# Two-Dimensional Linear Inversion of GPR Data with a Shifting Zoom along the Observation Line
## 3 The Effect of the Extension of the Observation Line beyond the Investigation Domain  
This section is devoted to dealing with the advantages and drawbacks of the extent of the observation domain, with regards to having a fixed extent for the investigation domain. As will be shown, these advantages (if any) are practically quite limited when the observation line becomes longer relative to that of the investigation domain, and when the investigation domain is in the order of a few wavelengths. This provides the basis for the shifting zoom shown in the next section, which is essentially based on the choice of equally long investigation and observation domains. In particular, the theoretical improvement achievable by prolonging the observation domain over the investigation one, can be seen in terms of the increase in the angle of view, under which the targets are seen, with a consequent improvement in resolution limits [6]. This can be quantified by evaluating the following figure of merit, which represents the energy kernel of the integral equation, integrated across the investigation and observation domains. \[En=\int\limits_  
As shown in Appendix A, a physical interpretation can be provided for the summation of the squared singular values; in particular, the quantity Equation (6) accounts for the integral over the investigation domain, for the energy of the images in the Point Spread Function, for each generic point in the investigation domain. Therefore, such a figure of merit allows us to account (in a global way) for the effect of prolongation of the observation domain, in terms of the possibility of capturing the scattered field from the targets in the investigation domain. Here, we consider an evaluation of the figure of merit (in Equation (6)), for the same matrix exploited later on to process the experimental data. In particular, we are concerned with a propagation medium with a relative dielectric permittivity equal to 2.37. The working frequency band is 800-4000 MHz. The investigation domain has a horizontal extent equal to 0.8 m and a vertical depth of 0.8 m (starting from zero). For the considered depth of the investigation domain, and dielectric permittivity of the medium, it is possible to show that a frequency step of 100 MHz is sufficient, in order to avoid aliasing problems in the inversion [6], and to provide a sufficient time range. The spatial step of the data is 0.01 m.  
Now, we consider four different observation domains, with extents of 0.8 m, 1.2 m, 1.6 m and 2.4 m. The relative investigation domains are centred with respect to the observation domain and are sized 0.8 \(\times\) 0.8 m\({}^{2}\) at all times. The singular values' behaviours, relative to the four observation domains, are depicted in Figure 1. It can be noted that there is only a smooth increase in the significant number of singular values for the considered threshold values of TSVD in the reconstruction, and that the curve tends to \"saturate\". The quantities (in Equation (6)) relative to four different cases are reported in Table 1. The second column reports the sum of all squared singular values for the four observation domains. The other two columns report the summation of a finite number of terms; in particular, we consider singular values larger than \(-20\) dB and \(-30\) dB, relative to the maximum one. All the figures are normalised relative to the quantity evaluated for the largest observation domain in Equation (6) (240 cm). Table 1 shows us that when making the observation domain at hand three times larger, the energy kernel of the operator increases by only about 20%. The physical reason for this is that when we progressively enlarge the observation line in the parity of the investigation domain, this entails a progressively marginal (and vanishing) increase of the maximum geometrical view angle, under which the investigation domain is seen, and on the other hand there is also a progressive decrease  
Figure 1: Singular values for increasing observation lines in the parity of the investigation line.