1. P. Wielgosz and A. Krankowski IGS AC Workshop Miami Beach, June 2-6, 2008 University of Warmia and Mazury in Olsztyn, Poland pawel.wielgosz@uwm.edu.pl Real-time Kinematic GPS Positioning Supported by Predicted Ionosphere Model

2. Outline Research objectives Research objectives ARMA method ARMA method RTK positioning model RTK positioning model Experiment design Experiment design Test results and analysis Test results and analysis Conclusion Conclusion

3. Research Objectives Develop and evaluate methodology and algorithms for OTF-RTK positioning technique suitable for medium and long ranges 10-100 km Develop and evaluate methodology and algorithms for OTF-RTK positioning technique suitable for medium and long ranges 10-100 km Test applicability of predicted ionosphere models to support medium range OTF-RTK positioning Test applicability of predicted ionosphere models to support medium range OTF-RTK positioning Evaluate prediction model based on ARMA method Evaluate prediction model based on ARMA method Study the impact of the model accuracy on the ambiguity resolution (speed and reliability) Study the impact of the model accuracy on the ambiguity resolution (speed and reliability)

4. Methodology – ARMA prediction of real-valued time series Let y t for t =1, 2, …., n be an equidistant stationary stochastic time series and y t+1 be the prediction at time t+1. The autoregressive-moving average process ARMA(p,q) is defined by the formula: where:  i are autoregressive coefficients,  i are the moving average coefficients, p and q are the autoregressive and moving average orders,  i is a white noise process After introducing the backshift operator B K the process can be converted to :

5. Methodology – ARMA prediction of real-valued time series The ARMA forecast L steps ahead - the part of the operator containing only nonnegative powers of B * 10 previous days of the TEC values were taken for the prediction computation

6. Methodology – ARMA prediction of real-valued time series Our previous studies showed that the TEC prediction for 1- to 3 hours ahead yields values very close to real, observed TEC (under quiet to moderate geomagnetic conditions) Our previous studies showed that the TEC prediction for 1- to 3 hours ahead yields values very close to real, observed TEC (under quiet to moderate geomagnetic conditions) After 3 hours the quality of the forecast diminishes very quickly After 3 hours the quality of the forecast diminishes very quickly ARMA forecasting method is very simple and does not need any a-priori information about the process nor additional inputs such as, e.g., solar or geomagnetic activity indices ARMA forecasting method is very simple and does not need any a-priori information about the process nor additional inputs such as, e.g., solar or geomagnetic activity indices Reference: Krankowski A., Kosek W., Baran L.W., Popiński W., 2005, Wavelet analysis and forecasting of VTEC obtained with GPS observations over European latitudes, Journal of Atmospheric and Solar-Terrestrial Physics, 67 (2005), pp. 1147 – 1156

7. Methodology – ARMA prediction of real-valued time series GPS data from European IGS stations were used for TEC calculations GPS data from European IGS stations were used for TEC calculations 10 previous days of the TEC values were taken for the prediction computation 10 previous days of the TEC values were taken for the prediction computation Prediction for May 8, 2007 Prediction for May 8, 2007 Ionospheric conditions with max Kp=4o and sum of Kp = 22+ Ionospheric conditions with max Kp=4o and sum of Kp = 22+ Test network area http://igscb.jpl.nasa.gov

8. Methodology – Positioning Adjustment Model All parameters in the mathematical model are considered pseudo-observations with a priori information (σ = 0 ÷  ) All parameters in the mathematical model are considered pseudo-observations with a priori information (σ = 0 ÷  ) Sequential Generalized Least Squares (GLS) - instantaneous parameters (e.g., DD ionospheric delays) - accumulated parameters (e.g., DD ambiguities) Two characteristic groups of interest: Two characteristic groups of interest: Flexibility, easy implementation of: Flexibility, easy implementation of: stochastic constraints stochastic constraints fixed constraints fixed constraints weighted parameters weighted parameters

9. Methodology – Positioning MPGPS software was used for all calculations MPGPS software was used for all calculations Mathematical model uses dual-frequency code and phase GPS data Mathematical model uses dual-frequency code and phase GPS data Unknowns: DD Ionospheric delays, Tropospheric TZD per station, DD ambiguities, rover coordinates Unknowns: DD Ionospheric delays, Tropospheric TZD per station, DD ambiguities, rover coordinates Tropospheric TZD calculated at the reference stations and interpolated to the rover location, tightly constrained in GLSTropospheric TZD calculated at the reference stations and interpolated to the rover location, tightly constrained in GLS DD Ionospheric delays obtained from the ARMA forecast, constrained to 10-20 cm in GLSDD Ionospheric delays obtained from the ARMA forecast, constrained to 10-20 cm in GLS Ambiguity resolution: Least square AMBiguity Decorrelation Algorithm (LAMBDA) Ambiguity resolution: Least square AMBiguity Decorrelation Algorithm (LAMBDA) Validation: W-test - minimum of 3 observational epochs (for 5-second sampling rate) and W-test > 4 required for validation Validation: W-test - minimum of 3 observational epochs (for 5-second sampling rate) and W-test > 4 required for validation

10. Experiment 67 km 50 km 25 km Map: www.asg-pl.pl GPS data from ASG-EUPOS and EPN networksGPS data from ASG-EUPOS and EPN networks 24-hour data set collected on May 8, 2007 with 5-second sampling rate24-hour data set collected on May 8, 2007 with 5-second sampling rate KATO station selected as a simulated user receiver (rover)KATO station selected as a simulated user receiver (rover) Ambiguity resolution was restarted every 5 minutes (288 times)Ambiguity resolution was restarted every 5 minutes (288 times) Maximum 5 minutes (60 epochs) for initialization allowedMaximum 5 minutes (60 epochs) for initialization allowed

11. Experiment 67 km 50 km 25 km Map: www.asg-pl.pl 3 baselines of different length were processed independently (single baseline mode) and also in a multi-baseline mode (all baselines together)3 baselines of different length were processed independently (single baseline mode) and also in a multi-baseline mode (all baselines together) predicted iono model was applied (1-2 hour forecast)predicted iono model was applied (1-2 hour forecast) Time-to-fix was analyzedTime-to-fix was analyzed Ambiguity resolution success rate was analyzedAmbiguity resolution success rate was analyzed Ambiguity validation failure ratio was analyzedAmbiguity validation failure ratio was analyzed ”True” reference coordinates derived using Bernese software”True” reference coordinates derived using Bernese software IGS predicted orbits and clocks used (ultra-rapid)IGS predicted orbits and clocks used (ultra-rapid)

12. Test results DD Ionospheric correction residuals, KATO-TARG baseline – 25 km

13. Test results DD Ionospheric correction residuals, KATO-WODZ baseline – 50 km

14. Test results DD Ionospheric correction residuals, KATO-KRAW baseline – 67 km

15. Test results Kinematic position residuals (NEU), KATO-TARG baseline – 25 km

16. Test results Kinematic position residuals (NEU), KATO-WODZ baseline – 50 km

17. Test results Kinematic position residuals (NEU), KATO-KRAW baseline – 67 km

18. Kinematic position residuals (NEU), multi-baseline 25, 50 and 67 km Test results

19. Test results and analysis Ambiguity resolution success rate [%] Ambiguity validation failure rate [%] [%]AverageTime-to-fix [epochs (s)] KATO-TARG 25 km 99.60.43.1 (15.5) KATO-WODZ 50 km 98.21.83.5 (17.6) KATO-KRAW 67 km 92.61.17.4 (36.7) Multi-baseline100.00.03.2 (16.3) *minimum 3 epochs (15 seconds) required for validation Ambiguity resolution statistics

20. Conclusions Cm-level horizontal kinematic position accuracy can be achieved using proposed methodology with dual-frequency GPS data over distances of tens of kmCm-level horizontal kinematic position accuracy can be achieved using proposed methodology with dual-frequency GPS data over distances of tens of km When the ionospheric correction accuracy is better that ½ cycle of L1 signal, fixed solution is possible just after a few observational epochs onlyWhen the ionospheric correction accuracy is better that ½ cycle of L1 signal, fixed solution is possible just after a few observational epochs only The ionosphere forecast model reduce ~ 40% of the ionospheric delay (its accuracy is limited by the base model)The ionosphere forecast model reduce ~ 40% of the ionospheric delay (its accuracy is limited by the base model) The applicability of the presented forecast model is limited to the distances of 25-50 km in a single-baseline mode and to 60-70 km in a multi-baseline modeThe applicability of the presented forecast model is limited to the distances of 25-50 km in a single-baseline mode and to 60-70 km in a multi-baseline mode

21. Future Developments Research on the level of stochastic constraints imposed on the ionospheric corrections Research on the level of stochastic constraints imposed on the ionospheric corrections Too tight constraints cause false fixes Too tight constraints cause false fixes Too loose constraints make time-to-fix longer Too loose constraints make time-to-fix longer Test prediction of more accurate ionospheric (base) models Test prediction of more accurate ionospheric (base) models Higher accuracy base models will also improve accuracy of the prediction, and hence, the predicted TEC level will be more beneficial to RTK positioning Higher accuracy base models will also improve accuracy of the prediction, and hence, the predicted TEC level will be more beneficial to RTK positioning