## 3 Empirical Stochastic Modeling Methodology  
In Section 2 we showed that the nominal measurement noise could be significantly different for different types of GNSS signals. To determine the actual value of the measurement noise and to create a realistic stochastic model of observations, the empirical stochastic modeling method based on the observation residuals was used. The empirical models of observation variance, as well as the time- and cross-correlation parameters, were derived using double-differenced (DD) and triple-differenced (TD) measurements from short baseline (SB) and zero-length baseline (ZB) conducted by two receivers of the same type. The time series of the mentioned measurement combinations allow to precisely separate the influence of particular correlations from each other. Thus, the SB TD combination was used to model the dependence of the carrier-phase and code variances on the C/N\({}_{0}\) parameter. The time- and cross-correlation were determined based on the ZB DD observation time series analysis, as this combination is not burdened with the influence of multipath, does not de-correlate the measurements and allows to determine both parameters directly. This section presents the methodology of determination of individual parameters of the empirical stochastic model, the accuracy analysis of the created variances functions and the values of correlation coefficients based on daily observations for all systems and signals. The simplified code (C) and carrier-phase (\(L\)) observation equations in units of length for a receiver \(a\) and a satellite \(i\) generalized for frequency \(f\) are given by:  
\[C_{a}^{i} =R_{a}^{i}+\delta T_{a}^{i}+\delta I_{a}^{i}+K_{a,C}-K_{C}^{i}+ \delta M_{a,C}^{i}+c(\delta t_{a}-\delta t^{i})+\epsilon_{a,C}^{i} \tag{4}\] \[L_{a}^{i} =R_{a}^{i}+\delta T_{a}^{i}-\delta I_{a}^{i}+K_{a,L}-K_{L}^{i}+ \delta M_{a,L}^{i}+c(\delta t_{a}-\delta t^{i})+\lambda N_{a}^{i}+\epsilon_{a, L}^{i} \tag{5}\]  
where: \(R\) is the geometric satellite-receiver range, which includes satellite orbit error, satellite and receiver antenna phase center errors and carrier-phase wind-up bias, \(\delta T\) and \(\delta I\) are troposphere and ionosphere delays, \(K_{C}\) and \(K_{L}\) are hardware biases for a receiver (with a subscript \(a\)) and a satellite (with a superscript \(i\)) for code and phase measurements respectively, \(\delta M_{C}\) and \(\delta M_{L}\) are code and phase multipath, \(\delta t\) is a satellite or receiver clock error, \(N\) denotes integer carrier-phase ambiguity and \(\epsilon_{C}\) and \(\epsilon_{L}\) are code and carrier-phase random errors. All quantities, except for clock errors given in seconds and ambiguity given in carrier wave cycles, are expressed in units of length. Additionally, \(c\) is the speed of light, and \(\lambda\) denotes the carrier wavelength. Assuming the same atmospheric delays affect observations from two receivers for SB and ZB setup, the DD code and carrier-phase residuals can be calculated by subtracting from the observations the double differences of geometric distance and carrier-phase ambiguity, which can be easily determined, if the coordinates of both receivers are known with high accuracy. The TD residuals, derived as the time difference of two DD for consecutive observation epochs (\(t_{1}\) and \(t_{2}\)), also significantly reduce the multipath error as it has almost the same value for two close epochs (e.g., for 1 Hz data). Table 2 shows the calculation formula, the expectation (E\(\{\cdot\}\)) and dispersion (D\(\{\cdot\}\)) values for two linear residuals combination (DD and TD) of carrier-phase observations for the short and zero-length baseline, where \(L(t_{1})\) is the phase residual at epoch \(t_{1}\) and symbol \(\
abla\Delta\) denotes the double-differencing operator. This example illustrates only phase observation residuals, but the formulas are the same for the code residuals (except for the carrier-phase ambiguity which is not existent in code observations).