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Estimation of DGPS Carrier-Phase Errors Using a Reference Receiver Network

Estimation of DGPS Carrier-Phase Errors Using a Reference Receiver Network. Maj John Raquet John.Raquet@afit.af.mil Air Force Institute of Technology (and The University of Calgary). Overview. Motivation Setting up the problem NetAdjust solution Implementation issues

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Estimation of DGPS Carrier-Phase Errors Using a Reference Receiver Network

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  1. Estimation of DGPS Carrier-Phase Errors Using a Reference Receiver Network Maj John Raquet John.Raquet@afit.af.mil Air Force Institute of Technology (and The University of Calgary)

  2. Overview • Motivation • Setting up the problem • NetAdjust solution • Implementation issues • Putting this approach in context • Covariance function description • NetAdjust test results • Covariance analysis technique • Summary/Conclusion

  3. Overview • Motivation • Setting up the problem • NetAdjust solution • Implementation issues • Putting this approach in context • Covariance function description • NetAdjust test results • Covariance analysis technique • Summary/Conclusion

  4. Reference Receiver Network Motivation (1/3) • Single reference receiver coverage Desired Coverage Area 100 80 60 40 20 Ref. Northing (km) 0 -20 -40 -60 -80 -100 -100 -80 -60 -40 -20 0 20 40 60 80 100 Easting (km)

  5. Reference Receiver Network Motivation (2/3) • One (poor) solution Desired Coverage Area 100 Ref. Ref. Ref. Ref. Ref. 80 60 Ref. Ref. Ref. Ref. Ref. 40 20 Ref. Ref. Ref. Ref. Ref. Northing (km) 0 -20 Ref. Ref. Ref. Ref. Ref. -40 -60 Ref. Ref. Ref. Ref. Ref. -80 -100 -100 -80 -60 -40 -20 0 20 40 60 80 100 Easting (km)

  6. Reference Receiver Network Motivation (3/3) • Better solution: use a network

  7. Phase Measurements • Measurement with errors • Double-differencing

  8. Double-Difference Phase Errors • Highest positioning accuracy obtained by differential carrier-phase ambiguity resolution • If “close” to reference receiver, then correlated errors are removed

  9. Why Reducing Errors Helps Ambiguity Resolution • Almost all ambiguity resolution routines use some sort of residual analysis to determine integer ambiguities

  10. Why Reducing Errors Helps Ambiguity Resolution • Almost all ambiguity resolution routines use some sort of residual analysis to determine integer ambiguities Goal

  11. Overview • Motivation • Setting up the problem • NetAdjust solution • Implementation issues • Putting this approach in context • Covariance function description • NetAdjust test results • Covariance analysis technique • Summary/Conclusion

  12. Ref 1 Ref 2 Ref 3 Ref 4 Ref 5 Setting Up the Problem Sample 5-receiver network: + Computation Point x Measurement-minus-range observable: Measurements:

  13. Setting Up the Problem Measurement errors: Double-difference errors: Errors to be eliminated: Measurements available:

  14. Overview • Motivation • Setting up the problem • NetAdjust solution • Implementation issues • Putting this approach in context • Covariance function description • NetAdjust test results • Covariance analysis technique • Summary/Conclusion

  15. NetAdjust Solution • Use a linear minimum variance of error estimator • Generic case (to estimate x given measurements Y) • Assumption: x and Y are jointly Gaussian • Our case (to estimate given measurements ) • Assumes and are zero-mean

  16. Are the Assumptions Valid? • Assumption 1: and are jointly Gaussian • Each individual error source tends to be Gaussian • Central limit theorem strengthens assumption • Assumption 2: and are zero-mean • Reasonable for uncorrelated errors (multipath and noise) • Reasonable for correlated errors, if systemic biases removed by a model

  17. Cleaner Statement of NetAdjust Solution • Corrections to apply to measurements from reference receiver network • Corrections to apply to mobile receiver measurements • Minimizes trace --the ultimate goal! (1) (2)

  18. Overview • Motivation • Setting up the problem • NetAdjust solution • Implementation issues • Putting this approach in context • Covariance function description • NetAdjust test results • Covariance analysis technique • Summary/Conclusion

  19. ˆ ˆ é ù é ù l d d l l n n n ê ú 1 ê ú 1 1 ˆ l d l ê ú ê ú n = n l d = 2 l 2 ê ú ê ú n n M l M ê ú ê ú n 1 ˆ l ê d ú ê ú l ë û n ë û n n n ˆ ˆ l d l n cp 1 l cp Implementation Approach NetAdjust Equation 1 + + Computation Point Equation 2 Mobile Receiver (at Computation Point) Ambiguity Resolution and Positioning Algorithm Mobile Receiver Position

  20. ˆ ˆ é ù é ù l d d l l n n n ê ú 1 ê ú 1 1 ˆ l d l ê ú ê ú n = n l d = 2 l 2 ê ú ê ú n n M l M ê ú ê ú n 1 ˆ l ê d ú ê ú l ë û n ë û n n n ˆ l n 1 l cp Alternate Implementation Approach NetAdjust Equation 1 + + Computation Point + - Equation 2 Includes all corrections Mobile Receiver (at Computation Point) Ambiguity Resolution and Positioning Algorithm Mobile Receiver Position

  21. How Do You Transmit for Mobile User at Any Location? • Question that must be answered for multi-user one-way-broadcast network • Corrections vary with location (as they should) • Variation is not easily modeled • Different approaches can be taken using grid • Nearest point • Interpolation (linear, quadratic) • Update rates • See ION AM 2000 paper by Fotopoulos

  22. Calculation of Network Ambiguities • Algorithm requires no initialization per se • Ambiguities between reference receivers must be known • Best if all fixed • Will work (slightly less well) with floating ambiguity estimates • Can account for a mix of fixed and floating • Real-time estimation of ambiguities between network reference stations is one of the largest implementation challenges

  23. Overview • Motivation • Setting up the problem • NetAdjust solution • Implementation issues • Putting this approach in context • Covariance function description • NetAdjust test results • Covariance analysis technique • Summary/Conclusion

  24. Three “Views” of the NetAdjust Approach • Linear Minimum of Variance Estimator • Explicitly minimizes squared error Bayes’ risk • Estimation of one variable using observables • Least-Squares Condition Adjustment • Apply condition to measurements • Condition is that all double-differenced measurement-minus-range observables within network are zero • Explains the “data encapsulation” effect • Least-Squares Collocation • Interpolation • Use of covariance kernel

  25. Three Classes of Approaches to This Problem • Error Mitigation Approach • Explicitly estimate individual error sources • Gao, vanderMarel, etc. • Polynomial Fit Approach • Assume differential errors can be expressed as a particular functional form of position • Calculate coefficients for the specified function • Varner, Wubbena, etc. • Covariance Fit Approach (NetAdjust) • Assume error covariance can be expressed as a functional form • Use functionally generated covariance with NetAdjust

  26. Overview • Motivation • Setting up the problem • NetAdjust solution • Implementation issues • Putting this approach in context • Covariance function description • NetAdjust test results • Covariance analysis technique • Summary/Conclusion

  27. Covariance Function Concept Data from test network Information about error characteristics (i.e., covariance matrix) Express covariance in functional form: Example: function of: - Distance - SV elevation - Rcvr-specific multipath/noise levels Use covariance function to generate predicted covariance matrix for new configuration Calculate corrections Predict performance

  28. 0.5 0.035 0.4 0.03 ) 0.025 2 Zenith DD Err Variance 0.3 ) 2 Zenith DD Err Variance (L1 cycles 0.02 0.2 (WL cycles 0.015 0.1 0.01 0.005 0 0 100 200 300 400 500 600 700 0 Distance Between Receivers (km) 0 100 200 300 400 500 600 700 Distance Between Receivers (km) Zenith Phase Covariance Functions(Based on 55 Baselines Between 11 Receivers) L1 WL

  29. Example of How Covariance Function Can Change

  30. Overview • Motivation • Setting up the problem • NetAdjust solution • Implementation issues • Putting this approach in context • Covariance function description • NetAdjust test results • Covariance analysis technique • Summary/Conclusion

  31. 400 TRON 300 ALES 200 TRYR TRYM 100 Region of Interest GEIR Northing (km) BERG 0 GEIM -100 STAV ARER -200 AREM KRIS -300 -200 0 200 400 Easting (km) Norway Test Network

  32. Norway Network • 24 hours of data at 2 second intervals • Ambiguities calculated • Between every pair of reference receivers • Over 24 hour period • Receiver positions calculated • Based on ionospheric-free carrier-phase observable (requires L1 and L2 ambiguities) • Network adjustment procedure • Relative positioning accuracy: 2-3mm horizontal, 5-7mm vertical

  33. Seven Test Networks 400 TRON 300 Test Networks ARER-0 GEIR-29 ARER-67 STAV-143 GEIR-164 GEIR-223-sparse ALES-242 ALES 200 TRYR TRYM 100 Northing (km) GEIR BERG 0 GEIM -100 STAV ARER -200 AREM KRIS -300 -200 0 200 400 Easting (km)

  34. Testing NetAdjust on Norway Network • Improvement in double-difference measurement error • Improvement in differential positioning accuracy (using correct integer ambiguities) • Improvement in carrier-phase ambiguity resolution

  35. Raw 0.8 NetAdjust 0.6 (L1 cycles) Double Difference Meas Error RMS 0.4 0.2 GEIR-223-sparse STAV-143 ALES-242 GEIR-164 ARER-67 GEIR-29 ARER-0 0 0 50 100 150 200 250 Distance To Nearest Reference Receiver (km) Improvement in DD Measurement ErrorL1 Phase

  36. 0.25 Raw NetAdjust 0.2 0.15 (WL cycles) Double Difference Meas Error RMS 0.1 0.05 GEIR-223-sparse STAV-143 ALES-242 GEIR-164 ARER-67 GEIR-29 ARER-0 0 0 50 100 150 200 250 Distance To Nearest Reference Receiver (km) Improvement in DD Measurement ErrorWL Phase

  37. 0.4 Raw 0.35 NetAdjust 0.3 0.25 0.2 3-D RMS Position Error (m) 0.15 0.1 0.05 0 0 50 100 150 200 250 Distance To Nearest Reference Receiver (km) Improvement in Positioning AccuracyL1 Phase (Fixed Integer Ambiguities)

  38. 0.4 Raw 0.35 NetAdjust 0.3 0.25 0.2 3-D RMS Position Error (m) 0.15 0.1 0.05 0 0 50 100 150 200 250 Distance To Nearest Reference Receiver (km) Improvement in Positioning AccuracyWL Phase (Fixed Integer Ambiguities)

  39. Improvement in Ambiguity Resolution • University of Calgary’s FLYKINTM software • Run iteratively, start times staggered by 10 minutes (138 runs over 24 hours) • Stopped immediately if integer ambiguities determined • Three performance criteria • Percentage of correct fixes • Percentage of incorrect fixes • Average time to resolve ambiguities

  40. 100% Raw Code, Raw L1 Phase Raw Code, NetAdjust L1 Phase NetAdjust Code, NetAdjust L1 Phase 80% 60% Percentage Correct Fixes 40% GEIR-223-sparse STAV-143 ALES-242 GEIR-164 ARER-67 GEIR-29 ARER-0 20% 0% 0 50 100 150 200 250 Distance to Nearest Reference Receiver (km) Improvement in Ambiguity ResolutionPercentage of Correct Fixes - L1 Phase

  41. 100% 80% 60% Percentage Correct Fixes Raw Code, Raw WL Phase Raw Code, NetAdjust WL Phase 40% NetAdjust Code, NetAdjust WL Phase 20% GEIR-223-sparse STAV-143 ALES-242 GEIR-164 ARER-67 GEIR-29 ARER-0 0% 0 50 100 150 200 250 Distance to Nearest Reference Receiver (km) Improvement in Ambiguity ResolutionPercentage of Correct Fixes - WL Phase

  42. Improvement in Ambiguity ResolutionMean Time to Fix - WL Phase 7 Raw Code, Raw WL Phase Raw Code, NetAdjust WL Phase 6 NetAdjust Code, NetAdjust WL Phase 5 4 Mean Time to Resolve Ambiguities (minutes) 3 2 GEIR-223-sparse 1 STAV-143 ALES-242 GEIR-164 ARER-67 GEIR-29 ARER-0 0 0 50 100 150 200 250 Distance to Nearest Reference Receiver (km)

  43. Overview • Motivation • Setting up the problem • NetAdjust solution • Implementation issues • Putting this approach in context • Covariance function description • NetAdjust test results • Covariance analysis technique • Summary/Conclusion

  44. Motivation • It’s difficult and costly to deploy a reference receiver network • Differential network performance varies with • Number/location of reference receivers • Number/geometry of visible satellites • Type of measurement used (e.g., L1 or WL) • Characteristics (especially correlations) of DGPS errors • May be possible to test small subset of network configurations • Desirable to predict performance for other (untested) network configurations • “What if” scenarios • Based upon test results • Critical for final network design

  45. Covariance Analysis Procedure • Straightforward propagation of DGPS measurement error covariance into double-difference space:

  46. Validation of Covariance Function and Analysis Procedure • Seven “test networks” selected • One receiver selected as “mobile” receiver • Remaining (or subset) form network • Closest reference receiver identified (for single reference case) • Double difference errors predicted by covariance analysis • Single reference (raw) case • Multiple reference (NetAdjust) case • Prediction compared with actual results

  47. Raw from Data Raw from Cov. Analysis NetAdjust from Data NetAdjust from Cov. Analysis Validation: Predicted and ActualL1 Phase 0.8 0.7 0.6 0.5 0.4 Double Difference Error RMS (L1 cycles) 0.3 0.2 GEIR-223-sparse 0.1 STAV-143 ALES-242 GEIR-164 ARER-67 GEIR-29 ARER-0 0 0 50 100 150 200 250 Distance to Nearest Reference Receiver (km)

  48. 0.25 Raw from Data Raw from Cov. Analysis 0.2 NetAdjust from Data NetAdjust from Cov. Analysis 0.15 Double Difference Error RMS (WL cycles) 0.1 0.05 GEIR-223-sparse STAV-143 ALES-242 GEIR-164 ARER-67 GEIR-29 ARER-0 0 0 50 100 150 200 250 Distance to Nearest Reference Receiver (km) Validation: Predicted and ActualWL Phase

  49. Development of Network Performance Specification • Primary emphasis is carrier-phase ambiguity resolution • Develop relationship between double difference measurement error and distance between mobile and reference receivers • Specification made in terms of distance from reference receiver (under “normal” conditions) • More intuitive than pure error statistics • Typically, already is distance specification established • Convert distance specification into measurement error specification

  50. WL Phase 0.2 0.15 Zenith Double Difference Meas Error Standard Deviation (WL cycles) 0.1 0.05 0 0 100 200 300 400 500 600 700 Baseline Distance (km) Zenith DD Measurement Error vs. Baseline Distance L1 Phase 0.7 0.6 0.5 0.4 Standard Deviation (L1 cycles) Zenith Double Difference Meas Error 0.3 0.2 0.1 0 0 100 200 300 400 500 600 700 Baseline Distance (km)

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