×

zbMATH — the first resource for mathematics

Depth sounding: An illustration of some of the pitfalls of inverse scattering problems. (English) Zbl 1109.62367
Summary: The principal objective of this work is to show how various ”connections” between the estimator and the predictor affect the solution of an inverse scattering problem as it is formulated in the frequency domain. We show that when there is little or no connection, it is impossible to obtain a solution. The other extreme, i.e., identity of the estimator and predictor, enables solutions to be obtained, whatever the particular choices of the estimator or predictor, but these solutions are not trivial, as is written by D. Colton and R. Kress, Inverse acoustic and elecromagnetic scattering theory. 2 nd ed. (1998; Zbl 0893.35138), in that they are not unique. Moreover, we show that by a suitable change of external variables (e.g., frequency), one can lift the degeneracy and thereby spot the correct solution, which is unique. In this respect, the inverse crime turns out to be useful in that it enables one to devise methods for resolving the nonuniqueness issue of inverse problems. More generally, we show that successful inversion, in both the frequency and time domains, can be accomplished only when the discrepancy between the estimator and the predictor is small.

MSC:
62P35 Applications of statistics to physics
62P99 Applications of statistics
35R30 Inverse problems for PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Colton, D.; Kress, R.; Colton, D.; Kress, R., (), 289
[2] Wirgin, A., ()
[3] Bui, H.D., ()
[4] Tikhonov, A.; Arsénine, V., ()
[5] Engl, H.W.; Hanke, M.; Neubauer, A., ()
[6] Li, P.; Ramm, A.G., Numerical recovery of the layered medium from the surface data, J. comput. appl. math, 25, 267-276, (1989) · Zbl 0673.65091
[7] Ramm, A.G., Stability estimates in inverse scattering, Acta appl. math, 28, 1-42, (1992) · Zbl 0756.35116
[8] Ramm, A.G., Stability of the solution to 3D inverse scattering problems with fixed-energy data, Proc. ASME meeting on inverse problems in mechanics, AMD-186, 99-102, (1994)
[9] Cheney, M.; Isaacson, D.; Lassas, M., Optimal acoustic measurements, SIAM J. appl. math, 61, 1628-1647, (2001) · Zbl 1006.74055
[10] Sabatier, P.C., Past and future of inverse problems, J. math. phys, 41, 4082-4124, (2000) · Zbl 0982.34010
[11] Banks, H.T.; Bihari, K.L., Modelling and estimating uncertainty in parameter estimation, Inverse probs, 17, 95-111, (2001) · Zbl 1054.35121
[12] Buchanan, J.L.; Gilbert, R.P.; Wirgin, A.; Xu, Y., Identification by the intersecting canonical domain method of the size, shape and depth of a soft body of revolution located within an acoustic waveguide, Inverse probs, 16, 1709-1726, (2000) · Zbl 0968.35123
[13] E. Ogam and A. Wirgin, Shape and size retrieval of obstacles by nonlinear inversion of measured scattered field data in response to audible acoustic probe radiation, In Acoustical Imaging, Volume 23, (Edited by P. Tortoli et al.), Plenum, New York (to appear).
[14] Ogam, E.; Wirgin, A., (), 447-454, C2I 2001, Volume 2
[15] Ogam, E.; Scotti, T.; Wirgin, A., Non-ambiguous boundary identification of a cylindrical object by acoustic waves, C.R. acad. sci. Paris iib, 329, 61-66, (2001) · Zbl 1072.76058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.