EESC Geodesy with the Global Positioning System. Class 7: Relative Positioning using Carrier-Beat Phase

Transcription

1 EESC 9945 Geodesy with the Global Positioning System Class 7: Relative Positioning using Carrier-Beat Phase

2 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

3 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

4 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

5 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

6 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

7 Single-difference geometry: Short b limit Range Difference (km) Exact Short BL Approx Baseline Length (km) 4000 Difference is km 7

8 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

9 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

10 Errors in IGS Orbit Products IGS Product Latency Nominal Nominal Name Accuracy (cm) Accuracy (ppb) Broadcast Ultra-Rapid 0 9 h Rapid h Final d ppb orbit error 1 mm baseline 1000 km 10

11 Error History for IGS Orbit Products 11

12 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

13 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

14 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

15 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

16 Clock solution showing clock steering Clock Error (km) Clock Error (km) Epoch (fraction of day) km 1 msec 16

17 Pseudorange Residuals LC Pseudorange Residual (m) Epoch (fraction of day) 1.0 RMS = 5.4 m 17

18 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

19 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

20 L1 DD Residuals (b = 10 km) Units: cycles/hours 20

21 L1 DD Residuals (b = 10 km) with cycle slip Units: cycles/hours 21

22 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

23 Melbourne-Wübbena Wide-Lane MW_WL_07_cycles MW_WL_07_cycles Cycle slip Time_Hrs From Herring, Principles of the Global Positioning System 23

24 Next time Regional solutions using fixed orbits Orbit integration Orbit parameter estimation The IGS network 24