## 3 Empirical Stochastic Modeling Methodology
The created empirical model of variance is based on fitting a function that approximates the variance depending on the C/N0 parameter. The advantage of choosing the C/N0 parameter, unlike the elevation angle, is a close theoretical relationship with the measurement noise (expressed by Equations (2) and (3)) and the possibility of including the actual received signal power. The relationship with the elevation is not direct [80]. Most of the modern precise GNSS receivers output C/N0. It is then available to users via RINEX files or RTCM and NMEA data streams. Unfortunately, for some types of receivers the parameter is unavailable. Based on a general form of tracking error, Equations (2) and (3) can be represented as:  
\[\sigma=a_{1}\cdot 10^{-C/N_{0}/20}+a_{2}\cdot 10^{-C/N_{0}/10} \tag{6}\]  
where: coefficients \(a_{1}\) and \(a_{2}\) are related to tracking loop bandwidths, predetection integration intervals and correlator spacing separately for code and phase measurements; C/N0 is expressed in dB-Hz. The developed empirical variance model uses a modified Equation (6) which extends to the form of a \(n=4\) degree series, in the form:  
\[\sigma=\sum_{i=1}^{n}a_{i}\cdot 10^{(1-i)C/N_{0}/40} \tag{7}\]  
Function (7) is fitted to the standard deviation of the SB TD code and phase observations residuals determined for the adopted time series segments, and the coefficients \(a_{i}\) are determined by the Least-Squares Method. To assess the variance of the observation noise (\(\sigma_{L}^{2}\) and \(\sigma_{C}^{2}\)), the time-correlation of observation residues should be taken into account (according to Table 2)--the method of determining \(\rho_{t}\) is described in the further part of this subsection. Figures 5-12 show the results of the approximation of the standard deviation of SB TD code and phase observations for all analyzed GNSS signals using the function (7). Standard deviations were determined for daily observations with an interval of 1 Hz for 120-s observation segments and recalculated to a UD level. The argument of the approximating function C/N0 was calculated as a mean value of carrier-to-noise density ratios of the reference satellite and the second satellite in a pair. A detailed description of the data used can be found in Section 5. Aforementioned Figures present the modeling results for all observations for a given signal with no distinction between satellite blocks. The exception is the BeiDou system, for which the \(n=2\) degree model was adopted for observations from geostationary (GEO) satellites. Based on the presented figures, it can be concluded that the selected approximating function accurately reflects the dependence of measurement noise standard deviation on the C/N0 parameter. The Figures also show the coefficient of determination \(R^{2}\), which is within the range of 0.80-0.96, proving that the model replicates the empirical standard deviation values very well. The only exception is the model for GPS C2W code observations, for which the standard deviation of the noise has a non-standard pattern--it increases along with the value of C/N0 to about 25 dB-Hz and then decreases. This effect may be related to a signal power loss while tracking the P(Y)-code using the Z-tracking technique [83]. Nevertheless, this hypothesis cannot be confirmed unequivocally. The matter should be investigated in more detail. In this case, we use the same approximating function as for the other signal modeling only for observation with C/N0 \(>\) 25 dB-Hz. It should be noted, however, that the observation error is much smaller than for other code observations (in all the graphs, the same scale was adopted for the code noise variance with a maximum of 0.4 m, except for C2W observations for which the maximum is 0.1 m), and the impact on the final results of the positioning model solution is significantly smaller than the one of the carrier-phase measurements. The second exception is the model for BeiDou GEO observations.