## 4 Research Results
### Designation of the CBA and the Distance Between the Geoid and Quasigeoid
If DEM is expressed in rectangular coordinates, the topographic reduction will take the form:  
\[\delta g_{Top}\equiv G\left(\sum\
olimits_{i=1}^{n}p_{i}\int_{z_{i1}}^{z_{i2}} \int_{y_{i1}}^{y_{i2}}\int_{x_{i1}}^{x_{i2}}\frac{\left(z_{i}-H_{P}\right)}{l_ {i}^{3}}dx_{i}dy_{i}dz_{i}\right)=G\left(\sum\
olimits_{i=1}^{n}\rho_{i}K_{i}\right) \tag{26}\]  
where \(n\) is the number of rectangular prisms of DEM; \(x_{i1}\), \(x_{i2}\), \(y_{i1}\), \(y_{i2}\), \(z_{i1}\), \(z_{i2}\) are the coordinates defining the rectangular prism \(i\) of DEM; \(\rho_{i}\) is its constant density; \(l_{i}=\sqrt{\left(x_{i}-X_{P}\right)^{2}+\left(y_{i}-Y\right)^{2}+\left(z_{i} -Z_{P}\right)^{2}}\) and \(K_{i}=\int_{z_{i1}}^{z_{i2}}\int_{y_{i1}}^{y_{i2}}\int_{x_{i1}}^{x_{i2}}\frac{ \left(z_{i}-H_{P}\right)}{l_{i}^{3}}dx_{i}dy_{i}dz_{i}\). The solutions of [PERSON]'s integral (26) are presented by [18; 19]. If we have the mean square errors (standard deviations) of densities \(\left(\sigma_{\rho_{i}}\right)\), we will write on the basis of the law of covariance propagation:  
\[\sigma_{\delta g_{Top}}=G\sqrt{\sum\
olimits_{i=1}^{n}K_{i}^{2}\sigma_{\rho_{ i}}^{2}} \tag{27}\]  
Further, we assume that the impact of the density errors on the topographic reduction is equal to the impact of these errors on the CBA, so:  
\[\sigma_{CBA_{\rho}}=\sigma_{\delta g_{Top}} \tag{28}\]  
Value \(\sigma_{N-\zeta_{\rho}}\) can be found based on Equation (18), where \(\overline{g}\) is determined, as mentioned in Section 2.2.4, by Poincare and Prey reduction according to the procedure presented:  
\[\overline{g}=g_{P}-\delta g_{F}-\delta g_{Top}+\delta g_{Top}^{\prime} \tag{29}\]  
where \(\delta g_{F}\) is free-air reduction from point \(P\) down to point \(P^{\prime}\) (Figure 2), \(\delta g_{Top}\) is given by Equation (26), and \(\delta g_{Top}^{\prime}\) is topographic reduction calculated for point \(P^{\prime}\):  
\[\delta g_{Top}^{\prime}\equiv G\left(\sum\
olimits_{i=1}^{n}p_{i}\int\limits _{z_{i1}}^{z_{i2}}\int\limits_{y_{i1}}^{y_{i2}}\int\limits_{x_{i1}}^{x_{i2}} \frac{\left(z_{i}-H_{P^{\prime}}\right)}{l_{i}^{3}}dx_{i}dy_{i}dz_{i}\right)=G \left(\sum\
olimits_{i=1}^{n}\rho_{i}K_{i}^{\prime}\right) \tag{30}\]  
When topographic reductions \(\delta g_{Top}\) and \(\delta g_{Top}^{\prime}\) are calculated based on the same DEM, Equation (29) can be written in the form:  
\[\overline{g}=g_{P}-\delta g_{F}-G\left(\sum\
olimits_{i=1}^{n}\rho_{i}K_{i} \right)+G\left(\sum\
olimits_{i=1}^{n}\rho_{i}K_{i}^{\prime}\right)=g_{P}- \delta g_{F}-G\sum\
olimits_{i=1}^{n}\rho_{i}(K_{i}-K_{i}^{\prime}) \tag{31}\]  
Based on this, with given values \(\sigma_{\rho_{i}}\), we can calculate the mean square error of the value \When analyzing the presented results, it should be noted that in most of the study area, the formal errors of the topography density used in calculations are small (up to about 250 \(\frac{\text{kg}}{\text{m}^{3}}\)). This suggests a very precise determination of \(\text{CBA}_{\textit{LINB}}\) and \((N-\zeta)_{\textit{LINB}}\) values (formal errors mostly at the level of 1-3 mGal for \(\text{CBA}_{\textit{LINB}}\) and 0.1-0.4 cm for \((N-\zeta)_{\textit{LINB}}\) (Figure 9)). Therefore, it should be considered that in the regions where the differences \(\delta\text{CBA}\) and \(\delta(N-\zeta)\) are greater (Figure 7), the \(\text{CBA}_{\textit{GGI}}\) and \((N-\zeta)_{\textit{GGI}}\) values are inappropriate (also remember that the \(\text{CBA}_{\textit{GGI}}\) values we previously considered low quality). On the other hand, as was shown previously, the gravity prediction results using the UNB_TopoDens model are clearly not as good as the other analyzed versions. These results do not follow to the principle that with well-chosen densities of topography, the CBA and topographic-isostatic anomalies should be smoother, which should result in better prediction.