## 1 Introduction  
Layered scenarios constitute a topic of large interest within Ground-Penetrating Radar (GPR) prospecting. Common situations are asphalt or concrete pavements, within and under which metallic rods, pipes, voids, and several kinds of discontinuities could be displaced [1, 2, 3, 4]. In other contexts, the internal coverage of the road tunnels also generates an intrinsically layered medium [5, 6]. Another relevant case of large interest is the imaging of buried cavities [7, 8], which is of interest for cultural and/or safety purposes. In particular, a cavity can be seen locally as a particular three-layered medium, according to the sequence \"soil-air-soil\". Further cases of interest are archaeological heritage buried beyond the first layer of the soil or through-the-wall imaging, where the first layer of the investigated scenario is constituted by masonry. In many cases, the inhomogeneity of the background scenario is neglected. However, a good choice of the background scenario provides better results. Two relevant challenges to this are the nonlinearity of the underlying inverse scattering problem and the computational burden required to afford it in a rigorous way. More specifically, while slow variations of the properties of the soil can be accommodated by linear inverse scattering algorithms [9, 10] or reverse migrations [11; 12; 13], abrupt variations are more challenging. This can be the case of cavities and layered media with strong discontinuity of the electromagnetic characteristics between the layers. In particular, we consider here the case of a two-layered medium with a non-flat interface between the layers. Unlike the case of layers with a flat interface parallel to the air-soil interface [14; 15; 16; 17], in this case, there is no analytical solution for the inverse scattering problem. Two issues consequently arise, namely, the focusing of the buried targets and the time-depth conversion to apply to the achieved image. In homogeneous soil, these two issues are strictly linked to each other because the \"true\" value of the propagation velocity is the value driving both the best focusing and the most correct time-depth conversion [18; 19; 20]. However, things are trickier in stratified media, especially when a curved interface separates the layers. In particular, modeling is complicated and, in general, the commercial codes available for data processing are not able to accommodate such situations [21; 22]. Moreover, we have to consider the fact that in GPR prospecting, the extension of the inversion domain can be electrically very large (even thousands of central wavelengths or more), so there is an obvious need for fast and efficient algorithms. This excludes, in most cases, the concrete possibility of exploiting nonlinear full wave inversions [23; 24], which are incidentally also prone to the problem of the local minima [25]. For all these reasons, we have introduced two simple algorithms, aimed at the focusing and time-depth conversion in layered media, which we have labeled \"joined migration\" and \"combined time-depth conversion\" [4; 26]. These algorithms correspond to a rough model of the propagation of the waves and of the scattering, but on the other hand, are computationally non-demanding, easy to implement, and do not require more than a commercial code for the post-processing plus a home-made code that is quite easy to implement. In spite of the simplicity of the model, in several situations of interest, the imaging of the background scenario can be meaningfully improved and the spatial relationship between the buried targets can be better reproduced with respect to the results achievable from more customary algorithms. With regard to this aspect, let us outline the fact that the customary non-invasive methods (diffraction hyperbolas, common mid-point) for retrieving the propagation velocity of the waves work correctly with regard to the shallower layer of a stratified soil, but they do not work satisfyingly any longer with regard to the deeper layers.