CiteULike: norris's inverse

The linear sampling method in a waveguide: a modal formulation

L Bourgeois — 2008-06-10T14:50:10-00:00

Inverse Problems, Vol. 24, No. 1. (2008), 015018.

This paper concerns the linear sampling method used to retrieve obstacles in a 2D or 3D acoustic waveguide. The classical mathematical results concerning the identifiability of the obstacle and the justification of the inverse method are established for this particular geometry. Our main concern is to derive a modal formulation of the linear sampling method that is well adapted to the waveguide configuration. In particular, thanks to such formulation, we highlight the fact that finding some obstacles from remote scattering data is more delicate in a waveguide than in free space. Indeed, the presence of evanescent modes increases the ill posedness of the inverse problem. However, we show that the numerical reconstruction of obstacles by using the far field is feasible, even by using a few incident waves.

Analytic solutions to the acoustic source reconstruction problem

C Maury — 2007-12-22T18:43:49-00:00

Passive seismic imaging in the presence of white noise sources

Deyan Draganov — 2007-07-18T17:16:34-00:00

The Leading Edge, Vol. 23, No. 9. (1 September 2004), pp. 889-892.

Passive seismic imaging is based on the relation between the reflection and the transmission responses of the subsurface. Let one have the transmission responses measured at surface points A and B of a 3D inhomogeneous medium in the presence of white noise sources in the subsurface. When these transmission responses are cross-correlated, one obtains the reflection response of the same medium as if measured at point A in the presence of an impulsive source at point B. The quality of the simulated reflection response strongly depends on the whiteness and the distribution of the noise sources. Reflectors present below the sources cause the appearance of some ghost events. Random distribution of the noise sources will, however, weaken these ghost events. 10.1190/1.1803498

Gaussian beams and Legendre polynomials as invariants of the time reversal operator for a large rigid cylinder

Alexandre Aubry — 2007-03-20T13:01:01-00:00

The Journal of the Acoustical Society of America, Vol. 120, No. 5. (2006), pp. 2746-2754.

The DORT method (French acronym for decomposition of the time reversal operator) is an active remote sensing technique using an array of antennas for the detection and localization of scatterers. This method is based on the singular value decomposition of the interelement response matrix. In this paper an analytical expression of the singular vectors due to the reflection from a large rigid cylinder is provided. Depending on the array aperture, two asymptotic regimes are described. It is shown that the singular vectors correspond to Hermite-Gaussian modes for large apertures and Legendre polynomials for small ones. Using perturbation theory, the corresponding singular values are deduced. Theoretical predictions are in good agreement with experimental results. ©2006 Acoustical Society of America

RECONSTRUCTION OF STEPLIKE POTENTIALS

PE Sacks — 2007-02-27T17:44:33-00:00

Wave Motion (1993)

Inverse Problems in Geophysics

R Snieder — 2007-02-25T19:41:26-00:00

(2000), pp. 1-73.

The role of nonlinearity in inverse problems

R Snieder — 2007-02-25T15:33:11-00:00

Inverse Problems, Vol. 14, No. 3. (1998), pp. 387-404.

Inverse Scattering for Discrete Transmission-Line Models

AM Bruckstein — 2007-02-25T15:21:37-00:00

SIAM Review, Vol. 29, No. 3. (1987), pp. 359-389.

Convergence of Numerical Inversion Methods for Discontinuous Impedance Profiles

KP Bube — 2007-02-23T22:49:39-00:00

SIAM Journal on Numerical Analysis, Vol. 22, No. 5. (1985), pp. 924-946.

Determination of Vocal-Tract Shape from Impulse Response at the Lips

MM Sondhi — 2007-02-22T15:48:34-00:00

The Journal of the Acoustical Society of America, Vol. 49 (1971)

Discrete inverse methods for elastic waves in layered media

James Berryman — 2007-02-19T00:31:38-00:00

Geophysics, Vol. 45, No. 2. (1980), pp. 213-233.

The seismic inverse problem for waves at normal incidence on horizontally layered media is discussed. The emphasis is theoretical rather than practical, but some long-standing questions concerning the general applicability of the often taught Goupillaud inverse method are answered. The main purpose is to demonstrate in detail the equivalence between the Goupillaud method of inversion for the wave equation and the Marchenko integral equation (inverse scattering) method for the Schroedinger equation. We show that the very simple method of solution due to Goupillaud for a specialized model (layers of equal traveltime) actually has a much wider significance. If seismic data are smoothed before sampling using a type of antialiasing filter, the Goupillaud method gives a valid approximate inversion for models with arbitrary layer thicknesses (or continuous impedance variation) when the "reflection coefficients" are appropriately reinterpreted. In all, three inverse methods are considered: (1) the Goupillaud method for the wave equation and both (2) continuous and (3) discrete inverse scattering methods for the Schroedinger equation. A computationally fast algorithm for solving the inverse scattering formulas is deduced from the equivalent Goupillaud method. By comparing the continuous and discrete formalisms in the continuum limit, a preferred form is found within the class of symmetric tridiagonal discretizations of the Schroedinger equation. For the elastic wave inverse problem, two cases are distinguished: (1) If the impedance is continuous, we show that both the Goupillaud method and the discrete inverse scattering method converge to the impedance when the equal-traveltime layer thickness goes to zero; and (2) if the impedance has a finite number of discontinuities, we show that the inverse scattering method assigns the arithmetic average across the discontinuity at the point of discontinuity, while the Goupillaud method assigns the value of the right-hand (spatially deeper) limit. Thus, in the continuum limit, both methods will reconstruct the same impedance except (possibly) for the values at a finite number of jump points in any finite span of traveltime.

Numerical Methods for Reflection Inverse Problems: Convergence and Nonimpulsive Sources

KP Bube — 2007-02-18T00:32:03-00:00

SIAM Journal on Numerical Analysis, Vol. 23, No. 2. (1986), pp. 227-258.

Characterization of the null space of a generally anisotropic medium in linearized cross-well tomography

Kennethp Bube — 2007-02-18T00:25:13-00:00

Geophysical Journal International, Vol. 133, No. 1. (1998), pp. 65-84.

Anisotropic traveltime tomography can potentially determine many useful rock properties, such as crack density or pore shape, that cannot be found from isotropic methods. Similar to isotropic cross-well transmission tomography, many features of an anisotropic model are poorly resolved owing to limited ray-path coverage; the lateral smearing of isotropic tomography occurs in each elastic parameter in the anisotropic problem. Unlike the isotropic problem, however, there is additional indeterminacy in the solution of the anisotropic tomography problem because of ambiguity amongst the several elastic parameters needed to describe anisotropic media. We investigate the nature of this indeterminacy by studying the null space for linearized tomography, that is the class of model perturbations of a background medium which, to first order, cause no perturbation at all in the cross-well transmission traveltimes. Such model perturbations cannot be determined from the traveltime perturbations; describing these null-space model perturbations gives insight into the indeterminacy in the anisotropic problem. Complementary to computational approaches towards identifying the null space for discrete formulations of tomography, we study a continuum formulation. As expected, the anisotropic null space is larger than the isotropic null space owing to the ambiguity amongst the elastic parameters. We identify three categories of model perturbations in the anisotropic null space. The first category consists of model perturbations for which the perturbation in each of the individual elastic parameters is itself in the isotropic null space. Elements in the second category are anisotropic versions of the most well-known isotropic null-space elements: perturbations which are polynomials in the depth variable with coefficients which are functions of the horizontal variable satisfying certain linear integral constraints; unlike the isotropic problem, the integral constraints in the anisotropic problem couple together the several elastic parameters. The third category consists of model perturbations satisfying zero boundary conditions in the wells for which a specific linear combination of integrals and derivatives of the several elastic parameters is in the isotropic null space. In particular, there are model perturbations in this third category which represent anomalies that are completely contained in the interior of the model and yet are in the null space; this behaviour is in marked contrast to the isotropic problem. These categories are sufficient to describe the anisotropic null space completely. We demonstrate that every model perturbation in the null space is the sum of an element in the first category (indicating an indeterminacy of the same nature as in the isotropic problem in each of the elastic parameters separately) and an element in the third category (indicating an ambiguity amongst the parameters). The second category gives a rich family of examples of sums of null-space elements in the first and third categories, and thereby gives a sense of just how large the anisotropic null space is. Moreover, we show that the traveltime perturbations determine only a small number of features of an elastic perturbation which distinguish between the several elastic parameters. We identify these features precisely: they are functions of depth representing horizontal averages of combinations of the elastic parameters and their derivatives. Elastic parameters influencing the horizontal velocity appear more prominently in these features than those influencing the vertical velocity. All other features of the anisotropic model are ambiguous amongst the elastic parameters: except for these features, it is completely impossible from the traveltime perturbations alone to determine which elastic parameter (or which combination of the elastic parameters) gives rise to given traveltime perturbations.

The null space of a generally anisotropic medium in linearized surface reflection tomography

KP Bube — 2007-02-18T00:10:34-00:00

Geophysical Journal International, Vol. 139, No. 1. (1999), pp. 9-50.

Time-reversed acoustics

M Fink — 2007-01-29T15:52:57-00:00

(2000), pp. 1933-1994.

The objective of this paper is to show that time reversal invariance can be exploited in acoustics to create a variety of useful instruments as well as elegant experiments in pure physics.Section 1 is devoted to the description of time reversal cavities and mirrors together with a comparison between time reversal and phase conjugation. To illustrate these concepts, several experiments conducted in multiply scattering media, waveguides and chaotic cavities are presented in section 2. Applications of time reversal mirrors (TRMs) in hydrodynamics are then presented in section 3. Section 4 is devoted to the application of TRMs in pulse echo detection. A complete theory of the iterative time reversal mode is presented. It will be explained how this technique allows for focusing on different targets in a multi-target medium. Another application of pulse echo TRMs is presented in this section: how to achieve resonance in an elastic target? Section 5 explores the medical applications of TRMs in ultrasonic imaging, lithotripsy and hyperthermia and section 6 shows the promising applications of TRMs in nondestructive testing of solid samples.

A survey on sampling and probe methods for inverse problems

Roland Potthast — 2007-01-29T15:49:37-00:00

Inverse Problems, Vol. 22, No. 2. (2006), pp. R1-R47.

The goal of the review is to provide a state-of-the-art survey on sampling and probe methods for the solution of inverse problems. Further, a configuration approach to some of the problems will be presented. We study the concepts and analytical results for several recent sampling and probe methods. We will give an introduction to the basic idea behind each method using a simple model problem and then provide some general formulation in terms of particular configurations to study the range of the arguments which are used to set up the method. This provides a novel way to present the algorithms and the analytic arguments for their investigation in a variety of different settings. In detail we investigate the probe method (Ikehata), linear sampling method (Colton–Kirsch) and the factorization method (Kirsch), singular sources method (Potthast), no response test (Luke–Potthast), range test (Kusiak, Potthast and Sylvester) and the enclosure method (Ikehata) for the solution of inverse acoustic and electromagnetic scattering problems. The main ideas, approaches and convergence results of the methods are presented. For each method, we provide a historical survey about applications to different situations.

The One-Dimensional Inverse Problem of Reflection Seismology

Kenneth Bube — 2006-10-28T12:59:12-00:00

SIAM Review, Vol. 25, No. 4. (1983), pp. 497-559.

An impulsive normal traction is applied uniformly over the surface of a perfectly stratified plane layered earth. The ensuing particle velocity at the surface is assumed to be measured. The problem of calculating the subsurface characteristic impedance from knowledge of the input pulse and the measured data is the one-dimensional inverse problem of reflection seismology. The problem is set up mathematically as an inverse problem for a first order 2 × 2 hyperbolic system. The role of propagation of singularities is explained and the relation between the solution of the inverse problem and the Cholesky factorization of certain matrices constructed from the data is elaborated for both the continuum problem and the related discrete problem. It is found that recursive numerical techniques such as the Levinson algorithm and certain other fast methods for Cholesky factorization are intimately related to explicit finite-difference schemes for solving hyperbolic systems. The order of approximation between the solutions of the discrete schemes and of the differential equations is discussed. Finally the favored discrete downward continuation (DC) algorithm is demonstrated by means of numerical examples, which are illustrated graphically.

Multiple scattering: The key to unravel the subwavelength world from the far-field pattern of a scattered wave

F Simonetti — 2006-10-09T16:24:09-00:00

Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), Vol. 73, No. 3. (2006)

For more than a century the possibility of imaging the structure of a medium with diffracting wave fields has been limited by the tradeoff between resolution and imaging depth. While long wavelengths can penetrate deep into a medium, the resolution limit precludes the possibility of observing subwavelength structures. Near-field microscopy has recently demonstrated that the resolution limit can be overcome by bringing a probing sensor within one wavelength distance from the surface to be imaged. This paper extends the scope of near-field microscopy to the reconstruction of subwavelength structures from measurements performed in the far-field. It is shown that the distortion undergone by a wave field as it travels through an inhomogeneous medium and the subsequent generation of local evanescent fields encode subwavelength information in the far-field due to multiple scattering within the medium. This argument is proved theoretically and supported by a limited view experiment performed with elastic waves in which an image with a resolution better than a third of the wavelength is achieved.

Microlocal analysis of seismic inverse scattering in anisotropic elastic media

Christiaan Stolk — 2006-06-19T14:52:00-00:00

Communications on Pure and Applied Mathematics, Vol. 55, No. 3. (2002), pp. 261-301.

Seismic data is modeled in the high-frequency approximation, using the techniques of microlocal analysis. We consider general, anisotropic elastic media. Our methods are designed to allow for the formation of caustics. The data is modeled in two ways. First, we give a microlocal treatment of the Kirchhoff approximation, where the medium is assumed to be piecewise smooth, and reflection and transmission occur at interfaces. Second, we give a refined view on the Born approximation based upon a linearization of the scattering process in the medium parameters around a smooth background medium. The joint formulation of Born and Kirchhoff scattering allows us to take into account general scatterers as well as the nonlinear dependence of reflection coefficients on the medium parameters. The latter allows the treatment of scattering up to grazing angles.The outcome of the analysis is a characterization of the singular part of seismic data. We obtain a set of pseudodifferential operators that annihilate the data. In the process we construct a Fourier integral operator and a reflectivity function such that the data can be represented by this operator acting on the reflectivity function. In our construction this Fourier integral operator becomes invertible. We give the conditions for invertibility for general acquisition geometry. The result is also of interest for inverse scattering in acoustic media. © 2002 John Wiley & Sons, Inc.

Inverse potential scattering in duct acoustics

Barbara Forbes — 2006-06-15T17:18:17-00:00

The Journal of the Acoustical Society of America, Vol. 119, No. 1. (2006), pp. 65-73.

The inverse problem of the noninvasive measurement of the shape of an acoustical duct in which one-dimensional wave propagation can be assumed is examined within the theoretical framework of the governing Klein–Gordon equation. Previous deterministic methods developed over the last 40 years have all required direct measurement of the reflectance or input impedance but now, by application of the methods of inverse quantum scattering to the acoustical system, it is shown that the reflectance can be algorithmically derived from the radiated wave. The potential and area functions of the duct can subsequently be reconstructed. The results are discussed with particular reference to acoustic pulse reflectometry. ©2006 Acoustical Society of America