A significant number of GNSS applications uses and will continue to use single-frequency GNSS receivers. Therefore the precise estimation of the ionospheric range error from single-frequency GNSS data is and remains to be an important issue.
We describe a method which allows to autonomously determine the ionospheric range error using single-frequency GNSS data of a single GNSS receiver. The main idea is not to consider each receiver-satellite link individually, but to use all information present in each epoch and between epochs. For the ionospheric range error there is a non-zero correlation, both between different links at a given time and between links at different times. By taking into account all these correlations, we are able to determine the (absolute) ionospheric range error from single-frequency phase and code measurements only.
Since the dispersive ionospheric delays affect phase and code measurements with opposite sign, it is possible to extract relative ionospheric range errors by forming the difference of code and phase measurements. Besides the (double) ionospheric range error, this difference contains the code and phase multipath noise, the instrumentation offsets, and the phase ambiguity. In order to suppress the dominant noise contribution, the code multipath noise, we use a Hatch-type filtering with a suitably chosen time constant. Note that there is a new phase ambiguity for each data arc, i.e., a continuous range of receiver-satellite measurements. Hence, we have to determine one new constant for each new data arc in order to calibrate the relative ionospheric range errors obtained from the filtered difference of phase and code observables.
We have developed a model-assisted TEC calibration and reconstruction technique, using a simple polynomial model of the ionosphere, which approximates the ionosphere in the vicinity of the GNSS receiver by a single layer in a local, sun-fixed coordinate system. This technique is capable of calibrating the relative ionospheric range errors in near-realtime by using a Kalman-filter-type weighted-least-squares algorithm with a priori knowledge; the input measurement errors are determined from the Hatch-type filtering, while the initial model covariances may be determined, e.g., by the start-up weighted-least-squares solution.
Without using any additional information the warm-up phase is found to be at least one hour. However, it is possible to shorten the warm-up phase by using, e.g., Klobuchar-model derived vertical TEC values as initial information. After the start-up phase, new model and ambiguity estimates are computed every couple of minutes by using both, the new measurements and the last model coefficients to update the model.
Along with the model coefficients describing the variation of vertical TEC around the receiver we obtain further statistical information, e.g., the relative chi2 value of the fit which is a measure of how good the model Ansatz is consistent with the measured data, given the measurement errors and model covariances. By monitoring the chi2 value, deficiencies in data preprocessing (cycle slips) and, more importantly, extreme ionospheric perturbations can be detected.
The comparison of vertical TEC derived from single-station single-frequency data obtained by the proposed method with TEC values obtained from European TEC maps under near-solar-minimum conditions shows a good agreement. This is encouraging, taking into account that we compare TEC valued determined from a single station using single-frequency GNSS data with TEC maps which are produced using dual-frequency phase and code GNSS data from a network of 20-30 GNSS receivers.
Since the algorithm is able to calibrate ionospheric range errors containing different phase ambiguities for each arc, we have applied the algorithm to the reconstruction of vertical TEC from dual-frequency phase-only GNSS data. Here, the relative chi2 values are quite high, signaling that the simple single-layer polynomial ionospheric model cannot fully describe the ionospheric information contained in the low-(multipath-)noise GNSS phase observables Nevertheless, calibrated TEC can be obtained with this method. A possible continuation along the lines of this route would be to consider more elaborated ionospheric models, possibly transcending the single-layer approximation.
We anticipate that the algorithm will work under other geophysical conditions, too, e.g., low-latitudes and/or high solar activity. The calibrated ionospheric range errors may be used in single-frequency point positioning. We compare the performance of our locally valid, near-real-time ionospheric corrections with the Klobuchar model, whose model coefficients are determined in intervals of days and are valid globally.
The proposed algorithm is capable to derive calibrated ionospheric range errors from single-frequency GNSS data. In addition, a model describing the ionosphere in the vicinity of the receiver is provided, along with various statistical quantities.
In the context of single-frequency point positioning, e.g. using low-cost GNSS receivers in quasi-static setups, this method is anticipated to provide autonomously determined, near-real-time ionospheric corrections comparable or better than the Klobuchar model. In (civil) aeronautical GBAS systems, which due to certification issues will continue to be restricted to use single-frequency GNSS equipment for some time, this method allows to detect ionospheric perturbations, including ionospheric gradient information, providing the necessary information for sigma monitors
