# GNSS-Based Vertically Monitoring of Super-Tall Buildings
## 2 Calculation of the Monitoring Point's Coordinate Time Series
### Criteria of Eliminating the Gross Errors  
Before eliminating the gross error of baseline vectors, we construct a closed loop with the reference network and the baseline vectors between the monitoring and reference stations at each epoch. In this paper, three-dimensional baseline vectors and a covariance matrix will be directly used for GNSS network adjustment according to the Specifications for global positioning system (GPS) surveys in China, and we can assume that the closed loop consists of n baselines. Then, the misclosures are computed in three coordinate components as:  
\[\left\{ \begin{array}{l}w_{X}=\sum\limits_{i=1}^{n}dX_{i}\\ w_{Y}=\sum\limits_{i=1}^{n}dY_{i}\\ w_{Z}=\sum\limits_{i=1}^{n}dZ_{i}\end{array} \right.. \tag{1}\]  
Figure 2: The flowchart diagram for calculating the monitoring point’s coordinate time series. Assuming that the baseline vectors with a different precision \(\sigma_{i}\) are not correlated, the tolerance of misclosures reads as follows [28]:  
\[\left\{ \begin{array}{l}w_{X}\leq 3\sqrt{\sum\limits_{i=1}^{n}\sigma_{i}^{2}}\\ w_{Y}\leq 3\sqrt{\sum\limits_{i=1}^{n}\sigma_{i}^{2}}\\ w_{Z}\leq 3\sqrt{\sum\limits_{i=1}^{n}\sigma_{i}^{2}}\\ w_{s}\leq 3\sqrt{\sum\limits_{i=1}^{n}\sigma_{i}^{2}}\end{array} \right. \tag{2}\]  
where \(w_{s}=\sqrt{w_{X}^{2}+w_{Y}^{2}+w_{Z}^{2}}\), \(\sigma_{i}=\sqrt{a^{2}+\left(bd\right)^{2}}\), \(a\) is the constant error of GNSS receivers in millimeters (\(a\) = 5 mm for static mode and \(a\) = 10 mm for kinematic mode), \(b\) is the coefficient of ratio error in ppm (\(b\) = 1 ppm for static and kinematic mode), and \(d\) is the length of the baseline in kilometers. According to the error-processing criterion, if the misclosure is 3 times larger than the STD of the misclosure, then all baselines in the closed loop can be considered unqualified. After rejecting the baselines by checking the formed closed loop, the remaining baselines are all qualified. Since the above process is based on the WGS-84 system, one can of course transform the baseline results as well as the closed loops into a Gaussian plane coordinate system (\(w_{x}\), \(w_{y}\), \(w_{H}\)) as follows [28]:  
\[\left[ \begin{array}{c}w_{B}\\ w_{L}\\ w_{H}\end{array} \right]=R_{B}\left[ \begin{array}{c}w_{X}\\ w_{Y}\\ w_{Z}\end{array} \right] \tag{3}\]  
\[\left[ \begin{array}{c}w_{x}\\ w_{y}\end{array} \right]=R_{S}R_{B}\left[ \begin{array}{c}w_{X}\\ w_{Y}\\ w_{Z}\end{array} \right] \tag{4}\]  
where the transformation matrices \(R_{B}\) and \(R_{S}\) are given by [28]  
\[R_{B}=\left[ \begin{array}{cc}-\frac{\sin(R_{b})\cos(L_{b})}{M+H}&-\frac{\sin (R_{b})\sin(L_{b})}{M+H}&\frac{\cos(B_{b})}{M+H}\\ -\frac{\sin(L_{b})}{(N+H)\cos(B_{b})}&\frac{\cos(L_{b})}{(N+H)\cos(B_{b})}&0 \\ \cos(B_{k})\cos(L_{k})&\cos(B_{k})\sin(L_{k})&\sin(B_{k})\end{array} \right] \tag{5}\]  
\[R_{g}=\left[ \begin{array}{cc}N\Big{[}\frac{1-e^{2}}{W^{2}}+\frac{e^{2}}{2}(1 -2\sin^{2}B_{k}+e^{2}\sin^{2}B_{k}\cos^{2}B_{k})\Big{]}&N\sin B_{k}\cos B_{k} \Big{[}l+\frac{1}{2}(5-6\sin^{2}B_{k})l^{3}\Big{]}&0\\ -N\sin B_{k}\Big{[}\frac{1-e^{2}}{W^{2}}l+\frac{1}{6}(5-6\sin^{2}B_{k})l^{3} \Big{]}&N\cos B_{k}\Big{[}1+\frac{l^{2}}{2}(1-2\sin^{2}B_{k}+e^{2}\cos^{2}B_{ k})\Big{]}&0\end{array} \right] \tag{6}\]  
where \(B_{k}\) and \(L_{k}\) are the latitude and longitude of station \(k\), respectively, \(l\) is the difference of longitude between station \(k\) and the central meridian used for projection, and \(M\) and \(N\) are the radii of curvature in the meridian and prime verticals, respectively. A minimally constrained least-squares adjustment (free network adjustment) is performed. The residuals that result from this adjustment are strictly related to the baseline vectors. These residuals are examined, and from them blunders that may have gone undetected through the analyses of misclosures can be found and eliminated. The baseline vector correction tolerances are as follows:  
\[\left\{ \begin{array}{l}V_{\Delta X}\leq 3\sigma_{i}\\ V_{\Delta Y}\leq 3\sigma_{i}\\ V_{\Delta Z}\leq 3\sigma_{i}\end{array} \right.