## 2 Data and Methods
### Semi-Analytical Model
\begin{split} P_{FS}^{k}(t,B)&\approx\frac{\lambda^{2 }G_{0}^{2}D_{0}c\sigma_{0}}{32\pi^{2}h^{3}\eta}H\Bigg{[}t+\frac{\eta hg_{bb}^{k ^{2}}}{c}\Bigg{]}e^{-ik_{0}B(\chi+\beta/\eta)}\\ &\cdot\int_{0}^{2\pi}d\phi d[\rho_{k}\cos\phi-\xi_{b}^{k}]e^{ik_{ 0}B_{0}\rho_{k}\sin\phi}\\ &\cdot\exp\Bigg{[}-2\Bigg{(}\frac{(\rho_{k}\cos\phi-\mu)^{2}}{ \gamma_{1}^{2}}\\ &+\frac{(\rho_{k}\sin\phi-\mu-\beta/\eta)^{2}}{\gamma_{2}^{2}} \Bigg{)}\Bigg{]}\end{split}  \tag{4}\]  
where \(\lambda\) is the carrier wavelength, \(G_{0}\) is the antenna boresight gain, \(D_{0}\) is the one-way gain of the synthetic beam, \(\sigma_{0}\) is the backscatter coefficient, \(\mu\), \(\chi\) and \(h\) are the satellite pitch, roll and altitude, \(\beta\) is the across-track surface vector gradient, \(\eta\) is the factor to account for the curvature of the Earth and \(H[t]\) is the Heaviside step function. Furthermore to allow for the slant range correction  
\[\rho_{k}=\sqrt{\frac{ct}{\eta h}+\xi_{b}^{k}}\]  
where \(\gamma_{1},\gamma_{2}\) are coefficients to account for the ellipticity of the SIRALs antennas with  
\[\gamma_{1}=\sqrt{\frac{2}{2/\gamma^{2}+2/\gamma^{2}}},\gamma_{2}=\sqrt{\frac {2}{2/\gamma^{2}-2/\gamma^{2}}}.\]  
The function \(d[\rho_{k}\cos\phi-\xi_{b}^{k}]\) in Equation (4) as described by [PERSON] et al. is used to describe the beam gain for the SAR rectangular footprint window and the LRM footprint. Taking the semi-analytical form of a multi-look idealised SAR or LRM power waveform from Equation (4), the echoes depend on up to nine different instrument and surface variables: \(\mu,\chi,h,\xi,\eta,\
u_{s},t_{0},\sigma_{0},\sigma_{s}\). In SAR mode, it is possible to reduce this to two parameters if one is only interested in SWH. Firstly, the antenna baseline pitch and roll can be taken from the CS2 product, as they are measured independently by three star trackers mounted on the interferometer bench, as well as the satellite altitude. Secondly, [PERSON] et al. [23] find that for all but the largest possible altitude rates, the effect of the radial component of velocity on the beam formation is negligible (see [29] for further description). Finally, if the backscatter is not of interest, the absolute power is not important and the waveform can be normalised in such a way to optimise the fitting procedure. \begin{table}
\begin{tabular}{l c c} \hline \hline  & **Description** & **Value** \\ \hline \(\Delta t_{p}\) & Pulse duration & 3.52 ms \\ \(G_{0}\) & Antenna gain & 42 dB \\ \(D_{0}\) & SAR synthetic beam gain & 36.12 dB \\ \(\xi_{b}^{k}\) & beam look angles & 51 \\ \(c\) & Speed of light & \(3\times 10^{8}\) \\ \(\eta\) & & \(1+h/R_{e}\) \\ \(R_{e}\) & Earth radius & \(6.38\times 10^{6}\) m \\ \(\lambda\) & Carrier wavelength & \(c/(13.58\times 10^{9})\) \\ \(k_{0}\) & Carrier wave number & \(2.\pi/\lambda\) \\ \(\tilde{\gamma}\) & Antenna shape parameter & 0.0122 \\ \(\tilde{\gamma}\) & Antenna shape parameter & 0.0382 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter values for the semi-analytical retracker. [PERSON] et al. [23] provide the equational form to take the full convolution into a usable numerical form. The full echo power convolution from Equations (1)-(4) is fully integrated over the beam count \(k\) and range ring angle \(\phi\). This integration results in a fully analytical form for the echo power for specific \(\sigma_{s}\)and \(t\) (amongst other parameters) that can then be repeated over the expected range of \(\sigma_{s}\)and over the echo range \(t\) to provide look up tables. We used the simplifications for the specifics of the CS2 instrument (ellipticity of the antennas, relative smallness of the satellite pitch \(\mu\), roll \(\chi\) and altitude \(h\), as discussed above) to give analytical forms of \(\mu\), \(\chi\) and \(h\) while containing \(\sigma_{s}\) and \(t\) within lookup tables that provide the four components of the cross product \(\mathbf{H}^{\mathbf{m}}[\sigma_{s},t]\) for \(\mathbf{m}=\text{SAR}\), LRM, such that  
\[