# Shipborne GNSS-Determined Sea Surface Heights Using Geoid Model and Realistic Dynamic Topography
## 5 Derivation and Validation of Shipborne GNSS-Based Sea Surface Heights
### Vessel Sailing-Related Corrections  
In Figure 11b, abnormal peaks can be detected in the C2 campaign mass center solution. These coincide with the occasions of the vessel's stopping; for example, to collect water samples. This phenomenon occurs due to the disappearance of the squat effect that causes a vessel to sail deeper than its nominal draft. The squat is a function of a vessel's velocity and dimensions, but is also influenced by depth in shallower, more confined waters [57]. Since the surveys were generally conducted in relatively deep and open marine areas, it is assumed that depth has a negligible influence on the squat. The only slight exception could be campaign C3, which also surveyed shallower regions of the Baltic Sea (notice the near-shore routes in Figure 2).  
Due to occasional stops, the velocity could be related to height changes between a moving and static vessel, by utilizing filtered residuals over distances up to 3 km. These estimates that represent squat values at various velocities are presented in Figure 13a as black dots. Since the vessel's dimensions are constant, the squat is approximately a quadratic velocity function [37; 57]. It was, thus, estimated as a least-squares fit of a quadratic function to the empirical data. A linear squat approximation, also shown in Figure 13a, is further discussed in Section 6.4.  
Figure 12: Statistical properties of differences between unfiltered and filtered residuals (filtered results were subtracted from unfiltered) for antennas A1–A4 and the mass center solution (denoted CoM). The black lines are mean values, and colored bars denote standard deviation estimates. Crosses show 99 th percentile minimum and maximum differences.  
The second vessel sailing-related correction that must be considered is the static draft. Due to fuel consumption, the vessel floats gradually higher. The lowest/highest estimates of the static draft are, correspondingly, at the time of harbor departure/return. Thus, the static draft was approximated from differences in tape measurements conducted before and after marine survey campaigns. Since tape measurements may contain errors in the range of static draft itself (e.g., notice the difference between the campaign C4 estimates in Figure 13b), the empirical data were considered altogether, whereas all negative values were excluded. The static draft was then estimated as a distance-related least-squares fit of a linear trend (Figure 13b).  
The empirically derived vessel sailing-related squat (a function of the vessel's velocity \(v\)) and static draft (a function of the total traveled distance \(s\)) corrections were subsequently applied to the filtered residuals (Equation (5) modification; also notice the functions presented in Figure 13):  
\[\begin{array}{l}r_{CoM}^{F+C}(\varphi,\lambda,t)=r_{CoM}^{F}(\varphi,\lambda,t)-\{squat(v)\}-\{static\ draft(s)\}\\ =r_{CoM}^{F}(\varphi,\lambda,t)-\Big{\{}(-0.0012)\cdot[v(t)]^{2}\Big{\}}-\{5.9 5\cdot 10^{-5}\cdot s(t)\},\end{array} \tag{21}\]  
where the first set of curly braces denotes squat correction, and the second set, static draft correction. The values of \(v\) and \(s\) are given at a GNSS observation epoch \(t\). Note that the vessel's velocity is considered in knots, and the total traveled distance is in kilometers from the beginning (i.e., home harbor) of the campaign. The term \(r_{CoM}^{F+C}\) denotes filtered and corrected residuals, which were validated as described in the following section.