Single-Frequency PPP

The main challenge with single-frequency PPP is to mitigate ionospheric effects. Global ionospheric maps (GIM) can reduce the contribution of this error source to some extent but residual errors often lead to meter-level positioning accuracies. A popular option is to use the GRAPHIC combination, an average of code and phase observations that “eliminates” ionospheric errors. This formulation is, however, not always your best option.


Following a concept explained in a previous post, forming a linear combination to eliminate a parameter is usually equivalent to estimating this parameter unconstrained. Hence, explicitly forming the GRAPHIC combination implies that the temporal variation of slant ionospheric delays is modeled as white noise.

The ionospheric delay between a satellite and a receiver varies at a rate of a few millimeters to a few centimeters per second. This important piece of information can be exploited when using uncombined signals, i.e. using directly the code and phase observations. The main advantages of using uncombined signals are that:

  • Modeling slant ionospheric delays as a random walk process can reduce noise.
  • External information on the ionosphere can improve position estimates and possibly reduce convergence times.

The following plot shows three “kinematic” single-frequency PPP solutions computed from data collected at station UNB3 (Trimble NetR9) on 15 March 2015.

  1. The solution labeled ‘GIM’ does not estimate any parameters related to the ionosphere and simply uses corrections from a GIM. As expected, errors in the model and a poor temporal resolution lead to meter-level accuracies.
  2. A second solution labeled as 'GRAPHIC' uses uncombined signals but models the temporal variation of the ionospheric parameters as white noise. For this reason, it is expected to be equivalent to explicitly forming the GRAPHIC combination. This approach leads to a much better accuracy, but is quite noisy.
  3. The last solution also uses uncombined (UC) signals but models the temporal variation of the ionosphere as a random walk process. As a result, the noise is greatly reduced.

In the last two cases, the singularities of the systems are handled by constraining ionospheric parameters using the GIM for the first epoch when a satellite is observed only.

In conclusion, using uncombined signals allows adding extra information on the temporal variability of the ionosphere. Since single-frequency receivers typically have large code noise, this approach should be beneficial for improving the precision of the position estimates.

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Comments: 6
  • #1

    Ahmed El-Mowafy, Australia (Friday, 08 May 2015 12:25)

    - You used a single frequency data extracted from Trimble NetR9, which is a high geodetic-grade receiver. The code noise could be much larger if you used true single frequency receivers, e.g. U-blox.
    - How about the use of ionospheric correctional data received from satellite-based augmentation systems, such as WAAS or EGNOS, which seems more realistic for single-frequency users.

  • #2

    Simon Banville (Friday, 08 May 2015 21:01)

    Dear Ahmed,

    I agree with you that "true" single-frequency receivers have larger code noise than geodetic-grade receivers which, as a consequence, may lead to larger positioning errors. But even in this case, modeling slant ionospheric delays as a random walk process would reduce the noise significantly, which is the aspect I wanted to emphasize.

    The best source of ionospheric correction would be slant ionospheric delays from nearby stations of course. I'm also hoping that more agencies will broadcast ionospheric corrections (both VTEC and STEC) using the new RTCM-SSR messages.

  • #3

    Lakshay Narula (Thursday, 31 May 2018 14:44)

    Hi Simon,

    I'm curious about the 'GIM' solution chart here. It seems that single-frequency PPP wouldn't converge within 24 hours when using raw ionospheric corrections from a model, even with a high-quality antenna. But this contradicts many research papers like [1] where a similar set up has been shown to provide better than 30 cm accuracy after a couple of hours. Note that [1] also makes no attempt to estimate or eliminate the ionospheric delay beyond using corrections from CODE.

    What are your thoughts on this? Am I missing some important detail?

    Thanks for the post!

    [1] van Bree, R.J.P. & Tiberius, C.C.J.M. GPS Solut (2012) 16: 259.

  • #4

    Simon Banville (Friday, 01 June 2018 08:04)

    @Lakshay Interesting remark! There are a few of things to clarify first:
    1) The 30 cm accuracy that you are referring to is for the horizontal component, while the plot above shows the 3D error (and the height is usually less accurate)
    2) The quality of ionospheric corrections from GIMs depends on many factors, such as ionospheric activity, time of day, time resolution and GIM provider.
    3) You suppose that no attempt is made to estimate the ionospheric delay in [1], although this is not explicitly stated in the paper. I looked for a reference to the authors' previous work (van Bree et al., 2009) and this aspect is not being discussed either in that paper.

    If the results in your reference were based on the estimation of additional parameters for the ionosphere, then the results obtained are not surprising and match roughly the ones presented above.

    If we suppose that no iono-related estimation was included in the filter then we would have to compare apples to apples and look at the same data sets and corrections. If ionospheric errors are not properly accounted for the by the GIM, then they have to propagate somewhere... carrier-phase ambiguities, assuming they are modelled as constant parameters, cannot account for time-varying errors and the only place these errors can go is in the position or receiver clock parameters. So, depending on the magnitude of the errors, position estimates could be impacted significantly.

    Thanks for pointing this out, it is something that might be worthy of further investigation. For my part, I'm still convinced that adding parameters to model residual ionospheric effects is a good way to cope with imperfections in the GIM corrections and it will work regardless of ionospheric conditions!

  • #5

    Lakshay Narula (Monday, 11 June 2018 11:48)

    Hi Simon,

    Thanks for the discussion, this is quite close to what I had in mind. I dug down in to the details of the SF-PPP method likely used in [1], and I believe it's based on the phase-adjusted pseudorange algorithm from [2]. Equation (8) from [2] would suggest to me that no ionosphere parameters are being estimated. However, since this is a recursive filter, the ambiguities are likely estimated as parameters with small but non-zero process noise, enabling them to absorb at least some time-varying GIM errors. Even if they were to be treated as constant unknowns in a batch estimator, they could still compensate for a fixed bias per satellite.

    Adding parameters to model residual ionospheric effects is obviously the way to go with PPP. I'm currently trying to analyze long-term position biases, if any, that may result from using ionospheric corrections from a GIM in a code-phase-based receiver for vehicular urban positioning, where PPP convergence could be harder to achieve. I'm seeing some interesting GIM characteristics that I'd be happy to share once I have a write up.

    Thanks again.

    [1] van Bree, R.J.P. & Tiberius, C.C.J.M. GPS Solut (2012) 16: 259.
    [2] Le, A. Q., and P. J. G. Teunissen. "Recursive least-squares filtering of pseudorange measurements." In European navigation conference, pp. 7-10. 2006.

  • #6

    Simon Banville (Thursday, 28 June 2018 21:05)

    @Lakshay Feel free to post a link to your findings once available!