Automatic Calibration of Process Noise Matrix and Measurement Noise Covariance for Multi-GNSS Precise Point Positioning

Xinggang Zhang, Pan Li, Rui Tu, Xiaochun Lu, Maorong Ge and Harald Schuh
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Xinggang Zhang: National Time Service Center, Chinese Academy of Sciences, Shu Yuan Road, Xi’an 710600, China
Pan Li: German Research Centre for Geosciences (GFZ), 14473 Potsdam, Germany
Rui Tu: National Time Service Center, Chinese Academy of Sciences, Shu Yuan Road, Xi’an 710600, China
Xiaochun Lu: National Time Service Center, Chinese Academy of Sciences, Shu Yuan Road, Xi’an 710600, China
Maorong Ge: German Research Centre for Geosciences (GFZ), 14473 Potsdam, Germany
Harald Schuh: German Research Centre for Geosciences (GFZ), 14473 Potsdam, Germany

Mathematics, 2020, vol. 8, issue 4, 1-20

Abstract: The Expectation-Maximization algorithm is adapted to the extended Kalman filter to multiple GNSS Precise Point Positioning (PPP), named EM-PPP. EM-PPP considers better the compatibility of multiple GNSS data processing and characteristics of receiver motion, targeting to calibrate the process noise matrix Q t and observation matrix R t , having influence on PPP convergence time and precision, with other parameters. It is possibly a feasible way to estimate a large number of parameters to a certain extent for its simplicity and easy implementation. We also compare EM-algorithm with other methods like least-squares (co)variance component estimation (LS-VCE), maximum likelihood estimation (MLE), showing that EM-algorithm from restricted maximum likelihood (REML) will be identical to LS-VCE if certain weight matrix is chosen for LS-VCE. To assess the performance of the approach, daily observations from a network of 14 globally distributed International GNSS Service (IGS) multi-GNSS stations were processed using ionosphere-free combinations. The stations were assumed to be in kinematic motion with initial random walk noise of 1 mm every 30 s. The initial standard deviations for ionosphere-free code and carrier phase measurements are set to 3 m and 0.03 m, respectively, independent of the satellite elevation angle. It is shown that the calibrated R t agrees well with observation residuals, reflecting effects of the accuracy of different satellite precise product and receiver-satellite geometry variations, and effectively resisting outliers. The calibrated Q t converges to its true value after about 50 iterations in our case. A kinematic test was also performed to derive 1 Hz GPS displacements, showing the RMSs and STDs w.r.t. real-time kinematic (RTK) are improved and the proper Q t is found out at the same time. According to our analysis despite the criticism that EM-PPP is very time-consuming because a large number of parameters are calculated and the first-order convergence of EM-algorithm, it is a numerically stable and simple approach to consider the temporal nature of state-space model of PPP, in particular when Q t and R t are not known well, its performance without fixing ambiguities can even parallel to traditional PPP-RTK.

Keywords: EM-algorithm; multi-GNSS; PPP; process noise; observation covariance matrix; extended Kalman filter; machine learning (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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