## 3 Derivation of the CRB for Short Observation Intervals
### Introducing Phase Coherence Using FIM's Parameter Transformation  
So far, we have derived an expression for the FIM of \(\xi\) given the scenario defined in Section 2. However, in the GNSS-R context, we are actually more interested in a different set of unknown parameters that can be related to the geometry of the scenario, as well as to the properties of the reflecting surface. Moreover, up to this point, we have not taken into account in our signal model the phase coherence presented by specular reflection. This implies that the LOS and the reflected signals maintain a fixed phase relationship with each other and with respect to the local oscillator clock, used as the reference, i.e., \(\phi_{1}=g(\phi_{0})\).  
In our approach, we have decided to compute the CRB for \(\xi\) and then transform it for a new set of parameters \(\mathbf{\Psi}\). This transformation of \(\mathbf{J}(\xi)\) allows us to conveniently introduce the phase coherence assumption. We define the new vector of unknown parameters as:  
\[\mathbf{\Psi}=\left[u_{0},\phi_{0},\rho_{0},\left|\Gamma\right|,\phi_{\Gamma},h\right]^{T}, \tag{29}\]  
where \(u_{0}\), \(\phi_{0}\), \(\rho_{0}\) are the amplitude, phase and propagation distance (\(\rho=\tau\,c\)) for the LOS signal component. \(\Gamma=\left|\Gamma\right|e^{i\phi_{\Gamma}}\) is the reflected signal coefficient, i.e., the complex attenuation experienced by the reflected signal when compared to the LOS. Its module is defined as:  
\[\left|\Gamma\right|=\frac{u_{1}}{u_{0}}. \tag{30}\]  
\(\Gamma\) takes into account the surface reflectivity coefficient, the antenna complex gain and the polarization mismatch. Thus, it will depend on the satellite elevation angle (\(\theta_{el}\)) and on the azimuth angle of arrival of the reflected signal to the antenna. Since \(\theta_{el}\) varies very slowly [67], we will assume it to remain constant for observation periods up to 1 s. \(h\) is the height of the receiver, defined as the orthogonal distance between the antenna phase center and the reflection surface. In our flat surface scenario, this height \(h\) relates to the propagation path difference between the LOS and the reflected component \(\Delta\rho\) as:  
\[h=\frac{\Delta\rho}{2\sin(\theta_{el})}, \tag{31}\]  
where \(\Delta\rho\) is defined as \(\Delta\rho=\left(\tau_{1}-\tau_{0}\right)c\). In addition, because of the coherence assumption on the reflected component, the phase difference between the two components, i.e., \(\Delta\phi=\phi_{1}-\phi_{0}\), can be expressed as:  
\[\Delta\phi=\phi_{\Gamma}-\frac{2\pi}{\lambda}\Delta\rho, \tag{32}\]  
where \(\lambda\) is the carrier wavelength.  
Following the approach described in [71], we can express the FIM for \(\mathbf{\Psi}\) as:  
\[\mathbf{J}\left(\mathbf{\Psi}\right)=\mathbf{G}\mathbf{J}\left(\xi\right)\mathbf{G}^{T}, \tag{33}\]  
where \(\mathbf{G}\) is the Jacobian transformation matrix defined as:  
\[\mathbf{G}=\left\{\frac{\partial\xi_{j}}{\partial\mathbf{\Psi}_{i}}\right\}. \tag{34}\]  
The expressions for each of the elements of \(\mathbf{G}\) are provided in Appendix A. The CRB for the parameters in \(\mathbf{\Psi}\) can be directly obtained as the diagonal elements of \(\mathbf{J}^{-1}\left(\mathbf{\Psi}\right)\). This two-step approach, i.e., first computing the FIM for \(\mathbf{\xi}\) and then transforming it, has been used for its simplicity. It allows us to redefine the bound for a different set of parameters and to introduce more assumptions into our signal model without having to derive the FIM expression from scratch.