# 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  
Determination of accurate SSH requires considering the vessel's pitch and roll motions (Figure 7), where the yaw motion can be neglected, as the focus is on heights. In this study, the approach developed by [11] was improved using an additional GNSS antenna. By computing instantaneous SSH from multiple antennas jointly to the vessel's stable mass center, the method eliminates (or, at the very least, significantly reduces) the vessel's high-frequency attitude changes from the joint SSH solution. The distances between GNSS antennas relative to the vessel's mass center (Figure 7) were derived from total station measurements (see also [23]). The GNSS determined instantaneous SSH (cf. Equation (8)) from three antennas were first interpolated linearly to the location of the vessel's center of mass in two steps:  
\[iSSH_{X\alpha}(\varphi,\lambda,t,t_{0})=iSSH_{A\gamma}(\varphi,\lambda,t,t_{0} )+c_{1}\Big{[}iSSH_{A(\gamma+1)}(\varphi,\lambda,t,t_{0})-iSSH_{A\gamma}( \varphi,\lambda,t,t_{0})\Big{]} \tag{13}\]  
\[iSSH_{CoM}^{\xi}(\varphi,\lambda,t,t_{0})=iSSH_{X\alpha}(\varphi,\lambda,t,t_{ 0})+c_{2}\Big{[}iSSH_{A(\{1,2,3,4\}-\{\gamma,\gamma+1,\xi\})}(\varphi,\lambda,t,t_{0})-iSSH_{X\alpha}(\varphi,\lambda,t,t_{0})\Big{]}, \tag{14}\]  
where the index \(\alpha\) denotes the number of an imaginary intersection point, the index \(\gamma\) number of an antenna and \(\xi\) solution numbered according to an excluded fourth antenna (also refer to Figure 7 and Table 2). Coefficients \(c_{1}\) and \(c_{2}\) are determined from total station measurement derived distances. The geodetic coordinates (\(\varphi\), \(\lambda\)) represent the general location of the vessel (e.g., chosen according to the best performing antenna; in this study, coordinates of the antenna A4 were consistently used), and \(t\) is an observation epoch of a GNSS measurement. These calculations resulted in four \(iSSH_{CoM}^{\xi}\) solutions (i.e., instantaneous SSH at the vessel's mass center). Unfortunately, such calculations were not possible for the C4 campaign, due to the malfunctioning of two instruments [23]. For the C4 campaign, the instantaneous SSH of antennas A1 and A2 were averaged at observation epochs \(t\). Figure 6: The (**a**) predicted DB and (**b**) corrected HDM-based DT during marine survey campaigns. Since the four antennas form a rigid system, the instantaneous SSH at the vessel's mass center can be further used for validation purposes. Hence, the instantaneous SSH estimates at the previously excluded fourth antenna location were interpolated similarly: \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Index/Coefficient**} & **Solution** & **Solution** & **Solution** & **Solution** \\  & \(\xi\)=1 & \(\xi\)=2 & \(\xi\)=3 & \(\xi\)=4 \\ \hline \(\alpha\) & 3 & 4 & 1 & 2 \\ \(\beta\) & 4 & 3 & 2 & 1 \\ \(\gamma\) & 3 & 3 & 1 & 1 \\ \(c_{1}\) & \(2.05+1.48\) & \(2.05+1.48+1.68\) & \(4.59+4.32\) & \(8.05\) \\ \(c_{2}\) & \(4.45+13.63\) & \(4.08+13.70\) & \(14.92\) & \(17.11\) \\ \(c_{3}\) & \(2.05+1.48\) & \(2.05+1.48\) & \(8.05\) & \(4.59+4.32\) \\ \(c_{4}\) & \(4.08+13.20\) & \(2.05+1.48+1.68\) & \(7.16+4.59\) & \(4.59+4.32\) \\ \(c_{4}\) & \(4.08+13.20\) & \(4.45+13.63\) & \(12.11+4.441\) & \(14.92+5.37\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The solutions’ (\(\xi\) ) associated indexes (\(\alpha\), \(\beta\), and \(\gamma\) ) and coefficient values (\(c_{1}\), \(c_{2}\), \(c_{3}\), and \(c_{4}\)) also see Figure 7). Figure 7: Pitch, roll, and yaw motions of a moving vessel and relative locations of the used GNSS antennas (colored blue). The red lines are the principal axes of the vessel, whereas the black lines denote total station measurement derived distances between antennas and imaginary (magenta colored) intersection points. All points are shown in horizontal projection (i.e., as reduced to the sea surface).