Global Positioning System: what it is and how we use it for measuring the earth s movement. May 5, 2009

Transcription

1 Global Positioning System: what it is and how we use it for measuring the earth s movement. May 5, 2009

2 References Lectures from K. Larson s Introduction to GNSS asen5090/ Strang, G. and K. Borre Linear Algebra, Geodesy, and GPS, Wellesley Cambridge Press, 1997 Blewitt, G., Basics of the GPS Technique: Observation Equations, in Geodetic Applications of GPS GPS/GPS_Tutorial_2.pdf Lecture notes from G. Mattioli (comp.uark.edu/ ~mattioli/geol_4733/gps_signals.ppt)

3 Basics of how it works Trilateration GPS gives distance to 4 satellites x,y,z,t Earth centered, Earth Fixed Why t? What are some of reasons why measuring distance is difficult? How do we know x,y,z,t of satellites?

4 GPS: Space segment Several different types of GPS satellites (Block I, II, II A, IIR) All have atomic clocks Stability of at least sec 1 sec every ~300,000 yrs Dynamics of orbit? Reference point?

5 Orbital Perturbations (central force is 0.5 m/s 2) Source Acceleration Perturbation Type m/s 2 3 hrs Earth oblateness (J 2 ) 5 x hrs secular + 6 hr Sun & moon 5 x hrs secular + 12hr Higher Harmonics 3 x hrs Various Solar radiation pressure Ocean & earth tides Earth albedo pressure 1 x days Secular + 3 hr 1 x 10-9 days secular + 12hr 1 x 10-9 days From K. Larson

6 GPS: Space Segment 24+ satellites in orbit Can see 4 at any time, any point on earth Satellites never directly over the poles For most mid latitude locations, satellites track mainly north south

7 GPS: Satellite Ground Track

8 Satellite transmits on two carrier frequencies: L1 (wavelength=19 cm) L2 (wavelength=24.4 cm) Transmits 3 different codes/ signals P (precise) code Chip length=29.3 m C/A (course acquisition) code Chip length=293 m Navigation message Broadcast ephemeris (satellite orbital parameters), SV clock corrections, iono info, SV health GPS Signal

9 GPS Signal Signal phase modulated: vs Amplitude modulation (AM) Frequency modulation (FM)

10 C/A and P code: PRN Codes PRN = Pseudo Random Noise Codes have random noise characteristics but are precisely defined. A sequence of zeros and ones, each zero or one referred to as a chip. Called a chip because they carry no data. Selected from a set of Gold Codes. Gold codes use 2 generator polynomials. Three types are used by GPS C/A, P and Y

11 PRN Codes: first 100 bits

12 PRN Code properties High Autocorrelation value only at a phase shift of zero. Minimal Cross Correlation to other PRN codes, noise and interferers. Allows all satellites to transmit at the same frequency. PRN Codes carry the navigation message and are used for acquisition, tracking and ranging.

13 PRN Code Correlation

14 Non PRN Code Correlation

15 Schematic of C/A code acquisition Since C/A code is 1023 chips long and repeats every 1/1000 s, it is inherently ambiguous by 1 msec or ~300 km.

16 BASIC GPS MEASUREMENT: PSEUDORANGE Receiver measures difference between time of transmission and time of reception based on correlation of received signal with a local replica ρ = s ( u t ) c t t t u s = time of reception as observed by the receiver = time of transmission as generated by the satellite The measured pseudorange is not the true range between the satellite and receiver. That is what we clarify with the observable equation.

17 PSEUDORANGE OBSERVABLE MODEL ( u s ) ( u s ) ρ = R + c δt δt + T + I + M + ε 1 ρ1 ρ1 ρ1 ρ = R + c δt δt + T + I + M + ε 2 ρ 2 ρ 2 ρ 2 ρ ρ 1 2 = pseudorange measured on L1 frequency based on code = pseudorange measured on L2 frequency based on code R = geometrical range from satellite s to user u δt δt u s ρ1/ 2 ρ1/ 2 ρ1/ 2 = user/receiver clock error = satellite clock error T = tropospheric delay I M = ionospheric delay in code measurement on L1/2 = multipath delay in code measurement on L1/2 ε = other delay/errors in code measurement on L1/2

18 CARRIER PHASE MODEL ( u s ) ( u s ) φ λ = R + c δt δt + T I + M + N λ + ε 1 1 ρ1 φ1 1 1 φ1 φ λ = R + c δt δt + T I + M + N λ + ε 2 2 ρ 2 φ φ 2 φ φ 1 2 = carrier phase measured on L1 frequency (C/A or P(Y) parts) = carrier phase measured on L2 frequency R = geometrical range from satellite s to user u δt δt u s = user/receiver clock error = satellite clock error T = tropospheric delay I, I = ionospheric delay in code measurement on L1/2 ρ1 ρ 2 M N, M = multipath delay in carrier phase measurement on L1/2 φ1 φ 2, N = carrier phase bias or ambiguity 1 2 λ, λ = carrier wavelength ε 1 2, ε = other delay/errors in carrier phase measurement on L1/2 φ1 φ 2

19 COMPARE PSEUDORANGE and CARRIER PHASE ( u s ) ( u s ) ρ = R + c δt δt + T + I + M + ε 1 ρ1 ρ1 ρ1 φ λ = R + c δt δt + T I + M + N λ + ε 1 1 ρ1 φ1 1 1 φ1 bias term N does not appear in pseudorange ionospheric delay is equal magnitude but opposite sign troposphere, geometric range, clock, and troposphere errors are the same in both multipath errors are different (phase multipath error much smaller than pseudorange) noise terms are different (factor of 100 smaller in phase data)

20 Atmospheric Effects Ionosphere ( km) Delay is proportional to number of electrons Troposphere (~16 km at equator, where thickest) Delay is proportional to temp, pressure, humidity.

21 Vertical Structure of Atmosphere

22 Tropospheric effects Lowest region of the atmosphere index of refraction = ~ at sea level Neutral gases and water vapor causes a delay which is not a function of frequency for GPS signal Dry component contributes 90 97% Wet component contributes 3 10% Total is about 2.5 m for zenith to 25 m for 5 deg

23 Tropospheric effects At lower elevation angles, the GPS signal travels through more troposphere.

24 Dry Troposphere Delay 3 Saastamoinen model: Tz d = ( cos 2φ ) P 0 is the surface pressure (millibars) φ is the latitude h is the receiver height (m) h P, 0 Hopfield model: Tz, d h d is 43km T 0 is temperature (K) = P h 6 0 T 0 d 5 ~2.5 m at sea level Mapping function: E satellite elevation m d = sin E tan E (zenith) 10 (5 deg)

25 Wet Troposphere Correction Less predictable than dry part, modeled by: Saastamoinen model: T 1255 = e T 3 z, w 0 Hopfield model: T = z, w 2 T0 h w is 12km e 0 is partial pressure of water vapor in mbar e h w cm Mapping function: m d = sin E tan E

26 Examples of Wet Zenith Delay

27 Ionosphere effects Pseudorange is longer group delay Carrier Phase is shorter phase advance s ( ) s ( ) s ( ) s ( ) ρ = R + c δt δt + I + T + MP + ε L1 u ρl1 ρl1 ρl1 ρ = R + c δt δt + I + T + MP + ε L2 u ρl2 ρl2 ρl2 λ φ = R λ N + c δt δt + I + T + MP + ε 1 L1 1 1 u φl1 φl1 φl1 λ φ = R λ N + c δt δt + I + T + MP + ε 1 L2 2 2 u φl2 φl2 φl2 I ρ 2 s ( ) s ( ) 1 L1 1 1 u ρl1 φl1 φl1 1 I φ 40.3 TEC f λ TEC = Total Electron Content φ = R λ N + c δt δt I + T + MP + ε λ φ = R λ N + c δt δt I + T + MP + ε L2 2 2 u ρl2 φl2 φl2

28 Determining Ionospheric Delay I I f 2 ρl1 2 = 2 2 L2 f1 f2 L1 f ( ρ ρ ) 2 ρl2 1 = 2 2 L2 f1 f2 L1 ( ρ ρ ) 2 2 f f TEC = ( ) ( ρ ) 2 2 L2 ρl 1 f1 f2 Ionospheric delay on L1 pseudorange Ionospheric delay on L2 pseudorange Where frequencies are expressed in GHz, pseudoranges are in meters, and TEC is in TECU s (10 16 electrons/m 2 ) 28

29 Ionosphere maps

30 Ionosphere free Pseudorange f I ρ ρ ρ 2 L1 2 = 2 2 L2 f1 f2 L1 ( ) f f ρ ρ ρ ρ IF = " L3" = 2 2 L1 2 2 L2 f1 f2 f1 f2 ρ = 2.546ρ 1.546ρ IF L1 L2 Ionospheric delay on L1 pseudorange Ionosphere-free pseudorange Ionosphere free pseudoranges are more noisy than individual pseudoranges. 30

31 Multipath Reflected signals Can be mitigated by antenna design Multipath signal repeats with satellite orbits and so can be removed by sidereal filtering

32 Standard Positioning Error Budget Single Frequency Double Frequency Ephemeris Data 2 m 2 m Satellite Clock 2 m 2 m Ionosphere 4 m m Troposphere m m Multipath 0 2 m 0 2 m UERE 5 m 2 4 m UERE = User Equivalent Range Error

33 Intentional Errors in GPS S/A: Selective availability Errors in the satellite orbit or clock Turned off May 2, 2000 With SA 95% of points within 45 m radius. SA off, 95% of points within 6.3 m Didn t effect the precise measurements used for tectonics that much. Why not?

34 Intentional Errors in GPS A/S: Anti spoofing Encryption of the P code (Y code) Different techniques for dealing with A/S Recover L1, L2 phase Can recover pseudorange (range estimated using Pcode) Generally worsens signal to noise ratio

35 AS Technologies Summary Table Ashtech Z 12 & µz Trimble 4000SSi From Ashjaee & Lorenz, 1992

36 PSEUDORANGE OBSERVABLE MODEL ( u s ) ( u s ) ρ = R + c δt δt + T + I + M + ε 1 ρ1 ρ1 ρ1 ρ = R + c δt δt + T + I + M + ε 2 ρ 2 ρ 2 ρ 2 ρ ρ 1 2 = pseudorange measured on L1 frequency based on code = pseudorange measured on L2 frequency based on code R = geometrical range from satellite s to user u δt δt u s ρ1/ 2 ρ1/ 2 ρ1/ 2 = user/receiver clock error = satellite clock error T = tropospheric delay I M = ionospheric delay in code measurement on L1/2 = multipath delay in code measurement on L1/2 ε = other delay/errors in code measurement on L1/2

37 EXAMPLE OF PSEUDORANGE (1) s ( u ) ρ = R + c δt δt + T + I + M + ε 1 ρ1 ρ1 ρ1

38 EXAMPLE OF PSEUDORANGE (2)

39 GEOMETRIC RANGE Distance between position of satellite at time of transmission and position of receiver at time of reception s s s ( u ) ( u ) ( u ) R = x x + y y + z z

40 PSEUDORANGE minus GEOMETRIC s ( u ) Difference is typically dominated by receiver clock or satellite clock. RANGE ρ R = c δt δt + T + I + M + ε 1 ρ1 ρ1 ρ1

41 L1 PSEUDORANGE L2 PSEUDORANGE ( u s ) ( u s ) ρ = R + c δt δt + T + I + M + ε 1 ρ1 ρ1 ρ1 ρ = R + c δt δt + T + I + M + ε 2 ρ 2 ρ 2 ρ 2 ρ ρ = I I + M M + ε ε 1 2 ρ1 ρ 2 ρ1 ρ 2 ρ1 ρ 2 Differencing pseudoranges on two frequencies removes geometrical effects, clocks, troposphere, and some ionosphere

42 Geometry Effects: Dilution of Precision (DOP) Good Geometry Bad Geometry

43 Dilution of Precision VDOP = σ h HDOP = σ 2 + σ 2 n e PDOP = σ 2 + σ 2 + σ 2 n e h TDOP = σ t GDOP = σ 2 + σ 2 + σ 2 + c 2 σ 2 n e h t Covariance is purely a function of satellite geometry

44 Dilution of Precision (VDOP) Wuhan, China, 30 lat Casey station, Antarctica, 66.3 latitude

45 Positioning Most basic: solve system of range equations for 4 unknowns, receiver x,y,z,t P 1 = ( (x 1 x) 2 + (y 1 y) 2 + (z 1 z) 2 ) 1/2 + ct ct 1 P 4 = ( (x 4 x) 2 + (y 4 y) 2 + (z 4 z) 2 ) 1/2 + ct ct 4 Linearize problem by using a reference, or a priori, position for the receiver Even in advanced software, need a good a priori position to get solution.

46 Positioning vs. Differential GPS By differencing observations at two stations to get relative distance, many common errors sources drop out. The closer the stations, the better this works Brings precision up to mm, instead of m.

47 Single Differencing j ΔL AB j = Δρ AB j + cδτ AB + ΔZ AB j ΔI AB j + ΔB AB Removes satellite clock errors Reduces troposphere and ionosphere delays to differential between two sites Gives you relative distance between sites, not absolute position

48 Double Differencing j ΔL AB j = Δρ AB j + cδτ AB + ΔZ AB j ΔI AB j + ΔB AB k ΔL AB k = Δρ AB k + cδτ AB + ΔZ AB k ΔI AB k + ΔB AB jk ΔL AB = Δρ jk AB + ΔZ jk AB ΔI jk jk AB + λ ΔN AB Receiver clock error is gone Random errors are increased (e.g., multipath, measurement noise) Double difference phase ambiguity is an integer

49 High precision GPS for Geodesy Use precise orbit products (e.g., IGS or JPL) Use specialized modeling software GAMIT/GLOBK GIPSY OASIS BERNESE These software packages will Estimate integer ambiguities Reduces rms of East component significantly Model physical processes that effect precise positioning, such as those discussed so far plus Solid Earth Tides Polar Motion, Length of Day Ocean loading Relativistic effects Antenna phase center variations

50 High precision GPS for Geodesy Produce daily station positions with 2 3 mm horizontal repeatability, 10 mm vertical. Can improve these stats by removing common mode error.