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 Qt and observation matrix Rt, 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 Rt 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 Qt 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 Qt 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 Qt and Rt are not known well, its performance without fixing ambiguities can even parallel to traditional PPP-RTK.
Highlights
Since Precise Point Positioning (PPP) emerged [1,2], people are primarily focusing on improving precise orbit and clock products, developing new algorithms to solve for ambiguities, to accelerate its convergence and expand its applications such as PPP-real-time-kinematic (PPP-real-time kinematic (RTK)), triple frequency
In the following paper, the authors will only consider the Kalman filter for PPP data processing. Both the process noise Qt and observation covariance matrix Rt are the key to Kalman filter, limited attention is paid to the fundamental problem for multi-GNSS PPP
EM-algorithm, maximum likelihood estimation (MLE) and restricted maximum likelihood (REML) are built upon a certain distribution, which explains why when applying least-squares (co)variance component estimation (LS-variance component estimation (VCE)), it is necessary for users to set weight matrix on their own
Summary
Since Precise Point Positioning (PPP) emerged [1,2], people are primarily focusing on improving precise orbit and clock products, developing new algorithms to solve for ambiguities, to accelerate its convergence and expand its applications such as PPP-real-time-kinematic (PPP-RTK), triple frequency. M-estimator is introduced into an adaptive Kalman filter to increase its resistance to outliers, where an adaptive factor α to state error covariance matrix is constructed [17,18]. Is developed to the extended Kalman filter to estimate PPP states, x t , together with a large number of Qt and Rt. The EM-algorithm, which can be classified as the first scheme, works in an iterative procedure to locate maximum likelihood estimates of parameters. It is capable of finding Kalman parameters even if we have missing data It can detect outliers by introducing small weights for large outliers and can even estimate the outliers [21].
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