Imaging of buried objects from multi-frequency experimental data using a globally convergent inversion method

Abstract This paper is concerned with the numerical solution to a three-dimensional coefficient inverse problem for buried objects with multi-frequency experimental data. The measured data, which are associated with a single direction of an incident plane wave, are backscatter data for targets buried in a sandbox. These raw scattering data were collected using a microwave scattering facility at the University of North Carolina at Charlotte. We develop a data preprocessing procedure and exploit a newly developed globally convergent inversion method for solving the inverse problem with these preprocessed data. It is shown that dielectric constants of the buried targets as well as their locations are reconstructed with a very good accuracy. We also prove a new analytical result which rigorously justifies an important step of the so-called “data propagation” procedure.

[1]  Guy Chavent,et al.  Nonlinear Least Squares for Inverse Problems , 2010 .

[2]  Stanley Osher,et al.  A survey on level set methods for inverse problems and optimal design , 2005, European Journal of Applied Mathematics.

[3]  R. Novikov,et al.  An iterative approach to non-overdetermined inverse scattering at fixed energy , 2015 .

[4]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[5]  Armin Lechleiter,et al.  A trigonometric Galerkin method for volume integral equations arising in TM grating scattering , 2012, Adv. Comput. Math..

[6]  David Colton,et al.  Qualitative Methods in Inverse Scattering Theory , 1997 .

[7]  K. S. Kölbig.  Table errata: Table of integrals, series, and products by I. S. Gradshteĭn and I. M. Ryzhik [Academic Press, Boston, 1994; MR1243179 (94g:00008)] , 1995 .

[8]  Larisa Beilina,et al.  Domain decomposition finite element/finite difference approach for the Maxwell's system in time domain , 2011 .

[9]  Jun Zou,et al.  Locating Multiple Multiscale Acoustic Scatterers , 2014, Multiscale Model. Simul..

[10]  H. Bateman,et al.  Tables of Integral Transforms [Volumes I & II] , 1954 .

[11]  Frédéric Hecht,et al.  New development in freefem++ , 2012, J. Num. Math..

[12]  H. Ammari,et al.  Reconstruction of Small Inhomogeneities from Boundary Measurements , 2005 .

[13]  Fioralba Cakoni,et al.  Qualitative Methods in Inverse Scattering Theory: An Introduction , 2005 .

[14]  Lea Fleischer,et al.  Regularization of Inverse Problems , 1996 .

[15]  A. V. Goncharsky,et al.  Supercomputer technologies in inverse problems of ultrasound tomography , 2013 .

[16]  V. Vladimirov.  Generalized functions in mathematical physics , 1979 .

[17]  Michael V. Klibanov,et al.  A new approximate mathematical model for global convergence for a coefficient inverse problem with backscattering data , 2012 .

[18]  A. Bakushinsky,et al.  Iterative Methods for Approximate Solution of Inverse Problems , 2005 .

[19]  Lam H. Nguyen,et al.  Quantitative Image Recovery From Measured Blind Backscattered Data Using a Globally Convergent Inverse Method , 2013, IEEE Transactions on Geoscience and Remote Sensing.

[20]  Abdigany D. Satybaev,et al.  Direct Methods of Solving Multidimensional Inverse Hyperbolic Problems , 2004 .

[21]  B. Hecht,et al.  Principles of nano-optics , 2006 .

[22]  Xiaodong Liu,et al.  Recovering multiscale buried anomalies in a two-layered medium , 2015 .

[23]  Michael V. Klibanov,et al.  Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems , 2012 .

[24]  Michael V. Klibanov,et al.  Picosecond scale experimental verification of a globally convergent algorithm for a coefficient inverse problem , 2010 .

[25]  L. Debnath.  Tables of Integral Transforms , 2012 .

[26]  Andreas Kirsch,et al.  Characterization of the shape of a scattering obstacle using the spectral data of the far field operator , 1998 .

[27]  Alexey Agaltsov,et al.  Riemann–Hilbert problem approach for two-dimensional flow inverse scatteringa) , 2014 .

[28]  Hui Liu,et al.  Numerical solution of a coefficient inverse problem with multi-frequency experimental raw data by a globally convergent algorithm , 2016, J. Comput. Phys..

[29]  Michael V. Klibanov,et al.  Reconstruction of the Refractive Index from Experimental Backscattering Data Using a Globally Convergent Inverse Method , 2013, SIAM J. Sci. Comput..

[30]  L. Beilina.  Energy estimates and numerical verification of the stabilized Domain Decomposition Finite Element/Finite Difference approach for time-dependent Maxwell’s system , 2013 .

[31]  Kerstin Vogler,et al.  Table Of Integrals Series And Products , 2016 .

[32]  Lam H. Nguyen,et al.  Blind backscattering experimental data collected in the field and an approximately globally convergent inverse algorithm , 2012 .

[33]  Michael V. Klibanov,et al.  A globally convergent numerical method for a 3D coefficient inverse problem with a single measurement of multi-frequency data , 2016, 1612.04014.

[34]  A. Tikhonov,et al.  Numerical Methods for the Solution of Ill-Posed Problems , 1995 .

[35]  J. Scales,et al.  Global optimization methods for multimodal inverse problems , 1992 .

[36]  Michael V. Klibanov,et al.  Single measurement experimental data for an inverse medium problem inverted by a multi-frequency globally convergent numerical method , 2016, 1610.08018.

[37]  A. Kirsch,et al.  A simple method for solving inverse scattering problems in the resonance region , 1996 .

[38]  Karl K. Sabelfeld,et al.  Numerical solution of the multidimensional Gelfand–Levitan equation , 2015 .