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.

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Inverse problems in elasticity

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.

https://iopscience.iop.org/article/10.1088/0266-5611/19/6/R01/pdf

Inverse scattering series and seismic exploration

Optical tomography has been widely investigated for biomedical imaging applications. In recent years optical tomography has been combined with digital holography and has been employed to produce high-quality images of phase objects such as cells. In this paper we describe a method for imaging 3D phase objects in a tomographic configuration implemented by training an artificial neural network to reproduce the complex amplitude of the experimentally measured scattered light. The network is designed such that the voxel values of the refractive index of the 3D object are the variables that are adapted during the training process. We demonstrate the method experimentally by forming images of the 3D refractive index distribution of Hela cells.

/pdf/learning-approach-to-optical-tomography-571t3nwj7t.pdf

Learning approach to optical tomography

Seismic data regularization, which spatially transforms irregularly sampled acquired data to regularly sampled data, is a long-standing problem in seismic data processing. Data regularization can be implemented using Fourier theory by using a method that estimates the spatial frequency content on an irregularly sampled grid. The data can then be reconstructed on any desired grid. Difficulties arise from the nonorthogonality of the global Fourier basis functions on an irregular grid, which results in the problem of “spectral leakage”: energy from one Fourier coefficient leaks onto others. We investigate the nonorthogonality of the Fourier basis on an irregularly sampled grid and propose a technique called “antileakage Fourier transform” to overcome the spectral leakage. In the antileakage Fourier transform, we first solve for the most energetic Fourier coefficient, assuming that it causes the most severe leakage. To attenuate all aliases and the leakage of this component onto other Fourier coefficients, the data component corresponding to this most energetic Fourier coefficient is subtracted from the original input on the irregular grid. We then use this new input to solve for the next Fourier coefficient, repeating the procedure until all Fourier coefficients are estimated. This procedure is equivalent to “reorthogonalizing” the global Fourier basis on an irregularly sampled grid. We demonstrate the robustness and effectiveness of this technique with successful applications to both synthetic and real data examples.

Antileakage Fourier transform for seismic data regularization

/pdf/marchenko-imagingmarchenko-imaging-2mlkjgjqc1.pdf

Marchenko imagingMarchenko imaging

We present a multidimensional multiple‐attenuation method that does not require any subsurface information for either surface or internal multiples. To derive these algorithms, we start with a scattering theory description of seismic data. We then introduce and develop several new theoretical concepts concerning the fundamental nature of and the relationship between forward and inverse scattering. These include (1) the idea that the inversion process can be viewed as a series of steps, each with a specific task; (2) the realization that the inverse‐scattering series provides an opportunity for separating out subseries with specific and useful tasks; (3) the recognition that these task‐specific subseries can have different (and more favorable) data requirements, convergence, and stability conditions than does the original complete inverse series; and, most importantly, (4) the development of the first method for physically interpreting the contribution that individual terms (and pieces of terms) in the inv...

An inverse-scattering series method for attenuating multiples in seismic reflection data

Nonlinear Inverse Scattering For Multiple Suppression: Application to Real Data. Part I

Seismic prospecting can be viewed as a process of extracting subsurface information from seismic measurements. Today’s interpretation and inversion technologies demand sophisticated pre-stack amplitudepreserving processing technologies that are effective in complex geologic environments. For example, the advent and increased use AVO require advanced multiple suppression technology in seismically challenging areas where conventional ‘tried and true’ methods are no longer adequate.

Inverse Scattering Internal Multiple Attenuation: Results From Complex Synthetic And Field Data Examples

The inverse-scattering series is a multidimensional inversion procedure that directly determines physical properties using only reflection data and a reference medium. This inversion process can be thought of as performing the following four tasks: (1) free surface multiple removal, (2) internal multiple removal, (3) location of reflectors in space, and (4) identification of medium property changes across reflectors. Since the entire process requires only reflection data and reference medium information, it is reasonable to assume that intermediate steps that are associated with achieving that objective would also be attainable with only the reference medium and reflection data. Subseries have been identified that exhibit this property for the tasks of free-surface and internal multiple attenuation (Weglein et al., 1997) and algorithms resulting from these subseries have been successfully applied to field data. Here we report our initial research efforts to address the third and fourth stages: imaging and parameter estimation. The objective is to be able to determine the precise reflector position in space and a map of earth properties, without the traditional need for the exact velocity model. Two distinct approaches are described: the first uses a series that computes the actual wavefield at depth (to be used for subsequent imaging and inversion) and the second is a series that directly computes earth physical properties. Neither approach requires or determines the actual velocity model or ever updates the reference medium. Numerical and analytical examples will be used to exemplify these concepts and to define the current state of understanding of these fundamentally new approaches to imaging and inverting seismic data. Open issues and future plans will be discussed.

Paulo M. Carvalho

Papers

An inverse-scattering series method for attenuating multiples in seismic reflection data

Inverse scattering series and seismic exploration

Nonlinear Inverse Scattering For Multiple Suppression: Application to Real Data. Part I

Inverse Scattering Internal Multiple Attenuation: Results From Complex Synthetic And Field Data Examples

Imaging And Inversion At Depth Without a Velocity Model: Theory, Concepts And Initial Evaluation