# Shipborne GNSS-Determined Sea Surface Heights Using Geoid Model and Realistic Dynamic Topography
## 2 Theoretical Principles  
With the knowledge of the vertical range (\(R\)) between a reference point on a vessel (e.g., GNSS antenna's reference point) and the sea surface (e.g., determined by a total station survey), instantaneous SSH can be calculated relative to a geodetic reference ellipsoid (e.g., GRS80), using GNSS measured ellipsoidal heights \(h\):  
\[iSSH(\varphi,\lambda,t)=h(\varphi,\lambda,t)-R, \tag{1}\]  
where \(\varphi\) and \(\lambda\) are the measurement points' geodetic latitude and longitude at a GNSS observation epoch \(t\), respectively. Instantaneous SSH is only an approximation of the actual SSH, as it contains the vessel's high-frequency attitude changes (e.g., pitch and roll motions) and the impact of sea state (i.e., waves). In addition, vessel sailing-related squat and static draft corrections must be applied, since these cause the vessel to sail with an offset relative to the reference level at which \(R\) is usually determined. Accurate derivation of shipborne GNSS-based SSH requires consideration of all these factors. During GNSS data sampling, an inertial measurement unit can be deployed for specifying the vessel's attitude [12; 32]. However, utilizing an inertial measurement unit may be costly and require dedicated software for data processing. As an alternative, the vessel's high-frequency attitude changes can be estimated and eliminated from GNSS measurements by using data from at least three GNSS antennas [11; 33]. The elimination of  
Figure 1: Interrelations between the used datasets. The solid lines denote geometry at a reference epoch, whereby the dashed lines show the vertical land motion (VLM) affected geometry at a GNSS observation epoch. Global sea level rise and geoid change trends are neglected for simplification (i.e., the sea’s surface is assumed to be unchanged). Notice that the depicted dynamic bias (DB) is negative, cf. Equation (3). the vessel's high-frequency attitude changes, by computing a joint height solution from multiple antennas to the vessel's mass center, is further discussed in Section 5.1, with a complementing data example. Note that such a joint height solution from multiple antennas retains the impact of waves, which manifests itself as heave motion (i.e., vertical movements of the vessel). The required SSH data can be separated from these height estimates by applying a low-pass filter [11, 32]. The filtering window (spatial) length can be several kilometers long, depending on the GNSS data sampling rate, implying that the filter may propagate errors to the resulting SSH data; for instance, in steep geoid gradient areas. Therefore, the heights should be reduced to residual values prior to low-pass filtering:  
\[r^{UF}(\varphi,\lambda,t)=iSSH(\varphi,\lambda,t)-\hat{DT}(\varphi,\lambda,t)- N(\varphi,\lambda), \tag{2}\]  
where \(r^{UF}\) denotes unfiltered residual values. The term \(\hat{DT}\) denotes the ever-changing DT during a shipborne GNSS survey and \(N\) geoidal heights. The latter can be obtained from a suitable geoid model, whereas DT must be estimated. As already discussed in the Introduction, the combination of TG and HDM datasets allows DT estimation [15, 30]. First, the DB values of an HDM are determined at the locations of TG stations:  
\[DB\Big{(}\varphi^{TG},\lambda^{TG},t\Big{)}=DT_{HDM}\Big{(}\varphi^{TG}, \lambda^{TG},t\Big{)}-DT_{TG}\Big{(}\varphi^{TG},\lambda^{TG},t\Big{)}, \tag{3}\]  
where \(DT_{HDM}\) and \(DT_{TG}\) are HDM- and TG-based DT, respectively (see also Figure 1). With a suitable spatial interpolation method, the DB estimates are then gridded with HDM resolution. These predicted offshore DB (\(\hat{DB}\)) provide correction to the initial HDM:  
\[\hat{DT}\Big{(}\varphi^{HDM},\lambda^{HDM},t\Big{)}=DT_{HDM}\Big{(}\varphi^{ HDM},\lambda^{HDM},t\Big{)}-\hat{DT}\Big{(}\varphi^{HDM},\lambda^{HDM},t \Big{)}, \tag{4}\]  
where \(\hat{DT}\) denotes the corrected HDM-based DT. Dynamic topography at the GNSS measurement locations can finally be estimated via bilinear interpolation.