# 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 Vessel's High-Frequency Attitude Changes
\[iSSH_{X\beta}(\varphi,\lambda,t,t_{0})=iSSH_{A\gamma}(\varphi,\lambda,t,t_{0})+c_{ 3}\Big{[}iSSH_{A(\gamma+1)}(\varphi,\lambda,t,t_{0})-iSSH_{A\gamma}(\varphi, \lambda,t,t_{0})\Big{]} \tag{15}\]  
\[iS\hat{S}H_{A\xi}(\varphi,\lambda,t,t_{0})=iSSH_{X\beta}(\varphi,\lambda,t,t_{0} )+c_{4}\Big{[}iSSH_{CoM}^{\xi}(\varphi,\lambda,t,t_{0})-iSSH_{X\beta}(\varphi, \lambda,t,t_{0})\Big{]}, \tag{16}\]  
where the index \(\beta\) denotes the number of an imaginary intersection point, and coefficients \(c_{3}\) and \(c_{4}\) are determined from total station measurement derived distances (Figure 7 and Table 2). The conducted calculations were validated by comparing the estimated (\(iS\hat{S}H_{A\xi}\)) and measured (\(iSSH_{A\xi}\)) heights:  
\[d_{A\xi}(\varphi,\lambda,t)=iS\hat{S}H_{A\xi}(\varphi,\lambda,t,t_{0})-iSSH_{ A\xi}(\varphi,\lambda,t,t_{0}), \tag{17}\]  
where \(d_{A\xi}\) denotes discrepancies at the location of the initially excluded fourth antenna. Statistics of discrepancies are summarized in Figure 8, indicating the expected GNSS height determination accuracy. It can be noticed that, generally, the discrepancy mean values are sub-centimeter, suggesting the successful reduction of ellipsoidal heights to the sea surface using Equation (8) (i.e., all four antennas are approximately on the same plane). Note that the weighted mean (according to campaign distances presented in Table 1) standard deviation estimate (by averaging the values shown in Figure 8) of 4.1 cm showed a good performance for height determination. Ideally, all four \(iSSH_{CoM}^{\xi}\) solutions should be free of vessel high-frequency attitude changes and have matching results considering the successful reduction of ellipsoidal heights. The performance of \(iSSH_{CoM}^{\xi}\) was evaluated as:  
\[\overline{\sigma}_{CoM}=\frac{1}{I}\sum_{i=1}^{I}\Bigg{\{}\sqrt{\frac{1}{3} \sum_{\xi=1}^{4}\Bigg{[}iSSH_{CoM}^{\xi}(\varphi,\lambda,i,t_{0})-\frac{1}{4} \sum_{\xi=1}^{4}iSSH_{CoM}^{\xi}(\varphi,\lambda,i,t_{0})\Bigg{]}^{2}\Bigg{\}}, \tag{18}\]  
where \(i=1,\ 2,\ \ldots,\ I\) is the number of a GNSS observation. These evaluation results vary between 0.4 cm (C5 campaign) and 1.1 cm (C1 campaign), indicating excellent agreement between the four solutions for all campaigns. Even though the C2 campaign validation suggests more significant (than usual) discrepancies at the antenna locations (Figure 8), the solutions agree well at the vessel's mass center (\(\overline{\sigma}_{CoM}\) is 0.8 cm). In practical applications use of three GNSS antennas, thus, appears sufficient, and any of the four \(iSSH_{CoM}^{\xi}\) solutions would be equally suitable for further use. Due to data availability, the final instantaneous SSH solution was calculated as:  
\[iSSH_{CoM}^{Final}(\varphi,\lambda,t,t_{0})=\frac{1}{4}\sum_{\xi=1}^{4}iSSH_ {CoM}^{\xi}(\varphi,\lambda,t,t_{0}) \tag{19}\]  
and used in subsequent data filtering. The temporal resolutions of \(iSSH_{CoM}^{Final}\) were 15 s for the first C1 campaign and 30 s for subsequent campaigns. Recall that the \(iSSH_{CoM}^{Final}\) of campaign C4 is a simple average of the A1 and A2 antenna heights.