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/N)-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.
Highlights
The stochastic model of observations describes the dispersion of the measurement random errors
The stochastic model is of particular importance in the case of Global Navigation Satellite Systems (GNSS) data adjustment, for which precise definition of a fully propagated VC matrix is difficult to obtain due to a multi-dimensional dependency of GNSS observations quality and the complex correlations existing between the measurements
Based on the theoretical analysis of the GNSS receiver tracking errors, which depend on the carrier-to-noise density ratio parameter, a variance model was proposed
Summary
The stochastic model of observations describes the dispersion of the measurement random errors ( referred to as measurement noise). Apart from the numerous works related to the refinement of stochastic models for GPS and GLONASS observations, research on stochastic models of state-of-the-art Galileo and BeiDou systems has been conducted in the recent years These studies include analyses of single systems (e.g., variance of dual-frequency BeiDou B1/B2 code and phase observations [60], variance-covariance components as well as cross- and time-correlation parameters for dual-frequency [61] and triple-frequency BeiDou signals [4], a combination of BeiDou-2 and BeiDou-3 stochastic modeling [62], noise characteristic for five Galileo signals [63]), combined dual systems: GPS/Galileo [22], GPS/BeiDou [64], GPS/IRNSS [65].
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