EESC 9945 Geodesy with the Global Positioning System Class 7: Relative Positioning using Carrier-Beat Phase
GPS Carrier Phase The model for the carrier-beat phase observable for receiver p and satellite r we developed, ignoring atmospheric and ionospheric propagation delays, was φ r p(t) = 1 λ ρr p(t) + φ r φ p + N r p + f [δ r (t) δ p (t)] We ve left off the since too many Subscript letters are different 2
Single Difference The single difference (or single difference phase ) is the difference between phase observations from two sites for a single satellite: φ r pq(t) = φ r p(t) φ r q(t) = 1 [ ρ r λ p (t) ρ r q(t) ] φ p + φ q + Np r Nq r +f [δ q (t) δ p (t)] Satellite initial phase and clock error have differenced out 3
Baseline Vector Let b pq be the vector between two GPS site located at x p and x q : bpq = x q x p b pq is called the baseline vector (or inter site vector ) between sites p and q 4
Single-difference geometry: Short baseline limit The geometry term in the single-difference phase is ρ r pq = ρ r p ρ r q = x r x p x r x q Substituting x q = x p + b pq we have ρ r pq = ρ r p x r x p b pq = ρ r p ρ r p b pq The length of the vector ρ r p b pq is [ ρ r p b pq = ρ r p 1 2ˆρr p b pq ρ r p + ( bpq ρ r p ) 2 ] 1/2 5
Single-difference geometry: Short b limit Expanding for small b pq gives [ ρ r p b pq ρ r p 1 ˆρr p b pq ρ r + o p ( (bpq ρ r p ) 2 )] Dropping terms of order (b pq /ρ r p) 2 and smaller gives ρ r p b pq ρ r p ˆρ r p b pq The geometry term in the single-difference phase for b pq ρ r p is thus ρ r pq = ρ r p ρ r p b pq ˆρ r p b pq 6
Single-difference geometry: Short b limit Range Difference (km) 3000 2500 2000 1500 1000 500 Exact Short BL Approx. 0 0 1000 2000 3000 Baseline Length (km) 4000 Difference is 10 cm @ 100 km 7
Implications of differencing Single-difference phase observations are much less sensitive to position and much more sensitive to difference of position In other words, sensitivity to coordinate origin has been significantly decreased 8
Implications of differencing Sensitivity of phase to baseline error (as rule of thumb) is δφ δb + bδˆρ δb + b δx sat x sat Thus, an error in the orbital position of a GPS satellite can compensated (i.e., Φ 0) if δb b δx sat x sat This is why orbit errors are often spoken of in terms of fractional errors (e.g., parts-per-billion) 9
Errors in IGS Orbit Products IGS Product Latency Nominal Nominal Name Accuracy (cm) Accuracy (ppb) Broadcast 0 100 40 Ultra-Rapid 0 9 h 3 5 0.1 0.2 Rapid 17 41 h 2.5 0.09 Final 12 18 d 2.5 0.09 1 ppb orbit error 1 mm baseline error @ 1000 km 10
Error History for IGS Orbit Products 11
Double Difference Single difference for sites p and q to satellite r: φ r pq (t) = φr p (t) φr q (t) = 1 [ ρ r λ p (t) ρ r q (t)] φ p + φ q + N p r N q r +f [δ q (t) δ p (t)] Double difference formed from two single differences to two satellites: φ rs pq (t) = φr pq (t) φs pq (t) = 1 λ [ ρ r p (t) ρ r q (t) ρs p (t) + ρs q (t)] + N r p N r q N s p + N s q = 1 λ ρrs rs pq (t) + Npq All site-only- and satellite-only-dependent terms have canceled 12
Working with Double Differences Some advantages: 1. No clocks, initial phases 2. Simple model Some disadvantages: 1. Bookkeeping 2. Estimating ambiguities 3. Further loss of sensitivity 4. Unable to reconstruct one-way phases 13
Triple Difference Double difference formed from two single differences to two satellites: φ rs pq (t) = φr pq (t) φs pq (t) = 1 λ [ ρ r p (t) ρ r q (t) ρs p (t) + ρs q (t)] + N r p N r q N s p + N s q = 1 λ ρrs rs pq (t) + Npq Triple difference formed from time differences of double difference φ rs pq (t j, t k ) = φ rs pq (t k) φ rs pq (t j) 1 [ ρ r λ p (t) ρ r q (t) ρs p (t) + ρs q (t)] (t k t j ) Useful for cycle-slip detection 14
GPS Data Analysis: Outline 1. Perform clock solution (a) Estimate receiver clock errors (b) Estimate site position if no prior better (< 1 m) is available 15
Clock solution showing clock steering Clock Error (km) 300 250 200 150 100 50 0 0.0 0.2 0.4 0.6 0.8 1.0 300 250 Clock Error (km) 200 150 100 50 0 0.00 0.02 0.04 0.06 Epoch (fraction of day) 0.08 0.10 300 km 1 msec 16
Pseudorange Residuals LC Pseudorange Residual (m) 40 30 20 10 0-10 -20-30 -40 0.0 0.2 0.4 0.6 0.8 Epoch (fraction of day) 1.0 RMS = 5.4 m 17
GPS Data Analysis: Outline 1. Perform clock solution (a) Estimate receiver clock errors (b) Estimate site position if no prior better (< 1 m) is available 2. Using best model, repair cycle slips 18
Data Processing: Cycle Slips We have assumed that the integer bias N in the phase model is not a function of time In fact, cycles slips can occur in systems such as GPS receivers that employ phase-locked loops A phase-locked loop is a feedback system that uses the phase difference of the received and internal signals to modify the internal frequency in an attempt to match frequency variations of the incoming signal and to keep the signal locked Such systems are highly sensitive to SNR to be able to separate the signal (which should be locked onto) from the noise (which should be ignored) Also, if the phase varies rapidly in a noise-like manner (e.g., ionosphere, multipath), the receiver can lose lock 19
L1 DD Residuals (b = 10 km) Units: cycles/hours 20
L1 DD Residuals (b = 10 km) with cycle slip Units: cycles/hours 21
Rectifying cycle slips These days, cycle slip detection and removal is done using automatic processing In old days, done by hand One useful data combination is the Melbourne- Wübbena wide lane: mw = φ 1 φ 2 1 c (f 1 f 2 ) (f 1 + f 2 ) [f 1R 1 + f 2 R 2 ] (1) 22
Melbourne-Wübbena Wide-Lane -23500000-24000000 MW_WL_07_cycles -24500000-25000000 MW_WL_07_cycles Cycle slip -25500000 18.0 19.0 20.0 21.0 Time_Hrs 22.0 23.0 24.0 From Herring, Principles of the Global Positioning System 23
Next time Regional solutions using fixed orbits Orbit integration Orbit parameter estimation The IGS network 24