## 1. Introduction

[2] As the physical features of an unknown target are contained in its scattered field, it is possible in theory to retrieve these characteristics thanks to the resolution of an inverse scattering problem. In practice, there exists some difficulties in particular when dealing with measurements. Indeed, as the scattered field can not be measured on an entire closed surface englobing the target, the data are truncated [*Bucci and Isernia*, 1997; *Snieder and Trampert*, 1999]. The measurements are also inevitably disturbed by some noise, moreover this noise a priori does not follow an uniform distribution [*Geffrin et al.*, 2009].

[3] Recent works in microwave imaging, in particular for buried targets reconstructions, are concerned with the development of fast linear algorithms allowing the detection and the localization of the scatterers [see, e.g., *Soldovieri*, 2010; *Litman et al.*, 2010; *Soldovieri et al.*, 2011]. These methods which present a low computational time and a relatively low memory requirement allow exploring large spatial regions. Unfortunately, except for scatterers with small dimensions and/or with low contrasts, these procedures do not lead to quantitative reconstructions.

[4] A complete non-linear formulation is therefore necessary to reach such quantitative values. In these cases, global optimizations as in the works by *Garnero et al.* [1991] and *Zigliavsky* [1991] should be preferable due to the possible presence of several local minima, but these methods require generally high computational time. Some approaches have also been developed to reduce the non-linearity degrees of the problem [*Brancaccio et al.*, 1995; *Isernia et al.*, 1997]. Here, we choose to regularize our problem by using available a priori information related to the effective measurement noise and to the physical properties of the scatterer.

[5] Another difficulty of the quantitative inversion procedures concerns their large memory storage requirement. To overcome this difficulty, the investigation domain is usually reduced to few wavelengths [see, e.g., *Semenov et al.*, 2000; *Abubakar and van den Berg*, 2004; *de Zaeytijd et al.*, 2009]. Indeed, these procedures are performed with an a priori knowledge on the position and on the dimension of the target. In some cases, this information is provided, as for example in the last 3D database of experimental scattered fields proposed by the Institut Fresnel (*3D Fresnel database*) [*Geffrin and Sabouroux*, 2009; *Litman and Crocco*, 2009]. Otherwise, these geometrical features must be directly deduced from the measurements. In the work by *Catapano et al.* [2009], the authors have employed the Linear Sampling Method to pre-localize the targets. Our approach, detailed here, is to directly exploit the scattered field spectral behavior to estimate the investigation area.

[6] In this paper, we explain the complete imaging strategy that we have constructed in order to retrieve the target's characteristics directly from the measured scattered fields and without any a priori geometrical information related to the target in the measurement scene. In a previous study [*Eyraud et al.*, 2011], we have thoroughly investigated the influence of the polarization effects on the quantitative inversion procedure. In that former study, we have used the a priori knowledge of the position and the dimensions of the target. Herein, we will focus on the retrieval of the quantitative characteristics of the scatterer without any knowledge on its position and on its approximate extent using the most favorable case of polarization.

[7] The adopted strategy is summarized here. First, the measurements are performed and the fields are post-treated for drift correction and calibration, before being inverted. The imaging procedure is then decomposed into two steps. The first step consists in the localization of the target and the estimation of its size by exploiting the spatial spectrum of the scattered field while taking into account the measurement accuracy. In the second step, a quantitative inversion procedure based on a Bayesian approach, which also takes into account the effective measurement noise, produces the permittivity map within the pre-selected investigation zone.

[8] The present paper is organized as follows. In section 2, the physical characteristics of the target are described. The experimental work is presented in section 3. In section 4, the inversion process is detailed step by step and associated results are presented. Concluding remarks follow in section 5.