## 1 Introduction
Therefore, the world network of such facilities is limited. Using the GNSS data can also provide estimates for the electron density profile. This information may be obtained by using 4D spatio-temporal ionospheric tomography suggested by [PERSON] and [PERSON] [23]. GNSS-tomography results agree with ionosphere models and ionosonde data [24]. The tomography algorithms have been developed over the last decades. Additional data (such as ionosonde and Langmuir probes [25]) and new reconstruction algorithms [26] enhance GNSS-tomography. However, these techniques require complicated calculation and large computation resources. Also, for precise measurements, it is necessary to have a very dense network, as is available in Japan or in the United States. When using the data from sparse networks of stations, a more promising way is to build TEC maps. The International GNSS Service (IGS) Working Group on Ionosphere [27] provides the most common TEC maps. Such global TEC maps are used to analyze the regional or global electron content [28]. For regions with a small number of stations, it makes sense to estimate the vertical TEC over the station. Currently, there are several methods to estimate the vertical TEC. These methods are based on various TEC models [29; 30]. One usually uses the spherical harmonic expansion to build global vertical TEC maps. Estimation of the vertical TEC over the station by using spherical harmonics is not appropriate, it is better to use other expansions. For example, [PERSON] and [PERSON] [31] use a TECmodel based on expansion of basic functions that are the products of univariate B-splines with different scales and variables. There are also methods based on simple polynomial expansion. For the first time, this method was offered by [PERSON] and [PERSON] [2] and by [PERSON] and [PERSON] [32], and then was used by [PERSON] et al. [33], who studied the nature of seasonal and solar cycle bias variability in the polar cap region and showed the influence of user's choice of shell-height on standard DCB estimation procedure. Also, the Taylor series expansion (as a generalized expansion) of TEC is used as a TEC model; this expansion on space was mentioned by [PERSON] [34] for regional vertical TEC modeling. We should note that [PERSON] et al. [35] mentioned a problem with unstable DCB at some BeiDou satellites and suggested two-step technique to reduce effects of such satellites on overall solution. Different techniques for TEC estimation do not solve the problem of negative TEC values, especially negative slant TEC on some line of sights. [PERSON] [PERSON] [36] and [PERSON] [37] solved the mathematical problem for constrained (or bounded-variable) least squares. [PERSON] et al. [38] used such an approach to improve GIMs. We developed an algorithm to recover the absolute vertical TEC, its gradients, and its time derivatives from a single GNSS station data, as well as slant TEC along all lines of sight and DCB. The procedure uses the Taylor space-and-time expansion and constrained least squares. It provides both non-negative vertical TEC and non-negative slant TEC along all lines of sight. The algorithm can use data from MEO GPS/GLONASS/Galileo/BeiDou satellites. A combined solution can also use the GEO satellite data. However, here, we show only the GPS and GLONASS data. We termed the algorithm TuRBTEC: TayloR-series and bounded-variable-least-squares-based iOnosphere TEC.