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Global Positioning System: what it is and how we use it for measuring the earth’s movement. April 21, 2011 References • Lectures from K. Larson’s “Introduction to GNSS” http://www.colorado.edu/engineering/ASEN/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” • http://www.kowoma.de/en/gps/index.htm • http://www.kemt.fei.tuke.sk/predmety/KEMT559_SK/ GPS/GPS_Tutorial_2.pdf • Lecture notes from G. Mattioli’ (comp.uark.edu/~mattioli/geol_4733/GPS_signals.ppt) 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? GPS: Space segment • Several different types of GPS satellites (Block I, II, II A, IIR) • All have atomic clocks – Stability of at least 10-13 sec 1 sec every ~300,000 yrs • Dynamics of orbit? • Reference point? Orbital Perturbations – (central force is 0.5 m/s2) Source Acceleration Perturbation m/s2 3 hrs Earth oblateness (J2 ) 5 x 10-5 2 km @ 3 hrs secular + 6 hr Sun & moon 5 x 10-6 5-150 m @ 3 hrs secular + 12hr Higher Harmonics 3 x 10-7 5-80 m @ 3 hrs Various Solar radiation pressure 1 x 10-7 100-800 m @2 days Secular + 3 hr Ocean & earth tides 1 x 10-9 0-2m @2 days secular + 12hr Earth albedo pressure 1 x 10-9 1-1.5m @2 days From K. Larson Type 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 GPS: Satellite Ground Track 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 GPS Signal • Signal phase modulated: vs Amplitude modulation (AM) Frequency modulation (FM) 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 PRN Codes: first 100 bits 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. PRN Code Correlation Non-PRN Code Correlation 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. 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 c t u t s t u = tim e o f receptio n as ob served b y the receiver s t = tim e o f transm ission as gen erated b y the satellite The measured pseudorange is not the true range between the satellite and receiver. That is what we clarify with the observable equation. PSEUDORANGE OBSERVABLE MODEL 1 R c t u t s 2 R c t u t s T I T I 1 2 M M 1 2 1 2 1 = pseu d o ran ge m easu red on L 1 frequ en cy based o n co d e 2 = p seu do rang e m easu red o n L 2 freq u ency b ased o n cod e R = g eom etrical rang e fro m satellit e s to u ser u t u = user/receiver clo ck erro r t = satellite clo ck erro r s T = tro p osph eric d elay I 1 / 2 = io no sph eric d elay in co d e m easu rem ent o n L 1/2 M 1/ 2 = m u ltip ath d elay in cod e m easurem en t o n L 1 /2 1 / 2 = oth er d elay/erro rs in cod e m easu rem en t o n L 1 /2 CARRIER PHASE MODEL 1 1 R c t u t 2 2 R c t u T I t T I s 1 s 2 M 1 N 1 1 1 M 2 N 2 2 2 1 = carrier phase m easu red on L 1 freq uency (C /A or P (Y ) parts) 2 = carrier phase m easured on L 2 frequency R = g eo m etrical ran ge fr om satellite s to user u t u = u ser/receiver clo ck error t = satellite clock erro r s T = tro po sp heric del ay I 1 , I 2 = ion osph eric d elay in co de m easurem ent o n L1 /2 M 1 , M 2 = m ultipath delay in carrier phase m easurem ent o n L 1/2 N 1 , N 2 = carrier phase b ias or am b ig uity 1 , 2 = carrier w avelen gth 1 , 2 = other delay/errors in carrier p hase m easu rem en t on L 1 /2 COMPARE PSEUDORANGE and CARRIER PHASE 1 R c t u t 1 1 R c t u T I t T I s 1 M s 1 1 1 M 1 N 1 1 1 • 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) 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. Vertical Structure of Atmosphere Tropospheric effects • • • • • Lowest region of the atmosphere – index of refraction = ~1.0003 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 Tropospheric effects At lower elevation angles, the GPS signal travels through more troposphere. Dry Troposphere Delay Saastamoinen model: T 2.277 10 1 0.0026 cos 2 0.00028 h P • P0 is the surface pressure (millibars) • is the latitude • h is the receiver height (m) 3 z ,d Hopfield model: T 77.6 10 • hd is 43km • T0 is temperature (K) 0 6 z ,d Mapping function: • E – satellite elevation P0 h d ~2.5 m at sea level T0 5 1 (zenith) – 10 (5 deg) 1 md sin E 0.00143 tan E 0.0445 Wet Troposphere Correction Less predictable than dry part, modeled by: Saastamoinen model: Hopfield model: T z , w 2 .2 7 7 1 0 T z , w 0.373 3 1255 0 .0 5 e0 T e0 hw 2 T0 0 – 80 cm 5 • hw is 12km • e0 is partial pressure of water vapor in mbar Mapping function: 1 md sin E 0.00035 tan E 0.0 17 Examples of Wet Zenith Delay Ionosphere effects • Pseudorange is longer – “group delay” • Carrier Phase is shorter – “phase advance” L 1 R c t u t s L 2 R c t u t s I I L1 I I T M P L 2 L 2 L2 1 L 1 R 1 N 1 c t u t 1 L 2 R 2 N 2 c t u T M P L 1 L 1 I t I s L1 T M P L 1 L 1 s L2 T M P L 2 L 2 4 0.3 T E C f 2 TEC R 1 N 1 c Content t u t 1 L 1 = Total Electron 1 L 2 R 2 N 2 c t u I t I s L1 T M P L 1 L 1 s L2 T M P L 2 L 2 Determining Ionospheric Delay 2 I L1 IL2 f2 f1 f 2 f1 2 2 2 f1 f 2 2 2 L 2 L1 Ionospheric delay on L1 pseudorange L 2 L1 Ionospheric delay on L2 pseudorange 2 TEC 2 f1 f 2 40.3 f 1 f 2 2 2 L 2 L1 Where frequencies are expressed in GHz, pseudoranges are in meters, and TEC is in TECU’s (1016 electrons/m2) 28 Ionosphere maps Ionosphere-free Pseudorange 2 I L1 f2 f1 f 2 2 IF " L 3 " 2 L 2 f1 Ionospheric delay on L1 pseudorange 2 f1 f 2 L1 2 2 2 L1 f2 f1 f 2 2 2 L2 Ionosphere-free pseudorange IF 2.546 L 1 1.546 L 2 Ionosphere-free pseudoranges are more noisy than individual pseudoranges. 30 Multipath • Reflected signals – Can be mitigated by antenna design – Multipath signal repeats with satellite orbits and so can be removed by “sidereal filtering” Standard Positioning Error Budget Single Frequency Double Frequency Ephemeris Data 2m 2m Satellite Clock 2m 2m Ionosphere 4m 0.5 – 1 m Troposphere 0.5 – 1 m 0.5 – 1 m Multipath 0-2 m 0-2 m UERE 5m 2-4 m UERE = User Equivalent Range Error 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? 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 AS Technologies Summary Table Ashtech Z-12 & µZ Trimble 4000SSi From Ashjaee & Lorenz, 1992 PSEUDORANGE OBSERVABLE MODEL 1 R c t u t s 2 R c t u t s T I T I 1 2 M M 1 2 1 2 1 = pseu d o ran ge m easu red on L 1 frequ en cy based o n co d e 2 = p seu do rang e m easu red o n L 2 freq u ency b ased o n cod e R = g eom etrical rang e fro m satellit e s to u ser u t u = user/receiver clo ck erro r t = satellite clo ck erro r s T = tro p osph eric d elay I 1 / 2 = io no sph eric d elay in co d e m easu rem ent o n L 1/2 M 1/ 2 = m u ltip ath d elay in cod e m easurem en t o n L 1 /2 1 / 2 = oth er d elay/erro rs in cod e m easu rem en t o n L 1 /2 EXAMPLE OF PSEUDORANGE (1) 1 R c t u t s T I 1 M 1 1 EXAMPLE OF PSEUDORANGE (2) GEOMETRIC RANGE • Distance between position of satellite at time of transmission and position of receiver at time of reception R x s xu y 2 s yu z 2 s zu 2 PSEUDORANGE minus GEOMETRIC RANGE 1 R c t u t s T I • Difference is typically dominated by receiver clock or satellite clock. 1 M 1 1 L1 PSEUDORANGE - L2 PSEUDORANGE 1 R c t u t s 2 R c t u t s T I T I 1 2 M M 1 2 1 2 1 2 I 1 I 2 M 1 M 2 1 2 • Differencing pseudoranges on two frequencies removes geometrical effects, clocks, troposphere, and some ionosphere Geometry Effects: Dilution of Precision (DOP) Good Geometry Bad Geometry Dilution of Precision VDOP h HDOP PDOP 2 n 2 n 2 e 2 e 2 h TDOP t GDOP c 2 n 2 e 2 h 2 Covariance is purely a function of satellite geometry 2 t Dilution of Precision 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. 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. Single Differencing L j AB j AB c AB Z j AB I j AB B • Removes satellite clock errors • Reduces troposphere and ionosphere delays to differential between two sites • Gives you relative distance between sites, not absolute position j AB Double Differencing L AB AB c AB Z AB I AB B AB j j j j j L AB AB c AB Z AB I AB B AB k k k k k L AB AB Z AB I AB N AB jk jk jk jk jk • Receiver clock error is gone • Random errors are increased (e.g., multipath, measurement noise) • Double difference phase ambiguity is an integer 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 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.