## 6 Analysis of the CRB(\(\boldsymbol{\
### CRB(\(\boldsymbol{\
abla}_{long}\)) vs. \(\Delta\theta_{el}\)  
In this first case, we compute the CRB for different \(h\) values and elevation spans \(\Delta\theta_{el}\), covering from \(\theta_{el}=10\,^{\circ}\)-\(35\,^{\circ}\). In the results obtained, the first individual \(\Delta\theta_{el}\) value represents a special case, since it corresponds to a single short observation interval of \(T_{obs}=1\) s, with no change in \(\theta_{el}\). In this case, the bound matches the results obtained using the short-period CRB derived in Section 3. Figure 6 shows the CRB of \(h\) results for the isotropic antenna and the AR25 antenna cases, with the GPS L1 C/A signal. In the figure, the highest CRB values are located at the left-upper corner of the plot, which correspond to the lower values of antenna heights (below 5 m) and elevation spans (below \(15^{\circ}\)). The CRB values decrease almost monotonically with both the increase of \(h\) and the elevation span covered. As expected, it becomes clear from the plots that the antenna pattern has a very significant impact on the CRB of \(h\) given the same \(h\) and \(\Delta\theta_{el}\). Figure 6a shows that sub-millimeter precision could be achieved in theory in the ideal isotropic antenna case. On the other hand, in the AR25 antenna case, the plot in Figure 6b shows that now, only centimeter precision can be achieved. These CRB values now obtained are significantly lower than the CRB values computed previously in Section 3. Firstly, given the satellite elevation rate of \(5\times 10^{-4}\,\circ\)/s considered, the observation time is above the hour duration for \(\Delta\theta_{el}>18^{\circ}\). The longer the observation time considered, the more independent samples we collect, thus the lower CRB values obtained. Secondly, the CRB computed now takes into account that the phase difference information can be used for the estimation of \(h\) in this case. Intuitively, this can be explained by the different rate of change of the two contributions to the phase of the reflected component: the phase shift due to the scenario geometry (related to \(h\)) and the phase shift due to the polarization change due to the reflecting surface properties, i.e., to \(\Gamma\).  
Having a higher true \(h\) has two main implications. First, the higher the \(h\), the faster the oscillation rate of the interference pattern observed in the composite amplitude of the received signal with the change of satellite elevation. Second, the \(\Delta\rho\) will be higher, which makes it easier to differentiate between the two signal components, given the code autocorrelation properties, and the resulting CRB can be lower, as we have shown in Section 3. However, techniques like the IPT only use the output of the estimated SNR or the prompt correlator (single output), and they can only work if there is interference between the LOS and the reflected signal, i.e., if \(\Delta\rho<\rho_{chip}\)[21]. Thus, with IPT, we face a trade-off: we require low \(\Delta\rho\) values in order to observe the interference pattern, but according to the CRB expressions obtained, low \(\Delta\rho\) values degrade the achievable estimation precision.