# GNSS-Based Vertically Monitoring of Super-Tall Buildings
## 3 Matching the Results of Static GNSS Measurements and Laser Plumbing
### Determination of Deflection of the Vertical Components  
As shown in Figure 3, the deflection of the vertical \(\mathrm{u}\) is the angle between the gravity vector at a reference point and the ellipsoidal normal through the same point for a particular ellipsoid. It is conventionally decomposed into two perpendicular components: a north-south meridional component \(\xi\) and an east-west prime vertical component \(\eta\). In this paper, the deflections of vertical are determined from an Earth gravity field model and a geoid model.  
By using a set of Earth gravity field coefficients (e.g., EGM2008), one can calculate the deflection of the vertical component \(\xi\) along the meridian and \(\eta\) along the prime vertical as: [25]  
\[\xi(\varphi,\lambda,r)=-\tfrac{1}{\gamma r}\tfrac{\partial T}{\partial\varphi} =-\tfrac{GM}{r^{2}\gamma}\sum\limits_{m=0}^{N_{\mathrm{max}}}\left[\cos m \lambda\sum\limits_{n=\mu}^{N_{\mathrm{max}}}\left(\tfrac{\mu}{r}\right)^{n} \overline{\zeta}_{nm}\overline{P}_{nm}(\sin\varphi)+\sin m\lambda\sum\limits_{ n=\mu}^{N_{\mathrm{max}}}\left(\tfrac{\mu}{r}\right)^{n}\overline{S}_{nm}\overline{P}_{nm}( \sin\varphi)\right] \tag{8}\]  
\[\eta(\varphi,\lambda,r)=-\tfrac{1}{\gamma r\cos\varphi}\tfrac{\partial T}{ \partial\lambda}=-\tfrac{GM}{r^{2}\gamma\cos\varphi}\sum\limits_{m=0}^{N_{ \mathrm{max}}}m\left[\cos m\lambda\sum\limits_{n=\mu}^{N_{\mathrm{max}}} \left(\tfrac{\mu}{r}\right)^{n}\overline{S}_{nm}\overline{P}_{nm}(\sin\varphi )+\sin m\lambda\sum\limits_{n=\mu}^{N_{\mathrm{max}}}\left(\tfrac{\mu}{r} \right)^{n}\overline{C}_{nm}\overline{P}_{nm}(\sin\varphi)\right] \tag{9}\]  
where \(N_{\mathrm{max}}\) is the maximum degree of the Earth gravity field model, \(n=\mu=\mathrm{max}\) (2, \(\mathrm{m}\)), \(GM\) is the geocentric gravitational constant; \(a\) is the specific scale parameter of the Earth global gravitational model. \(\varphi\), \(\lambda\), and \(r\) are the latitude, longitude, and geocentric radius of the spherical coordinates for the computation point, respectively, \(\gamma\) is the normal gravity at (\(\varphi\),\(\lambda\), \(r\)), \(\overline{P}_{nm}(\sin\varphi)\) are the fully normalized associated Legendre functions of degree \(n\) and order \(m\), and \(\overline{C}_{nm}\) and \(\overline{S}_{nm}\) are the fully normalized spherical harmonic coefficients referred to some normal gravity field.  
Assume that we have two closely spaced locations \(A\) and \(B\) on the surface of the earth with their geoid height difference \(\Delta N_{AB}\) and geodetic azimuth \(\alpha_{AB}\). One computes the deflection of the vertical component \(\xi\) along the meridian and \(\eta\) along the prime vertical as [26]  
\[-\frac{\Delta N_{AB}}{\Delta s_{AB}}\approx\xi\cos\alpha_{AB}+\eta\sin\alpha _{AB}=u \tag{10}\]where \(u\) is the deflection of the vertical and \(\Delta s_{AB}\) is the length of the corresponding geodesic arc. \(\Delta N_{AB}\) can be obtained from the precise leveling \(\Delta H_{AB}\) and GNSS measurements \(\Delta h_{AB}\). As an alternative, the geoid height difference can be calculated by a geoid model.  
When we apply an error distribution to Equation (10), and omit the effect of distance \(\Delta s_{AB}\), the error formula can be written as follows:  
\[\sigma_{u}^{2}=\frac{1}{\Delta s_{AB}^{2}}(\sigma_{\Delta N_{AB}}^{2}). \tag{11}\]  
For instance, for \(\sigma_{\Delta N_{AB}}=8\,\)mm and \(\Delta s_{AB}=1.6\,\)km, the standard deviation of the calculated deflection of the vertical would be about \(\sigma_{u}\approx 1.0\) arc seconds.