Properties of the spectral response cost functional employed in the inverse wave-scattering problem of the retrieval of the shear body wavespeed of the homogeneous solid occupying a half space bounded by a plane stress-free surface

Parseval's theorem leads to the finding that the minima of a least-squares spectral response cost functional K are at the same positions as the minima of a least squares signal response functional C. We describe the useful functional properties of K, in the context of a simple geophysical inverse problem pertaining to the retrieval of the shear wavespeed in a homogeneous underground, notably for enabling the location of its global minimum and dealing with the secondary minima. We show how the width of the search interval, the number and positions of the sensors, as well as the central frequency and bandwidth of the spectrum of the probe radiation, condition the aspect of the cost functional, particularly as regards the number of secondary minima and the depth of the trough associated with the global minimum. Finally, we evaluate the influence of prior uncertainties on the accuracy of the retrieval (via K) of the shear body wavespeed of the underground

Multifrequency mismatch functions for nonlinear parametric identifications

International audienceThis investigation is concerned with the ill-posed nature of (non-linear) inverse problems, in the frequency domain, for which the unknown parameters are determined, in iterative manner, by seeking the minima of a mismatch function quantifying the discrepancy between given data and the output of a numerical model. One might think that the identification can be done solely by the search of the minima of the mismatch function, but this problem is mathematically ill-posed and it is illusory to try to identify a satisfactory solution until the problems of stability (noisy data) and non-uniqueness (local minima) are resolved. This paper shows that a mismatch function resulting from summing up mono frequency mismatch functions corresponding to several different frequencies turns the inverse problem into a global optimization problem and overcomes the problem of the sensitivity of solutions to noisy data, without requiring a priori information on the probed medium

Acoustic identification of a poroelastic cylinder

We show how to cope with the acoustic identification of poroelastic materials when the specimen is in the form of a cylinder. We apply our formulation, based on the Biot model, approximated by the equivalent elastic solid model, to a long bone-like or borehole sample specimen probed by low frequency sound

Efficient shape reconstruction of non-circular tubes using broadband acoustic measurements

International audienceWe propose an algorithm for reconstructing the boundaries of a non-circular cylindrical tube from broadband acoustic measurements. This algorithm is based on the minimization of a cost function, which is the average over frequency of the absolute difference between the estimated and the measured scattered field. The estimated field is computed efficiently (very fast) using ICBA, an analytic method that provides an approximate solution of the forward problem. Numerical results show that our algorithm is robust and provides an accurate reconstruction without any explicit regularization

Multiparameter identification of a lossy fluid-like object from its transient acoustic response

International audienceA transient acoustic wave multiparameter inverse problem is studied. The aim is to simultaneously retrieve three constitutive parameters and one geometrical parameter of a lossy fluid-like cylinder by nonlinear full-wave inversion of synthetic data. Contrary to most publications treating this subject, it is assumed herein that the retrieval model is not identical to the data model, this being so because some of the parameters (priors) in the retrieval model are different (by the simple fact of being more or less unknown) from their true counterparts in the data model. This so-called model discordance, which is the usual situation in real-world inverse problems, is a source of errors for the retrieval of the other parameters. Moreover, it is a source of non-uniqueness and instability, a fact revealed by the employment of a properly-tuned Nelder-Mead downhill Simplex optimization algorithm, and which requires a second-order regularization for its resolution. Retrieval errors as a function of prior uncertainty are computed, and found to be large, for various model discordance scenarios involving one or two uncertain priors, for data with and without noise

Reconstruction temps-réel de la taille position orientation et forme d'un objet à partir de champs diffractés acoustiques

poste

Reconstruction of the three mechanical material constants of a lossy fluid-like cylinder from low-frequency acattered acousti

The inverse medium problem for a circular cylindrical domain is studied using low-frequency acoustic waves as the probe radiation. It is shown that to second order in $k_{0}a$ ($k_{0}$ the wavenumber in the host medium, $a$ the radius of the cylinder), only the first three terms (i.e., of orders 0, -1 and +1) in the partial wave representation of the scattered field are non-vanishing, and the material parameters enter into these terms in explicit manner. Moreover, the zeroth-order term contains only two of the unknown material constants (i.e., the real and imaginary parts of complex compressibility of the cylinder $\kappa_{1}$) whereas the $\pm 1$ order terms contain the other material constant (i.e., the density of the cylinder $\rho_{1}$). A method, relying on the knowledge of the totality of the far-zone scattered field and resulting in explicit expressions for $\rho_{1}$ and $\kappa_{1}$, is devised and shown to give highly-accurate estimates of these quantities even for frequencies such that $k_{0}a$ is as large as 0.1.Comment: submitted to C.R.Acad.Sc

Retrieval of the physical properties of an anelastic solid half space from seismic data

International audienceIn recent years, due to the rapid development of computation hard- and software, time domain full-wave inversion, which makes use of all the information in the seismograms without appealing to linearization, has become a plausible candidate for the retrieval of the physical parameters of the earth's substratum. Retrieving a large number of parameters (the usual case in a layered substratum comprising various materials, some of which are porous) at one time is a formidable task, so full-wave inversion often seeks to retrieve only a subset of these unknowns, with the remaining parameters, the priors, considered to be known and constant, or sequentially updated, during the inversion. A known prior means that its value has been obtained by other means (e.g., in situ or laboratory measurement) or simply guessed (hopefully, with a reasonable degree of confidence). The uncertainty of the value of the priors, like that of data noise, and the inadequacy of the theoretical/numerical model employed to mimick the seismic data during the inversion, is a source of retrieval error. We show, on the example of a homogeneous, isotropic, anelastic half-plane substratum configuration, characterized by five parameters: density, P and S wavespeeds and P and S quality factors, when a perfectly-adequate theoretical/numerical model is employed during the inversion and the data is free of noise, that the retrieval error can be very large for a given parameter, even when the prior uncertainty of another single parameter is very small. Furthermore, the employment of other load and response polarization data and/or multi-offset data, as well as other choices of the to-be-retrieved parameters, are shown, on specific examples, not to systematically improve(they may even reduce) the accuracy of the retrievals when the prior uncertainty is relatively-large. These findings, relative to the recovery, via an exact retrieval model processing noiseless data obtained in one of the simplest geophysical configurations, of a single parameter at a time with a single uncertain prior, raises the question of the confidence that can be placed in geophysical parameter retrievals: 1) when more than one parameters are retrieved at a time, and/or 2) when more than one prior are affected by uncertainties during a given inversion, and/or 3) when the model employed to mimick the data during the inversion is inadequate, 4) when the data is affected by noise or measurement errors, and 5) when the parameter retrieval is carried out in more realistic configurations