## 4 Numerical Results  
We have considered a three-dimensional two-layer scenario, simulated by means of the open access GPRMax code [31], based on the method of the finite difference in time domain FDTD [35]. The simulated scenario is shown in Figure 3.  
In particular, the observation surface is extended to 4.2 \(\times\) 3 m, and three pipes are buried and labeled P\({}_{1}\), P\({}_{2}\), and P\({}_{3}\), respectively. Pipe P\({}_{1}\) is the upper one: it is parallel to the long side of the box and is buried at a depth of 45 cm. Pipes P\({}_{2}\) and P\({}_{3}\) are both at the depth level of 90 cm. Pipe P\({}_{2}\) follows a direction orthogonal to the upper pipe P\({}_{1}\)  
Figure 3: 3D Simulation scenario. (**A**) Geometric details of the perspective view scene. (**B**) Geometric details of the frontal view scene. (**C**) Geometric details of the top-down view scene. Figure 2: Flux diagram of the algorithm. whereas P\({}_{3}\) is slightly tilted (7.91 degrees) with respect to P\({}_{2}\). The background medium is composed of two layers separated by a flat but slanted interface that starts from the air-soil interface on the left-hand side and arrives at a depth of 1 m on the right-hand side. The dielectric permittivity is equal to 4 in the upper layer and 16 in the lower one, which means that the propagation velocity of the electromagnetic waves is equal to 15 cm/ns in the upper layer and 7.5 cm/ns in the lower layer. No magnetic properties are present, and a small electrical conductivity equal to 0.005 S/m has been introduced in both layers. The data have been taken along an orthogonal grid with interline spacing of 30 cm and in-line spacing of 2 cm. The time bottom scale is 80 ns and the time step is 0.0192 ns. The pixels used for implementing the FDTD were sized 1 cm in any direction. The transmitting and receiving antennas are two dipoles located 1 cm from the air-soil interface and displaced at negligible distances from each other. The radiated signal is a Ricker wavelet whose 3 dB band ranges from 137.5 MHz to 450 MHz. The data have been processed with the Reflexo code [36] through zero timing, background removal up to 5 ns (in order to avoid distortion of the shallowest pipe), linear and exponential gain vs. depth, Butterworth filtering, and Kirchoff migration (with two different velocities as mentioned). We have put into the inversion algorithm c\({}_{1}\) = 16 cm/ns for the upper layer, as retrieved from the diffraction hyperbola of pipe P\({}_{3}\) (the exact value is 15 cm/ns instead). With regard to the second layer, we have exploited a value of 12 cm/ns for the focusing, retrieved by trials, and then we have exploited a value of 8 cm/ns for the time-depth conversion, voluntarily introducing a small error with respect to the exact value of 7.5 cm/ns. In this way, some uncertainty in the measurement of the propagation velocity of both layers has been implicitly accounted for. In Figure 4, a Bscan processed according to customary data elaboration is shown. In particular, the 6 th (out of 11) Bscan taken along the long side of the grid is shown. The length of the Bscan is 4.2 m.  
In Figure 4, we have also converted the return time into the spatial depth according to the estimated propagation velocity in the upper medium, namely 16 cm/ns. From Figure 4, we can see that we achieved good focusing of the pipe in the upper medium, namely P\({}_{3}\). However, the focusing of pipe P\({}_{2}\), embedded entirely in the lower layer, is not satisfying at all. Moreover, while the depth of P\({}_{3}\) is reliably retrieved (indeed, its depth appears equal to 100 cm instead of the actual value of 90 cm because of the overestimation of the propagation velocity), the apparent depth of P\({}_{2}\) is 150 cm, which is meaningfully larger than the actual value. This happens because the propagation velocity of the waves in the deeper layer is lower than that in the upper layer. We also appreciate the signature of pipe P\({}_{1}\), crossing the two layers.