## 6 Conclusions and Future Developments  
In this contribution, we have shown some potentialities of the application of joined migration and a combined time-depth conversion to GPR data, which has allowed us to improve the imaging in a two-layered medium. The algorithm improved both the vertical Bscans and the slices because the different propagation velocities in the two layers have been accounted for. In particular, we have proposed a simplified algorithm that neglects the vectorial refraction of the waves at the interface but can account for the relatively complex shapes of the interface between the two layers, which is a case for which no analytic solution is currently available. Theoretically, the exposed considerations also easily extend to the case of N-layered media, but it is naturally progressively more difficult (and less reliable) to retrieve the electromagnetic characteristics of the subsequently deeper layers. Whether this is practically possible depends on the particular application, the losses, the thickness of the layers, and the difference between the electromagnetic characteristics of the adjacent layers, namely the amplitude of the first kind of discontinuity jump shown by the propagation velocity of the waves. Several possible future developments are being thought of. First of all, the mathematical description of the interface between two different layers (or the top and bottom of a buried cavity) can be made faster by means of some computer graphic algorithm enabling the user to rapidly pick up these curves in the time domain by clicking points on their computer screen with the mouse and/or by moving the mouse along the interface on the screen. This is especially important if the interface between two buried layers does not have a simple shape. A further development related to the efficiency of the algorithm is the automatization of the exportation of the Bscans from a commercial code (we exploited the Reflexw, version 9.1.3) to an external code for the interpolations and the zero paddings (we dialed a home-made code in the MATLAB (version 9.3.0) environment) and then the re-importation into commercial code for the slicing. In particular, we exported the Bscans in ASCII format from Reflexw to MATLAB in order to apply joined migration and the combined time-depth conversion and then reimported the combined files again in ASCII format from MATLAB to Reflexw in order to perform the slicing. This implied the loss of the headers of the files each time (essentially the length of the Bscans and their bottom scale) that have been manually rewritten each time. Another aspect of interest is the retrieval of the permittivity of the second layer from the same GPR data. This aspect has been studied in the case of a sequence of flat layers [15; 16; 17], where analytical calculations are possible. In the case of a curved interface, no analytical solution is available, but notwithstanding, suitable forward modeling could help in retrieving the propagation velocity of the waves in the second layer. Forward modeling can be implemented, making use of the above-quoted open-access code GPRMax [31], using trials to determine the value of permittivity of the second medium and allowing the best graphical matching of the diffraction curves visible in the second layer (and presumably attributable to small targets). This analysis might also provide an evaluation of the available precision on the basis of the sensibility of the diffraction curve to the trial value of dielectric permittivity. This point is relevant because we might not have the possibility of performing local ground truthing and we might not have a TDR probe at our disposal or, even with it, the depth of the first layer could result in a difficult local excavation up to the second layer. Future work is also necessary with regard to a specific comparison of this method with more analytic algorithms, like time reverse migration or regularized linear inverse scattering methods.