## 7 Conclusions and Future Work
We are considering as future work the derivation of CRB expressions for different kinds of ground-based receiver scenarios, which is not straightforward from the analytic point of view. **Author Contributions:** [PERSON] and [PERSON] conceived of the idea of deriving CRB within this context. [PERSON] carried out the theoretical derivations, designed and implemented the MATLAB software, performed the simulations and wrote the paper. [PERSON] contributed the analysis and the discussion of the results. All of the authors contributed to the revision of the manuscript. **Conflicts of Interest:** The authors declare no conflict of interest. Appendix A Expressions for the Transformation Matrices Used and Analytic Expressions for the CRB in Short Observation Times  
Here, we provide more detail about how the Jacobian transformation matrices, \(\mathbf{G}\) and \(\mathbf{A}\), which were described in Sections 3.3 and 5, are computed. In addition, we show how the CRB expressions for short observation times discussed in Section 4 were obtained. As discussed in Section 4.1, the FIM \(\mathbf{J}(\xi)\), which is expressed using Equations (20) and (26), becomes a diagonal matrix when \(\Delta\rho>\rho_{clip}\), i.e. :  
\[\mathbf{J}\left(\xi\right)=\frac{2N}{\sigma_{w}^{2}}\text{diag} \left\{\left[ \begin{array}{cc}R\left(0\right),&u_{0}^{2}R(0),&-u_{0}^{2}R^{ \prime\prime}(0),&R\left(0\right),&u_{1}^{2}R(0),&-u_{1}^{2}R^{\prime\prime}(0 )\end{array} \right]^{T}\right\}. \tag{11}\]  
However, we are interested in computing the CRB for the set unknown of parameters \(\mathbf{\Psi}\), already defined in Equation (29) as:  
\[\mathbf{\Psi}=\left[u_{0},\phi_{0},\rho_{0},\left|\Gamma\right|,\phi_{\Gamma},h\right]^{T}, \tag{12}\]  
by transforming \(\mathbf{J}(\xi)\) into \(\mathbf{J}(\mathbf{\Psi})\) (which is not diagonal anymore), as described by Equation (33), i.e. :  
\[\mathbf{J}\left(\mathbf{\Psi}\right)=\mathbf{G}\mathbf{J}\left(\xi\right) \mathbf{G}^{T}. \tag{13}\]  
As discussed in Section 3.3, the Jacobian transformation matrix \(\mathbf{G}\) for short observation intervals and this specific case with phase coherence between both signal components is expressed as:  
\[\mathbf{G}=\left\{\frac{\partial\xi_{j}^{}}{\partial\mathbf{\Psi}_{i}}\right\} =\left[ \begin{array}{cccccc}1&0&0&|\Gamma|&0&0\\ 0&1&0&0&1&0\\ 0&0&c^{-1}&0&0&c^{-1}\\ 0&0&0&u_{0}&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&-2k\sin\left(\theta_{el}\right)&2\sin\left(\theta_{el}\right)c^{-1} \end{array} \right]. \tag{14}\]  
With some algebra, the CRB for \(\mathbf{\Psi}\) is then obtained as:  
\[\text{CRB}\left(\mathbf{\Psi}\right)=\left[\mathbf{J}^{-1}\left(\mathbf{\Psi }\right)\right]_{ii}=\frac{\sigma_{w}^{2}}{2N}\left[ \begin{array}{c}\frac{ \frac{1}{R(0)}}{\frac{1}{u_{0}^{2}R(0)}}\\ -\frac{c^{2}}{u_{0}^{2}R^{\prime\prime}(0)}\\ \frac{|\Gamma|^{2}+1}{u_{0}^{2}R(0)}\\ \frac{\left(R^{\prime\prime}(0)-0)(b^{2}c^{2})(|\Gamma|^{2}+1)\right.}{u_{0}^{ 2}|\Gamma|^{2}|\Gamma|^{2}|\Gamma|^{2}|\Gamma|^{2}|\Gamma|^{2}|\Gamma|^{2}| \Gamma|^{2}|\Gamma|^{2}|\Gamma|^{2}|\Gamma|^{2}|\Gamma|^{2}|\Gamma|^{2}\left( \theta_{el}\right)R^{\prime\prime}(0)}\\ \frac{-c^{2}(|\Gamma|^{2}+1)}{4u_{0}^{2}|\Gamma|^{2}|\Gamma|^{2}\sin^{2} \left(\theta_{el}\right)R^{\prime\prime}(0)}\end{array} \right]. \tag{15}\]  
\(\text{CRB}(h)\), \(\text{CRB}(|\Gamma|)\), \(\text{CRB}(\phi_{\Gamma})\), defined in Expressions (35), (36) and (39), are obtained in a straightforward manner from the 3 rd, 4 th and 5 th elements of Expression (15), respectively. The CRB expression presented in Equation (41) for the phase altimetry case, described in Section 4.1.1, is obtained by removing the fifth row and the fifth column (both related to \(\Phi_{\Gamma}\)) from \(\mathbf{J}(\mathbf{\Psi})\) and then computing the inverse of this matrix. Now, the \(\text{CRB}_{all}(h)\) is the element related to \(h\) in the main diagonal, i.e., the fifth element.