## 4 Methods and Data Analysis
### GPS-Derived tropospheric Parameters
ZWD is mainly a function of partial water vapour pressure (\(P_{w}\)) and the temperature (T), then it can be calculated [3] with a numerical integration through a full atmospheric profile starting from the orthometric height (\(H\)) of the observation point using the two refractivity constants (\(K_{2^{\prime}}\), \(K_{3}\))  
\[\text{ZWD}=10^{-6}\int\limits_{H}^{\infty}\left(K_{2}^{2}\frac{P_{w}}{T}+K_{3} \frac{P_{w}}{T^{2}}\right) \tag{4}\]where, \(K_{2^{\prime}}\) =22.1 \(\pm\) 2.2 K/hPa. In fact, in most of the available processing procedures, the tropospheric zenith delay (ZTD = ZHD + ZWD) is estimated jointly with position coordinates using a mapping function such as the Vienna Mapping Function [25; 26] and consequently ZWD is derived as:  
\[\text{ZWD}=\text{ZTD}-\text{ZHD} \tag{5}\]  
A key parameter used to describe the water vapor content of the atmosphere is precipitable water vapour (PWV). This is a measure of the equivalent height of the column formed if all the water vapor was condensed and collected at the ground surface. According to [6], it can be computed as:  
\[\text{PWV}=\frac{\text{ZWD}}{10^{-6}\left(K_{2}^{\prime}+\frac{K_{3}}{T_{m}} \right)R_{v}\rho} \tag{6}\]  
where, in addition to the atmospheric refractivity constants (\(K_{2^{\prime}}\), \(K_{3}\)) given in Equation (4), \(R_{v}=461\) (expressed in J kg/K) stands for the ideal gas constant for water vapour, \(\rho\) is the density of the water vapour; \(T_{m}\), the atmospheric column mean weighted temperature, is an essential parameter for retrieving PWV from the ZWD of GPS signal propagation. It is site-dependent and varies seasonally and diurnally. Routinely, in GPS analysis, \(T_{m}\) is derived from models (e.g., GPT2 [27]) or is tabulated in grid format (e.g., the European Center for Medium-Range Weather Forecasts (ECMWF) reanalysis dataset) or can be well-approximated by means of empirical relationships [6] from surface temperature observed with meteo stations. The high spatial and temporal variability of \(PWV\) calls for its regular and accurate 4D monitoring for climatological and meteorological purposes. Indeed, knowing the total vertical column of this parameter, also called integrated water vapour (IWV), and its variations, is particularly more useful than surface humidity in understanding water vapour feedback on global warming and rain forecast. Assuming a stratified model for the troposphere with respect to refractivity (\(N\)), which is variable with height along the distance \(L\) of the ray path from the satellite (\(sat\)) to the receiver (\(rec\)), the IWV can be expressed as  
\[\text{IWV}=\Pi\rho\int\limits_{sat}^{rec}NdL \tag{7}\]  
In SI units, IWV is expressed in kg/m\({}^{2}\); a PWV value of 1 mm is equivalent to an IWV value of 1 kg/m\({}^{2}\). When the vertically integrated water vapor is stated in terms of PWV, namely as the length of an equivalent column of liquid water, then this quantity can be related to the ZWD by means of an empirical dimensionless coefficient, \(\Pi=PWV/ZWD\)[28; 29], which can be derived from the Equation (6). \(\Pi\) is a function of the atmospheric column mean temperature (\(T_{m}\)), which is site-dependent too and might depend on the local climate (e.g., geography, season and weather) (\(\Pi\approx 0.15\div 0.16\)).