## 2 DoV Estimation: Description of the Methods
### Determination of the DoV Using the GNSS-LEV Technique  
Geoid undulation and DoV values can be estimated with the combined use of GNSS techniques and geometric leveling. The theoretical foundation of the method is based on the relation [27]:  
\[\varepsilon\approx-\frac{\Delta N}{\Delta S}=-\frac{\Delta h-\Delta H}{\Delta S} \tag{4}\]  
where \(\varepsilon\) is the DoV value in the direction of the GNSS baseline, \(\Delta S\) is the length of the geodesic line and \(\Delta N\) is calculated by the difference between the ellipsoidal height difference (\(\Delta h\)), measured using GNSS, and the height difference (\(\Delta H\)) obtained from geometric leveling (Figure 3). Combining Equation (4) with Equation (2), we get:  
\[-\frac{\Delta N}{\Delta S}\approx\xi\ cos\alpha+\eta\ sin\alpha \tag{5}\]  
where \(\alpha\) is the ellipsoidal azimuth of the GNSS baseline and \(\Delta S\) is the length of the corresponding geodesic arc. Measuring the geoidal and the ellipsoidic height differences in, at least, two directions, the two unknown components of the DoV can be calculated using Equation (5). Figure 2: Area of the IRA-INAF observatory of Noto (Sicily) (satellite image courtesy of Google Earth®). The topographic surveys were planned to estimate the components of the DoV with a precision comparable to that of the angular readings of high-end total stations, _i.e._, \(\approx 0.5\) arcsec. Nevertheless, this level of precision is hardly preserved, even when high precision accessories and refined operative arrangements are adopted. A 1.5 arcsec accuracy level is more likely achieved under real operational conditions, particularly on networks of limited size. Considering that, as a rule of thumb, the precision on DoV estimate can affect the angle measurement standard deviation up to 40%, we fixed a threshold of 1 arcsec as maximum uncertainty on the DoV estimate. This threshold is compatible with an overall angular readings standard deviation lower than 2 arcsec. Assuming a standard deviation of 3 mm in the determination of the height differences on short baselines measured with static GNSS surveying (using high-quality antennas and state-of-the-art scientific software for post-processing) and a standard deviation of 1 mm/\(\sqrt{\text{km}}\) for geometric leveling, the desired 1 arcsec DoV standard deviation over a 1 km baseline can be obtained. Therefore, two approximately orthogonal GNSS baselines with lengths varying between 0.9 and 1.3 km were measured at both observatories. Within this distance range, the DoV value \(\varepsilon\) can be referred to a unique average value along the baseline [27]. Following this approach, we estimated the DoV components \(\xi\), and \(\eta\) with respect to the point of intersection between the two GNSS baselines, as shown in Figures 1 and 2. These values, although not being determined at the exact position of the tie vector, can be easily referred to it, with a good approximation: the geoid model at the two sites varies by 0.1-0.2 arcsec over the baseline, a variation which is much smaller than our working precision and that can therefore be disregarded. At Medicina observatory, the measurements were carried out in March 2014. In Figure 1, the four endpoints of the GNSS baselines are identified by red dots. They were actually materialized with a metallic hemispherical nail head. Due to the lack of stable manmade objects (having sufficiently deep foundations and being suitable for installing the nails) nearby, the East and South endpoints for the leveling markers were materialized with leveling bolts screwed on top of metal rods (see the inset of  
Figure 3: Relation between the geoid undulation N and the deflection of the vertical \(\varepsilon\). (Adapted from [27]). Figure 4). The rods were hammered down to a depth of about 1.5 m, this being sufficient to avoid short term height changes caused by the plastic variations of the silty clay cohesive soil in response to precipitation [25].