# Shipborne GNSS-Determined Sea Surface Heights Using Geoid Model and Realistic Dynamic Topography
## 3 Shipborne GNSS Surveys, Data, and Other Used Information
By employing the principles outlined in [26], the GNSS measurements were corrected retrospectively for VLM occurring at the GNSS-CORS:  
\[h(\varphi,\lambda,t,t_{0})=h_{0}\Big{(}\varphi^{RS},\lambda^{RS},t_{0}\Big{)}+ dh(t)+\Big{[}VLM_{geometric}\Big{(}\varphi^{RS},\lambda^{RS}\Big{)}-GC(\varphi, \lambda)\Big{]}\cdot(t-t_{0}), \tag{7}\]  
where \(h_{0}\) is the ellipsoidal height of a reference station at a reference epoch \(t_{0}\), and \(dh\) represents the estimated height difference between a reference station and remote GNSS measurement at an observation epoch \(t\). The term \(VLM_{geometric}\) denotes the geocentric VLM rate at a GNSS reference station, and \(GC\) is the geoid change rate at a GNSS measurement location. The VLM and geoid change rates were obtained from the NKG2016 LU VLM model [36]. It became evident during GNSS post-processing that some of the vessel's routes were too distant from the GNSS-CORS (westernmost routes in the Baltic Proper shown in Figure 2), resulting in reduced post-processed data quality. Previous studies [24; 37; 38] have shown that the Canadian CSRS-PPP online global precise point positioning service [39] can provide reliable post-processing for remote shipborne GNSS measurements. In particular [37], demonstrated that CSRS-PPP is a viable option for post-processing GNSS data, even in a transoceanic scenario for determining SSH. The results in [23] indicate that CSRS-PPP-based data are consistent, regardless of the distance from the coast and, thus, suitable for complementing poor-performing Trimble Business Centre post-processed route sections. Therefore, the poor-performing sections were replaced by CSRS-PPP solutions. The results of a total station survey and tape measurements (in the harbor) allowed reducing the post-processed and VLM corrected GNSS ellipsoidal heights to the sea surface. The four GNSS antennas' reference points and three benchmarks on the vessel's railing were assigned coordinates in an arbitrary local system, whereby tape measurements (conducted separately for each campaign) determined the vertical distances between benchmarks and the sea surface. The ellipsoidal heights \(h\) obtained using Equation (7) were transformed into instantaneous SSH as (Equation (1) modification):  
\[iSSH(\varphi,\lambda,t,t_{0})=h(\varphi,\lambda,t,t_{0})-H^{ARP}_{TS}+H^{BM}_ {TS}-H^{BM}_{tape}, \tag{8}\]  
where \(H^{ARP}_{TS}\) and \(H^{BM}_{TS}\) denote the total station determined heights of an antenna's reference point and a benchmark on the railing, respectively, and \(H^{BM}_{tape}\) is a distance measured by tape. Additional details about the total station survey and instantaneous SSH calculations can be found in [23]. This contribution now aims to further process the instantaneous SSH data, so that these are suitable for subsequent applications (e.g., validation of SA results, marine geoid models, HDMs' performance). However, Section 4 first examines the derivation of offshore DT, since SSH computations were performed jointly with data validation, which also requiresgeoidal heights. The latter were obtained from the high-resolution (0.01 \(\times\) 0.02 arc-deg, i.e., approximately 0.6 nautical miles) NKG2015 quasigeoid (over marine areas, the geoid coincides with the quasigeoid; henceforth, the shorter term will be used) model [40] (cf. Figure 2). Note that NKG2015 represents a geoid model using the zero-tide permanent tide concept, to which a correction from zero-tide to the tide-free concept has been applied (see [41; 42] for details about permanent tide concepts). The correction provides consistency with GNSS measurements (using the tide-free concept). In other words, by subtracting NKG2015 heights from the GNSS determined ellipsoidal heights, normal heights using the zero-tide concept are obtained.