## 3 The Combined Time-Depth Conversion  
After achieving joined migration in the time domain, we have to convert it into a graph represented by the abscissa and the spatial depth. The schematic of Figure 1 displays how this can be achieved. In particular, Figure 1A schematizes a result in abscissa \"x\" and time \"t\", and the (not-flat) buried interface is represented in red: the representation is piecewise constant because the pixels are put into evidence. Three targets are embedded in the soil, represented by three small circles. After performing the joined migration, or let us say more generally, any elaboration on the data that provides a focused result in the (x-t) plane, we have to represent the buried scenario in the abscissa and spatial depth, i.e., in the (x-z) plane. The essential point is that the vertical size of the pixels in the x-t plane is the same in the two layers, but the corresponding vertical size in the x-z domain is not the same because it is given by the well-known round-trip formula, which should, however, be applied separately to the upper and lower layers. Therefore, we have:  
\[\left\{ \begin{matrix}\Delta z_{1}=\frac{c_{1}\Delta t}{2}\\ \Delta z_{2}=\frac{c_{2}\Delta t}{2}=\frac{c_{2}}{c_{1}}\Delta z_{1}\end{matrix} \right. \tag{1}\]where \(c_{1}\) is the actual propagation velocity of the electromagnetic waves in the upper layer and \(c_{2}\) is the actual propagation velocity in the lower layer. \(\Delta t\) is the time step of the data, \(\Delta z_{1}\) is the ideal vertical thickness of the pixel in the upper layer, and \(\Delta z_{2}\) is the homologous quantity in the lower layer. So, the converted pixels should appear as in Figure 1B. More precisely, Figure 1B represents the case \(c_{1}>c_{2}\). In the opposite case, the pixels of the lower layer should be vertically dilatated with respect to those of the upper layer. As can be seen from Figure 1A, in the time domain, the central and left-hand targets appear at the same depth, whereas the left-hand target appears deeper. Instead, in Figure 1B, i.e., \"in reality\", the left-hand and central targets are at the same depth, whereas the right-hand one is shallower. Whatever the case, the final result has to be represented by a matrix, where the values are necessarily associated with a regular rectangular grid, and this is not the case in Figure 1B. In order to achieve pixels of the same size, interpolation is needed so that the cells of the upper medium and those of the lower medium have the same size. With some easy algebra, this means that in the case of \(c_{1}>c_{2}\), for each column of the \"matrix\", starting from the air-soil interface and ending at the interface between the two layers, the initial number of pixels \(n(j)\) (indexed after the column index j) should be interpolated into \(N(j)=\frac{c_{1}}{c_{2}}n(j)\) pixels. In the dual case \(c_{2}>c_{1}\), for each column of the \"matrix\", starting from the interface between the two layers and ending at the (local) spatial bottom scale, the initial number of pixels \(n(j)\) should be interpolated into \(N(j)=\frac{c_{2}}{c_{1}}n(j)\) pixels. In general, the ratio \(\frac{c_{1}}{c_{2}}\left(\frac{c_{2}}{c_{1}}\right)\) is not an integer number, and so the integer number closest to the quantity \(\frac{c_{1}}{c_{2}}n(j)\left(\frac{c_{2}}{c_{1}}n(j)\right)\) should be chosen. The resulting residual distortion is customarily negligible. After interpolation, the pixels will appear as in Figure 1C. This set of pixels can be completed with zero padding as shown in Figure 1D. In this way, we compose a complete matrix that depicts the scenario in the x-z domain. The necessity of zero padding corresponds to the fact that the time bottom scale of the data is constant vs. the abscissa but the corresponding spatial bottom scale is not, and this is natural because the average velocity along any vertical path is not the same vs. the abscissa.