Empirical Stochastic Model of Multi-GNSS Measurements

Abstract

The stochastic model, together with the functional model, form the mathematical model of observation that enables the estimation of the unknown parameters. In Global Navigation Satellite Systems (GNSS), the stochastic model is an especially important element as it affects not only the accuracy of the positioning model solution, but also the reliability of the carrier-phase ambiguity resolution (AR). In this paper, we study in detail the stochastic modeling problem for Multi-GNSS positioning models, for which the standard approach used so far was to adopt stochastic parameters from the Global Positioning System (GPS). The aim of this work is to develop an individual, empirical stochastic model for each signal and each satellite block for GPS, GLONASS, Galileo and BeiDou systems. The realistic stochastic model is created in the form of a fully populated variance-covariance (VC) matrix that takes into account, in addition to the Carrier-to-Noise density Ratio (C/N0)-dependent variance function, also the cross- and time-correlations between the observations. The weekly measurements from a zero-length and very short baseline are utilized to derive stochastic parameters. The impact on the AR and solution accuracy is analyzed for different positioning scenarios using the modified Kalman Filter. Comparing the positioning results obtained for the created model with respect to the results for the standard elevation-dependent model allows to conclude that the individual empirical stochastic model increases the accuracy of positioning solution and the efficiency of AR. The optimal solution is achieved for four-system Multi-GNSS solution using fully populated empirical model individual for satellite blocks, which provides a 2% increase in the effectiveness of the AR (up to 100%), an increase in the number of solutions with errors below 5 mm by 37% and a reduction in the maximum error by 6 mm compared to the Multi-GNSS solution using the elevation-dependent model with neglected measurements correlations.

Analysis of Different Weighting Functions of Observations for GPS and Galileo Precise Point Positioning Performance

“…These satellites belong to the new satellite blocks (IIF and IIIA), which is in line with the goal of GPS modernization, i.e., improved accuracy, signal strength and observation quality [73]. also noticed that Galileo observations, at lower elevation angles, have significantly higher SNR than GPS observations, due to the multipath effects (Figure 3) [71], whereas Galileo observations have higher performance with lower noise [20,72]. In addition, for L2 observations, it can be seen that some of the GPS satellites have a higher SNR.…”

“…Defining the full VC matrix is very difficult since it depends on used models, used GNSS systems (also blocks of satellites) and their observations (frequencies and signals), used equipment (receiver and antenna types), noise, multipath and other mis-modeled errors. VC matrix also needs spatial-and time-correlation between observations [20]. Consequently, it is impossible to determine one universal model that fits all GNSS measurements and equipment.…”

Phase Centre Corrections of GNSS Antennas and Their Consistency with ATX Catalogues

“…For the analysed station KRAW (antenna ASH70195C_M SNOW), a full upgrade had the advantage of introducing four GNSS systems compared to the existing GPS-only. Especially, introduction of the Galileo system may benefit the station solution [38,39], e.g., decreasing the positioning error statistics for GPS + Galileo combinations [40] despite Galileo still not providing the same satellite availability as GPS [41]. Concluding the results and achievements in the investigation of PCC by other authors [16,19,21,24], we are inclined to say that the time of antennas from the 1990s should be completed and full modernization carried out, despite the higher costs.…”

A Stochastic Model Based on Optimal Satellite Subset Selection Strategy for Smartphone Pseudorange Relative Positioning

“…This includes methods like the Helmert variance component estimation and the minimum norm quadratic unbiased estimation methods [ 18 , 19 ]. Given these traditional models, many researchers have optimized them in tandem with practical applications, yielding notable results in GNSS positioning with geodetic receivers [ 20 , 21 , 22 , 23 ]. Nonetheless, smartphones exhibit distinct characteristics compared to geodetic receivers, with GNSS observations that are prone to noise and markedly affected by multipath and NLOS errors.…”