# Consolidating ICESat-2 Ocean Wave Characteristics with CryoSat-2 during the CRYO2 ICE Campaign
## 3 Methods
### Atl03
#### 3.3.1 Histogram Model  
The significant wave height approximately follows a Rayleigh distribution (\(p_{H}(H)\)) in deep water scenarios, with a probability density function (PDF) of  
\[p_{H}(H)=\frac{H}{4m_{0}}\exp\left(-\frac{H^{2}}{8m_{0}}\right), \tag{1}\]  
where \(H\) is the wave height and \(m_{0}\) is the variance of the ocean surface [1]. As the SWH is defined as the average of the highest one-third of the waves, calculating it requires the individual surface waves to be distinguished. This is not always possible; therefore, the corresponding cumulative distribution function (CDF) of the Rayleigh distribution can be evaluated to provide the relationship \(\mathrm{SWH}=4\sqrt{m_{0}}\), and therefore only the variance of the surface needs to be known [1].  
The high-resolution laser data from ATLAS enables us to observe the individual surface waves of the ocean and thereby establish an assessment of the SWH using the distribution of the individual surface waves; this method is described in [3]. This method of using each return photon event as an individual surface measurement requires robust filtering and model fitting to reconstruct the measured \"ground\" surface, as seen in Figure 4.  
The ground tracks are divided into 10 m along-track segments and are thereafter filtered with a hard limit of \(\pm 20\) m from the EGM2008 geoid, and photon heights that are \(\pm 3\)\(\sigma\) from the mean in each 10 m bin are also filtered out. The data are subsequently fitted with a smoothing Savitzky-Golay filter, which preserves the peak amplitudes of the signal while smoothing the noisy data. If there are missing data in less than four consecutive 10 m bins, the data are fitted with a local spline before the smoothing filter is applied. The crests and troughs are thereafter located while correcting for the distribution of photons around the mean (to establish a 0 m height that represents a theoretical calm sea). In 10 km segments with 500 m of overlap, the wave heights are then sorted by height, and the mean of the highest one-third of the waves is then used to estimate the SWH in that section. To check for low amounts of data, a lower limit of at least 15 waves in a segment of 10 km is set; this provides an amount of data coverage comparable to that of ATL12.  
As the histogram model derives the SWH from the individual observed surface waves, the wave distribution along the ground track is observed and provides an additional  
Figure 4: (**Left**) Scatterplot of the geolocated surface photons along a segment of a ground track (\(\sim\)2.5 km along the surface) of ICESat-2 granule 10460805 beam GT3L. (**Right**) Histogram of the observed ocean wave heights showing the locations of the highest 1/3 (\(H_{1/3}\)) of wave heights and the corresponding SWH for the 10 km segment starting at the latitude 63.17\({}^{\circ}\) N.  
ocean observation output. For the single granule of ICESat-2 data (ID: 10460805) shown in Figure 4, that is, for a segment of 10 km (only \(\sim\)2.5 km of which is shown in the left plot), 54 individual waves were found from a single beam; they have an SWH of 2.66 m, a maximum wave height of 3.26 m, and a minimum wave height of 0.9 m.