## 1 Introduction  
Non-invasive concealed object detection, identification, and discrimination have been of interest to the research community for decades due to the need for preserving infrastructures or ancient ruins and artifacts from invasive excavations, with a further advantage in terms of time and costs reduction; or, in war or post-war scenarios, to guarantee safety conditions for the detection and location of landmines in order to avoid devastation and human tragedy caused by explosions due to accidental contact, to name some examples. Among the wide number of approaches developed, those that deserve to be cited are electromagnetic induction, nuclear quadrupole resonance, thermal imaging, and, above all, the techniques based on ground penetrating radar (GPR), which has been widely used in landmine detection (see [1] and the references therein for an overview). A modern and very effective means to address this problem is based on the use of an unmanned aerial vehicle, i.e., a drone, equipped with GPR [2, 3, 4, 5], which, unlike classic techniques based on a hand-operated metal detector or GPR mounted on a dragged trolley, has the advantage of not requiring direct contact with the ground. Moreover, high-resolution underground images can be obtained by coherently combining the measurements by means of a synthetic aperture radar algorithm [6], and it can be easily conceived that the adoption of modern array configurations for the transmitting and receiving antennas could further improve system performance [7]. Due to the complexity of real scenarios, suitable modeling of the problem is needed to perform numerical simulations that could efficiently provide useful information to best utilize the potential of such kinds of non-invasive techniques. Despite the huge number of techniques based on the finite element method [8], the finite-difference time-domain method [9; 10], and the conjugate gradient method [11; 12], to cite the most important ones, when dealing with finite scatterers in unbounded media, integral equation formulations are largely preferred because the unknowns and the boundary conditions from which the equations originate are defined on finite supports [13; 14; 15; 16]. Since no closed form solution is available for the obtained integral equation, discretization of the integral operator is needed. However, due to the singular nature of the integral equation at hand, the existence of the solution cannot be generally stated, and even more, if such a solution exists, the convergence of a general discretization scheme cannot be demonstrated. Several techniques, collectively called methods of analytical regularization [17], have been developed to overcome these problems. The rationale is to convert the integral equation at hand to a Fredholm second-kind equation by suitably inverting the most singular part of the integral operator. Of course, based on the problem at hand, the integral operator to be inverted can be conveniently identified with the static part, the high-frequency contribution, a suitable canonical problem, and so on. The well-behaved nature of the equation thus obtained is demonstrated by the Fredholm theory, which allows to establish the existence of a solution from its uniqueness and, moreover, the asymptotic convergence of a discretization scheme preserving the nature of the equation to the exact solution of the problem [18]. In principle, there are no reasons to think of regularization and discretization as two consecutive steps. As a matter of fact, they can proceed \"hand to hand\" by adopting, for example, the Nystrom discretization technique [19; 20; 21], which leads to guaranteed-convergence numerical codes founded on the theorems of the quadrature approximations of singular integrals; or the method of analytical preconditioning (MAP) [22], in which Galerkin projection acts as a perfect preconditioner in case a set of orthonormal eigenfunction of the most singular part of the integral operator is selected as expansion basis, thus leading to a Fredholm second-kind matrix equation.