# A New Model of Quasigeoid for the Baltic Sea Area
## 4 Quasigeoid Computation
### Qusigeoid Validation  
To estimate the absolute accuracy of the gravity model the difference  
\[\delta_{i}=(h_{i}-H_{i})-\xi_{i}^{grav} \tag{13}\]  
was computed in each point of the satellite GNSS network, where \(\xi_{i}^{grav}\) is quasigeoid from gravity data, and the mean value from  
\[\hat{x}=\frac{1}{n}\sum_{i=1}^{n}\delta_{i} \tag{14}\]  
The empirical standard deviation is computed from  
\[\hat{v}=\pm\sqrt{\frac{\sum v_{i}^{2}}{n-1}} \tag{15}\]  
which gives the numerical description of the accuracy of the tested quasigeoid model (Table 4). Such an assessment of the accuracy of quasigeoids is obtained on the assumption that the difference (h-H) is determined without errors. Since \(\sigma_{\text{h}}\) is a few millimeters (2-4 mm) [44], \(\sigma_{\text{H}}\) is of the order of 6 mm [45], so the empirical standard deviation of \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Mean** & **Std dev** & **Min** & **Max** \\ \hline GOCE-DIR6 & 0.35 & 0.04 & 0.27 & 0.40 \\ GOCO06s & 0.36 & 0.04 & 0.27 & 0.40 \\ EIGEN-6C4 & 0.34 & 0.04 & 0.24 & 0.40 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Assessment of the accuracy, in meters, of the subsequent quasigeoid models at the points of the ASG-EUPOS network and Kronstadt86 vertical system. Figure 9: The final model of quasigeoid, in meters, from the GOCO model. the difference (h-H) is in the order of \(\sqrt{4^{2}+6^{2}}\approx 7\) mm. This means that gravimetric quasigeoids are computed with accuracy \(\sqrt{40^{2}-7^{2}}\approx 3.9\) cm. This means that the ASG-EUPOS network is suitable for testing gravimetric quasigeoids. The results in the table show very good compliance in the assessment of quasigeoid accuracy calculated from three different geopotential models. The analysis shows that the quasigeoids calculated in this work are 35 cm below the reference surface of the Kronstadt86 height system and their accuracy is in the order of \(\pm 4\) cm. A similar analysis can be carried out using the heights of the Polish leveling network in the EVRF2007 height system. The physical heights in the PL-KRON86-NH (Kronstadt86) system can be converted to EVRF2007 using the transformation proposed in the publication [47]. Transformational formulas were developed in two variants: in the form of polynomials approximated by the least-squares method and in the form of an interpolative grid. Basic empirical relationships were implemented among others in the program TRANSPOL v. 2.06 [48], elaborated according to the assumptions of the Head Office of Geodesy and Cartography in Poland. As a result of this analysis, the displacement of the calculated quasigeoid models relative to the reference area of EVRF2007 was 18 cm and the accuracy of the quasigeoid models' estimate remained unchanged at \(\pm 4\) cm (see Table 5). All three quasigeoid models were calculated using the same gravimetric data but using different geopotential models. Comparing the calculated quasigeoids with each other can provide interesting information on geopotential models. For this purpose, the difference between the different quasigeoid models and the statistical properties of these differences are calculated in Table 6 below. The spatial distribution of differences between models (GOCE-EIGEN) is shown in Figure 10. From this figure, it appears that in the area of the Baltic Sea, the differences are in the order of a few centimeters (up to 10 cm). In contrast, the land area is contaminated with much larger discrepancies from \(-0.5\) to \(+0.5\) m. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Mean** & **Std dev** & **Min** & **Max** \\ \hline GOCE-GOCO & 0.01 & 0.02 & \(-0.20\) & 0.15 \\ GOCE-EIGEN & 0.00 & 0.05 & \(-0.51\) & 0.49 \\ GOCO-EIGEN & 0.00 & 0.05 & \(-051\) & 0.52 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Difference (in meters) between computed quasigeoid.