## 2 DoV Estimation: Description of the Methods
### Determination of DoV from the ITALGEO2005 Model  
The \(\xi\) and \(\eta\) components of the DoV can be also obtained by gravity measurements, for instance using the approaches described in [20; 21], based on high resolution Global Geopotential Model (GGM). However, in the interested area (Noto and Medicina), the authors have available the residual gravity anomaly dataset \(\Delta g_{r}\) used for the estimate of the current Italian quasi-geoid model, ITALGEO2005 ([24; 35]), whereby these high resolution observations, based on terrestrial gravity measures, have been used for the gravimetric estimate of the deflection of the vertical. The residual gravity anomalies \(\Delta g_{r}\) have been obtained by the ITALGEO2005 authors removing the long wavelength of the gravity field \(\Delta g_{GGM}\), predicted by a Global Geopotential Model and the high frequency component \(\Delta g_{RTC}\), due to the gravity effect of the masses, which are between the actual surface and the surface accounted by the GGM used, that is the so-called Residual Terrain Correction (RTC). In physical geodesy, using the Least-squares Collocation approach, any linear functional of the anomalous potential T can be estimated if observed values are available, which are themselves linear functionals of T [36]. Therefore, knowing the residual gravity anomalies \(\Delta g_{r}\), which are related to the anomalous potential \(T\) by the fundamental equation of physical geodesy [27] by the following equation  
\[\Delta g=-\frac{\partial T}{\partial t^{\prime}}+\frac{1}{\gamma}\frac{ \partial\gamma}{\partial n^{\prime}}T \tag{7}\]  
instead of computing the geoid undulation values, the components of the DoV can be obtained, because the following equation exist  
\[ \begin{array}{l}\xi=-\frac{1}{\gamma_{r}T}\frac{\partial T}{\partial\varphi }\\ \eta=-\frac{1}{\gamma_{r}\cdot c\cdot cos\varphi}\frac{\partial T}{\partial\chi} \end{array}  \tag{8}\]  
where \(\gamma\) is the normal gravity, \(r\) is the Earth radius and \(n^{\prime}\) the normal line of the reference ellipsoid and Equation (8) represents the spherical approximation [37]. Hence, as a further product of the ITALGEO2005 model, we can estimate the vertical deflection components (\(\xi\), \(\eta\)). The quasi-geoid ITALGEO2005 model has been performed using the remove-restore approach with Least-Square Collocation [38], so starting from the results of the ITALGEO2005 remove phase, that is the gravity residuals \(\Delta g_{r}\), the residual vertical deflection components \((\xi,\eta)_{r}\) were obtained using the Fast Collocation approach [39]. The restore phase was performed adding the long-wavelength component in term of deflection of the vertical \((\xi,\eta)_{GGM}\) and the high frequency \((\xi,\eta)_{RTC}\) due to the topography to the residual values of \(\xi\) and \(\eta\). This procedure can be summarized as follows:  
\[ \begin{array}{l}\text{Remove : }\Delta g_{\text{oss}}-\Delta g_{GGM}- \Delta g_{RTC}=\Delta g_{r}\\ S(\Delta g_{r})=(\xi,\eta)_{r}\,,\text{ where }\text{S}(\cdot)\text{ is a solution operator, e.g., Least Squares Collocation}\\ \text{Restore : }(\xi,\eta)_{r}+(\xi,\eta)_{GGM}+(\xi,\eta)_{RTC}=(\xi_{ Ital},\eta_{ Ital})\end{array}  \tag{9}\]  
As the gravity residuals \(\Delta g_{r}\) of ITALGEO2005 have been obtained removing the long-wavelength signals, \(\Delta g_{GGM}\) predicted using the Global Geopotential Model GPM98 CR up to degree 720 [40], we are forced to use the same model in the restore phase, although this GGM is outdated, otherwise distortions could be introduced in the data, due to not coherence between models. For the aim of this work, where three different geodetic methodologies for the estimate of the deflection of the vertical are compared, we will demonstrate (next sections) that this updated model is sufficient, but we are confident that next Italian geoid model (under processing, private communication with Prof. [PERSON]), based on one of the new GGM will give better performance.