# Analytical Computation of the Spatial Resolution in GNSS-R and Experimental Validation at L1 and L5
## 3 Results
### Experimental Validation  
Figure 11 shows the measured reflectivity (\(\Gamma\)) in a transition between land and calm water in the Devilbend Reservoir, south of Melbourne, Victoria, Australia. The data was acquired during one of the test flights of the MIR [8] instrument on 30 April 2018. The top plot corresponds to one of the four beams at GPS L1, and the bottom plot at L5. Flight speed was 75 m/s, flight height was \(h_{R}=1000\) m, and GNSS-R data were coherently integrated for 1 ms (GPS L1 C/A code), and incoherently for T\({}_{\rm inc}=20\), 40, 100, and 200 ms. At 20 ms, the blurring produced by the aircraft movement is 1.5 m, which is negligible compared to the best spatial resolution computed in Section 3.1. Inspecting Figure 11, at 20 and 40 ms incoherent integration time, one can clearly observe some reflectivity ripples that are reminiscent of those seen in Figure 9. At 100 ms, the blurring is 7.5 m, which is now comparable to the best spatial resolution, and the ripples can no longer be appreciated. At 200 ms the reflectivity plot is even smoother. In the land part (left part of the plots), note that the L5 reflections exhibit a more marked fading pattern than the L1 ones. This may be because L5 penetrates a bit more in the land (and vegetation), and it is moister soil underneath the surface. Although it is not very clear, the width of the reflectivity step is approximately 0.3 s, which corresponds roughly to 22.5 m, as compared to the 20.7 m predicted value using Equation (9) and Figure 10 for a \(\sim\)\(-15\) dB reflectivity. Figure 10: Width of the \(\Delta v\) parameter for a transition from 10% above \(\rho_{min}\) to 90% of \(\rho_{max}\). Taking a closer look to the periodicity of the ripples, it can be observed that the period in Figure 11b is larger than that in Figure 11a, as it can be inferred from Equation (5), and Figures 6 and 9, parameterized in terms of \(v\). In Figure 11a, the separation between consecutive peaks in time is ~100 ms, while in Figure 11b it is ~130 ms, which, for an aircraft speed of \(v\) = 75 m/s corresponds to 7.50 and 9.75 m, respectively. More detailed results are shown in Table 1 for several peaks. Replacing \(\Delta x(t)\) by \(\
u_{\perp}\cdot\Delta t\) in Equation (10), where \(\
u_{\perp}=\
u\cdot\cos(\varphi)\), \(\varphi\) being the angle formed between the aircraft ground-track and the coastline, one obtains:  
\[\Delta v=\
u_{\perp}\cdot\Delta t\sqrt{\frac{2\cdot\cos(\theta_{i})}{\lambda \cdot h_{R}}}. \tag{10}\]  
Assuming \(h_{R}\) = 1000 m, \(\theta_{i}\) = 45\({}^{\circ}\), and substituting \(\Delta t\) for the time difference between consecutive peaks in Figure 11 (except for roundoff errors due to the discrete sampling), this ratio closely matches the squared root of the ratio of the wavelengths at L1 and L5: \(\sqrt{1575.42\;\mathrm{MHz}}/1176.45\;\mathrm{MHz}=1.16\). Other sources of error are variations of the flight height (approximately \(h_{R}\) = 1000 m), variations of the incidence angle, and most likely the fact that the velocity vector is not perpendicular to the coastline (as implicitly assumed in Figure 5). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(v_{peaks}\) & \(\mathbf{|\Delta v_{peaks}|}\) & \(\mathbf{t_{peaks,\,L1}}\) & \(\mathbf{\Delta t_{peaks,\,L1}}\) & \(\mathbf{\Delta t_{peaks,\,L1}}\) & \(\mathbf{t_{peaks,\,L5}}\) & \(\mathbf{\Delta t_{peaks,\,L5}}\) & \(\mathbf{\Delta t_{peaks,\,L5}}\) \\ \hline \(-1.22\) & - & Not visible & - & - &  
These results are corroborated by many other transitions. In what follows a few more examples are presented to illustrate this effect. A minimum incoherent integration time of 40 ms is used, since blurring is still negligible, and these results are almost identical to when 20 ms is used. Figure 12 shows a geo-located double transition from land-calm water-land, the temporal evolution of the reflection coefficient at L1, and two zooms around the transitions for different incoherent integration times. Figure 13 shows another example at L5.