Time Correlation

The precision associated with position estimates obtained from GNSS processing software is often too optimistic. The culprit is an improper definition of the stochastic model, neglecting time-correlated errors such as multipath or satellite orbit and clock errors. In situations where single-epoch ambiguity resolution is not possible, such as PPP or long-baseline RTK, time correlation can become a serious concern for ambiguity validation.

 

The ambiguity covariance matrix is basically a reflection of the position covariance matrix in the sense that if the true position is not within its associated error ellipsoid, then the “true” vector of integer ambiguities will not be in the ambiguity search space either. This means that, regardless of the ambiguity validation method used (ratio test, difference test, projector test, BIE, etc.), ambiguity validation will always fail. While this aspect was often stated in “old” papers on ambiguity resolution, I noticed that it is frequently omitted in more recent papers involving PPP.

 

As an example, let us consider the PPP convergence of station DRAO, located in western Canada, at 11:00 GPST on 2 March 2008. The following plot shows the latitude, longitude and height errors, along with 2-sigma error bars. Due to poor geometry and possible biases in code observations, positioning errors clearly exceed the specified precision. Attempting ambiguity resolution during the first 55 minutes of this one-hour session can only lead to disastrous results (unless a better stochastic model is used).

PPP convergence and 2-sigma error bars at DRAO, on 2 March 2008.

There are several ways of dealing with time-correlated errors:

 

Time-differenced observations: by reformulating the Kalman filter equations, it is possible to obtain absolute values for parameters using (partially) time-differenced measurements. The downside of this approach is that the filter needs to be redesigned.

 

Augmented-state approach: additional states can be added to the filter to model time-correlated errors, typically one state per observation. This approach significantly increases the number of parameters to be estimated, especially with several GNSS constellations and multiple frequencies.

 

FAMCAR: this approach was proposed by people at Trimble, and essentially consists of modeling time-correlated errors such as code multipath using one-dimensional geometry-free filters which are much easier to handle. Those filters are combined with an augmented-state “geometry” filter using only carrier-phase observations.

 

Unfortunately, all of those approaches assume that you have a basic knowledge of the correlation time of the errors. Although this information could be obtained through empirical testing, I have yet to find a one-value-fits-all solution.

 

Further Reading

 

Brown, R.G. and P.Y.C. Hwang (1992). Introduction to random signals and applied Kalman filtering, 2nd edn. Wiley, New York

 

Petovello, M.G., K. O’Keefe, G. Lachapelle, and M.E Cannon (2009). "Consideration of time-correlated errors in a Kalman filter applicable to GNSS," Journal of Geodesy, Vol. 83, No. 1, pp.51–56.

 

Vollath, U.. (2004). "The Factorized Multi-Carrier Ambiguity Resolution (FAMCAR) approach for efficient carrier-phase ambiguity estimation," Proceedings of the 17th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS 2004), Long Beach, CA, September 2004, pp. 2499-2508.



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