An adaptive Kalman filter based on variance component estimation for a real-time ZTD solution

Abstract

The Global Navigation Satellite System (GNSS) precise point positioning (PPP) technology is currently used to process GNSS water vapor observations in real time or near real time. Further developments are required to improve the accuracy and real-time performance of processing tropospheric delays from which the water vapor observations are extracted. In real-time BDS/GPS precise clock correction estimation with square-root information filtering and PPP solution with Kalman filtering, a fixed variance is often assigned for the process noise of the troposphere dynamics model. However, this fixed value may deviate from reality as the weather conditions change, especially for extreme weather. In this paper, a new adaptive Kalman filter is proposed for tropospheric delay processing, in which the variance of the process noise for the zenith tropospheric delay (ZTD) dynamics model is tuned in real time using the least-squares variance component estimation technique. MGEX/IGS data of 15 consecutive days were processed in a stepwise manner. Namely: Real-time BDS/GPS precise clock corrections were estimated firstly, followed by comparison among ZTD solutions of four schemes based on these real-time clocks of first step and GFZ multi-GNSS precise clock (GBM) final clocks: (1) real-time solution of ZTD (G(GPS), GC(GPS and BDS)) without VCE; (2) real-time solution of ZTD (G, GC) with VCE; (3) final solution of ZTD (G, GC, GR (GPS and GLONASS), GE (GPS and GALILEO), GRC (GPS, GLONASS and BDS), GREC (GPS, GLONASS, GALILEO and BDS) without VCE; and (4) final solution of ZTD (G, GC, GR, GE, GRC, GREC) with VCE. The performance of the ZTD and positioning solution was analyzed. Results showed that the accuracy of estimated real-time satellite clock correction was 0.27, 1.31, 0.29, and 0.21 ns for GPS, BDS/GEO, BDS/IGSO, and BDS/MEO, respectively. For the ZTD solutions, the results of schemes 1 and 2 were 12.8 mm (GC) and 10.1 mm (GC) in terms of mean root-mean-square (RMS) values and 2.1 mm (GC) and 1.6 mm (GC) in terms of minimum RMS values, respectively, thereby showing an improvement for scheme 2 of 1.8–81.4% over scheme 1 with average increasing rates of 20.7% (GC) and 20.2% (G). The results for schemes 3 and 4 were 7.6 mm (G) and 6.3 mm (GRC) in terms of mean RMS values and 2.1 mm (G) and 1.9 mm (GRCE) in terms of minimum RMS values, respectively, thereby showing an improvement in scheme 4 over scheme 3 by 1.3–85.6% with average increasing rates of 22.1% (GRCE), 21.9% (GRC), 18.4% (GR), 15.9% (GC), 15.2% (GE), and 12.1% (G). Similar results can be observed for the positioning solution, especially in the height component. These findings clearly show the advantage of the proposed method, which is consistent with the theoretical analysis. Notably, the advantage of the adaptive VCE becomes significant with the inclusion of additional satellite systems.