Inverse Scattering

Fig11-TSWaveform tomography and in particular inverse scattering are essential to a broad spectrum of science and technology disciplines including geophysics, oceanography, medical imaging, energy production, and non-destructive material testing. In general the relationship between the wavefield scattered by an anomaly and its geometry (or physical characteristics) is nonlinear, which invites two overt solution strategies: i) linearization via e.g. Born approximation, or ii) pursuit of the nonlinear minimization approach. Over the past two decades, however, a number of sampling methods have emerged that both consider the nonlinear nature of the inverse problem and dispense with iterations. Commonly, these techniques deploy an indicator functional that varies with coordinates of the trial i.e. sampling point, and projects observations of the scattered field onto a functional space reflecting the baseline wave motion in a reference (“anomaly-free”) medium. Such indicator functional, designed to reach extreme values when the sampling point strikes the anomaly, accordingly provides a tomogram via its (thresholded) spatial distribution. Examples of such imaging paradigm include the Linear Sampling Method and the Method of Topological Sensitivity described below. In situations involving distinct anomalies, a tractable solution to the waveform tomography problem can also be sought via the boundary integral equation (BIE) framework and the concept of shape and material sensitivities. Our work also entails studies of the interior transmission problem which, despite its central role in the theory of inverse scattering, is not covered by the existing theories in partial differential equations.

Method of Topological Sensitivity

Since its birth in the context of shape optimization, the idea of Topological Sensitivity (TS) has been generalized and applied to tackle a variety of inverse scattering problems. Our research in this area is largely focused on the inverse problems in acoustics [27345361] and elastodynamics [171922293335374851565758]. In the imaging approach the TS, which quantifies the perturbation of a given cost functional due to nucleation of an infinitesimal defect in the (reference) background medium, is used as an effective anomaly indicator through an assembly of sampling points where it attains extreme negative values. Typically, the TS formulas are amenable to explicit representation in terms of the wavefields computed for the reference domain, which is the root of computational efficiency of this class of waveform tomography solutions. Our recent work has involved both experimental verification [56] and theoretical justification [61] of the TS approach to inverse scattering.

TS reconstruction of a crescent-shaped anomaly from sparse cross-hole seismic measurements [35]. Arranged clockwise: images obtained using low, intermediate, high, and multi- frequency illumination.

Linear Sampling Method

The Linear Sampling Method is a non-iterative technique for reconstructing obstacles from the scattered field data. This method entails the solution of a linear Fredholm equation of the first kind. Here, the linearity of the equation does not stem from an approximation of the physical conditions (e.g. the weak scatterer hypothesis); rather, it is the result of an equivalence between the nonlinear inverse scattering problem and the linear integral equation. Theory shows that the norm of a solution to the germane (Fredholm) equation becomes unbounded for sampling points outside of the support of a scatterer, which de facto yields a characteristic function of the anomaly. Notable advantages of this approach to inverse scattering include computational efficiency and insensitivity to the boundary conditions present on the surface of a scatterer. Our work in this area [182631324253] is focused on extending the Linear Sampling Method to inverse problems involving elastic waves, near-field sensory data, heterogeneous reference domains, variety of anomalies (inclusions, fractures, interfaces), and more recently unknown background media.

Three-dimensional LSM reconstruction of a tunnel in a layered half-space [31] from surface seismic measurements.

Obstacle sensing by shape-material sensitivity framework

This facet of our research [151636] focuses on the elastic-wave reconstruction and characterization of discrete heterogeneities (cavities, inclusions) in an otherwise homogeneous solid from the limited-aperture waveform observations taken on its surface. On adopting the boundary integral equation (BIE) framework as basis for describing the forward scattering problem, the inverse query is cast as a minimization task involving sensory data and their simulations for a trial anomaly that is defined through its boundary, elastic moduli, and mass density. For an optimal performance of the gradient-based search methods suited to solve the problem, explicit expressions for the shape (i.e. boundary) and material sensitivities of the misfit functional are obtained via an adjoint field approach and direct differentiation of the governing BIEs. Making use of the message-passing interface, the featured sensitivity formulas are implemented in a data-parallel code and integrated into a nonlinear optimization framework based on the direct BIE method and an augmented Lagrangian — whose inequality constraints are employed to avoid solving forward scattering problems for physically inadmissible (or overly distorted) trial inclusion configurations. Numerical results for the reconstruction of an ellipsoidal defect in a semi-infinite solid show the effectiveness of the proposed shape-material sensitivity formulation, which constitutes an essential component of the defect identification algorithm.


Shape sensitivity-driven evolution of a trial subterranean void in the boundary-only imaging process [15]: iterations k=0 and k=26 denote respectively the initial “guess” and final shape, the latter overlapping with the “true” defect. The ground i.e. testing surface is indicated in dark blue.

Interior transmission problem

This facet of our research deals with the theory of the interior transmission problem (ITP) for heterogeneous and anisotropic (visco-) elastic solids [4050]. The subject is central to the so-called qualitative methods (e.g. Linear Sampling Method) for inverse scattering involving penetrable obstacles. Although simply stated as a coupled pair of elastodynamic wave equations, the ITP for elastic bodies is neither self-adjoint nor elliptic. The aim of this work is to provide a systematic treatment of the ITP for elastic bodies, considering a broad range of material-contrast configurations. In particular, we investigate questions of the existence and uniqueness of a solution to the ITP, the discreteness of its eigenvalues, and the existence of such eigenvalue spectrum. Necessitated by the breadth of material configurations studied, the relevant claims are established via a suite of variational formulations, each customized to meet the needs of a particular subclass of eigenvalue problems. For instance the analysis shows that in the case of a viscoelastic scatterer embedded in an elastic solid, the ITP eigenvalue spectrum is empty.

Elastodynamic ITP configuration [40]: scatterer composed of inclusions D∗ and cavities Do (left) and scatterer support, D, occupied by the background material (right)

Elastodynamic ITP configuration [40]: scatterer composed of inclusions D∗ and cavities Do (left) and scatterer support, D, occupied by the background material (right)

Experimental Verification

We are particularly interested in validating our theories by way of carefully designed laboratory experiments. In this vein the method of topological sensitivity (TS) for solving inverse scattering problems, previously supported by a multitude of numerical simulations, is put to test experimentally [56] with the focus on 2D obstacle reconstruction in a thin aluminum plate using elastic waves. To this end, in-plane measurements of transient elastodynamic waveforms along the edges of the plate are captured in a non-contact fashion by a 3D Scanning Laser Doppler Vibrometer (SLDV). Using an elastodynamic (time-domain) finite element model as a computational platform, the TS reconstruction maps are obtained and analyzed under varying experimental conditions. The results show significant agreement between the defect geometry and its reconstruction, thus demonstrating the utility of the TS approach as an efficient and robust solution tool for this class of inverse problems. For completeness, the experimental investigation includes a set of parametric studies geared toward exposing the effect of key problem parameters on the quality of obstacle reconstruction such as the (dominant) excitation frequency, the source aperture, and duration of the temporal records. On the analytical front, it is shown that the use of a suitable temporal windowing function in specifying the L2 cost functional (that underpins the TS formulation) is essential from both theoretical and computational points of view.

Elastodynamic scattering experiment [56] featuring a 3D Scanning Laser Doppler Vibrometer (top), and experimental TS maps of an aluminum plate containing a hole and a slit (bottom).

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