# Shipborne GNSS-Determined Sea Surface Heights Using Geoid Model and Realistic Dynamic Topography
## 5 Derivation and Validation of Shipborne GNSS-Based Sea Surface Heights
### Reducing the Effects of Sea State Conditions  
The computed instantaneous SSH, now with a substantially reduced impact from the vessel's high-frequency attitude changes (i.e., pitch and roll motions), still contains some influence from sea state conditions (i.e., waves), which primarily manifest as the vessel's up-down direction heave motion. As a result, instantaneous SSH contains high-frequency variations that should be eliminated from the expected SSH; for instance, by applying a low-pass filter. This study used the approach developed by [24] and successfully employed by [11]:  
\[OUT(i)=\frac{1}{F}\sum_{m_{2}=m_{1}-\frac{F-1}{2}}^{m_{1}+\frac{F-1}{2}}Median_{ m_{2}}\bigg{\{}IN_{m_{1}}(i)\bigg{|}i-\frac{F-1}{2}\leq m_{1}\leq i+\frac{F-1}{2} \bigg{\}}, \tag{20}\]  
where \(IN\) and \(OUT\) denote input and output data, respectively, and \(F\) is the filtering window size (i.e., a certain number of measurements). As in Equation (18), \(i\) represents a GNSS observation number. Refs. [11; 24] used the low-pass filter directly on SSH data. Recall the temporal data resolution of 15 or 30 s, which at a vessel velocity of around 9 knots results in a filtering window (spatial) length of several kilometers. With such lengthy window sizes, low-pass filtering may contaminate results, due to the gradients of DT and geoid. Thus, (unfiltered) residual estimates were first calculated from \(iSSH_{CoM}^{Final}\) using Equation (2) (note that the residuals are not dependent on the reference epoch \(t_{0}\), since this dependency disappears using a compatible geoid model NKG2015). A sum of absolute differences (between unfiltered and filtered signals) function was then compiled for each marine survey campaign, by applying the Equation (20) low-pass filter on unfiltered residuals. Only odd filtering window sizes were considered, and the optimal windows were estimated as sizes where the functions became roughly linear. The results are summarized in Figure 9. For comparability, the functions are normalized according to the maximum values. Figure 8: Statistical properties of discrepancies \(d_{A\xi}\) for antennas A1–A4. The black lines are mean values, and colored bars denote standard deviation estimates. Crosses show 99 th percentile minimum and maximum discrepancies. Notice that the functions are similar in shape, except for the C5 campaign, which is much shorter than the others (i.e., containing fewer measurements; cf. Table 1). However, the optimal filtering window sizes vary, likely due to approximations assumed during the size determination. For consistency, the final filtering window size (used for all data filtering) was estimated as a weighted mean (according to the campaign distances presented in Table 1) of an individual campaigns' optimal window sizes. The resulting 53 measurements are almost the same as the window size (51 measurements, at which the filtered data had a similar signal frequency to the geoid) suggested by [11]. Notably, such a filtering approach is independent of temporal data resolution (unless the GNSS sampling rate is very high, i.e., sub-second), since sea state conditions cause random data noise; hence, allowing investigation of the filtering window (spatial) length dependency on the GNSS sampling rate and vessel velocity (Figure 10). Ideally, the filtering window length should approximately match the spatial resolution of the employed model datasets (i.e., geoid model and HDM), to decrease error propagation. At a vessel velocity of 9 knots, this implies a GNSS sampling rate of around 5 s. In this study, the sampling rates were 15 or 30 s, and the filtering window lengths were, correspondingly, roughly 4 and 8 km. Future studies and marine survey campaigns should consider the results in Figure 10.  
Low-Pass Filtering Results  
The Equation (20) low-pass filter, with the filtering window size of 53 measurements, was applied on the unfiltered residuals, resulting in the filtered residuals denoted as \(r^{F}_{CoM}(\varphi,\lambda,t)\) in the following.