## 1 Introduction  
Microwave inverse scattering is a suitable method for non-destructive testing (NDT) applications. It is based on the ability of the electromagnetic field to penetrate and propagate inside dielectric materials, at frequencies in the microwave band, and on the effects on the propagation of the objects embedded inside the background medium [1]. Ground-penetrating radar (GPR) is one of the most interesting microwave-based techniques, and it is applied in many fields, ranging from geophysics to archeology and civil engineering [2, 3, 4, 5, 6]. The study of electromagnetic scattering from cylindrical objects embedded inside a homogeneous dielectric is useful for schematizing many different situations, such as the localization of buried pipelines [7], tree root reconstruction [8], grid diagnostics [9, 10], and reinforced concrete monitoring [11, 12]. In particular, the diagnosis and monitoring of reinforced concrete structures are of great interest within civil engineering. The bars' number, diameter, and deployment, together with the concrete cover thickness, are important parameters attesting to the correct structural behavior towards external mechanical stress and long-lasting structures. However, the details of the structural design in terms of such parameters can be lost and must be recovered. Moreover, the structure can be damaged during its life by concrete corrosion or if some bars oxidize, with a diameter smaller than the nominal one. Many destructive and non-destructive methods already exist for acquiring information about materials and structures within civil engineering. Conventional destructive methods usually require samples of the structure under test to be taken, for instance by drilling and coring, or removal of some external layers in order to allow visual inspection of the internal status of the structure. Such methods have the disadvantage of being invasive and providing only punctual information. On the contrary, methods based on NDT techniques do not damage or modify the structure under test and make in situ diagnostics and monitoring possible. The currently available NDT methods for the investigation of reinforced concrete structures are based on a variety of different physical phenomena. Acoustic methods (sonic and ultrasonic frequencies) [13], infrared thermography [14], magnetic flux leakage (MLF) [15], eddy currents [16], optical fibers [17; 18], and electromagnetic radar-based methods [5] have been extensively studied. In [15] and the references therein, MLF is discussed, and the experimental results pave the way for its application in concrete bar detection. In [19], a comparison between a destructive method and a non-destructive method based on an ultrasonic pulse for concrete strength measurement is presented, and, in the same paper, GPR measurements are used as a complementary tool to investigate the structure. Among the NDT methods for reinforced concrete investigation, those based on electromagnetic propagation and scattering, such as radar and GPR, allow contactless measurements to be performed. This feature is very attractive because makes it possible to perform flexible measurements, and it even allows for operation in remote sensing configurations [20]. On the other hand, data processing is challenging for several reasons. Contactless measurements are affected by the strong reflection from the interface between the air and concrete. Additionally, the final result of a commercial GPR is a radargram whose interpretation is not immediate, except in very simple cases, and the need to properly model the electromagnetic scattering phenomenon and formulate an inverse problem arises [21; 22]. The focus of this paper is on the mathematical modeling of the scattering phenomenon and on the study of the related inversion algorithm. Microwave imaging requires the modeling and solving of an inverse electromagnetic scattering problem, which is intrinsically non-linear.