# GNSS-Based Vertically Monitoring of Super-Tall Buildings
## 3 Matching the Results of Static GNSS Measurements and Laser Plumbing
### Bias Correction of GNSS Measurements and Laser Plumbing  
To make use of GNSS surveying to check the plumbing result of a laser plummet, we need to eliminate the systematic bias of GNSS affected by deflections of the vertical. In other words, the GNSS coordinates should be transformed into a system with the same datum.  
As shown in Figure 3b, the deflection of the vertical u is decomposed into two perpendicular components: a north-south meridional component \(\zeta\) (the positive direction is toward the south) and an east-west prime vertical component \(\eta\) (the positive direction is toward the west). The geodetic coordinates \(P\)-\(xyz\) are related to the normal ellipsoid. The \(z\)-axis points toward the interior of the ellipsoid along the ellipsoid normal. The \(z^{\prime}\)-axis points toward the interior of the geoid along the plumb line. The \(x\)-axis points toward the north (N) and the \(y\)-axis points east (E) to form an orthogonal, right-handed rectangular coordinate system. The points \(P_{0}\) and \({P_{0}}^{\prime}\) are the projections from the point P along the \(z\)-axis and the \(z^{\prime}\)-axis, respectively. The projection heights of the points \(P_{0}\) and \({P_{0}}^{\prime}\) are expressed by \(h\) and \(h^{\prime}\), respectively. In general, the deflection of the vertical is so small (e.g., 10 arc seconds) that the projection heights \(h\) and \(h^{\prime}\) (with a building height of 432 m, the difference between \(h\) and \(h^{\prime}\) is about 2 cm) are approximately equal; therefore, the effect of curvature of the plumb line is neglected in this paper.  
The plane-coordinate difference between the points \(P_{0}\) and \({P_{0}}^{\prime}\) can be approximately calculated as  
\[\Delta x\approx-h\tan\zeta \tag{12}\]  
Figure 3: Difference between the Global Navigation Satellite System (GNSS) measurement and laser plumbing. (**a**) The relationship between plumb line and ellipsoid normal during monitoring; (**b**) Decomposition of the deflection of the vertical \(u\).  
\[\Delta y\approx-h\tan\eta. \tag{13}\]  
Therefore, the authors can assess the systematic bias between the coordinates of the GNSS measurement and laser plumbing using Equations (12) and (13).  
In order to analyze the systematic bias between the coordinates of the GNSS measurement and laser plumbing at different building heights, the authors can assume that a building (\(23^{\circ}02^{\prime}24.17311^{\prime\prime}\) N, \(113^{\circ}23^{\prime}43.95410^{\prime\prime}\) E) located at Guangzhou city has different heights. Then, the deflection of the vertical components can be calculated by Equations (8) and (9) with the Earth Gravitational Model EGM2008 (to degree 2160) [31] at different heights (e.g., \(H\) = building height). Finally, an approximation of the systematic bias between the coordinates of the GNSS measurement and laser plumbing can be calculated by Equations (12) and (13). Table 1 lists the approximations of the systematic bias in the \(x\) and \(y\) directions at different building heights.  
Referring to the values listed in Table 1, one may see that the systematic bias between the coordinates of the GNSS measurement and laser plumbing is small when the building height is smaller than 100 m for a building in Guangzhou. However, the systematic bias increases with an increase in building height when the building height exceeds 100 m. Therefore, the systematic bias cannot be ignored in verticality monitoring of super-tall buildings.