Adjustment of GPS Surveys

1. Adjustment of GPS Surveys

2. Global Positioning System • Reference – www.trimble.com (tutorial) • 24 satellites – altitude ~20,000 km, period ~12 hr, velocity ~14,000 km/hr • Broadcast: L1 – 1575.42 MHz (λ=19cm) and L2 – 1227.60 MHz (λ=24cm) • Modulated C/A code (chip rate 1.023 MHZ, 293m) and P code (chip rate 10.23 MHz, 29.3m)

3. Signal • Carrier • C/A code • P code • Ephemeris • Timing signal • Miscellaneous information (satellite health, etc.)

4. GPS Modes • Carrier phase positioning – high precision • Code phase positioning – lower precision, but can use a single receiver

5. Carrier phase • Good for large, unobstructed areas • Need two or more receivers (relative positioning) • Single or dual frequency (L1, L2) • Think of wavelengths (19cm and 24cm) as graduations of a scale

7. Relative positioning – Carrier • Determines vector from point to point • Relative accuracy of the vectors is a fraction of the carrier wavelength (~1 cm) • We need to connect vectors to control points in order to get coordinates

8. Code Phase • Phase modulation – gives pseudorandom digital signal • C/A code – “chip” length = 293m • P code – “chip” length = 29.3m • Like a ruler with larger graduations

10. Primary Code Phase Method • Single receiver navigation • With selective availability (pre May 2000), accuracy ±100m • Without SA (now) accuracy ±15m • Computed by pseudorange solution • Receiver clock error corrected

12. Other Issues • Dilution of precision (DOP) • Multipath • Atmospheric and ionospheric effects • Cycle slips

15. Coordinate System • Geocentric X,Y,Z used for computation – earth-centered Cartesian coordinate system • Conversion to latitude, longitude, and height • Conversion to map projection coordinates

17. Application of Least Squares for GPS • Least squares adjustment shows up in many areas • Code phase pseudo-range solution • Relative positioning by carrier phase measurements • Step 1 – determine ΔX,ΔY,ΔZ between receivers • Step 2 – use control points and network to compute X,Y,Z coordinates

18. Code Phase Pseudo-Range Observation Equation (nonlinear) Unknowns: XA, YA, ZA, Δt (4 total) Observations: Ranges (ρ) to visible satellites for each epoch 5 or more ranges (satellites) result in least squares solution for each epoch

19. Other Systematic Errors • Speed of light – affected by atmosphere and ionosphere as well as relativity • Satellite time – even atomic clocks have errors • Satellite position – broadcast ephemeris is predicted • Multipath • Other

20. GPS Vector Networks

21. Coordinates from a Single Vector If A is a control point, then B can be determined by:

22. Adjustment of GPS Baselines

23. Observation Equations For vector from I to J : Observations are very similar to those for differential leveling. The weight matrix is different due to covariance between the X,Y,Z components of the vector.

24. Observations

25. AX = L + V

26. Observation Covariance Matrix

27. Weight Matrix

28. Least Squares Adjustment • Form normal equations and solve • Linear equations – no iteration • Compute residuals, standard deviation of an observation of unit weight, and statistics as before

29. Results

30. Results - Continued

31. Results - Continued