Single Point Positioning with Vertical Total Electron Content

This paper proposes a new mathematical method of ionospheric delay estimation in single point positioning (SPP) 6 using a single-frequency receiver. The proposed approach focuses on the ΔVTEC component estimation (MSPPwithdVTEC) 7 with the assumption of an initial and constant value equal to 5 in any observed epoch. The principal purpose of the study is 8 to examine the reliability of this approach to become independent from the external data in the ionospheric correction 9 calculation process. To verify the MSPPwithdVTEC, the SPP with the Klobuchar algorithm was employed as a reference 10 model, utilizing the coefficients from the navigation message. Moreover, to specify the level of precision of the 11 MSPPwithdVTEC, the SPP with the IGS TEC map was adopted for comparison as the high-quality product in the 12 ionospheric delay determination. To perform the computational tests, real code data was involved from three different 13 localizations in Scandinavia using two parallel days. The criterion were the ionospheric changes depending on geodetic 14 latitude. Referring to the Klobuchar model, the MSPPwithdVTEC obtained a significant improvement of 15 – 25% in the 15 final SPP solutions. For the SPP approach employing the IGS TEC map and for the MSPPwithdVTEC, the difference in 16 error reduction was not significant, and it did not exceed 1.0% for the IGS TEC map. Therefore, the MSPPwithdVTEC can 17 be assessed as an accurate SPP method based on error reduction value, close to the SPP approach with the IGS TEC map. 18 The main advantage of the proposed approach is that it does not need external data. 19


Introduction 20
Single point positioning (SPP) allows of the indication of an autonomous position of a receiver using code data from 21 the Global Positioning System (GPS). Code ranges are not ambiguous and do not require to apply the precise method of 22 ambiguity initialization (Bakuła, 2020). The principal problem of SPP stems from different types of errors degrading the 23 GPS signal between a rover and a specified satellite in a given epoch. Ionospheric delay contributes to the general GPS error 24 budget by its volatility in the range of 40 -60 m during daytime and 6 -12 m at night (US Army Corps of Engineers, 2003). 25 The ionosphere consists of charged particles that appear because of the ionization process (El-Rabbany, 2002; Awange, 26 2012). Problems with ionosphere modeling come from difficulties between solar activity and the geomagnetic field 27 interactions (Xu and Xu, 2016). The basic concepts of the GPS signals delay were briefly considered by Golubkov  4 autonomous method to estimate the ionospheric delay. This is followed by the appropriate algorithm presentations with 96 suitable explanations. In addition, the accuracy analysis criteria will be described in view of models credibility procedure. 97

SPP with ionospheric corrections using Klobuchar algorithm and IGS TEC map 98
In this study, the Klobuchar  (1) 104 where A1 is a constant value of 5 ns. In turn, A2 is a sum of multiplying four α coefficients and the geomagnetic latitude of 105 where the adopted ' z angle is equivalent to the zenith angle at the piercing point in (4).
Therefore, to briefly explain the mathematical model of SPP with utilized ionospheric corrections, the code observation 149 equation was adapted based on Strang and Borre (2008) with complementary changes: 150 where the first equation is concerning on the SPP approach with Klobuchar algorithm and the second one is referring 152 to the IGS TEC map. The left side is the measured pseudo-range. On the right side are the model and estimated magnitudes: 153 the geometrical distance between rover and satellite (satellite coordinates computed by utilization of the ephemeris 154 information -SP3 file), speed of light, receiver and satellite clock biases, tropospheric delay, ionospheric delay (computed 155 using Klobuchar algorithm (eight coefficients from navigation message) or IGS TEC map utilizing IONEX file) and pseudo-156 range remaining error, respectively. In the research, the tropospheric corrections were obtained based on Hopfield (see 157 Hopfield, 1969) using model values of the dry and the wet subcomponents. Additionally, the clock bias of satellites has been 158 received by the utilization of satellites' ephemeris data and the relativistic improvements. Continuing, to simplify successive descriptions of the modified SPP approach, the mapping coefficient is denoted: 180 The system of code equations (11) after linearization can be introduced in the matrix notation: 182 The vector of unknowns receives an additional parameter in the adjustment process: 189 https://doi.org/10.5194/gi-2020-28 Preprint. Discussion started: 30 September 2020 c Author(s) 2020. CC BY 4.0 License.
The disclosure vector is: 191 The last entry amounts to zero because of assumption (12).

193
The weight matrix has been prepared based on pseudo-range measurement error which was assumed as a 2.00 m and 194 appropriate satellite elevation angle. The criterion of the minimal mask was implemented as a 10 degree. After where subscript "r" means calculated rover's coordinates and "t" regarding to the actual position. (25) 218 The NEU (North East Up) coordinates system was used in the comparative analysis, where the calculated rover's 219 position is compared to the actual position. Therefore, the rotation matrix was used to convert the covariance matrix (21) of 220 the parameters to the NEU system: 221  In this section, the explanation of the research concept will be done. Next, the appropriate numerical experiment in 232 view of graphics and numeric settings. The parallel discussion about obtained results for appropriate interpretation will be 233 made. 234

Research concept 235
The numerical experiment is based on real single frequency code pseudorange observations. Namely, C1C code data on    The error quantities of the difference in the positions were achieved for MSPPwithdVTEC and SPPwithITM on a 371 close level. Separating the horizontal and the vertical components of the position, the MSPPwithdVTEC is characterized by 372 improved precision compared to SPPwithKM in the North and East direction, therefore, the additional estimated parameter 373 in the code equation does not change the SPP model enough to reduce its quality. The case is repeated in the context of the 374 vertical component U, the MSPPwithdVTEC is again profitable to SPPwithKM and achieves the similar values of the 375 mean errors to SPPwithITM. In general, the values of mean errors are close to each other and the differences are not as clear 376 in the context of the code data use. Therefore, the quantities of average errors demonstrate that MSPPwithdVTEC is the 377 approach of the closest precision to the SPP method with an IGS TEC map, specified as a high-quality product, which is the 378 most important from the authors' point of view. 379

Conclusions and future perspectives 380
The main idea of this paper was to introduce the new method to estimate the ionospheric delay in the SPP without using 381 the external data. Moreover, in the case of comparative analysis, two common approaches in SPP was employed: SPP with 382 Klobuchar algorithm and SPP with IGS TEC map. The first one was treated as a reference one. The SPP model with IGS 383 TEC map was utilized to authenticate the proposed model in view of IGS TEC map use -defined as a high-quality product. 384 The explanation of mathematical models and appropriate accuracy analysis criteria was done. Next, the numerical 385 experiment using real code data from three different GNSS stations with discussion to interpret the obtained results. 386 https://doi.org/10.5194/gi-2020-28 Preprint. Discussion started: 30 September 2020 c Author(s) 2020. CC BY 4.0 License.