0018-926X (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAP.2015.2446995, IEEE Transactions on Antennas and Propagation 1 Abstract—An electromagnetic inverse scattering approach for imaging of shallow subsurface objects is reported. It extends to multifrequency processing an efficient method previously developed for single frequency imaging. The considered approach is an iterative procedure based on an inexact-Newton method developed in L p Banach spaces, which exhibits effective regularization capabilities and reduced over-smoothing effects. The approach is validated by using numerical simulations in which cylindrical scatterers are reconstructed in a homogeneous lossy medium. Results are compared with those obtained by using standard single frequency operating conditions. Index Terms—Electromagnetic scattering inverse problems, Buried object detection, Newton method, Banach spaces. I. INTRODUCTION The use of microwave systems for imaging of unknown targets has been recently considered in several applications [1], ranging from industrial and civil nondestructive evaluation and testing [2]–[4], to biomedical diagnostics [5]–[8], and shallow subsurface prospecting [9]–[12]. In the latter case, the main aim is to extend the capabilities of the classical ground penetrating radars (GPRs) [13]–[15]. In such systems, the information about the targets is retrieved by processing the diffracted wavefield values collected in proper observation domains. In particular, it is necessary to solve an electromagnetic inverse problem, which turns out to be nonlinear and strongly ill-posed. Moreover, if strong scatterers have to be imaged, the “exact” equations of the electromagnetic scattering problem must be solved without any simplifying assumption. Another important limiting factor is related to the reduced information content of the data, which can be usually collected only in a limited set of measurement points. In fact, in buried object detection it is not possible to fully exploit angular diversity as in a tomographic arrangement. To address the previous problems, two ways can be essentially followed. The first one concerns the combination of nonlinear inverse scattering methods with focusing approaches, in order to reduce the number of unknowns and progressively “spend” the imaging capabilities to reduced inspection regions (regions of interest, ROI) including the target [16]. The second one concerns the exploitation of multifrequency input data [17]. Recently, several deterministic and stochastic imaging methods have been proposed in the scientific literature [18]–[23]. Among them, the authors proposed in [24] a quite efficient iterative approach for inspecting strong scatterers in free space, based on an inexact- Newton strategy. The above method, extended to the framework of Banach spaces in [25], has also been applied to retrieve shallow buried objects [26], but its capabilities have been limited by the single frequency processing. In this paper, the improvements obtainable by using multifrequency input data with the inversion scheme are discussed by means of numerical simulations. II. MULTI-FREQUENCY FORMULATION Let us consider a two-dimensional configuration in which a buried object (modeled as an infinite cylinder directed along the axis) is enclosed in a subsurface investigation domain Σ⊂ ℝ . A set of antennas, located in an observation domain Ω⊂ ℝ , illuminates the investigation domain with known TM- polarized electromagnetic fields. The lossy soil, in which the object is immersed, is characterized by a relative dielectric permittivity and a conductivity . Our goal is to find the unknown dielectric properties and of the investigation domain Σ by combining the scattered field data collected when the target is illuminated with a finite set of time- harmonic electromagnetic fields with angular frequencies ,…, (a frequency domain formulation is assumed in the following and the dependence is omitted). To this end, it is convenient to work with frequency-independent unknowns. Let us define as = − , = − (1) the normalized dielectric permittivity and electric conductivity functions, respectively, being the vacuum dielectric permittivity. The unknown functions and (both real- valued) can be represented as a vector function given by = ! ". (2) By extending to multifrequency data the formulation in [25], [26], the problem solution can be written in an operator form as # = $, (3) where # = % & & & & & & & ’ ℜ)ℱ + , ℑ)ℱ + , ℜ)ℱ . , ℑ)ℱ . , ⋮ ℜ)ℱ 0 , ℑ)ℱ 0 , 1 2 2 2 2 2 2 2 3 , $= % & & & & & & & ’ ℜ)5 6 + , ℑ)5 6 + , ℜ)5 6 . , ℑ)5 6 . , ⋮ ℜ)5 6 0 , ℑ)5 6 0 , 1 2 2 2 2 2 2 2 3 , (4) in which ℱ 7 denotes the nonlinear scattering operator mapping the unknown on the scattered electric field at the angular frequency 8 , and 5 6 7 is the measured - component of the scattered electric field at the same frequency. In order to solve equation (3), the two-step inexact Newton method in Banach spaces developed in [25] has been used. A Multi-Frequency Inexact-Newton Method in L p Banach Spaces for Buried Objects Detection Claudio Estatico, Alessandro Fedeli, Matteo Pastorino, and Andrea Randazzo Manuscript received November 27, 2014; revised April 16, 2015. C. Estatico is with the Department of Mathematics, University of Genoa, 16146 Genova, Italy (email: estatico@dima.unige.it). A. Fedeli, M. Pastorino, and A. Randazzo are with the Department of Electrical, Electronic, Telecommunication Engineering, and Naval Architecture, University of Genoa, 16145 Genova, Italy (e-mail: alessandro.fedeli@edu.unige.it; matteo.pastorino@unige.it; andrea.randazzo@unige.it).