two parts: (1) finding a solution, and (2) representing, in a meaningful way, the degree of non-Present address: lnstitut fur

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“…Finally, mention should be made of 6he use of Monte Carlo inversion techniques for solving similar problems. Keilis-Borok and Yanovskaya [I967], Press [1968, 1970a, b], Wiggins [1969], and Anderssen [1970] There are two interesting variations on equation 2 that have been examined In •his example we consider only the fundamental Rayleigh mode between periods of 100 sec and 350 sec. This period range corresponds to 78 spheroidal freeoscillation modes, which are labeled oS97 (100-see period) to oS2o (350-sec period).…”

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confidence: 99%

The general linear inverse problem: Implication of surface waves and free oscillations for Earth structure

Wiggins¹

1972

Reviews of Geophysics

711

431

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The discrete general linear inverse problem reduces to a set of m equations in n unknowns. There is generally no unique solution, but we can find k linear combinations of parameters for which restraints are determined. The parameter combinations are given by the eigenvectors of the coefficient matrix. The number k is determined by the ratio of the standard deviations of the observations to the allowable standard deviations 2. Particular solution. We seek a particular set of parameters P• + AP'• that will satisfy the observations within their variance.3. Resolving power. By using the reparameterization determined in par• 1 we wish to find the resolution of the observations in parameter space. Since we cannot determine the corrections for individual parameters uniquely, our approach to studying resolution is to try to find the smallest groups of parameters for which the average value can be determined. Iniormation distribution. Generally, each particular observation doesno• contribute independent information about the model. We shall determine the distribution of information among the observations. Such knowledge can then be used as a constrain• on data acquisition and smoothing operations. THEORETICAL DISCUSSION In the introduction, I indicated how the general inverse problem can be reduced to the solution of a set of linear simultaneous equations. I shall begin the discussion of the properties of the solution by introducing a matrix notation' Ap • : ß ß ß ß ß ß •b,C t = ---var {AP', } var {Ap'} = vat { AP', } ß ß v•r { AP', } _vat (3) ]c,/oP, ocdoP, ... oc/oP. ß OC'•/OP• ' LOC•/OP• ... Notice that each row of A' corresponds to one particular observation and that each column corresponds to a particular parameter. 254 RALPH A. WIGGINS Example: To help fix ideas, I shall illustrate •he s•eps for constructing the solution to a general inverse problem by considering the specific problem of determining the shear-wave velocity structure of the earth from Rayleigh wave phasevelocity observations. There are two principal types of surface waves generated by earthquakes and explosions: Love waves and Rayleigh waves. The depth of penetration of such waves depends on their period; longer-period waves penetrate deeper than shorter-period waves. In a radially heterogeneous medium, both types of waves are dispersed. The amount of dispersion for Rayleigh waves depends on the shear-wave velocity/•(z), the compressional-wave velocity a(z), and the density p(z) over the range of depths z to which the wave penetrates. The dispersion for Love waves is controlled by only/•(z) and p(z). This example assumes that a(z) and p(z) are known and that we are seeking to determine/•(z). When surface waves travel around the earth several times, various frequencies interfere constructively and destructively to produce discrete modes of free oscillation. Oscillations corresponding to Love-wave interference are called toroidal; oscillations corresponding to Rayleigh-wave interference are called spheroidal. (See Bullen [1965., chapters 5 and 8] or Ga...

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The character of non-uniqueness in the conductivity modelling problem for the earth

Cited by 17 publications

References 22 publications

Monte Carlo Methods in Geophysical Inverse Problems

Monte Carlo Methods in Geophysical Inverse Problems

A multi-station magnetotelluric study in southern Scotland -- II. Monte-Carlo inversion of the data and its geophysical and tectonic implications

The general linear inverse problem: Implication of surface waves and free oscillations for Earth structure

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