## 5 Derivation of the CRB for Long Observation Intervals: CRB(\(\boldsymbol{\
abla}_{long}\))  
So far, we have assumed total observation times of up to one second. However, the IPT uses much longer observation times, usually on the order of several minutes. Can we use the CRB expressions obtained in Sections 3.3 and 4 for such long observation times? Unfortunately, we cannot use them for observation times \(T_{obs}\) longer than one second. This is due to the fact that, for \(T_{obs}>1\) s, the approximation described in Section 2, of considering the parameters in \(\boldsymbol{\
abla}\) to be constant over \(T_{obs}\), is no longer valid. In this case, the change in the delay of the received LOS signal cannot be well approximated anymore as a linear expression with an initial value of \(\tau_{0}\) s. In addition, neither the Doppler shift observed, nor the phase shift \(\phi_{0}\) with respect to the receiver oscillator can be considered as constant. That is due to the non-linear relative movement of the satellite and the receiver over long observation times and due to the signal propagating through the atmosphere, which effects the signal as it varies with time [62]. In addition, the direction in which the signals impinge on the antenna will vary over time as a consequence of the variation of the satellite's elevation, i.e., \(\theta_{el}\), and the azimuth angles observed. Subsequently, this variation can result in a change in the LOS signal amplitude \(u_{0}\) and an extra phase shift, i.e., a variation of \(\phi_{0}\), since these two parameters also model the effect the receiver's antenna radiation pattern. In realistic scenarios, modeling the reflection coefficient \(\Gamma\), even for homogeneous and horizontal flat surfaces, can be a complex matter. Nonetheless, we still can characterize \(\Gamma\) as a function of the  
Figure 5: \(\sqrt{\text{CRB}(h)}\) for different \(|\Gamma|^{2}\) and \(\Delta\rho\) values for the considered GNSS signals. Red lines are used to represent the level curves, each one labeled with its corresponding value of \(\sqrt{\text{CRB}(h)}\) (in m). \(B_{fe}\) represents the assumed front-end’s one-sided 3-dB bandwidth. vector \(\mathbf{s}_{\Gamma}\), containing the set of parameters related to the reflection surface physical properties; the signal's impinging angle on the surface; and the antenna gain towards the specular reflection point [61]. The two last terms can be expressed as a function of \(\theta_{el}\). Thus, we have that:  
\[\Gamma(\theta_{el})=f(\theta_{el};\mathbf{s}_{\Gamma}). \tag{44}\]  
If we consider \(h\) and the parameters in \(\mathbf{s}_{\Gamma}\) constant over \(T_{obs}\gg 1\) s, then we can compute the CRB for a new vector of unknown parameters \(\mathbf{\Psi}_{long}\), defined as:  
\[\mathbf{\Psi}_{long}\triangleq\left[\mathbf{u}_{0}^{T},\boldsymbol{\phi}_{0 }^{T},\mathbf{\tau}_{0}^{T},\mathbf{s}_{\Gamma}^{T},h\right]^{T}, \tag{45}\]  
where \(\mathbf{u}_{0}\), \(\boldsymbol{\phi}_{0}\) and \(\mathbf{\tau}_{0}\) are column vectors of \(L\) elements, which we refer to as \"fast-varying\" parameters. In our analysis, these parameters are associated with the values of \(u_{0}\), \(\phi_{0}\) and \(\tau_{0}\) at the \(l\)-th time interval of \(T_{coh}=\)1 s duration, with \(L=T_{obs}/T_{coh}\) being the total number of these intervals for \(T_{obs}\). We assume that \(\theta_{el}\) remains constant over each \(T_{coh}\) interval. The previous assumption is based on the fact that the rate of change of \(\theta_{el}\) over time observed by ground-based static receivers is very low, i.e. on the order of \(10^{-3}\circ/\)s [67]. We can now use the same FIM parameter transformation approach to compute the CRB\((\mathbf{\Psi}_{long})\), which was introduced in Section 3.3. This approach allows us to obtain \(\mathbf{J}(\mathbf{\Psi}_{long})\) as a function of the \(\mathbf{J}(\mathbf{\Psi})\) for each of the different \(L\) short time intervals \(T_{coh}\).