1. Global Positioning System: what it is and how we use it for measuring the earth’s movement.
April 21, 2011

2. References
Lectures from K. Larson’s “Introduction to GNSS”
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”
Lecture notes from G. Mattioli’ (comp.uark.edu/~mattioli/geol_4733/GPS_signals.ppt)

3. Basics of how it works Trilateration GPS positioning requires
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/s2)
Source
Acceleration
m/s2
Perturbation
3 hrs
Type
Earth oblateness (J2 )
5 x 10-5
2 3 hrs
secular + 6 hr
Sun & moon
5 x 10-6
hrs
secular + 12hr
Higher Harmonics
3 x 10-7
hrs
Various
Solar radiation pressure
1 x 10-7
days
Secular + 3 hr
Ocean & earth tides
1 x 10-9
days
Earth albedo pressure
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. GPS Signal 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
Say something about PRN numbers?

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
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

18. CARRIER PHASE MODEL

19. COMPARE PSEUDORANGE and CARRIER PHASE
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 (50-1000 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 Saastamoinen model:
P0 is the surface pressure (millibars)
f is the latitude
h is the receiver height (m)
Hopfield model:
hd is 43km
T0 is temperature (K)
Mapping function:
E – satellite elevation
~2.5 m at sea level
1 (zenith) – 10 (5 deg)

25. Wet Troposphere Correction
Less predictable than dry part, modeled by:
Saastamoinen model:
Hopfield model:
hw is 12km
e0 is partial pressure of water vapor in mbar
Mapping function:
0 – 80 cm

26. Examples of Wet Zenith Delay

27. Ionosphere effects • Pseudorange is longer – “group delay”
• Carrier Phase is shorter – “phase advance”
TEC = Total Electron Content

28. Determining Ionospheric Delay
Where frequencies are expressed in GHz, pseudoranges are in meters, and TEC is in TECU’s (1016 electrons/m2)

29. Ionosphere maps

30. Ionosphere-free Pseudorange
Ionosphere-free pseudoranges are more noisy than individual pseudoranges.

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
Satellite Clock
Ionosphere
4 m
0.5 – 1 m
Troposphere
Multipath
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 P-code)
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

37. EXAMPLE OF PSEUDORANGE (1)

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

40. PSEUDORANGE minus GEOMETRIC RANGE
Difference is typically dominated by receiver clock or satellite clock.

41. L1 PSEUDORANGE - L2 PSEUDORANGE
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
Covariance is purely a function of satellite geometry

44. Dilution of Precision

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

46. Positioning
• Most basic: solve system of range equations for 4 unknowns, receiver x,y,z,t
P1 = ( (x1 - x)2 + (y1 - y)2 + (z1 - z)2 )1/2 + ct - ct1 …
P4 = ( (x4 - x)2 + (y4 - y)2 + (z4 - z)2 )1/2 + ct - ct4
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.

47. 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.

48. Single Differencing • Removes satellite clock errors
• Reduces troposphere and ionosphere delays to differential between two sites
• Gives you relative distance between sites, not absolute position

49. Double Differencing • Receiver clock error is gone
• Random errors are increased (e.g., multipath, measurement noise)
• Double difference phase ambiguity is an integer

50. 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

51. 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.