## 3 TuRBOTEC Algorithm  
To obtain absolute TEC, one estimates the preset measurement model parameters by using experimental data. As a rule, one uses the following model of TEC:  
\[I_{M}=I_{S}+I_{DCB}=I_{V}\cdot S+I_{DCB} \tag{1}\]  
Figure 1: IONOLAB TEC (**a**), SEEMALA-TEC (**b**), Madrigal TEC (**c**). Grey dots on panels (**a,b**) show absolute slant TEC from different satellites. Grey dots on panel (**c**) show the vertical TEC for different cells (\(50^{\circ}\)–\(54^{\circ}\) N, \(102^{\circ}\)–\(106^{\circ}\) E). Orange dots show the median along the \(5^{\circ}\times 5^{\circ}\) region, while the orange line shows the spline interpolation regarded as the absolute vertical TEC. Panel (**d**) shows all the mentioned vertical TECs along with the TEC from CODE GIMs (black line). The data are for 1 January 2014.  
where \(I_{M}\) is the model (expected) value of the recorded slant TEC, \(I\)s is the slant TEC along the line of sight without a differential code bias, \(I_{V}\) is the vertical TEC estimate at a point corresponds with the measurement (ionospheric pierce point), \(S\) is the function that converts the slant TEC into the vertical one (mapping function), and \(I_{DCB}\) is an error related to the DCB of the satellite and the receiver. For geostationary Earth orbit (GEO) satellites it is difficult to estimate DCB (or similar error in single frequency data) from (1), because \(S\) in Equation (1) is a constant. The corresponding error can be regarded as a TEC offset [45]. However, GEO satellites' DCB could be estimated along with DCB of medium Earth orbit (MEO) satellites [46]. [PERSON] and [PERSON] [47] used de facto the approximation \(I_{V}\)=\(I_{V}\) (\(\phi_{0}\), \(l_{0}\), \(t_{0}\)), where \(\phi_{0}\), \(l_{0}\), and \(t_{0}\) are the station latitude, longitude, and the time, for which the calculation is performed. This implies that all the TEC measurements were supposed to be performed over the station. However, such an approach does not allow one to obtain spatial gradients. Such an approximation is correct in a limited number of cases when the spatial gradients and the time derivative can be neglected. In the most studies, a spatial-temporal expansion of the \(I_{V}\) function is used. The spherical harmonic expansion is often used. [PERSON] et al. [48] increased the capacity of GIM technique to simultaneously treat data from more than 1000 dual-frequency GPS receivers. [PERSON] et al. [49] used the DCB estimation technique based on spherical expansion to evaluate BeiDou DCBs. According to [PERSON] [34], the Taylor expansion is more appropriate than the spherical harmonic expansion for regional modeling. [PERSON] [34] mentioned the \(I_{V}\) Taylor series expansion in (1) only on space. In our research, we expand the vertical TEC function \(I_{V}\) (\(\phi\), \(l\), \(t\)) into the Taylor series on space and time:  
\[I_{V}(\phi,l,t)=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac {\left(\Delta\phi\right)^{m}\left(\Delta l\right)^{n}\left(\Delta t\right)^{ k}}{m!n!k!}\frac{\partial^{m+n+k}I_{V}}{\partial\phi^{m}\partial^{ln} \partial^{k}} \tag{2}\]  
where \(\Delta\phi\), \(\Delta l\) are the difference between latitude, longitude of the ionospheric pierce point and the point of station (\(\phi_{0}\), \(l_{0}\)), \(\Delta t\) is the difference between time of measurement and \(t_{0}\). We developed the following algorithm to estimate the vertical TEC, the TEC gradients, the time derivative, and DCBs. The algorithm involves:  
1. Calculating TEC based on the pseudorange \(I_{P}\) and phase \(I_{\varphi}\) measurements. For the analysis, we use the data with elevations greater than \(10^{\circ}\). 2. Dividing the data into continuous samples. 3. Detecting and eliminating outliers and cycle slips in the TEC data [50] (Figure 2). 4.