# SNR-Based GNSS-R for Coastal Sea-Level Altimetry
## 3 Geodetic GNSS-R Model  
SNR observations that are reported by GNSS receivers are a function of interferometric power \(\left(P_{1}=\frac{P_{r}}{P_{d}}\right)\) and interferometric phase (\(\phi_{1}=\phi_{r}-\phi_{d}\)) [19]:  
\[SNR=\frac{P_{d}(1+P_{i})}{P_{n}}+\frac{2P_{d}\surd{P_{i}}}{P_{n}}\cos\phi_{i} \tag{2}\]  
The incoherent power is neglected in geodetic GNSS-R, as it only impacts the SNR trend and not the interference fringes [20]. The first term on the right-hand side is the SNR trend, and the second term represents detrended interference fringes. \(P_{r}\), \(P_{d}\), and \(P_{n}\) are reflected, direct, and noise powers, respectively. \(P_{r}\) and \(P_{d}\) can be expressed as a function of direction dependent Right-Handed Circular Polarization (RHCP) power component (\(P_{d}^{R}\)) and the complex-valued surface/antenna coupled coefficient (X) [20]:  
\[P_{d}=P_{d}^{R}G_{d}^{R}P_{r}=P_{d}^{R}X^{2}S^{2}X=R^{S}\sqrt{G_{r}^{R}}\exp \left(\imath\Phi_{r}^{R}\right)+R^{X}\sqrt{G_{r}^{L}}\exp\left(\imath\Phi_{r}^{ L}\right) \tag{3}\]  
where \(G_{d,r}^{R,L}\) represents the partial antenna power gains for RHCP and LHCP (Left-Handed Circular Polarization) evaluated in the direct and reflected directions. \(\Phi_{r}^{R,L}\) is the respective antenna phase component. \(R^{S}\) and \(R^{X}\) are the complex-valued Fresnel reflection coefficients for same-sense and crossed polarization, respectively. The coherent power loss factor (\(S^{2}\)) is a function of the carrier wavenumber (\(k=2\pi/\lambda\)) and can be directly linked to the standard deviation of surface height (s) [21]:  
\[S^{2}=\exp\left(-k^{2}s^{2}\sin^{2}e\right) \tag{4}\]  
Figure 2: Occurrences of FFZs in January 2021 at the POPR. Each color depicts a GNSS satellite, and the location of the Port-au-Prince tide gauge station is shown as a yellow pin. The interferometric phase can be modeled as a function of coupled surface/antenna properties (\(\phi_{X}\)), the wavenumber and interferometric propagation delay (\(\tau_{i}\)), and the direct signal antenna phase contribution (\(\Phi_{d}^{R}\)) [22]:  
\[\phi_{i}=\phi_{X}+k\tau_{i}-\Phi_{d}^{R} \tag{5}\]  
The latter can be approximated as a function of the height component of the Antenna Phase Center (APC) relative to the Antenna Reference Point (ARP) [20]. The interferometric propagation delay (Figure 3), while assuming a horizontal surface in vacuum, can be expressed as a function of antenna-to-surface reflector height (\(H\)) [23]:  
\[\tau_{i}=2H\sin e \tag{6}\]  
Therefore, environmental variables such as snow depth and sea level can be retrieved using periodograms of the interference patterns observed by GNSS stations [24; 25]. Including biases to compensate for deficiencies in the GNSS-R model, the forward model of SNR can be formulated as:  
\[SNR=P_{d}(1+\frac{P_{i}}{\left|B\right|^{2}}+\frac{2\sqrt{P_{i}}}{\left|B \right|}\cos\phi_{i})/P_{n}K^{2}\;\phi_{i}=\phi_{i}-\phi_{B} \tag{7}\]  
where \(K\) is the noise power bias, and \(B\) and \(\phi_{B}\) are interferometric power and phase biases, respectively. Figure 3: Geometry of a geodetic GNSS-R in vacuum. The red lines show excess path delay traveled by the sea-surface reflection relative to the direct signal (blue). The biases in the inverse model are fixed to their optimal a priori values and corrections for each parameter, and corresponding covariance matrices are estimated [25]. They can be expanded as a polynomial in terms of sine of elevation angle, e.g., [9],  
\[\phi_{B}=\sum_{j=0}^{n}\phi^{(j)}\sin^{j}e=\phi_{B}^{(0)}+\phi_{B}^{(1)}\sin e+ \phi_{B}^{(2)}\sin^{2}e+\cdots \tag{8}\]  
where the superscript in parentheses (\(j\)) is an index; otherwise, it is a power exponent. The linear term of the interferometric phase bias that can be converted to a quantity with units of meters, known as height bias (\(H_{B}\)), accounts for errors of the forward model reflector height a priori value (\(H_{A}\)) [19].