Realtime Kinematic GPS Positioning Supported by Predicted Ionosphere
Real-time Kinematic GPS Positioning Supported by Predicted Ionosphere Model P. Wielgosz and A. Krankowski University of Warmia and Mazury in Olsztyn, Poland pawel. wielgosz@uwm. edu. pl IGS AC Workshop Miami Beach, June 2 -6, 2008
Outline n Research objectives n ARMA method n RTK positioning model n Experiment design n Test results and analysis n Conclusion
Research Objectives n n 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 Evaluate prediction model based on ARMA method Study the impact of the model accuracy on the ambiguity resolution (speed and reliability)
Methodology – ARMA prediction of real -valued time series Let yt for t =1, 2, …. , n be an equidistant stationary stochastic time series and yt+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 BK the process can be converted to :
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
Methodology – ARMA prediction of real -valued time series n n n 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 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
Methodology – ARMA prediction of real -valued time series n n n http: //igscb. jpl. nasa. gov Test network area n GPS data from European IGS stations were used for TEC calculations 10 previous days of the TEC values were taken for the prediction computation Prediction for May 8, 2007 Ionospheric conditions with max Kp=4 o and sum of Kp = 22+
Methodology – Positioning Adjustment Model Sequential Generalized Least Squares (GLS) n n All parameters in the mathematical model are considered pseudo-observations with a priori information (σ = 0 ÷ ) Two characteristic groups of interest: - instantaneous parameters (e. g. , DD ionospheric delays) - accumulated parameters (e. g. , DD ambiguities) n Flexibility, easy implementation of: ü stochastic constraints ü fixed constraints ü weighted parameters
Methodology – Positioning n n n MPGPS software was used for all calculations Mathematical model uses dual-frequency code and phase GPS data 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 GLS • DD Ionospheric delays obtained from the ARMA forecast, constrained to 10 -20 cm in GLS n n 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
Experiment • GPS data from ASG-EUPOS and EPN networks • 24 -hour data set collected on May 8, 2007 with 5 -second sampling rate 25 km 50 km • KATO station selected as a simulated user receiver (rover) 67 km • Ambiguity resolution was restarted every 5 minutes (288 times) • Maximum 5 minutes (60 epochs) for initialization allowed Map: www. asg-pl. pl
Experiment • 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) 25 km • Time-to-fix was analyzed 50 km 67 km • Ambiguity resolution success rate was analyzed • Ambiguity validation failure ratio was analyzed • ”True” reference coordinates derived using Bernese software Map: www. asg-pl. pl • IGS predicted orbits and clocks used (ultra-rapid)
Test results DD Ionospheric correction residuals, KATO-TARG baseline – 25 km
Test results DD Ionospheric correction residuals, KATO-WODZ baseline – 50 km
Test results DD Ionospheric correction residuals, KATO-KRAW baseline – 67 km
Test results Kinematic position residuals (NEU), KATO-TARG baseline – 25 km
Test results Kinematic position residuals (NEU), KATO-WODZ baseline – 50 km
Test results Kinematic position residuals (NEU), KATO-KRAW baseline – 67 km
Test results Kinematic position residuals (NEU), multi-baseline 25, 50 and 67 km
Test results and analysis Ambiguity resolution statistics Ambiguity resolution success rate [%] Ambiguity validation failure rate [%] Average Time-to-fix [epochs (s)] KATO-TARG 25 km 99. 6 0. 4 3. 1 (15. 5) KATO-WODZ 50 km 98. 2 1. 8 3. 5 (17. 6) KATO-KRAW 67 km 92. 6 1. 1 7. 4 (36. 7) Multi-baseline 100. 0 3. 2 (16. 3) *minimum 3 epochs (15 seconds) required for validation
Conclusions • Cm-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 L 1 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 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
Future Developments • Research on the level of stochastic constraints imposed on the ionospheric corrections • Too tight constraints cause false fixes • Too loose constraints make time-to-fix longer • 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
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