A drift line bias estimator: ARMA-based filter or calibration method, and its application in BDS/GPS-based attitude determination

Abstract

The multi-antenna synchronized receiver (using a common clock) is widely applied in GNSS-based attitude determination (AD) or terrain deformations monitoring, and many other applications, since the high-accuracy single-differenced carrier phase can be used to improve the positioning or AD accuracy. Thus, the line bias (LB) parameter (fractional bias isolating) should be calibrated in the single-differenced phase equations. In the past decades, all researchers estimated the LB as a constant parameter in advance and compensated it in real time. However, the constant LB assumption is inappropriate in practical applications because of the physical length and permittivity changes of the cables, caused by the environmental temperature variation and the instability of receiver-self inner circuit transmitting delay. Considering the LB drift (or colored LB) in practical circumstances, this paper initiates a real-time estimator using auto regressive moving average-based (ARMA) prediction/whitening filter model or Moving average-based (MA) constant calibration model. In the ARMA-based filter model, four cases namely AR(1), ARMA(1, 1), AR(2) and ARMA(2, 1) are applied for the LB prediction. The real-time relative positioning model using the ARMA-based predicting LB is derived and it is theoretically proved that the positioning accuracy is better than the traditional double difference carrier phase (DDCP) model. The drifting LB is defined with a phase temperature changing rate integral function, which is a random walk process if the phase temperature changing rate is white noise, and is validated by the analysis of the AR model coefficient. The auto covariance function shows that the LB is indeed varying in time and estimating it as a constant is not safe, which is also demonstrated by the analysis on LB variation of each visible satellite during a zero and short baseline BDS/GPS experiment. Compared to the DDCP approach, in the zero-baseline experiment, the LB constant calibration (LBCC) and MA approaches improved the positioning accuracy of the vertical component, while slightly degrading the accuracy of the horizontal components. The ARMA(1, 0) model, however, improved the positioning accuracy of all three components, with 40 and 50 % improvement of the vertical component for BDS and GPS, respectively. In the short baseline experiment, compared to the DDCP approach, the LBCC approach yielded bad positioning solutions and degraded the AD accuracy; both MA and ARMA-based filter approaches improved the AD accuracy. Moreover, the ARMA(1, 0) and ARMA(1, 1) models have relatively better performance, improving to 55 % and 48 % the elevation angle in ARMA(1, 1) and MA model for GPS, respectively. Furthermore, the drifting LB variation is found to be continuous and slowly cumulative; the variation magnitudes in the unit of length are almost identical on different frequency carrier phases, so the LB variation does not show obvious correlation between different frequencies. Consequently, the wide-lane LB in the unit of cycle is very stable, while the narrow-lane LB varies largely in time. This reasoning probably also explains the phenomenon that the wide-lane LB originating in the satellites is stable, while the narrow-lane LB varies. The results of ARMA-based filters are better than the MA model, which probably implies that the modeling for drifting LB can further improve the precise point positioning accuracy.