## 3 TuRBOTEC Algorithm
\(G_{l}=\partial I_{V}\partial l\), \(G_{q,\phi}=\partial^{2}I_{V}\partial\phi^{2}\), and \(G_{q,l}=\partial^{2}I_{V}\partial l^{2}\) are the linear and quadratic spatial TEC gradients, and \(G_{l}=\partial I_{V}\partial l\) and \(G_{q,t}=\partial^{2}I_{V}\partial t^{2}\) are the first and second time derivatives. Equation (3) represents the second-order Taylor series expsion of (1). We neglect the mixed derivatives. In the literature different mapping functions are used. Selecting the most correct one is a challenge. In most papers, one uses the one-layer approximation of the ionosphere as a thin layer. This approach was suggested by [PERSON] [51]. [PERSON] et al. [52] used the two-layer approximation of the ionosphere. Based on a tomographic procedure, [PERSON] et al. [53] proposed a climatological mapping function--the Barcelona Ionospheric Mapping Function. There is another approach: [PERSON] and [PERSON] [54] suggested one use the NeQuick model instead of the thin layer approximation to convert the slant TEC into the vertical one. However, such an approach did not provide an essential accuracy increase [54]. We use the modified single-layer model mapping function, and coefficient \(\alpha\) is introduced to more correctly account for the ionospheric layer height [34]:  
\[S_{j}^{i}=\left[cos\left(arcsin\left(\frac{R_{E}}{R_{E}+h_{max}}sin\left[ \alpha\big{(}90-\theta_{j}^{i}\big{)}\big{)}\right]\right)\right]^{-1}, \tag{4}\]  
where \(R_{E}=6371\) km is the Earth radius, \(h_{max}=450\) km is the ionospheric point height, and \(\alpha\) is the correcting coefficient. Figure 4 presents the slant TEC dependence, \(I_{S}\), on the elevation when converting \(I_{S}=S\cdot I_{V}\) for different values \(\alpha\). Simulation used the IRI-2012. By its physical implication, the coefficient \(\alpha\) adjusts a sharp non-physical TEC growth at low elevations, less than \(30^{\circ}\). It is worth noting that due to the ionosphere peculiarities, the coefficient \(\alpha\)should be latitude-dependent. Also, the \(\alpha\) parameter should affect the ionosphere disturbance level that may dramatically vary the ionization global distribution [55]. We estimated the \(\alpha\) values for several points on the earth surface. Table 1 presents the results. We obtain the set of equations by minimizing the functional \(U=\sum U^{k}\) (5) for the set of selected instant \(t^{k}\), for which we estimate the parameters. For computation, we represent (5) in the form (7):  
\[U^{k}=\sum\
olimits_{j=1}^{N^{k}}\sum\
olimits_{i=1}^{N^{k}_{j}}\omega^{k,i} \Big{(}I_{M_{j}^{i}}-I_{Exp_{j}^{i}}\Big{)}^{2}, \tag{5}\]  
Figure 4: TEC dependence on the elevation for different \(\alpha\). The simulation used the IRI-2012 model. The red solid line shows the IRI-2012 slant TEC data; the red dotted line exhibits the slant TEC converted from the vertical one by using \(\alpha\)=1; the blue solid line shows the slant TEC converted from the vertical one by using \(\alpha\) = 0.97; the green dotted line shows the same, but with the use of \(\alpha\) = 0.94; the black dotted line exhibits the same, but with the use of \(\alpha\) 0.87. The blue dot-and-dash line shows the elevation of the satellite.