# 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
The investigation domain is 80 \(\times\) 80 cm and is centered under the observation line. The observation lines are represented as follows: blue line, observation line, 80 cm long; red line, observation line, 120 cm long; green line, observation line, 160 cm long; cyan line, observation line, 240 cm long. The singular values’ curves are normalized with respect to the maximum singular value. in the received signal. In other words, it is illusory to consider achieving a meaningful quantity of extra information about a given target to prolong the observation line indefinitely, even in a lossless case, or even to neglect the effect of the radiation pattern of the antennas, as done in this analysis. Incidentally, to make the simulation more realistic, one should consider an effective or equivalent maximum view angle, customarily smaller than the geometrical maximum view angle (unless a very short observation domain is considered), and influenced by the losses and the directivity characteristics of the antennas; however, this would further enforce the retrieved conclusions. In order to show the negligible effect of an indefinite prolongation of the observation domain in terms of retrievable information, we consider a reconstruction of the central target, embedded in the investigation domain, sized 0.8 \(\times\) 0.8 m for different extents of the observation domain (0.8, 1.2, 1.6, 2.4 m), under a TSVD with a threshold of \(-\)30 dB. The simulated target is a circle made of a perfect electrical conductor, at a depth of 25 cm. The radius of the circle is 5 cm. Figures 2-5 depict the reconstruction of the target, with observation domain extents equal to 0.8, 1.2, 1.6 and 2.4 m, respectively. As can be seen, progressive enlargement of the observation domain does not improve the reconstruction of the target, which is predominantly the same for all the considered extents. On the other hand, the effects of possible targets outside the chosen investigation domain, in real cases, can produce artifacts. For this reason, from now on we will adopt observation domains of the same length as the relative investigation domains and superpose them. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Horizontal Size of the** **Investigation Domain** & \begin{tabular}{c} **Energy (Sum of** \\ **All the Squared** \\ **Singular Values)** \\ \end{tabular} & \
\begin{tabular}{c} **Sum of Squared** \\ **Singular Values for a** \\ **Threshold of \(-\)20 dB** \\ \end{tabular} &
\begin{tabular}{c} **Sum of Squared** \\ **Singular Values for a** \\ **Threshold of \(-\)20 dB** \\ \end{tabular} \\ \hline
0.8 m & 78.6\% & 78.1\% & 78.6\% \\
1.2 m & 89.8\% & 89.2\% & 89.8\% \\
1.6 m & 94.9\% & 94.1\% & 94.8\% \\
2.4 m & 100\% & 99.15 & 100\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Normalised energy (see Equation (6)) for four cases. Column 2 represents all the singular values. Column 3 represents singular values larger than \(-\)20 dB. Column 4 represents singular values larger than \(-\)30 dB. Figure 2: Target reconstruction with an observation domain equal to 0.8 m.