Location and size estimation of small rigid bodies using elastic far-fields

Abstract

We are concerned with the linearized, isotropic and homogeneous elastic scattering problem by (possibly many) small rigid obstacles of arbitrary Lipschitz regular shapes in 3D. Based on the Foldy-Lax approximation, valid under a sufficient condition on the number of the obstacles, the size and the minimum distance between them, we show that any of the two body waves, namely the pressure waves P or the shear waves S, is enough for solving the inverse problem of detecting these scatterers and estimating their sizes. Further, it is also shown that the shear-horizontal part SH or the shear vertical part SV of the shear waves S are also enough for the location detection and the size estimation. Under some extra assumption on the scatterers, as the convexity assumption, we derive finer size estimates as the radius of the largest ball contained in each scatterer and the one of the smallest ball containing it. The two estimates measure, respectively, the thickness and length of each obstacle.

The Foldy-Lax approximation of the scattered waves by many small bodies for the Lamé system

“…Indeed, C 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.mn-journal.com r For the bodies D j ∈ N m , j = m, we have the estimate (2.80) but for the bodies D j ∈ F m , we obtain the P r o o f o f L e m m a 3 . 2 We can factorize B as B = −(I + B n C)C −1 where C := Diag(C 1 ,C 2 , . .…”

“…Second, using formulas of the type (1.8) and (1.9), one can solve the inverse problems which consists of localizing the centers, z m , of the obstacles from the far-field measurements and also estimating their sizes from the computed capacitances C m , see [2], [15], [16], [19] for instance.…”