For low earth orbit (LEO) satellite GPS receivers, space-based augmentation system (SBAS) ephemeris/clock corrections can be applied to improve positioning accuracy in real time. The SBAS correction is only available within its service area, and the prediction of the SBAS corrections during the outage period can extend the coverage area. Two time series forecasting models, autoregressive moving average (ARMA) and autoregressive (AR), are proposed to predict the corrections outside the service area. A simulated GPS satellite visibility condition is applied to the WAAS correction data, and the prediction accuracy degradation, along with the time, is investigated. Prediction results using the SBAS rate of change information are compared, and the ARMA method yields a better accuracy than the rate method. The error reductions of the ephemeris and clock by the ARMA method over the rate method are 37.8% and 38.5%, respectively. The AR method shows a slightly better orbit accuracy than the rate method, but its clock accuracy is even worse than the rate method. If the SBAS correction is sufficiently accurate comparing with the required ephemeris accuracy of a real-time navigation filter, then the predicted SBAS correction may improve orbit determination accuracy.

A space-based augmentation system (SBAS) improves global navigation satellite system (GNSS) positioning accuracy by providing ephemeris/clock and ionospheric delay corrections. As with ground GNSS users, low earth orbit (LEO) satellites are able to use SBAS correction information if their GNSS receivers have the capability to receive SBAS signals. The corrections are transmitted in real time along with GNSS signals. The SBAS correction transmission does not require extra data links and is suitable for space applications.

LEO satellites may use the SBAS corrections inside the system service area, but the corrections are not available outside the service area. The service area of an SBAS is determined by the geographical location of its monitoring stations. Due to this limitation, SBAS coverage is mainly limited to land area and does not cover most of the ocean area, which occupies 70% of the Earth. To maximize the usability of the SBAS for LEO satellites, it is necessary to extend the area where the SBAS corrections are available. Prediction of the SBAS corrections during the outage period can be a solution.

Among the SBAS corrections, the ionospheric correction is a position-specific value, and its prediction outside the service area is not useful. However, the ephemeris/clock correction is a GNSS satellite-specific value. Preceding researches discussed SBAS correction accuracy and its applications [

The SBAS message provides the rate of change data for the corrections to fill the time gap between data transmit intervals. The use of this data to extend the correction period is tested. However, the rate of change data was designed for a short interval only, and another method for long interval prediction is required. There are various time series forecasting models, for example, simple polynomial extrapolation, autoregressive moving average (ARMA), autoregressive (AR), and neural network. We propose two forecast models, an ARMA model and an AR model, to predict the ephemeris/clock corrections outside the service area. These models are selected because they are two of the most widely used models. Other prediction techniques have been tested but are not included in this paper: a polynomial model and a neural network model. The polynomial model yielded much lower prediction accuracy than the rate model and is dropped from consideration. A preliminary analysis by the neural network model showed low prediction accuracy and frequent divergence. Using the neural network requires more extensive work and is not included in this paper. The SBAS ephemeris/clock correction is predicted by using the ARMA and AR models by simulating the GPS satellite visibility condition. The ARMA/AR prediction accuracy is compared with the results using the SBAS rate of change data.

SBAS messages include GPS satellite ephemeris/clock corrections, pseudorange fast corrections, and ionospheric delay corrections. The message is classified into message type (MT), and one MT is transmitted every second. The ephemeris/clock corrections are mainly included in MT 25. The corrections and their rate of change are also included in MT 25 [

Each MT 25 includes the issue of data (IOD). This IOD has to correspond with the IOD ephemeris (IODE) of the current GPS ephemeris for that satellite. If the IODE of the GPS broadcast ephemeris data does not match the IOD of the SBAS transmission, it is an indication that the GPS ephemeris data sets have changed. The user must continue to use the old matched data from the previous broadcasts, until a new MT 25 with matching IOD is broadcast for the new GPS ephemeris data set [

There is a difference between the SBAS coverage and service areas. The SBAS service area represents an area where SBAS provides precise differential corrections, while the SBAS coverage area represents an area where the SBAS signal covers. The service area is limited by ground based infrastructure such as GNSS monitoring stations. Since the SBAS signal is transmitted from a geostationary satellite in equatorial orbit, the coverage area is much wider than the service area. In the case of WAAS, both of North and South America are in the coverage area, but only a part of North America is in the service area.

An important aspect of the SBAS is the integrity function, which is given by providing the error bound of the corrections. The SBAS is primarily developed for aviation use, and the integrity function allows an aircraft user to estimate an error bound and to be alarmed on fault signals. The accuracy of the correction information degrades over time, and the error bound grows over time. MT 10 provides a degradation factor to adjust the integrity values. The SBAS equipment should only use data if it is current; that is, before it has timed out. The time-out intervals are different for each message type and are a function of the aviation flight phase. In case of MT 25, the time-out interval is different for the type of aircraft landing approach, 360 s for a nonprecision approach and 240 s for a precision approach.

The AR model predicts a future output as a combination of past inputs. A linear AR model can be represented as follows:

The ARMA model is a combination of the AR model and the moving average (MA) model. The MA model predicts a future output as a combination of past errors. ARMA is appropriate when a system is a function of a series of unobserved noise as well as its own behavior. The general ARMA model can be expressed as [

Four corrections, ECEF

Comparing with the rate model, the ARMA model may cause a divergence in its prediction. Unstable model coefficients may cause intermittent divergences of the prediction value. This is not a distinctive feature of the ARMA/AR model; any other prediction model may cause a divergence. Although the divergence rarely happens for the SBAS prediction, approximately less than once per day for each PRN, a counter measure should be prepared. In order to prevent the divergence problem, a simple monitoring algorithm is developed. The output of the ARMA/AR method is constantly compared with the output of the rate method for determining the divergence condition. If the magnitude of the ARMA/AR prediction output is greater than three times of the rate output, then the last output before the divergence is used instead. The factor three was determined, as a rule of thumb, from analyzing the prediction results.

A LEO satellite orbit is used to simulate the GPS and SBAS signal availability. The NASA/DLR Gravity Recovery and Climate Experiment (GRACE) satellite’s orbit is selected [

The first step of the simulation is to determine the GPS satellite visibility condition for the LEO satellite from the actual GPS measurements. This process determines which GPS satellite signal is available for the LEO satellite at each epoch. The second step is matching SBAS corrections to the GPS satellite at each epoch. The third step is to determine the SBAS signal availability using the LEO satellite location and the SBAS coverage area. This SBAS information is the input of the prediction process and the SBAS information of the second step is the reference information to match the prediction output. The reference SBAS signal is generated by applying the rate of change of the corrections up to 360 s, which is the time-out interval of the nonprecision approach. The data interval is 10 s as the GRACE GPS measurements.

Among the SBAS, that is, WAAS, MSAS, EGNOS, and GAGAN, one SBAS is used for the analysis. Combined use of different SBAS corrections can extend the signal availability, but this option is not adopted for this analysis due to different levels of correction characteristics and accuracies. The correction accuracy varies significantly with the systems [

Figure

Simulated WAAS availability using a LEO satellite trajectory (May 1, 2014).

Figure

Number of total and WAAS-available GPS satellites observed at GRACE satellite orbit (May 1, 2014).

Figure

WAAS correction time series on May 1, 2014 (PRN 22).

SBAS ephemeris/clock corrections are predicted with the ARMA and AR models. To determine an optimal ARMA order, a series of tests has been performed by changing the order. The orbit error differences among the different orders are less than 5 cm, far below the SBAS accuracy, and higher orders (>10) caused intermittent divergence. Differences among the

Figure

Predicted corrections by the ARMA, AR, and the rate of change methods (PRN22, May 1, 2014).

Figures

Prediction errors by the rate of change method (PRN 22, May 1, 2014).

Prediction errors by the ARMA method (PRN 22, May 1, 2014).

Prediction errors by the AR method (PRN 22, May 1, 2014).

Theoretically, the residuals from the ARMA or AR should be close to white noise if the model fits the time series perfectly. The fitting residuals from the ARMA and AR are analyzed in the frequency domain, but they are not shown as a figure. All the residual power spectral density (PSD), that is,

Figures

Orbit prediction RMS errors of all PRNs (May 1, 2014).

Clock prediction RMS errors of all PRNs (May 1, 2014).

The prediction accuracy depends on the length of the prediction time. Figure

RMS of orbit and clock prediction errors for different time intervals (May 1, 2014).

As mentioned earlier, the RMS of the correction magnitude is 4.276 m for the orbit and 2.436 m for the clock on this date. If we set a maximum allowable prediction error less than a half of the correction magnitude, they become approximately 2 m for the orbit and 1 m for the clock. With this justification, the ARMA prediction up to 1800 s can be feasible for use. If a single SBAS is used for the corrections, the 1800 s is not sufficient for covering most of the flight time. If multiple SBAS corrections, for example, EGNOS, MSAS, and GAGAN, are used, the 1800 s is sufficient for most of the flight time except in the South Pole region.

The prediction accuracy may depend on the correction data characteristics, and it is reasonable to analyze other days of data. Five days from 2014 are chosen to evaluate the accuracy, from January 1 to September 1 with two month interval. The latest 2014 dates are selected because the WAAS correction data accuracy and characteristics changed with the improvement of the system [

Figure

Orbit and clock prediction errors for different days in 2014.

The impact of the SBAS correction on the user position accuracy depends on the various effects, for example, navigation filters and other errors. In general, a high precision filter using carrier phase measurements can benefit from the SBAS [

The position accuracy of a GPS receiver is limited by the broadcast ephemeris/clock errors. For LEO satellite GPS receivers, SBAS ephemeris/clock corrections can be an alternative solution for real-time use. LEO satellites move at a high speed and the duration for LEO satellites to be inside the SBAS service area is relatively short. Prediction of the SBAS corrections during the outage period can extend the service area. Two widely used forecasting models, ARMA and AR models, are proposed to predict the ephemeris/clock corrections outside the coverage area. The simulated GPS satellite visibility is applied to the WAAS correction data, and the prediction accuracy degradation along with the time is investigated. A comparison with the prediction results using the SBAS rate of change information is performed. The ARMA method shows a greater accuracy than the rate method. The error reductions of the ephemeris and clock by the ARMA method over the rate method are 37.8% and 38.5%, respectively. For the prediction interval from 20 min. to 30 min., the RMS errors are 1.164 m for the orbit and 0.614 m for the clock. The AR method shows a slightly better orbit accuracy than the rate method, but its clock accuracy is even worse than the rate method. The positioning accuracy improvement by the raw or predicted SBAS correction is not covered in this paper. Impact of the SBAS corrections on real-time orbit determination mainly depends on the SBAS correction accuracy level. If the SBAS correction is sufficiently accurate comparing with the required ephemeris accuracy of a real-time navigation filter, then the raw or predicted SBAS correction may improve orbit determination accuracy.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work has been supported by the National GNSS Research Center program of Defense Acquisition Program Administration and Agency for Defense Development.