## 3 TuRBOTEC Algorithm
Eliminating the phase measurement ambiguity (\"leveling\", see Figure 2a): \(\mathit{const}=\frac{1}{N\sum_{i=1}^{N}S_{i}^{-1}}\cdot\sum_{i=1}^{N}\frac{ \left(I_{p}-I_{\varphi}\right)}{S_{i}}\), where \(N\) is the number of measurements over a continuous interval, \(S\) is the mapping function (see below). At this state we obtain experimental slant TEC \(I_{Exp}\). 5. Estimating DCBs by a simple measurement model and determining the model parameters based on minimizing the model data root-mean-square deviation. Although in some research the zero-order expansion of (2) is used (only term with \(m=0\), \(n=0\), \(k=0\) in Equation (2)), indeed, it is necessary to substantiate the selection of the expansion order in greater detail. We performed an analysis by using the IRI-2012 model. The TEC slant values, corresponding to real observed elevations and azimuths, as well as the vertical TEC, the time derivative, and the spatial gradients were modeled by latitude and by longitude. Then, we recovered the values of vertical TEC (\(I_{V}\)) from slant measurements by using expansions to the specified order of Taylor-expansion. At the first step, this recovery was performed based on the zero-order Taylor expansion (2) (leaving term with \(m=0\), \(n=0\), \(k=0\) in (2)). Next, TEC was recovered by using the first-order (leaving terms with \(n+m+k\leq 1\) in (2)) and second-order (leaving terms with \(n+m+k\leq 2\) in (2)) expansions. Figure 3 provides the results for recovery of the vertical TEC, the time derivative, and the latitudinal gradient. The solid black line denotes the model values on the plots. The recovered parameters obtained based on of the zero order are denoted by the black dash-dotted line, those based on the first order--by the blue dotted line, and those based on the second order--by the red solid line. From the plots, one can see that using the second order of the \(I_{V}\) expansion into the Taylor series, viz., using of time derivatives and square gradients on space, considerably improves the absolute vertical TEC recovery accuracy. Meanwhile, using the second order provides a sufficient accuracy to be able to neglect higher orders. Therefore, we use the expansion \(I_{V}\) into the Taylor series up to the second order:  
\[I_{M}=S_{j}^{i}\left[ \begin{array}{c}I_{V}(\phi_{0},I_{0},t_{0})+G_{\phi} \Delta\phi_{j}^{i}+G_{q\phi}\left(\Delta\phi_{j}^{i}\right)^{2}+\\ +G_{i}\Delta I_{j}^{i}+G_{q\phi}\left(\Delta I_{j}^{i}\right)^{2}+G_{t}\Delta I _{j}^{i}+G_{q\phi}\left(\Delta I_{j}^{i}\right)^{2}\end{array} \right]+I_{DCB,j}, \tag{3}\]  
where \(I_{V}\) is the absolute vertical TEC value, \(\Delta\phi\) (\(\Delta I\)) is the latitude (longitude) difference between the ionospheric point coordinate \(\phi(l)\) and that of the station \(\phi_{0}(l_{0})\), \(\Delta t\) is the difference between the measurement time \(t\) and the time \(t_{0}\) for which the calculation is performed. Further, \(G_{\phi}=\partial I_{V}\partial\phi\)  
Figure 3: Results of simulating the recovery of the absolute vertical TEC (**a**), the time derivative (**b**), and the latitudinal gradient (**c**) for 10 April 2012. Red solid line indicates values (vertical TEC, time derivative, and spatial gradient) recovered by using the second order \(I_{V}\) expansion; blue dotted line indicates values recovered by using the first order \(I_{V}\) expansion, dot-and-dash black line shows values recovered by using the zero order \(I_{V}\) expansion). The solid black exhibits the IRI-2012 data. Figure 2: Eliminating the phase ambiguity (**a**), outliers (**b**), and cycle slips (**c**). The panels show the numbers for the corresponding GNSS satellite. Red dots mark the uncorrected phase TEC, blue dots-corrected phase TEC, orange dots-uncorrected pseudorange TEC, grey dots-corrected pseudorange TEC.