## 2 Data and Methods
### Semi-Analytical Model  
To retrieve information on the SWH of the open ocean surface, we match the recorded radar reflection to an idealised semi-analytical waveform. The shape of the idealised waveform is a function of the reflected surface roughness, that for the open ocean surface is dependent on the height of ocean surface gravity waves. The waveform model is defined over the range of expected SWHs (0 to 10 m) and for the satellite's orientational parameters. [PERSON] et al. (2012) provide a model of a synthetic aperture, interferometric satellite radar altimeter echo power and echo cross product. This model depends on many instrument specific parameters (including the instrument transmit pulse \(p_{t}(t)\)) and the knowledge of the instrument location (range, inclination and velocity), which when combined with assumptions about the Gaussian nature of surface scattering \(s(z)\) gives a large semi-analytical convolutional form of the echo power cross product. For the retrieval of surface gravity waves the ocean surface is considered as a flat surface from which theoretical radar responses can be derived ([PERSON] et al., 2012). Individual recorded radar echo power waveforms \(P(t)\), where \(t\) is the time scale of the recorded echo power, can be described mathematically as the convolution  
\[P(t)=p_{t}(t)*P_{FS}(t)*s(ct/2), \tag{1}\]  
where \(p_{t}(t)\) is the transmit pulse, \(P_{FS}(t)\) is the flat surface impulse response and \(s(ct/2)\) is the distribution of surface scatterers for \(c\) the speed of light in a vacuum. The radar pulse transmitted from the SIRAL instrument has a shape that can be approximated as a Gaussian function  
\[p_{t}(t)=\frac{P_{0}}{\sqrt{\pi}}\exp\left[-\left(\frac{t}{\Delta t_{p}/2} \right)^{2}\right] \tag{2}\]  
where \(P_{0}\) is the transmitted pulse peak power (later solved for within Equation (6)) and \(\Delta t_{p}\) is the duration of the transmitted pulse (see Table 2 for values used). The probability distribution of the ocean surface with uniform waves is given as the Gaussian:  
\[s(z)=\frac{1}{\sqrt{2\pi\sigma_{\rm s}}}\exp\left[-\frac{z^{2}}{2\sigma_{\rm s }^{2}}\right] \tag{3}\]  
where is \(z=ct/2\) is the instrument altitude and \(\sigma_{\rm s}\) is the surface roughness standard deviation. As with measurement convention the significant wave height is taken as \(4\sigma_{\rm s}\). \begin{table}
\begin{tabular}{c c c c} \hline
**Buoy Name** & **Period** & **Location** & **Correlation** \\ \hline Swift array & 28 July 2014 to 28 September 2014 & \(-144.5^{\circ}\)E, 70.9\({}^{\circ}\)N & 0.77 \\  & & \(-159.9^{\circ}\)E, 74.6\({}^{\circ}\)N & \\ \hline Met Office K7 & 2014 & \(-4.5^{\circ}\)E, 60.7\({}^{\circ}\)N & 0.89 \\  & 2015 & & 0.91 \\  & 2016 & & 0.97 \\  & 2017 & & 0.96 \\  & 2018 & & 0.96 \\  & 2019 & & 0.96 \\ \hline
48,213 & 1 September 2012 to 7 October 2012 & \(164.1^{\circ}\)E, 71.5\({}^{\circ}\)N & \(-0.43\) \\  & 1 August 2013 to 31 October 2013 & & 0.14 \\ \hline
48,214 & 1 September 2014 to 11 October 2014 & \(165.2^{\circ}\)E, 70.9\({}^{\circ}\)N & \(-0.35\) \\  & 1 August 2014 to 8 October 2014 & & 0.51 \\  & 14 July 2015 to 12 October 2015 & & 0.5 \\ \hline \end{tabular}
\end{table} 
Table 1: Wave buoy data used in this study, plus correlation to our new SWH. All buoy locations are under the SAR mode operation of CS2. Pearson correlation coefficients between our new SWH and the wave buoy measurements are listed. See Section 3.5 for details. The flat surface response for a single beam \(P_{FS}^{k}\) (numbered \(k\)) of the synthetic aperture stack is given by [28] as a modified form of that of a pulse limited altimeter with  
\[