Localization of small obstacles from back-scattered data at limited incident angles with full-waveform inversion

Abstract

We investigate numerically the inverse problem of locating small circular obstacles in a homogeneous medium from multi-frequency back-scattered data limited to four angles of incidence. The main novelty of our paper is working with the position of the obstacles as parameter space in the frame work of full-waveform inversion (FWI) procedure. The computational cost of FWI is lowered by using a method based on single-layer potential. Reconstruction results are shown up to twenty-four obstacles, from initial guesses allowed to be far from the target. In experiments with six obstacles, we supplement the reconstruction with an analysis of the performance of the nonlinear conjugate gradient and quasi-Newton methods, in used with various line search algorithms.

Eigenvector models for solving the seismic inverse problem for the Helmholtz equation

“…Remark 1 (Multi-frequency algorithm). For the choice of frequency in Problem 5, applications commonly use a sequence of increasing frequencies during the iterative process (Bunks et al, 1995;Pratt & Worthington, 1990;Sirgue & Pratt, 2004;Brossier et al, 2009;Barucq et al, 2018;Faucher, 2017). Namely, one starts with a low frequency and minimize the functional for the fixed frequency content in the data.…”

“…The gradient of the cost function is computed using the first order adjoint-state method (Lions & Mitter, 1971;Chavent, 1974), which is standard in seismic application (Plessix, 2006). It avoids the formation of a dense Jacobian matrix and instead requires the resolution of an additional PDE, which is the adjoint of the forward PDE, with right-hand sides defined from the difference between the measurements and the simulations, see Plessix (2006); Faucher (2017); Barucq et al (2018Barucq et al ( , 2019b for more details.…”

“…Due to the large computational scale of the domain investigated, seismic experiments may have difficulties to incorporate second order (Hessian) information in the algorithm, and alternative techniques have been proposed, for example, in the work of Pratt et al (1998); Akcelik et al (2002); Choi et al (2008); Métivier et al (2013); Jun et al (2015). The quantitative (as opposed to qualitative) reconstruction methods based upon iterative minimization are naturally not restricted to seismic and we refer, among others, to Ammari et al (2015); Barucq et al (2018) and the references therein for additional applications using similar techniques.…”

Full Reciprocity-Gap Waveform Inversion in the frequency domain, enabling sparse-source acquisition

“…The method has its foundation in the body of work of [35], and is reviewed for seismic application in [40]. Application with complex fields is further described in [9,8]. The adjoint-state method for reciprocity-gap functional is briefly reviewed in Appendix B, see also [1].…”

An effective numerical strategy for retrieving all characteristic parameters of an elastic scatterer from its FFP measurements

“…More specifically, extensive efforts have been devoted during the past five decades to the determination of the shape of an unknown object from the knowledge of its corresponding FFP measurements and the nature of the object. Indeed, various computational procedures have been designed for this purpose (see, e.g., [8,9,10,11,12,13,14,15,16,17,18,19], and the references therein). Note that in spite of the absence of a rigorous proof of their convergence [20], regularized Newton-type methods have been among the primary candidates for solving inverse obstacle problems (IOP) (see, e.g., [9,21,22,23,24,25,26,27,28]).…”