Time Correlation in GNSS Precise Point Positioning (2013)

Precise point positioning (PPP) is a GNSS processing technique that can position to centimetre accuracy with a single receiver (Kouba and Heroux 2001). Applications of PPP in static positioning include reference frame determination, rapid determination of station coordinates following tectonic activity and precise determination of Low Earth Orbiters (Bisnath and Gao 2008). PPP processing capability has improved to such an extent that kinematic processing is undertaken in real-time and has become the dominant application in airborne sensor positioning and offshore marine positioning. For high precision real-time applications it is necessary to use precise orbit and clock corrections at a high rate in addition to high rate observation data which is most commonly processed using a recursive estimator, such as a Kalman filter.

In PPP processing both code and carrier phase observations are used as part of the observation model and their relative weighting in the stochastic model must also be considered. However, the usual assumption in GNSS processing that the observations are uncorrelated in time is often not true. Such an assumption is embedded within the observation stochastic model but frequently does not truly reflect reality. Several studies have been undertaken to establish the correlation between observations when forming the a priori stochastic model. In addition, when considering PPP processing it is important to consider that high rate clock corrections may also have an impact on time correlation.

Temporal correlation within the GNSS observations is propagated into the residuals. However, even when the observations are uncorrelated in time, non-zero autocorrelations are evident if the filter is sub-optimal. For a filter to be optimal the residuals should have zero mean and be uncorrelated in time (Salzmann 1993). An optimal filter requires knowledge of the true observation covariances and the process noise (covariances) in the dynamic model. In practice, as these are not known exactly, the innovation and post-fit residuals from a sub-optimal Kalman filter can be shown to exhibit non-zero autocorrelations. 

The objective of this research is to show that when processing high rate GNSS data with PPP using high rate orbit and clock corrections the residuals are highly correlated in time and to determine whether these time correlations can be modelled. Although much work has been done in identifying the correlation between observations at a single epoch (Teunissen et al. 1998, Amiri-Simkooei et al. 2009), studies so far in analysing the time correlation have been restricted to double differencing methodologies. This research investigates the time correlation present in zero difference processing.

A number of static and kinematic datasets with high rate observations were processed with high rate clock corrections using our PPP software in a real-time mode. Only GPS data was considered for this study. Analysis of the residuals using a variety of statistical methods reveals large time correlation between residuals over a short period. This has been observed for both code and carrier phase observations.
 
It is possible to model the time correlation in a residual time series as a series of autoregressive parameters (Priestley 1981). Following the evaluation of a number of autoregression methods, Burg’s method (Broersen 2006) was chosen to be most suitable for the level of time correlation observed as this produced numerically stable autoregressive parameters. We will show that for a given time series of residuals a first-order autoregressive process is sufficient to remove the majority of the time correlation, resulting in a time series of residuals much more closely characterised by white noise, creating the so called “whitened residuals”.

From this study we conclude that autoregression is an effective method of identifying time correlation in the residuals obtained from high rate data processed using PPP and that in most cases this time correlation can be characterised by a first-order autoregression AR(1) process. When developing a stochastic model it is therefore not only important to consider the correlations present between observations at a single epoch, but also the correlation between successive epochs. With the introduction of uncalibrated phase bias terms in order to facilitate integer ambiguity resolution in PPP the significance of time correlation on the accuracy of solutions will likely become increasingly significant.

References

Amiri-Simkooei AR, Teunissen PJG, Tiberius CCJM (2009) Application of Least-Squares Variance Component Estimation to GPS Observables. Journal of Surveying Engineering, Vol. 135(4), pp. 149–160

Baarda W (1968) A testing procedure for use in geodetic networks. vol 2. Netherlands Geodetic Commission, 

Bisnath S, Gao Y (2008) Current State of Precise Point Positioning and Future Prospects and Limitations. Proceedings of IUGG 24th General Assembly. 

Broersen PMT (2006) Automatic Autocorrelation and Spectral Analysis. Springer, 

Kouba J, Heroux P (2001) GPS Precise Point Positioning Using IGS Orbit Products. GPS Solutions 5 (2):12-28

Priestley MB (1981) Spectral Analysis and Time Series. Academic Press Limited, 

Salzmann MA (1993) Least squares filtering and testing for geodetic navigation applications. Department of Geodetic Engineering, Delft University of Technology.

Teunissen PJG, Jonkman NF, Tiberius CCJM (1998) Weighting GPS Dual Frequency Observations: Bearing the Cross of Cross-Correlation. GPS Solutions
Vol. 2(2), pp. 28-37