# Two-Dimensional Linear Inversion of GPR Data with a Shifting Zoom along the Observation Line
## 4 The Shifting Zoom  
In the previous section, we showed that the most convenient choice for performing the inversion of a buried area is, in most cases (except for an electrically short observation line that is customarily of no  
Figure 4: Target reconstruction with an observation domain equal to 1.6 m.  
Figure 5: Target reconstruction with an observation domain equal to 2.4 m.  
Figure 3: Target reconstruction with an observation domain equal to 1.2 m.  
practical interest), to adopt an investigation domain as large as the superposed observation line, and not shifted with respect to it. Consequently, in order to invert a long investigation domain, not viable for a \"one-step\" inversion, the most natural choice is to adopt subsequent adjacent investigation-observation domains along the radar profile, as schematically indicated in Figure 6. However, this introduces the problem of having targets that might be peripheral to the investigation domain wherein they are located, in particular, with regards to Figure 6 and the observation line (L\({}_{2}\)), target \(c\) is not seen under a symmetric maximum view angle (it is peripheral relative to the observation line, L\({}_{2}\)), whereas target \(a\) is able to cause artefacts in the investigation domain (D\({}_{2}\)), because it is quite probably perceived by the antennas when they pass the left edge of L\({}_{2}\). We also clearly understand that the problem of the peripheral targets, and the problem of the interference for external targets, are two sides of the same coin. In particular, target \(a\) is peripheral to D\({}_{1}\) and target \(c\) is a possible source of artefacts in D\({}_{3}\). This last effect is essentially due to the fact that when adopting any fixed investigation domain, we are implicitly assuming that no target is present outside it, which does not correspond, in general, to the ground truth. These problems are clearly related to the performed subdivision into adjacent subdomains. In fact, neither **a**, nor **c**, are peripheral targets to the comprehensively gathered profile. The proposed scheme also encompasses the case of a target that crosses the edge between two adjacent domains; in fact, such a target can be ideally seen as a couple of adjacent targets--one on the left-hand side and one on the right-hand side of the edge. Here, we propose a strategy to overcome this problem, resorting to a more refined definition of consecutive investigation domains. The proposed method is illustrated in Figure 7, and simply consists of retaining the central belt of the current investigation domain. Then, the subsequent investigation domain is shifted by the same amount, relative to the previous one, so that we do not have consecutive adjacent investigation domains, but rather, consecutive overlapping investigation domains. For example, in Figure 7, the generic \(k\)th investigation domain is indicated in red, and its central belt is coloured in red too. The homologous areas with respect to the \((k+1)\)th investigation domain are labelled in green. Also, the observation lines have to be coherently shifted, to keep them hanging on the relative underlying investigation domains, as indicated in Figure 7. In order to guarantee continuous hanging between the current investigation and observation domains, it is necessary for the horizontal extent of the pixels in the representation of the reconstruction to be an entire multiple, or an entire sub-multiple, of the spatial step of the measures. This can be achieved by means of a proper choice of the size of the pixels, or alternatively, by means of some interpolation, either on the measurement points, or in the representation of the current reconstruction. Please note that in Figure 7, the overall observation line is indicated with \(L\), and the indications of \(L_{k}\) and \(L_{k+1}\) indicate subsets of \(L\); they are represented at different heights just to allow us to distinguish them from each other.