Satellite Navigation: Lecture 5 -_gps_error_sources

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Satellite Navigation: Lecture 5 -_gps_error_sources

  1. 1. Satellite Navigation error sources and position estimation Picture: ESAAE4E08Sandra VerhagenCourse 2010 – 2011, lecture 5 1Satellite Navigation (AE4E08) – Lecture 4
  2. 2. Today’s topics• Recap: GPS measurements and error sources• Satellite clock and ephemeris• Assignments VISUAL and RINEX• Signal propagation errors: ionosphere and troposphere• Book: Sections 5.2 – 5.3 2Satellite Navigation (AE4E08) – Lecture 4
  3. 3. Recap: Code and Carrier Phase measurements ρ = r + I ρ + Tρ + c ⎡δ tu − δ t s ⎤ + ε ρ ⎣ ⎦ Φ = r + Iφ + Tφ + c ⋅ (δ tu − δ t s ) + λ ⋅ A + ε Φ 3Satellite Navigation (AE4E08) – Lecture 4
  4. 4. Recap: error sources• satellite: • receiver • orbit • clock • clock • instrumental delays • instrumental delays • other• signal path • spoofing • ionosphere • interference • troposphere • multipath 4Satellite Navigation (AE4E08) – Lecture 4
  5. 5. Satellite clock and ephemeris• determined by Control Segment based on measurements• parameters in navigation message• Kalman filter prediction • positions and velocities of satellites • phase bias, frequency bias, frequency drift rate clocks prediction errors (grow with age of data)Currently, ranging error < 3 meter rms 5Satellite Navigation (AE4E08) – Lecture 4
  6. 6. Satellite clock and ephemeris• Ephemeris: radial, along-track, cross-track errors. Which is of the three is principal error source? radial error true orbit predicted orbit along-track error cross-track error 6Satellite Navigation (AE4E08) – Lecture 4
  7. 7. Satellite clock and ephemeris Misra and Enge, Section 4.2.4Satellite clock correction:δt s = t s − t = a f 0 + a f 1 (t − t0 c ) + a f 2 (t − t0 c ) 2 + Δtrt0 c reference epochaf 0 clock offset [s] ~1 μs – 1 ms broadcast with navigationaf1 fractional frequency offset [s/s] ~10-11 s/s messageaf 2 fractional frequency drift [s/s2] ~0 s/s2Δtr relativistic correction 7Satellite Navigation (AE4E08) – Lecture 4
  8. 8. Signal propagation errors 20,000 height [km] • Above ~1,000 km: vacuum speed of light c • ~ 50 – 1,000 km: ionosphere • Below 40 km: troposphere 1,000 Atmosphere = ionosphere ionosphere + troposphere 40 Atmosphere changes speed and troposphere direction of propagation refraction Earth 8Satellite Navigation (AE4E08) – Lecture 4
  9. 9. Signal propagation errors c• Refractive index of a medium: n= v• Refractive index changes along path• Change in speed change in travel time of signal 9Satellite Navigation (AE4E08) – Lecture 4
  10. 10. Signal propagation errors c• Refractive index of a medium: n= v• Snell’s law: changing refractive index results in bending of path path longer than geometrical straight line n1 sin θ1 = n2 sin θ 2 n1 θ1 effect very small n2 θ2• Fermat’s principle of least time: transit time along curved path is shorter than for straight-line path 10Satellite Navigation (AE4E08) – Lecture 4
  11. 11. Signal propagation errors c• Refractive index of a medium: n = v• Refractive index changes along path: n(l )• Change in speed change in travel time τ of signal S R 1 dl τ = ∫ n(l )dl cS R 11Satellite Navigation (AE4E08) – Lecture 4
  12. 12. Signal propagation errors c• Refractive index of a medium: n = v• Refractive index changes along path: n(l )• Change in speed change in travel time τ of signal S R 1 dl τ = ∫ n(l )dl cS R 1 ⎡R R ⎤ R• Excess delay: Δτ = ⎢ ∫ n(l )dl − ∫ dl ⎥ Δρ = ∫ [ n(l ) − 1] dl c ⎣S S ⎦ S 12Satellite Navigation (AE4E08) – Lecture 4
  13. 13. Signal propagation errors: ionosphere• Dispersive medium: refractive index depends on frequency of signal• For GPS (L-band) signals: ionosphere is dispersive, troposphere is not• In dispersive medium: different phase (carrier) and group (code) velocities,v p and vg , resp. modulated carrier wave superposition of a group of waves of c c dn p different frequencies np = ng = = n p + f vp vg df 13Satellite Navigation (AE4E08) – Lecture 4
  14. 14. Signal propagation errors: ionosphere• ionosphere contains free electrons and ions• ionization caused by sun’s radiation state depends on solar activity• temporal variations: • during the day, peak at 2 PM local time • day to day due to solar activity, geomagnetic disturbances • seasonal • 11-year solar cycle • local short term effects due to traveling ionospheric disturbances 14Satellite Navigation (AE4E08) – Lecture 4
  15. 15. Signal propagation errors: ionosphere• propagation speed depends on total electron content (TEC) = number of electrons in tube of 1 m2 from receiver to satellite R TEC = ∫ ne (l )dl [TECU] S• with ne (l ) the electron density along the path• 1 TECU (TEC Unit) = 1016 electrons / m2• VTEC = TEC in vertical direction [in book TECV] 15Satellite Navigation (AE4E08) – Lecture 4
  16. 16. Signal propagation errors: ionosphereGlobal map of TEC (computed from global network of GPS receivers) Figure from S.M. Radicella – ARPL; Data Astronomical Institute University of Berne 16 Satellite Navigation (AE4E08) – Lecture 4
  17. 17. 17Satellite Navigation (AE4E08) – Lecture 4
  18. 18. Signal propagation errors: ionosphere• highest ionospheric delay within ±20o of magnetic equator• solar flares magnetic storms large and quickly varying electron densities, esp. polar regions rapid fluctuations in phase and amplitude of GPS signals, called scintillation and fading, resp. may cause losses of lock 18Satellite Navigation (AE4E08) – Lecture 4
  19. 19. Signal propagation errors: ionospherephase advance c 40.3ne R np = ≈ 1− TEC = ∫ ne (l )dl vp f2 S R 1 Δτ p = ∫ ( n p (l ) − 1) dl cS R 1 40.3ne (l ) 40.3 ⋅ TEC =− ∫ 2 dl = − 2 [s] cS f cf 40.3 ⋅ TEC Iφ = cΔτ p = − [m] f2 Φ = r + Iφ + Tφ + c ⋅ (δ tu − δ t s ) + λ 19 A + ε Φ ⋅Satellite Navigation (AE4E08) – Lecture 4
  20. 20. Signal propagation errors: ionospheregroup delay 40.3ne dn p ng = n p + f = 1+ df f2 40.3 ⋅ TEC I ρ = cΔτ g = 2 [m] f ρ = r + I ρ + Tρ + c ⎡δ tu − δ t s ⎤ + ε ρ ⎣ ⎦ I ρ = − Iφ = I 20Satellite Navigation (AE4E08) – Lecture 4
  21. 21. Signal propagation errors: ionosphere ionosphere ς′ IP : ionospheric pierce point IP hI : mean ionosphere height ς hI RE sin ς ′ = sin ς Earth RE + hI RE 1 I (ς ) = Iz cos ς ′ 21Satellite Navigation (AE4E08) – Lecture 4
  22. 22. Signal propagation errors: ionosphere obliquity factor 1 cos ς ′ zenith angle ς 22Satellite Navigation (AE4E08) – Lecture 4
  23. 23. Signal propagation errors: ionospherezenith delay mid-latitudes:• 1-3 m at night• 5-15 m mid-afternoonpeak solar cycle near equator:• max. ~36 m 23Satellite Navigation (AE4E08) – Lecture 4
  24. 24. Signal propagation errors: ionosphere 40.3 ⋅ TEC 40.3 ⋅ TEC f L21 I L1 = I L2 = = 2 I L1 f L21 2 fL2 fL2 ρ Li = r + I Li + T + c ⎣ ⎡δ tu − δ t s ⎤ + ε ρLi ⎦ bias removed;ionosphere-free combination: noise increased a ρ L1 − bρ L 2 ⎡δ tu − δ t s ⎤ + ε ρIC = r +T + c ⎣ ⎦ = ρ IF 24Satellite Navigation (AE4E08) – Lecture 4
  25. 25. Signal propagation errors: ionosphereKlobuchar modelA2 and A4 A2broadcasted withnavigation message~50% reduction RMS range error local 1 2 A4 time [h] 25Satellite Navigation (AE4E08) – Lecture 4
  26. 26. Signal propagation errors: ionosphere• NeQuick model (proposed for Galileo) • 3-D electron density model • One location dependent input parameter (Az) • Az is given for Galileo in broadcast message • Slant-TEC is compute by numerical integration along line-of- sight• Compute corrections from IGS Global Ionosphere Maps (GIM) • 2-D grid of VTEC (2.5o latitude x 5o longitude @ 2 hours) • Interpolate VTEC to ionospheric point at time of observation • Map VTEC to slant direction using mapping function 26Satellite Navigation (AE4E08) – Lecture 4
  27. 27. Signal propagation errors: troposphere• 9 km (poles) – 16 km (equator)• Dry gases and water vapor• Recall: non-dispersive, i.e. refraction does not depend on frequency• Propagation speed lower than in free space: apparent range is longer (~2.5 – 25 m)• Same phase and group velocities Tρ L1 = Tρ L 2 = TφL1 = TφL 2 = T 27Satellite Navigation (AE4E08) – Lecture 4
  28. 28. Signal propagation errors: troposphere• Refractivity N = ( n − 1) × 106 N = Nd + Nw T = 10−6 ∫ N (l )dl =10−6 ∫ [N d (l ) + N w (l )]dl =Td + Tw P N d ≈ 77.64 P : total pressure [mbar] T e T : temperature [K] N w ≈ 3.73 ⋅ 105 e : partial pressure water vapor [mbar] T2 if known refractivity known 28Satellite Navigation (AE4E08) – Lecture 4
  29. 29. Signal propagation errors: troposphere tropospheric delay Satellite computed hydrostatic delay T = md (el ) * Tz ,d + mw (el ) * Tz ,w mapping functions Receiver Unknown tropospheric zenith wet delay EarthFigure: H. van der Marel 29 29 Satellite Navigation (AE4E08) – Lecture 4
  30. 30. Signal propagation errors: troposphere• Saastamoinen model: zenith dry and wet delays calculated from temperature, pressure and humidity (measurements or standard atmosphere), height and latitude• Hopfield model: dry and wet refractivities calculated• Dry delay in zenith direction 2.3 – 2.6 m at sea level can be predicted with accuracy of few mm’s• Wet delay depends on water vapor profile along path, 0 – 80 cm accuracy of models few cm’s• If no actual meteorological observations available (standard atmosphere applied): total zenith delay error 5 – 10 cm 30Satellite Navigation (AE4E08) – Lecture 4
  31. 31. Signal propagation errors: summary ionosphere troposphereheight 50 – 1000 km 0 – 16 kmvariability diurnal, seasonal, solar low cycle (11 yr), solar flareszenith delay meters – tens of meters 2.3 – 2.6 m (sea level) 30o 1.8 2obliquity factor 15o 2.5 4 3o 3 10modeling error (zenith) 1 - >10 m 5 – 10 cm (no met. data)dispersive yes no all values are approximate, depending on location and circumstances Satellite Navigation (AE4E08) – Lecture 4
  32. 32. Signal propagation errorsHomework exercise:• make plots of the different mapping functions (page 173 Misra and Enge) as function of the elevation angle (ranging from 0 – 90o)• compare them with each other AND with the obliquity factor of the ionosphere delay (slide 22)• try to explain the differences• more details: see assignment on blackboard 32Satellite Navigation (AE4E08) – Lecture 4
  33. 33. Summary and outlook• GPS measurements and error sourcesNext:Position, Velocity and Time (PVT) estimation 33Satellite Navigation (AE4E08) – Lecture 4

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