Single-Frequency PPP-AR

The term “PPP-RTK” usually involves positioning using state-space corrections generated from a network of GNSS receivers. This flexible representation of error sources allows for a scalable solution to be deployed: a global PPP solution can be obtained with precise satellite orbit and clock corrections, and instantaneous convergence can also be obtained by providing local atmospheric augmentation. In this post, I apply this concept to achieve single-frequency PPP with ambiguity resolution (AR).

 

The solution developed herein is based on the decoupled-clock product generated at NRCan. To achieve ambiguity resolution for dual-frequency PPP, satellite phase-clock corrections are accompanied by additional code biases for the C1W and C2W signals. While single-frequency PPP can be performed using this product, two corrections are missing to achieve cm-level accuracies and ambiguity resolution: 1) precise slant ionospheric delays, and 2) geometry-free phase biases (or an equivalent representation).

 

Using a network of stations located in Canada, daily PPP-AR solutions were computed to obtain integer ambiguities on L1 and L2. These ambiguities were then used to remove the arc-dependency of the geometry-free carrier-phase combination, in a process called “integer leveling.” The resulting signal is a precise measure of slant ionospheric delay, but still contains receiver- and satellite-dependent biases. Coincidentally, it is these satellite biases that will enable single-frequency PPP-AR. Therefore, it is necessary to isolate these biases from TEC using a mathematical representation of the ionosphere, in exactly the same way that differential code biases (DCBs) are typically being estimated. More details regarding this process can be found in Banville et al. (2014). Once the geometry-free biases are estimated, they can be applied to single-frequency observations to maintain consistency in the propagation of satellite equipment delays and perform ambiguity resolution.

 

To demonstrate the validity of my implementation, I used data from station UNBJ on 8 March 2016. As discussed above, the inputs to my PPP-RTK solution were: precise orbit and clock corrections + code biases, geometry-free (L1-L2) phase biases, and slant ionospheric delays from nearby stations (the closest being FRDN, only 2 km away). In the filter, station coordinates, receiver clock offset, carrier-phase ambiguities and slant ionospheric delay parameters were estimated.  No assumptions were made regarding the dynamics of the station, leading to a so-called kinematic solution. The (smoothed) results are presented in the following figure.

Figure 1: Single-frequency PPP with ambiguity resolution for station UNBJ

on 8 March 2016

 

The results obtained are very close to a short-baseline RTK solution, although the height estimates are slightly biased. This can be explained by the fact that no troposphere augmentation was included in the solution, and no residual delay was estimated. Hence, modeling residual tropospheric zenith delay using nearby stations could lead to an RTK-like solution.

 

In a previous post on single-frequency PPP, I explained how to reduce the noise of position estimates through the use of uncombined signals and process noise. With exactly the same functional model, additional information on the ionosphere can now be simply integrated to produce very precise solutions.

 

Reference

Banville S., Collins P., Zhang W., Langley R. B. (2014). "Global and Regional Ionospheric Corrections for Faster PPP Convergence," NAVIGATION, Journal of The Institute of Navigation, Vol. 61, No. 2, Summer 2014, pp. 115-124.



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