Principles of the Global Positioning System - GPS Models and Processing

Principles of the Global Positioning System GPS Models and processing • Summary: – Finish up modeling aspects • Rank deficiencies – Processing methods: • Differencing of data • Cycle slip detection • Bias fixing and cycle slip repair 1 Rank deficiencies • Ranks deficiencies are combinations of parameters that can not be separately estimated. • In GPS, there are several rank deficiencies: – UT1, Longitudes of all the stations and the nodes of the satellite orbits, effectively can not be separated. – In theory, orbit perturbations by the moon/sun on the GPS orbits could be used to align the orbits in a solar system frame, but effect is too small to be useful (I think: never really tested) – Separation is solved by adopting UT1-AT from VLBI, and setting the mean longitude change of stations to ITRF coordinates. Longitude is standard problem because choice of Greenwich as origin is arbitrary. Rank deficiencies • Other rank deficiencies: – Pole position can not separated from over all rotation of coordinates. Again resolved either by adopting polar motions on one day or on average having zero rotation of the coordinates relative to an initial frame. – In principle could be separated by gravity field perturbations but effect is too small. – All station and satellite clocks can not be simultaneously estimated. Again there is sensitivity due moon/sun perturbations but these are too small. (Later we will see how differencing data, implicitly eliminates this problem). Solution, if clocks are explicitly estimated, is to adopt one clock as reference or set an average of the clock differences to be zero. 2 Rank deficiencies • Velocity rank deficiency: – It is not possible to separate “absolute” station motions from secular drift of pole and secular UT1AT changes. (Remember pole has drifted 10 meters in 100 years--10 cm/yr comparable to plate motions). – IERS polar motion is referred to a no-net-rotation geologic frame (Nuvel-1A). • There are some other rank deficiencies with nutations and orbits, but the apriori nutation series is very well defined by VLBI Subtle rank deficiencies • Phase center patterns for satellites and ground receivers can not separately determined using just GPS antennas. • Because the satellites point towards the center of the Earth; a given elevation angle at a GPS receiver can be mapped to an off-bore-sight angle on the satellite and two effects can not separated. • Interestingly, if the GPS satellites could be “rocked” (so no longer pointing at the center of the Earth), the two effects could be separated. • Even with low precision satellite phase center positions can be estimated assuming “point” antenna 3 Estimated Satellite Z-offsets Z-offset No Choke ring phase center Z offset Hannover Choke ring model 4.0 Satellite Z-phase center position (m) 3.0 2.0 Block IIR 1.0 Apriori Block II/IIA offset 0.0 -1.0 -2.0 Block IIR -3.0 5 10 15 PRN 20 25 30 Time series estimates Block IIR satellites 3.0 Z Phase center offset (m) 2.0 1.0 0.0 -1.0 PRN PRN PRN PRN PRN PRN 11 Abs 1.44±0.03 m 14 Abs 1.71±0.06 m 28 Abs 1.45±0.07 m 11 Rel -2.14±0.04 m 14 Rel -1.87±0.06 m 28 Rel -2.18±0.05 m 2000 2001 -2.0 -3.0 1999 2001 Year 2002 2002 2003 4 Zoom of Absolute series only Block IIR satellites. Absolute PC model only 3.0 PRN 11 Abs 1.44±0.03 m PRN 14 Abs 1.71±0.06 m PRN 28 Abs 1.45±0.07 m Z Phase center offset (m) 2.5 2.0 1.5 1.0 0.5 2000.8 2001.0 2001.2 2001.4 2001.6 Year 2001.8 2002.0 2002.2 2002.4 Effects on radial orbit position of satellite Apriori orbit: No satellite or choke ring PC, sites constrained Orbit Adjustment relative to constrained model DSemimajor Axis PRN 11 (m) 0.6 Abs Mean -0.01±0.03 RMS 0.15 ms 0.20 m Rel Mean -0.01±0.03 RMS 0.15 m s 0.21 m No Satellite PC Rel Mean 0.12±0.02 RMS 0.13 m s 0.18 m 0.4 0.2 0.0 -0.2 -0.4 2000.0 2000.4 2000.8 2001.2 Year 2001.6 2002.0 5 Summary of phase center • The effects of ground antenna phase center model only satellite phase center estimates are large (~3.6 meters) • Block II/IIA definitely different from Block IIR and some indication of differences between satellites within the same type (differences are a few centimeters) • Radial orbit changes are small (<1 cm on average). Interestingly better agreement of loose solution with constrained when satellite PC estimated (10 cm differences), Scale effects • From the different analyses and VLBI analysis we can estimate scale and its rate of change: Soln Abs Scale -6.04 +0.25 Srate -0.24 +0.06 Rel 11.99 0.25 -0.22 0.06 VLBI -0.21 0.04 -0.02 0.01 • Scale in ppb and scale rate ppb/yr (1ppb=6mm) 6 Processing methods • The clock and local oscillator phase variations are the biggest deviations in the model of GPS phase and range data. • These terms can be explicitly handled by estimation of clock variations (but if done brut-force in least squares is a very large estimation problem). Can be attacked with sequential LSQ or a Kalman filter. • When multiple sites see the same satellite, the satellite clocks can also be estimated, but at every epoch of measurement, one clock needs to be fixed, or an ensemble average of cocks set to have zero mean adjustment. Differencing • An alternative approach to explicit estimation is differencing data. • Single differences: two forms: – Difference measurements from two sites that see the same satellite. Eliminates error due to satellite clock. – Difference measurements from two satellites at the one site: Eliminates the ground receiver clock. • Double differences: – By differencing a pair a single differences, but the ground and satellites clocks are eliminated. – The local oscillator phases also cancel except the differences in the number of cycles of phase between the combination of two satellites and two stations. This difference should be an integer. 7 Differencing • There are subtle problems with the exact times that measurements are made with differencing. • In the receivers, the measurements of range and phase to all the satellites can be made at exactly the same (within electronics noise) • But signals measured at the same time receivers separated by large distances must have been transmitted from the satellite at different times due to the light propagation time.. Light propagation time and differencing • This effect can lead to 20 ms differences in the transmission times. When SA was on and satellite clocks had frequency drifts of ~1Hz, this lead to errors of 0.02 cycles (~4mm). Not such a problem anymore and even with SA was not severe. • Non-synchronized receiver sampling can cause problems. Normally receivers stay with in 1 ms of GPS time (by resetting their clock counters). Older receivers could be off by up to 80 ms: Feigl, K. L, R. W. King, T. A. Herring, and M. Rotchacher, A scheme for reducing the effect of selective availability on precise geodetic measurements from the Global Positioning System,
Geophys. Res. Lett., 1289–1292, 1991. 8 Cycle slip detection • When processing phase, cycles slips are a potential problem. You can look at this in HW2 data set. The L1 and L2 phase values are in the L1 and L2 slots in the rinex file. The have a large offset from the range values (initial ambiguity, which in double differences should be a integer value) • When the receiver looses lock (typically range will be missing but not always), a cycle slip occurs and this must be re-fixed to an integer or left as an unknown parameter Cycle slip detection • When o-minus-c is computed in one-ways for phase, variations are dominated by clocks in receiver. • Multiple techniques are used to detect cycle slips: – Ln phase - Ln range (n=1,2). Removes geometry but affected by ionospheric delay (opposite sign on phase and range) and noise in range measurements – L1 phase - L2 phase. Some times called wide-lane. Affected by ion-delay but is common detector – Double difference phase residuals: On short baselines, removes ionosphere and if good apriori positions are known, should be a smooth function of time. Often used to estimate number of cycles in sip and resolve to integer value. – Melbourne-Webena wide lane (ML-WL) (see over) 9 MW wide lane • Very useful combination of data that is often used in kinematic GPS where receiver coordinates are changing. • The MW WL should equal number of cycles of phase between between L1 and L2 and is calculated, effectively, be computing the expected L1 and L2 phase difference from the pseudorange data. MW Wide lane • From the equations for range and phase with the phase offsets for cycle offsets you can derive: MW - WL = N1 - N 2 = f L 2 - f L1 + (P1 + P2 ) f L1 - f L 2 f L1 + f L 2 • The MW-WL should be constant if there are no cycle slips. When the phase and range values are double differences, N2-N1 should be integer. • The factor that scales range is ~0.1 and so range noise is reduced. • Average values of the MW-WL are used to esimate L1/L2 phase difference independent of ion-delay and geometry changes. 10 Ambiguity resolution • The MW-WL is often used to get N1-N2 and then N1 is estimated, as non-integer value, from the leastsquares fit to the phase data. • If the sigma of the estimate is small, and the estimate is close to an integer then it can be resolved to an integer values. • Fixing ambiguities, improves the sigma of the east position estimate by typically a factor of two and makes it similar to the North sigma. Bias fixing has little effect on North and Up sigma except for short sessions. Summary • • • • Today’s lecture examined: rank deficiencies Differencing of data to eliminate clock errors Cycle slip detection and bias fixing (also called ambiguity resolution). 11