Obtaining mm-level positioning accuracies with GNSS requires modeling of all error sources such as higher-order ionospheric effects. As a part of an IAG working group, I collaborated with European colleagues to investigate how this error source could be estimated as a part of the PPP filter. The results were published last week in GPS Solutions (Banville et al. 2017).
The idea behind this project came from a paper by Zehentner and Mayer-Guërr (2016) who estimated higher-order ionospheric effects within their PPP filter for LEO orbit determination. Since both the first- and second-order effects are linearly dependent on the slant total electron content (STEC), it is possible to modify the partial derivatives of the STEC parameters in the PPP filter to effectively estimate both terms jointly into a single parameter. This new processing strategy was however not the focus of their paper and they provided very little insights into the method. Hence, we decided to verify if this methodology is indeed applicable and under which circumstances.
As an example, I will present results from station HERS with data collected on 8 November 2001, a day with high ionospheric activity. The first thing we were interested in was to measure the impact of higher-order ionospheric effects on kinematic PPP position estimates. The following figure shows position differences between two kinematic PPP solutions: one with second- and third-order ionospheric corrections applied (using the RINEX_HO utility) and another one without any of these corrections applied.
Fig 1 Impact of higher-order ionospheric effects on kinematic PPP
As we can see, differences are at the millimeter level and affect mainly the latitude component, with a peak error around 3 mm. I have to admit that this is quite tiny and that it is not perceptible in the original kinematic PPP time series since it is buried in the noise of the solution. It is only when differencing solutions that we can start to isolate the impact of this error source. In static mode, the impact is even less pronounced: it can be approximated by taking the mean of the time series in the above figure. Hence, we are talking about a latitude error of 1 mm or less over a 24-hour session.
In the next step, we modified the partial derivatives in the PPP filter to estimate both first- and second-order ionospheric effects, as described above. Fig 2 presents the difference in kinematic PPP solutions between this new approach and the RINEX_HO solution:
Fig 2 Errors introduced when neglecting the receiver DCB in the PPP filter
The new approach introduced biases of a few millimeters in the position estimates. The magnitude of these biases exceeds the position errors caused by higher-order ionospheric effects, which is quite concerning. These biases originate from the definition of the estimable parameters in the PPP filter: slant ionospheric delays in PPP are contaminated by the receiver DCB. Since every satellite now has slightly different partial derivatives for the STEC parameters, the receiver DCB propagates into the position estimates; the larger the receiver DCB, the larger the position biases.
To solve this issue, we need to estimate unbiased STEC. This can be achieved in two ways: either by using a mathematical representation of VTEC or by introducing external constraints from a global ionospheric map (GIM). With the latter approach, the two kinematic PPP solutions now agree fairly well:
Fig 3 With proper handling of the receiver DCB, the PPP solution performs to the same level as the RINEX_HO solution
In conclusion, it is indeed possible to estimate higher-order ionospheric effects directly within the PPP filter. However, we still need GIMs to isolate the receiver DCB which goes a little bit against the initial purpose of this approach which was to estimate STEC without relying on external inputs. For all the details, please read the full paper!
Banville S, Sieradzki R, Hoque M, Wezka K, Hadas T (2017) On the estimation of higher-order ionospheric effects in precise point positioning. GPS Solut doi:10.1007/s10291-017-0655-0
Zehentner N, Mayer-Gu¨rr T (2016) Precise orbit determination based on raw GPS measurements. J Geod 90:275–286. doi:10.1007/s00190-015-0872-7