## 2 Instruments and Methods
### Evaluation of the Atmospheric Opacity from Global Atmospheric Models  
Atmospheric transparency can be estimated on the basis of the approach used for modeling path delay in the neutral atmosphere [40]. The publicly available data provided by the NASA Global Modeling and Assimilation Office model GEOS-FPIT ([[http://gmao.gsfc.nasa.gov](http://gmao.gsfc.nasa.gov)]([http://gmao.gsfc.nasa.gov](http://gmao.gsfc.nasa.gov)), accessed on 1 September 2023) are used [9]. They evaluate atmosphere parameters (in particular, air temperature, total atmospheric pressure, and partial pressure of water vapor) using various ground, air-born and space-born measurements that are assimilated into a dynamic model. The current models have 72 levels in altitude, 0.25\({}^{\circ}\)\(\times\) 0.31\({}^{\circ}\) spatial grid and 3 h resolution in time. Atmospheric absorption at any frequency can be calculated for any selected location by using standard spectroscopic parameters, as described in [9]. The values of the zenith opacity obtained in this way are in good agreement with the results of sky dip measurements at the LMT and other sites, including measurements with the MIAP-2 radiometer at 2 mm on the Suffa plateau [41]. Similar data are provided by the Modern-Era Retrospective Analysis for Research and Applications, Version 2 (MERRA-2) [42; 43] and the European Center for Medium-Range Weather Forecast ReAnalysis (ERA5) [44]. Evaluation of the Precipitable Water Vapor (PWV) from the Global Navigation Satellite System (GNSS) Data  
The development of the Global Navigation Satellite Systems (GNSS) provides an opportunity for estimations of PWV from measurements of delays of the navigation signals [45; 46; 47]. Taking into account the surface temperature and pressure, the PWV values can be estimated from these delays with a low uncertainty. Nowadays, this method is widely used for this purpose, including evaluation of results obtained from the global atmospheric models mentioned above (e.g., [48; 49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59]). Evaluation of the Precipitable Water Vapor (PWV) from the Sky Dip Data by the Artificial Neural Network  
The direct comparison of the results obtained by the different methods may be problematic. The low PWV values in dry atmosphere are barely detectable by humidity sensors in aerosole. Wet conditions and high opacity lead to small variations in brightness temperature on different angles, leading to higher errors in \(\tau\) and PWV estimations. The spatial-temporal resolution of the dynamic atmospheric models is limited: it is lower than typical sizes of the topographic features such as mountains and typical times of the variations in PWV values. As a result, different methods should be cross-validated and extrapolated to the same scale. Recently, we proposed a statistical approach for the determination of the PWV values based on machine learning algorithms and MIAP-2 data [60]. The idea was to use nonlinear regression to calculate PWV values from the MIAP data using the receiver responses on different angles in the 2 mm and 3 mm bands as the input data, without addressing the physical model. The training set was based on a one-year monitoring session near the Badary observatory (2016-2017). The observatory was equipped with the water vapor radiometer (WVR) and GNSS receivers capable of tropospheric delay estimations. We found that the GNSS and WVR data were consistent. We used the GNSS data as the target values for the statistical model fitting and the k-nearest neighbors (kNN) [61] and artificial neural network (ANN) as the regression models. We utilized the individual component analysis to speed up the learning process. As a result, the regression model encapsulates the instrumental transfer function and the PWV fraction in the microwave absorption (Equation (1)). The bias introduced by the plane-parallel atmospheric model was also excluded.