The Global Positioning System

Principles of GPS positioning GPS signal and observables Errors and corrections Processing GPS data GPS measurement strategies Precision and accuracy

E. Calais Purdue University - EAS Department Civil 3273 ecalais@purdue.edu

The Global Positioning System (GPS)

A satellite-based positioning system available 24/24h everywhere on the globe with an accuracy better than 100 m. Originally designed for navigation and real-time positioning (meter-level accuracy): navigation (airplanes, ships, car, missiles, etc)

It is also capable of mm-level accuracy, with important scientific by-products: In geodesy: shape and rotation of the Earth, terrestrial reference frame In solid Earth geophysics: deformation of the Earths crust (earthquakes, volcanoes, plate tectonics) In atmospheric sciences: tropospheric water vapor, ionospheric electron content

Three segments
The space segment = satellites:
Broadcast radio signals toward users on the Earth Receive commands from the ground.

The control segment: monitors the space segment and send commands to satellites The user segment: receivers record and interpret the radio signals broadcast by the satellites

The GPS satellites

Four classes (=generations): blocks I, II, IIA, IIR,and IIF:
Block I:
11 satellites launched between 1978 and 1985 on Atlas F rockets Life expectancy = 4.5 years, actual mean life = 7.1 years Signal entirely accessible to civilian users Last block I satellite died on Feb. 28, 1994
Block II satellite

Block II (II-R and II-F):

Possibility to degrade the signal for civilian users 1 satellite ~ 25 million dollars Life expectancy = 10 years 5 m3, 2 tons, solar panels, boosters

New launches on a regular basis Monitored and controlled from the ground
Block IIR satellite

Orbital constellation
27 satellites (24 operational + 3 spares)
Quasi-circular orbits, mean altitude 20200 km 6 evenly spaced orbital planes (A to F), inclination 55 4-6 satellites per plane, spacing for optimized visibility Period = 12 sidereal hours (= 11h58mn terrestrial hours) in a terrestrial frame, the constellation repeats every 23h56mn. As Earth orbits around the Sun eclipse periods (solar radiation pressure = 0, transition to shadow difficult to model, often simply edited out)

In practice, 6-12 satellites are visible simultaneously, depending on:

Constellation geometry Elevation cut-off angle (chosen by the user)

Satellite transmissions
GPS satellites broadcast continuously on 2 frequencies in the L-band Future: GPS III, 3rd frequency GPS antennas point their transmission antenna to the center of the Earth Main beam = 21.4/23.4 (L1/L2) half width

Transmission antenna of a block II-R GPS satellite

GPS transmission beams

Satellite clocks
Frequencies broadcast by GPS satellites are derived from a fundamental frequency of 10.23 Mhz Fundamental frequency provided by 2 or 4 atomic clocks (Ce/Rb)
Clocks run on GPS time = UTC not adjusted for leap seconds Clock stability over 1 day = 10-13 (Rb) 10-14 (Ce), ~ 1 ns/jour Clocks synchronized between all satellites

Relativistic effects:
Clocks in orbit appear to run faster (38.3 sec/day = 11.5 km/day!) tuned at 10.22999999543 MHz before launching (g.) Clocks speed is a function of orbit eccentricity (45 nsec = 14 m) corrected at the data processing stage (s.): 2

t R =

a e sin E

GPS control segment

GPS control segment = 5 stations, master station at Colorado Springs Track satellites, computes and upload broadcast ephemerides into the satellites (broadcast ephemerides distributed to users via a navigation message included in the signal transmitted by the GPS satellites) Time synchronization on the satellites Monitors satellite health Decides and implements maneuvers when necessary

User segment

GPS receivers All sizes, all prices For and endless variety of applications

GPS positioning: A simple principle

satellite 2 satellite 3 satellite 1

Principle of GPS positioning:

Satellite 1 sends a signal at time te1 Ground receiver receives it signal at time tr The range measurement 1 to satellite 1 is:
1 = (tr-te1) x speed of light We are therefore located on a sphere with radius 1 centered on satellite 1

3 satellites intersection of 3 spheres

In simple mathematical terms:

rs = ( X s X r ) 2 + (Ys Yr ) 2 + ( Z s Z r ) 2
GPS receivers:
Measure tr Decode te Compute rs

2 2 3
You are here
x

Earth

If the position of the satellites in an Earth-fixed frame (Xs, Ys, Zs) is known, Then one can solve for (Xr, Yr, Xr) (if at least 3 simultaneous range measurements)

c = 299 792 458 m/s

Satellite-receiver time offset

The receiver clocks are:
1. Mediocre: stability ~10-5-10-6 (~ crystal wrist watch) 2. Not synchronized with the satellite clocks.

There is a time difference between the satellite clocks (ts) and the receiver clock (tr): t = tr ts The receivers therefore measures: = t + t In terms of distance: x c = (t + t) x c = r + r = The receiver actually measures = pseudorange Practical consequences:
The time offset between satellite and receiver clocks is an additional unknown We need 4 observations 4 satellites visible at the same time In order to compute a position, the receiver solves for t => GPS receivers are very precise clocks! (Timing is a very important application of GPS) t is used by the receiver to synchronize its clock with the satellite clocks. That sync is as good as t accuracy or ~ 0.1 sec: we will still need to solve for t

From the GPS signal to a position: Basic principle

Measure arrival time of GPS signals from several satellites simultaneously Decode the GPS signal and figure out the signal propagation time (tr-te), multiply by c = pseudoranges (= GPS data, or observables) Decode the navigation message and convert it into satellite positions Use at least 4 pseudoranges acquired at the same time from 4 different satellites to compute a position in an ECEF frame. Convert ECEF position into latitude-longitude-height in any geodetic system (for instance WGS84).

The GPS signal

The atomic clocks aboard the GPS satellites produce a fundamental frequency fo = 10.23 Mhz Two frequencies are derived from it: L1 (fo x 154) and L2 (fo
x 120):

L1: 1.57542 GHz, wavelength 19.0 cm L2: 1.22760 GHz, wavelength 24.4 cm L1 and L2 are the two carrier frequencies used to transmit timing information by the GPS satellites The information transmitted by the satellite is coded as a phase modulation of the carrier frequency

Phase modulation
Information is coded as a sequence of +1/-1 (binary values 0/1), shift in carrier phase when code state changes = biphase modulation Rate at which the phase shift occurs = chip rate Pseudorandom noise codes (= PRN codes):
Unique to each satellite Coarse Acquisition (C/A) code: L1 only Chip rate = 1023 MHz Precision (P) code: L1 and L2 Chip rate = 10.23 MHz Encryption (W) code: encrypts the P-code into the Y-code (highly classified)

Biphase modulation of the GPS carrier phase

Navigation message
Navigation message: ephemerides for all satellites, ionospheric correction parameters, system status, satellite clock offset and drift) Also coded by bi-phase modulation Chip rate = 50 bps 25 frames of 1500 bits each, divided into five 300 bits subframes 50 bps 300/50 = 6 sec to transmit one subframe, 6x5x25 = 750 sec (=12.5 min) to transmit an entire navigation message

Receiver start-up
General procedure:
1. 2. 3. 4. Acquire one satellite to get time and almanach Acquire 2 other satellites to get 2-D position Acquire 4th satellite to get 3-D position Acquire any other visible satellite

Time needed to get good position:

Hot start: few secs (rcv was off for a few secs: almanach ok, time ok, position close to last one) Warm start: few mins (rcv was off for less than a day: clock ~ok) Cold start: 10s of minutes (rvc was off for several days: time off, almanach expired, last position off)

Decoding in the receiver

Radio frequency (RF) part of the receiver processes incoming signals:
L1 only (single-frequency receivers) L1 and L2 (dual-frequency receivers)

RF unit:
Processes incoming signal from different satellites in different channels (multichannels receivers, 4 to 12 channels) Generates internal replica of the GPS signal:
Contains an oscillator (= clock) that generates L1 and L2 frequencies Knows each PRN code (almost)

Compares internally generated signal with incoming signal

Code measurements
Code-correlation:
Shift of the internally generated signal in time until it matches the incoming one (receiver locked on a satellite) Time shift needed = signal travel time from satellite to receiver

Other techniques to retrieve phase information, independent of PRN codes:

Squaring: autocorrelation of the incoming signal Cross-correlation: correlation between L1 and L2 using Y-code (Y-code is identical on L1 and L2) Z-tracking: correlation on L1 and L2 using the P-code to obtain W-code All these techniques have a lower SNR than the code-correlation:
Squaring: -30 dB Cross correlation: -27 dB Z-tracking: -14 dB

Code measurements
GPS receivers measure pseudoranges jRi(t), that can be modeled as:
j

Ri (t ) = j i (t ) + c ( j (t ) i (t )) + I (t ) + T (t ) + MP (t ) +
t = time of epoch jR = pseudorange measurement i j = satellite-receiver geometric distance i c = speed of light j = satellite clock bias i = receiver clock bias I = ionospheric propagation error T = tropospheric propagation error MP = multipath = receiver noise (ranges in meters, time in seconds)

I and T are correction terms because GPS signal propagation is not in a vacuum (more later) MP = multipath noise, reflection of GPS signal off surfaces near antenna (more later)

Pseudorange noise
Correlation function width: The width of the correlation is inversely proportional to the bandwidth of the signal.
C/A code = 1 MHz bandwidth correlation produces a peak 1 msec wide = 300 m P code = 10 MHz bandwidth correlation produces 0.1 msec peak = 30 m

Rough rule: Peak of correlation function can be determined to 1% of width (with care).
Range accuracy = 3 m for C/A code Range accuracy = 0.3 m for P code

Pseudorange measurements = low accuracy but absolute

Phase measurements
When a satellite is locked (at to), the GPS receiver starts tracking the incoming phase It counts the (real) number of phases as a function of time = (t) But the initial number of phases N at to is unknown However, if no loss of lock, N is constant over an orbit arc
S(t2) S(t1)

1
S(to)
or bi t

r(t2)

N N N

Earth

Phase measurements
Geometrical interpretation:
= phase measurement
R = pseudorange c = speed of light = geometric range = wavelength t = sat-rcv clock offset N = phase ambiguity

R = + ct

c = + t N

The phase equation (units of cycles):

ik (t ) = ik (t ) f + h k (t ) hi (t ) f + ionik (t ) + tropik (t ) N ik + c

t = time of epoch i = receiver, k = satellite ik = geometric range hk = satellite clock error, hi = receiver clock error ionik = ionospheric delay, tropik = tropospheric delay Nik = phase ambiguity, = phase noise

Phase measurements
Phase can be converted to distance by multiplying by the wavelength phase measurements are another way for measuring the satellite-receiver distance Phase can be measured to ~1% of the wavelength range accuracy 2 mm for L1, 2.4 mm for L2 Phase measurements are very precise, but ambiguous To fully exploit phase measurements, one must correct for propagation effects (several meters)

GPS observables
GPS receivers can record up to 5 observables :
1 and 2: phase measurements on L1 and L2 frequencies, in cycles C/A, P1, P2: pseudorange measurements, in meters

Plus Doppler phase = d/dt

GPS observables
GPS observables stored in receivers in binary proprietary format Receiver Independent Exchange format (RINEX) = ASCII exchange format Format description: ftp://igscb.jpl.nasa.gov/igscb/data/format/rinex2.txt Conversion from binary proprietary to RINEX:
Proprietary software Freewares: e.g. teqc (www.unavco.ucar.edu)

RINEX observation file

2.00 OBSERVATION DATA G (GPS) RINEX VERSION / TYPE teqc 1999Jul19 CNRS_UMRGA 20021201 12:04:20UTCPGM / RUN BY / DATE Solaris 2.3|S-Sparc|cc SC3.0|=+|*Sparc COMMENT BIT 2 OF LLI FLAGS DATA COLLECTED UNDER A/S CONDITION COMMENT SJDV MARKER NAME 10090M001 MARKER NUMBER ___ REGAL OBSERVER / AGENCY 845 ASHTECH Z-XII3 CD00 REC # / TYPE / VERS 317 ASH700936A_M NONE ANT # / TYPE 4433469.9683 362672.6919 4556211.6229 APPROX POSITION XYZ 0.0000 0.0000 0.0000 ANTENNA: DELTA H/E/N 1 1 WAVELENGTH FACT L1/2 5 L1 L2 C1 P1 P2 # / TYPES OF OBSERV 30.0000 INTERVAL Forced Modulo Decimation to 30 seconds COMMENT SNR is mapped to RINEX snr flag value [1-9] COMMENT L1: 1 -> 1; 90 -> 5; 210 -> 9 COMMENT L2: 1 -> 1; 150 -> 5; 250 -> 9 COMMENT 2002 11 30 0 0 30.0000000 GPS TIME OF FIRST OBS END OF HEADER 02 11 30 0 0 30.0000000 0 8G14G 7G31G20G28G 1G25G11 -7096034.24049 -5509904.97345 23971309.103 23971309.038 23971310.842 -12570276.74149 -9768618.40046 23379169.469 23379168.448 23379172.496 -4157689.84249 -3201324.38045 24195891.298 24195890.733 24195894.168 -25480193.34249 -19826614.77248 20670858.774 20670857.983 20670861.191 -5589280.20049 -4319738.39345 24553697.713 24553697.259 24553700.349 -10252537.24449 -7918950.15946 23060092.127 23060091.841 23060095.687 -4143445.15949 -2509987.53445 24581180.488 24581179.713 24581183.992 -29659606.34049 -23089397.33548 20312382.965 20312382.530 20312384.719 02 11 30 0 1 0.0000000 0 8G14G 7G31G20G28G 1G25G11 -7004806.32949 -5438818.30145 23988669.195 23988668.970 23988671.466 -12645245.09249 -9827035.30846 23364903.590 23364902.944 23364907.274 -4043324.79449 -3112208.77545 24217654.165 24217653.747 24217658.209 -25518762.53849 -19856668.69248 20663519.280 20663518.524 20663521.550 -5521754.77149 -4267121.22845 24566547.413 24566547.593 24566550.660 -10357839.61649 -8001003.94446 23040053.767 23040053.443 23040058.358 -4207531.87749 -2559925.21345 24568984.944 24568985.325 24568989.371 -29640011.07349 -23074128.30548 20316111.836 20316111.559 20316113.648

Header

Data blocks:
Range in meters Phase in cycles

GPS observables: Summary

Pseudorange measurements (C/A, P1, P2):
Geometric range + clock offset + noise: = r + t x c Accuracy of pseudorange measurements by GPS receivers ~ 1% of correlation peak width: 3 m with C/A code 0.3 m with P code Low accuracy but absolute measurements

Phase measurements (L1, L2):

Geometric range + clock offset - initial phase ambiguity N: = r x f/c + t x f N Accuracy of phase measurements in GPS receivers ~ 0.005 cycle (0.005 x 20 cm = 0.2 mm) millimeter accuracy theoretically possible Very accurate measurements but ambiguous