Cédric Bellis

Acoustic inverse scattering using topological derivative of far-field measurements-based L2 cost functionals

Originally formulated in the context of topology optimization, the concept of topological derivative has also proved effective as a qualitative inversion tool for a wave-based identification of finite-sized objects. This approach remains, however, largely based on a heuristic interpretation of the topological derivative, whereas most other qualitative approaches to inverse scattering are backed by a mathematical justification. As an effort toward bridging this gap, this study focuses on a topological derivative approach applied to the L2-norm of the misfit between far-field measurements. Either an inhomogeneous medium or a finite number of point-like scatterers are considered, using either the Born approximation or a full-scattering model. Topological derivative-based imaging functionals are analyzed using a suitable factorization of the far-field operator, for each of the considered cases, in order to characterize their behavior and assess their ability to reconstruct the unknown scatterer(s). Results include the justification of the usual sign heuristic underpinning the method for (i) the Born approximation and (ii) full-scattering models limited to moderately strong scatterers. Semi-analytical and numerical examples are presented. Within the chosen framework, the topological derivative approach is finally discussed and compared to other well-known qualitative methods.

Reconstruction of constitutive parameters in isotropic linear elasticity from noisy full-field measurements

Within the framework of linear elasticity we assume the availability of internal full-field measurements of the continuum deformations of a non-homogeneous isotropic solid. The aim is the quantitative reconstruction of the associated moduli. A simple gradient system for the sought constitutive parameters is derived algebraically from the momentum equation, whose coefficients are expressed in terms of the measured displacement fields and their spatial derivatives. Direct integration of this system is discussed to finally demonstrate the inexpediency of such an approach when dealing with noisy data. Upon using polluted measurements, an alternative variational formulation is deployed to invert for the physical parameters. Analysis of this latter inversion procedure provides existence and uniqueness results while the reconstruction stability with respect to the measurements is investigated. As the inversion procedure requires differentiating the measurements twice, a numerical differentiation scheme based on an ad hoc regularization then allows an optimally stable reconstruction of the sought moduli. Numerical results are included to illustrate and assess the performance of the overall approach.

Sensitivity of seismic measurements to frequency‐dependent attenuation and upper mantle structure: An initial approach

This study addresses the sensitivity of seismic attenuation measurements to dissipative mechanisms and structure in the Earth’s upper mantle. The Andrade anelastic model fits experimental attenuation data with a mild power law frequency dependence and can be scaled from laboratory to Earth conditions. We incorporate this anelastic model into 400 km 1-D thermal profiles of the upper mantle. These continuous-spectrum models are approximated by multiple relaxation mechanisms that are implemented within a finite-difference scheme to perform wave propagation simulations in 1-D domains. In two sets of numerical experiments, we evaluate the measurable signature of the intrinsic attenuation structure. The two sets are defined by thermal profiles with added step functions of temperature, varying in (i) amplitude and depth or (ii) amplitude and sharpness. The corresponding synthetic data are processed using both the conventional t∗ approach, i.e., a linear regression of the displacement frequency spectrum, and an alternative nonlinear fit to identify the integrated value of attenuation and its frequency dependence. The measured sensitivity patterns are analyzed to assess the effects of the anelastic model and its spatial distribution on seismic data (in the absence of scattering effects). We have two straightforward results: (1) the frequency dependence power law is recoverable from the measurements; (2) t∗ is sensitive to both the depth and the amplitude of the step, and it is insensitive to the sharpness of the step, in the 0.25 to 2 Hz band. There is much potential for gaining information about the upper mantle thermodynamic state from careful interpretation of attenuation.

Dynamical one-dimensional models of passive piezoelectric sensors

This study concerns the mathematical modeling of anisotropic and transversely inhomogeneous slender piezoelectric bars. Such rod-like structures are employed as passive sensors aimed at measuring the displacement field on the boundary of an underlying elastic medium excited by an external source. Based on the coupled three-dimensional dynamical equations of piezoelectricity in the quasi-electrostatic approximation, a set of limit problems is derived using formal asymptotic expansions of the electric potential and elastic displacement fields. The nature of these problems depends strongly on the choice of boundary conditions, therefore, an appropriate set of constrains is introduced in order to derive one-dimensional models that are relevant to the measurement of a displacement field imposed at one end of the bar. The structure of the first-order electric and displacement fields as well as the associated coupled limit equations are determined. Moreover, the properties of the homogenized material parameters entering these equations are investigated in various configurations. The obtained one-dimensional models of piezoelectric sensors are analyzed, and it is finally shown how they enable the identification of the boundary displacement associated with the probed elastic medium.

A non-iterative sampling approach using noise subspace projection for EIT

This study concerns the problem of the reconstruction of inclusions embedded in a conductive medium in the context of electrical impedance tomography (EIT), which is investigated within the framework of a non-iterative sampling approach. This type of identification strategy relies on the construction of a special indicator function that takes, roughly speaking, small values outside the inclusion and large values inside. Such a function is constructed in this paper from the projection of a fundamental singular solution onto the space spanned by the singular vectors associated with some of the smallest singular values of the data-to-measurement operator. The behavior of the novel indicator function is analyzed. For a subsequent implementation in a discrete setting, the quality of classical finite-dimensional approximations of the measurement operator is discussed. The robustness of this approach is also analyzed when only noisy spectral information is available. Finally, this identification method is implemented numerically and experimentally, and its efficiency is discussed on a set of, partly experimental, examples.

A full-field image conversion method for the inverse conductivity problem with internal measurements

This article investigates a Fourier-based algorithm for computing heterogeneous material parameter distributions from internal measurements of physical fields. Within the framework of the periodic scalar conductivity model, a pair of dual Lippmann–Schwinger integral equations is derived for the sought constitutive parameters based on full intensity or current density field measurements. A numerical method based on the fast Fourier transform and fixed-point iterations is proposed. Convergence, stability and approximation quality of the method are analysed. For materials with small contrast, a first-order Born-like approximation is also obtained. Overall, the proposed reconstruction approach enables a direct conversion of full-field measurement images, possibly noisy, into maps of material conductivity. A set of numerical results is presented to illustrate the performance of the method.