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Journal ArticleDOI

Inverse scattering series and seismic exploration

TL;DR: In this paper, the authors present an overview and a detailed description of the key logic steps and mathematical-physics framework behind the development of practical algorithms for seismic exploration derived from the inverse scattering series.
Abstract: This paper presents an overview and a detailed description of the key logic steps and mathematical-physics framework behind the development of practical algorithms for seismic exploration derived from the inverse scattering series. There are both significant symmetries and critical subtle differences between the forward scattering series construction and the inverse scattering series processing of seismic events. These similarities and differences help explain the efficiency and effectiveness of different inversion objectives. The inverse series performs all of the tasks associated with inversion using the entire wavefield recorded on the measurement surface as input. However, certain terms in the series act as though only one specific task, and no other task, existed. When isolated, these terms constitute a task-specific subseries. We present both the rationale for seeking and methods of identifying uncoupled task-specific subseries that accomplish: (1) free-surface multiple removal; (2) internal multiple attenuation; (3) imaging primaries at depth; and (4) inverting for earth material properties. A combination of forward series analogues and physical intuition is employed to locate those subseries. We show that the sum of the four task-specific subseries does not correspond to the original inverse series since terms with coupled tasks are never considered or computed. Isolated tasks are accomplished sequentially and, after each is achieved, the problem is restarted as though that isolated task had never existed. This strategy avoids choosing portions of the series, at any stage, that correspond to a combination of tasks, i.e., no terms corresponding to coupled tasks are ever computed. This inversion in stages provides a tremendous practical advantage. The achievement of a task is a form of useful information exploited in the redefined and restarted problem; and the latter represents a critically important step in the logic and overall strategy. The individual subseries are analysed and their strengths, limitations and prerequisites exemplified with analytic, numerical and field data examples.
Citations
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Journal ArticleDOI
TL;DR: In this paper, a review is devoted to some inverse problems arising in the context of linear elasticity, namely the identification of distributions of elastic moduli, model parameters or buried objects such as cracks.
Abstract: This review is devoted to some inverse problems arising in the context of linear elasticity, namely the identification of distributions of elastic moduli, model parameters or buried objects such as cracks. These inverse problems are considered mainly for three-dimensional elastic media under equilibrium or dynamical conditions, and also for thin elastic plates. The main goal is to overview some recent results, in an effort to bridge the gap between studies of a mathematical nature and problems defined from engineering practice. Accordingly, emphasis is given to formulations and solution techniques which are well suited to general-purpose numerical methods for solving elasticity problems on complex configurations, in particular the finite element method and the boundary element method. An underlying thread of the discussion is the fact that useful tools for the formulation, analysis and solution of inverse problems arising in linear elasticity, namely the reciprocity gap and the error in constitutive equation, stem from variational and virtual work principles, i.e., fundamental principles governing the mechanics of deformable solid continua. In addition, the virtual work principle is shown to be instrumental for establishing computationally efficient formulae for parameter or geometrical sensitivity, based on the adjoint solution method. Sensitivity formulae are presented for various situations, especially in connection with contact mechanics, cavity and crack shape perturbations, thus enriching the already extensive known repertoire of such results. Finally, the concept of topological derivative and its implementation for the identification of cavities or inclusions are expounded.

411 citations


Cites background from "Inverse scattering series and seism..."

  • ...medical imaging of tissues [11] or seismic exploration [98, 114, 123, 130, 136, 140]....

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Journal ArticleDOI
TL;DR: In this article, a physical interpretation of deconvolution interferometry based on scattering theory is presented, where the free-point or clamped-point boundary condition is circumvented by separating the reference waves from scattered wavefields.
Abstract: Interferometry allows for synthesis of data recorded at any two receivers into waves that propagate between these receivers as if one of them behaves as a source. This is accomplished typically by crosscorrelations. Based on perturbation theory and representation theorems, we show that interferometry also can be done by deconvolutions for arbitrary media and multidimensional experiments. This is important for interferometry applications in which (1) excitation is a complicated source-time function and/or (2) when wavefield separation methods are used along with interferometry to retrieve specific arrivals. Unlike using crosscorrelations, this method yields only causal scattered waves that propagate between the receivers. We offer a physical interpretation of deconvolution interferometry based on scattering theory. Here we show that deconvolution interferometry in acoustic media imposes an extra boundary condition, which we refer to as the free-point or clamped-point boundary condition, depending on the measured field quantity. This boundary condition generates so-called free-point scattering interactions, which are described in detail. The extra boundary condition and its associated artifacts can be circumvented by separating the reference waves from scattered wavefields prior to interferometry. Three wavefield-separation methods that can be used in interferometry are direct-wave interferometry, dual-field interferometry, and shot-domain separation. Each has different objectives and requirements.

191 citations


Cites background or methods from "Inverse scattering series and seism..."

  • ...Rodberg and Thaler, 1967; Weglein et al., 2003"....

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  • ...In fact, equation 13 is a Born-like approximation !e.g., Born and Wolf, 1959; Weglein et al., 2003" of equation 10....

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  • ...Forward- and inverse-scattering series serve, for instance, as the basis for methodologies in imaging and multiple suppression !e.g., Weglein et al., 2003; Malcolm et al., 2004"....

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  • ...Using a combination of perturbation theory and representation theorems !as in Vasconcelos, 2007", we first review interferometry by correlations....

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  • ...Weglein et al., 2003; Vasconcelos, 2007", such that the quantities G0 and G !...

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Journal ArticleDOI
TL;DR: In this paper, the main idea is that if the reconstruction is restricted to only the observable part, then the inversion will become stable, and the challenging task is how to design stable numerical methods for solving these inverse scattering problems inspired by the diffraction limit.
Abstract: This paper is concerned with computational approaches and mathematical analysis for solving inverse scattering problems in the frequency domain. The problems arise in a diverse set of scientific areas with significant industrial, medical, and military applications. In addition to nonlinearity, there are two common difficulties associated with the inverse problems: ill-posedness and limited resolution (diffraction limit). Due to the diffraction limit, for a given frequency, only a low spatial frequency part of the desired parameter can be observed from measurements in the far field. The main idea developed here is that if the reconstruction is restricted to only the observable part, then the inversion will become stable. The challenging task is how to design stable numerical methods for solving these inverse scattering problems inspired by the diffraction limit. Recently, novel recursive linearization based algorithms have been presented in an attempt to answer the above question. These methods require multi-frequency scattering data and proceed via a continuation procedure with respect to the frequency from low to high. The objective of this paper is to give a brief review of these methods, their error estimates, and the related mathematical analysis. More attention is paid to the inverse medium and inverse source problems. Numerical experiments are included to illustrate the effectiveness of these methods.

177 citations

Journal ArticleDOI
TL;DR: In this paper, the authors present an approach to use multiply scattered waves to illuminate structures not sensed by singly scattered waves, which can be viewed as a refinement of past work in which a method to predict artefacts in imaging with multiply-scattered waves was developed.
Abstract: SUMMARY If singly scattered seismic waves illuminate the entirety of a subsurface structure of interest, standard methods can be applied to image it. It is generally the case, however, that with a combination of restricted acquisition geometry and imperfect velocity models, it is not possible to illuminate all structures with only singly scattered waves. We present an approach to use multiply scattered waves to illuminate structures not sensed by singly scattered waves. It can be viewed as a refinement of past work in which a method to predict artefacts in imaging with multiply scattered waves was developed. We propose an algorithm and carry out numerical experiments, representative of imaging of the bottom and flanks of salt, demonstrating the effectiveness of our approach.

106 citations

Journal ArticleDOI
TL;DR: In this paper, new misfit functions for matching simulated and measured data have recently been introduced to enable the use of full waveform inversion in seismic imaging, which is a powerful computational tool for seismic imaging.
Abstract: Full-waveform inversion has evolved into a powerful computational tool in seismic imaging. New misfit functions for matching simulated and measured data have recently been introduced to avo...

96 citations


Cites background from "Inverse scattering series and seism..."

  • ...Several hierarchical methods that invert from low frequencies to higher frequencies have been proposed in literature to mitigate the cycle skipping of the inverse problem (Kolb et al., 1986; Pratt and Worthington, 1990; Bunks et al., 1995; Weglein et al., 2003; Sirgue and Pratt, 2004)....

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References
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Book
01 Jan 1966
TL;DR: In this paper, the authors present a survey of various approaches to the solution of three-particle problems, as well as a discussion of the Efimov effect, and the general approach to multiparticle reaction theory.
Abstract: Much progress has been made in scattering theory since the publication of the first edition of this book fifteen years ago, and it is time to update it. Needless to say, it was impossible to incorporate all areas of new develop- ment. Since among the newer books on scattering theory there are three excellent volumes that treat the subject from a much more abstract mathe- matical point of view (Lax and Phillips on electromagnetic scattering, Amrein, Jauch and Sinha, and Reed and Simon on quantum scattering), I have refrained from adding material concerning the abundant new mathe- matical results on time-dependent formulations of scattering theory. The only exception is Dollard's beautiful "scattering into cones" method that connects the physically intuitive and mathematically clean wave-packet description to experimentally accessible scattering rates in a much more satisfactory manner than the older procedure. Areas that have been substantially augmented are the analysis of the three-dimensional Schrodinger equation for non central potentials (in Chapter 10), the general approach to multiparticle reaction theory (in Chapter 16), the specific treatment of three-particle scattering (in Chapter 17), and inverse scattering (in Chapter 20). The additions to Chapter 16 include an introduction to the two-Hilbert space approach, as well as a derivation of general scattering-rate formulas. Chapter 17 now contains a survey of various approaches to the solution of three-particle problems, as well as a discussion of the Efimov effect.

4,044 citations


"Inverse scattering series and seism..." refers background in this paper

  • ...Although 2D and 3D closed form complete integral equation solutions exist for the Schrödinger equation (see [6]), there is no analogous closed form complete multi-dimensional inverse solution for the acoustic or elastic wave equations....

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Journal ArticleDOI
TL;DR: In this article, a solution to an inverse scattering problem that arises in the application of holography to the determination of the three-dimensional structure of weakly scattering semi-transparent objects is presented.

1,285 citations

01 Jan 2002

1,164 citations


"Inverse scattering series and seism..." refers methods in this paper

  • ...For the simple horizontal reflector between two elastic halfspaces, that forward non-linear relationship is expressed by the Zoeppritz equations (see [47])....

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Journal ArticleDOI
TL;DR: In this paper, a method for the elimination of all surface-related multiples by means of a process that removes the influence of the surface reflectivity from the data is proposed.
Abstract: The major amount of multiple energy in seismic data is related to the large reflectivity of the surface. A method is proposed for the elimination of all surface-related multiples by means of a process that removes the influence of the surface reflectivity from the data. An important property of the proposed multiple elimination process is that no knowledge of the subsurface is required. On the other hand, the source signature and the surface reflectivity do need to be provided. As a consequence, the proposed process has been implemented adaptively, meaning that multiple elimination is designed as an inversion process where the source and surface reflectivity properties are estimated and where the multiple-free data equals the inversion residue. Results on simulated data and field data show that the proposed multiple elimination process should be considered as one of the key inversion steps in stepwise seismic inversion.

740 citations

Book
07 Aug 1995
TL;DR: In this paper, the authors present a mathematical model of a GA multimodal fitness function, genetic drift, GA with sharing, and repeat (parallel) GA uncertainty estimates evolutionary programming -a variant of GA.
Abstract: Part 1 Preliminary statistics: random variables random nunmbers probability probability distribution, distribution function and density function joint and marginal probability distributions mathematical expectation, moments, variances and covariances conditional probability Monte Carlo integration importance sampling stochastic processes Markov chains homogeneous, inhomogeneous, irreducible and aperiodic Markov chains the limiting probability. Part 2 Direct, linear and iterative-linear inverse methods: direct inversion methods model based inversion methods linear/linearized inverse methods iterative linear methods for quasi-linear problems Bayesian formulation solution using probabilistic formulation. Part 3 Monte Carlo methods: enumerative or grid search techniques Monte Carlo inversion hybrid Monte Carlo-linear inversion directed Monte Carlo methods. Part 4 Simulated annealing methods: metropolis algorithm heat bath algorithm simulated annealing without rejected moves fast simulated annealing very fast simulated reannealing mean field annealing using SA in geophysical inversion. Part 5 Genetic algorithms: a classical GA schemata and the fundamental theorem of genetic algorithms problems combining elements of SA into a new GA a mathematical model of a GA multimodal fitness functions, genetic drift, GA with sharing, and repeat (parallel) GA uncertainty estimates evolutionary programming - a variant of GA. Part 6 Geophysical applications of SA and GA: 1-D seismic waveform inversion pre-stack migration velocity estimation inversion of resistivity sounding data for 1-D earth models inversion of resistivity profiling data for 2-D earth models inversion of magnetotelluric sounding data for 1-D earth models stochastic reservoir modelling seismic deconvolution by mean field annealing and Hopfield network. Part 7 Uncertainty estimation: methods of numerical integration simulated annealing - the Gibbs' sampler genetic algorithm - the parallel Gibbs' sampler numerical examples.

710 citations