## 5 Conclusions  
We have developed an algorithm to recover the absolute TEC, its gradients, its time derivative, and DCBs. The procedure is based on the space-and-time Taylor expansion and bounded-variable least squares. We termed it TayloR-series and Bounded-variable-least-squares based iOnosphere TEC (TuRBOTEC). We simulated the algorithm operation by using the IRI-2012 and the IRI-plas models. The absolute TEC values recovered through the developed algorithm were established to agree (qualitatively and quantitatively) with the IRI-2012 model-set values. The mean standard deviation for the mid-latitude IRKJ station is 0.09 TECU, for the equatorial NTUS is 0.35 TECU, for the high-latitude THU2 is 0.17 TECU. We did not find significant influence of the plasmosphere on TuRBOTEC estimations, except for high latitudes, where we should use another \(\alpha\) for the mapping function. About 9% of experimental vertical TECs from the TuRBOTEC estimates exceed (by more than one TECU) those from the same algorithm but without constraints.  
The recovered values of the TEC spatial gradients and of the TEC time derivative agree qualitatively with the model-set values. Also, we studied the accuracy of TEC estimates by means of the latitudinal and longitudinal gradients, for the ionosphere at a distance from a station. The latitude errors were established to grow more dramatically, than those of longitude. This could happen, because the \(S\) mapping function has an essential latitude dependence. Therefore, at a significant distance, the vertical TEC value converted from the slant TEC is not precisely determined. Hence, the gradients are not precisely determined, either. One should note that, in this case, this error would grow almost linearly, but this is not absolutely so. The real (including the model ones) latitude gradients have a considerably spatially non-uniform character at a distance from a station, and, at distances more than 10\({}^{\circ}\)-15\({}^{\circ}\), the TEC estimates based on gradients, start to surpass 10 TECU.  
Figure 16: Influence of intra-day DCB variations on pseudorange TEC (**a**, blue dots) and vertical TEC estimates (**b**, black line). Red line on panel (**b**) shows the phase-without-pseudorange solution. The data are for IRKJ 12 July 2009.  
The DCB values obtained through the developed algorithm for GPS satellites agree with the GIM and CODE data, but, for the GLONASS DCB values, the deviation from the CODE data is up to 17 TECU. At the same time, the recovered DCBs (at the IRI-2012 simulation) agree well with the initial data. At large errors in determining DCBs after correcting the slant TEC series, one may observe negative unphysical TEC values.  
The developed software may be used to calculate the vertical TEC from the local networks or to locally update ionosphere models [61]. The vertical TEC obtained through this procedure generally agrees with the TEC from the IONOLAB and CODE GIM. The TuRBOTEC data disagree with the Madrigal for a region with few stations and with the SEEMALA-TEC data.  
Conceptualization, Y.Y.; methodology, Y.Y., A.M., and A.V.; software, A.M., and A.V.; validation, Y.Y., A.M. and A.V.; formal analysis, A.M.; investigation, Y.Y., A.M., and A.V.; writing--original draft preparation, Y.Y., A.M., and A.V.; writing--review and editing, Y.Y., A.V.; visualization, A.M., Y.Y.; supervision, Y.Y.; project administration, Y.Y.; funding acquisition, Y.Y. All authors have read and agreed to the published version of the manuscript.  
The study was supported by the Russian Foundation for Basic Research (grant N 18-35-20038) and the Ministry of Education and Science (Basic Research program II.16).  
We acknowledge the IGS network for the GNSS data, as well as we do the IRI community and [PERSON] for the IRI-2012 and IRI-plas models' source code. The authors thank scipy.org for python library used in the study. The authors give a credit [PERSON] for the graphical abstract made for the paper.  
The authors declare no conflict of interest.