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Positioning with Carrier Phase

Positioning with Carrier Phase. Carrier Phase Tracking. Integer Ambiguity. Integer Ambiguity. Carrier Phase. Carrier Phase. At Lock-on. At a Later Epoch. Carrier Phase. Where e is measurement noise, ~mm GPS carrier phase potentially can achieve mm positioning accuracy

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Positioning with Carrier Phase

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  1. Positioning with Carrier Phase

  2. Carrier Phase Tracking Integer Ambiguity Integer Ambiguity Carrier Phase Carrier Phase At Lock-on At a Later Epoch

  3. Carrier Phase • Where e is measurement noise, ~mm • GPS carrier phase potentially can achieve mm positioning accuracy • Depends on if other errors can be removed

  4. Short Baselines • Ionospheric delay and Tropospheric delays are similar at both stations • They can be cancelled out by differencing • Orbit error effects are small • Common site movements (tides) • Integer nature of DD ambiguity reserved • Typical within 20km at mid latitudes • In high and low latitudes (i.e. in Hong Kong) • The distance is reduce to a few km

  5. Example in Hong Kong

  6. Double Difference • Double difference removes any common errors • Cancel out clock errors • Reduce ionospheric and tropospheric delays • Reduce orbit error • Measurement noise increases

  7. Typical Double Difference – between stations and satellites It is a function of receiver positions (two sites), satellite positions, and ambiguity (short baselines)

  8. Select satellite pairs • Base satellite • Observe 1, 2, 3,….satellites • Form double difference: 1-2, 1-3 • Based satellite: 1 • Sequential difference • 1-2, 2-3,……….

  9. Linear Model for double Difference • For relative positioning, point i is reference station with known coordinates • The troposphere and ionosphere errors are corrected (or ignored); orbital errors are ignored or precise orbits used; the coordinates of point j are to be determined • with the approximate coordinates of point j the linearized observation equation is

  10. If s satellites are tracked continuously for one epoch there are totally (s-1) independent observation equations and (2+s) unknown parameters (3 coordinates plus s-1 ambiguities) • k epochs: observation equations = k(s-1) • Observation equations read l + v= Ax +By with x coordinates corrections, y ambiguity unknown

  11. Variance-Covariance Matrix • Assume 3 satellites: 1,2,3=>2 double diff. • 2-1, 3-1 (satellite 1 is the base satellite)

  12. The V-C • In general with a base satellite • Assume the observations at different epochs independent S= diag{C, C, …, C} • GPS errors are correlated in time (i.e. tropospheric error) • Estimation is too optimal (variances are too small)

  13. Least Squares Solutions • Linear Model (k epoch, k(s-1) observables) • Ax+By=DZ=L+v • Covariance matrix: S= diag{C, C, …, C} • Least Squares Solution • Z=(ATS-1A)-1ATS-1L • Cov(Z)= (ATS-1A)-1

  14. ill-condition Problem • The partials of matrix A • Function of satellite and receiver positions • When the observation time is too short • Rows of matrix A similar • Ill condition equation • Solution • Extend observation time • Satellite position changes • Parameters become solvable

  15. Least Square Solution –ambiguity float solution • The LS solution for x and y obtained • Due to various sources of errors affecting phase observations the estimated ambiguities will not be integers  called floating DD solution

  16. Ambiguity Fixed Solution • To enhance solution one can enforce the ambiguities to integers  called fixed DD solution • Real numbers y => integer numbers y* • The fixed DD solution (l – By*)+ v = Ax*  x* • How to fix the ambiguity  ambiguity resolution techniques

  17. Ambiguity resolution techniques • Search techniques • If yi = 0.5 • Then yi*={-4, -3, -2, -1, 0, 1, 2, 3, 4, 5} • Test which one is correct • Suppose 8 satellites observed, 7 DD ambiguities • Each ambiguity has 10 possibilities • Total: 107 combinations • Which one is right? • Check them all!

  18. Ambiguity Validation • For all possible combinations (i.e. 107) • y1*, y2*,……. are integer combinations • Using (l – Byi*)+ v = Ax  xi, vi • Calculate sum of residuals • viTS-1vi, i=1,2,……….

  19. Ambiguity Validation • Rearrange all possible combinations in the order of vTS-1v (from smallest to largest) • y1*, y2*,……. • Then: y1* is the possible one, and xi is possible solutions • Under the condition: • v1TS-1v1 is ‘significantly’ smaller than • viTS-1vi, i=2, 3,……….

  20. What is ‘significantly’ smaller? • Ratio: • v2TS-1v2/ v1TS-1v1 > k, k=2 or 3 • Hypnosis Tests • H0: v1TS-1v1 < v2TS-1v2 • Or H0: v2TS-1v2/ v1TS-1v1 > 1 • Assume vTS-1v ~ distribution • Assume v2TS-1v2/ v1TS-1v1 ~ F distribution • Success rate

  21. How to compute quickly? • Large number of combinations needed to be tested (i.e. 107) • Reduce computation load

  22. How to compute quickly? • Let z = L(y*-y): a vector • The sum of residuals: • Do not need to calculate all the sum, if • viTS-1vi is too big already when add a few terms • Stop computation, and remove this combination

  23. How to compute quickly? • Reduce the size of searching • Combine pseudorange and carrier phase • Estimate the error ellipsoid of ambiguities • Only search inside k times of error ellipsoid Error ellipsoid

  24. Final Solution • If pass the validation: • Ambiguity fixed solution: y*, x*, S(x*) • If fix to wrong ambiguity, solution will be wrong!! • If not pass the validation: • Ambiguity float solution: y, x, S(x) • Observation time is too short • Other errors are too big • You may fix some ambiguities and leave others float • For short baselines: • Normally require ambiguity fixed solution • Extend observation time • Re-surveying

  25. Cycle Slips • Causes • Discontinuity of Satellite Signal tracking (Loss of Lock) • Signal to noise ratio too small • Receiver PLL error • Results • additional new ambiguities

  26. Effect of Cycle slips DN- an unknown integer

  27. Repair of Cycle Slips • Find where has a jump • Detection • Find the size of DN • Repair

  28. Repair of Cycle Slips • Introduce a new ambiguity • When there are a few cycle slips • Sufficient data before and after cycle slips • Editing Cycle Slips • polynomial fitting • triple differencing • double differencing residuals (outlier detection in LS)

  29. Positioning Methods • Static • occupy a position > 30 min • Fast Static • Occupy a position < 10 min • Kinematic • Determine position while the receiver is moving

  30. Data Processing (Static) • Pre-processing • approximate coordinates from pseudorange • editing cycle slips • Elevation angle not too low (i.e. 15 degrees) • Remove small pieces • least squares to solve for all parameters (float solution) • station coordinates • Ambiguities • Check cycle slips again, then repeat previous step

  31. Data Processing (Static) • Ambiguity fixing • Determine the integer values of ambiguities • Search method • Check the size of residuals • Ambiguity Fix solution • Input the fixed ambiguity as known values • Solve for station coordinates again

  32. Single or dual frequency? • Assuming ionospheric delay removed • Single frequency • Better precision • Receiver is cheaper • Long time for ambiguity resolution • Dual frequencies • Fast ambiguity resolution • More reliable

  33. RINEX Format • Receiver Independent Exchange Format • Two File Format • Ephemeris • Measurements • Each File • Title Section • Data Section

  34. NAVIGATION MESSAGE FILE - HEADER SECTION DESCRIPTION .......................................................................... HEADER LABEL DESCRIPTION FORMAT (Columns 61-80) ........................................................................... RINEX VERSION / TYPE Format version (2) I6,14X, File type ('W' for Navigation data) A1,19X ........................................................................... *COMMENT Comment lines (2) I6,14X ........................................................................... *ION ALPHA Ionosphere parameters A0-A3 of almanac 2X,4D12.4 (page 18 of subframe 4 ........................................................................... *ION BETA Ionosphere parameters B0-B3 of almanac 2X,4D12.4 ........................................................................... *DELTA UTC: A0,A1,T,W Almanac parameters to compute time in 3X,2D l9.12, UTC (page 18 of subframe 4) 219 AO,Al: term of polynomial T : reference time for UTC data W : UTC reference week number ........................................................................... *LEAP SECONDS Delta time due to leap seconds I6 ........................................ .................................. END OF HEADER Last record in the header section. 60X

  35. NAVIGATION MESSAGE FILE - DATA RECORD DESCRIPTION --------------------------------------------------------------------------- PRN / EPOCH / SV CLK - Satellite PRN number I2,513, F5.1,3D19.12 - Epoch: TOC - Time of Clock year (2 digits) month day hour minute second - SV clock bias (seconds) - Sv clock drift (sec/sec) - SV clock drift rate (sec/sec2) ........................................................................... BROADCAST ORBIT - 1 - AODE (age of data ephemeris) 3X.4Dl9.12 - Crs (meters) - [EQN "Delta n"] (radians/sec) - No (radians) ........................................................................... BROADCAST ORBIT - 2 - Cuc (radians) 3X,4D19.12 - Eccentricity - Cus (radians) - Al/2 (meter 1/2) ...........................................................................

  36. BROADCAST ORBIT - 3 - TOE Time of Ephemeris 3X,4D19.12 (seconds into GPS week) - Cuc (radians) - [EQN "Omega sub o"] (radians) - Cis (radians) ........................................................................... BROADCAST ORBIT - 4 - io (radians) 3X,4D19.12 - Crc (meters) - [EGN "omega:] (radians) - [EQN "omega dot"] (radians/sec) ........................................................................... BROADCAST ORBIT - 5 - IDOT (radians/sec) 3X.4D19.12 - Codes on L2 channel - GPS Week # (to go with TOE) - L2 P data flag ........................................................................... BROADCAST ORBIT - 6 - SV accuracy 3X,4D19.12 - SV health (MSB only) - TGD (seconds) - AODC (seconds) ........................................................................... BROADCAST ORBIT - 7 - Transmission time of message 3X,4D19.12 (seconds into GPS week, derived e.g. from Z-count in Hand Over Word (HOW) - spare - spare - spare

  37. 2 NAVIGATION DATA RINEX VERSION / TYPE XXRINEXN v2.0 IGGP 12-SEP-90 15:22 PGM / RUN BY / DATE EXAMPLE OF VERSION 2 FORMAT COMMENT .1676D-07 .2235D-07 -.1192D-06 -.1192D-06 ION ALPHA .1208D+06 .1310D+06 -.1310D+06 -.1966D+06 ION BETA .133179128170D-06 .107469588780D-12 552960 551 DELTA-UTC: AO,A1,T,W 6 LEAP SECONDS END OF HEADER 6 90 8 2 17 51 44.0 -.839701389031D-03 -.165982783074D-10 .000000000000D+00 .910000000000D+02 .934062500000D+02 .116040547840D-08 .162092304801D+00 .48410l474285D-05 .626740418375D-02 .652112066746D-05 .515365489006D+04 .09904000000OD+06 -.242143869400D-07 .329237003460D+00 -.596046447754D-07 .111541663136D+01 .326593750000D+03 .206958726335D+Ol -.638312302555D-08 .307155651409D-09 .OOOOOOOOOOOOD+00 .551OO0000000D+03 .000000000000D+00 .000000000000D+00 .000000000000D+00 .000000000000D+00 .910000000000D+02 .406800000000D+06

  38. 2 OBSERVATION DATA M (MIXED) RINEX VERSION / TYPE BLANK OR G = GPS, R = GLONASS, T = TRANSIT, M = MIXED COMMENT XXRINEXO V9.9 AIUB 12-SEP-90 12:43 PGM / RUN BY / DATE EXAMPLE OF A MIXED RINEX FILE COMMENT A 9080 MARKER NAME 9080.1.34 MARKER NUMBER BILL SMITH ABC INSTITUTE OBSERVER / AGENCY X1234A123 XX ZZZ REC # / TYPE / VERS 234 YY ANT # / TYPE 4375274. 587466. 4589095. APPROX POSITION XYZ .9030 .0000 .0000 ANTENNA: DELTA N/E/W 1 1 WAVELENGTH FACT L1/2 1 2 6 G14 G15 G16 G17 G18 G19 WAVELENGTH FACT L1/2 4 P1 L1 L2 P2 # / TYPES OF OBSERV 18 INTERVAL 1990 3 24 13 10 36.000000 TIME OF FIRST OBS Time of Last OBS Number of Sat PRN/No. of OBS END OF HEADER

  39. 90 3 24 13 10 36.0000000 0 3Gl2G 9G 6 -.123456789 23629347.915 .300 8 -.353 23629364.158 20891534.648 .120 9 -.358 20891541.292 20607600.189 .430 9 .394 20607605.848 90 3 24 13 10 50.0000000 4 3 1 2 2 G 9 G12 WAVELENGTH FACT L1/2 *** WAVELENGTH FACTOR CHANGED FOR 2 SATELLITES *** COMMENT COMMENT 90 3 24 13 1O 54.0000000 0 5Gl2G 9G 66R21R22 -.123456789 23619095.450 -53875.632 8 -41981.375 23619112.008 20886075.667 -28W.027 9 -22354.535 20886082.101 20611072.689 18247.789 9 14219.770 20611078.410 21345678.576 12345.567 5 22123456.789 23456.789 5 90 3 24 13 11 0.0000000 2 4 1 *** FROM NOW ON KINEMATIC DATA! *** COMMENT 90 3 24 13 11 48.000000 0 4Gl6G12G 9G 6 -.123456789 21110991.756 16119.980 7 12560.51O 21110998.441 23588424.398 -215050.557 6 -167571.734 23588439.570 20869878.790 -113803.187 8 -88677.926 20621649.276 20621643.727 73797.462 7 57505.177 20621649.276 3 4 A 9080 MARKER NAME 9080.1.34 MARKER NUMBER .9030 .0000 .0000 ANTENNA: DELTA H/E/N --> THIS IS THE START OF A NEW SITE <-- COMMENT

  40. Network Solution (1) • n > 2 stations observed at same time • Fix one station coordinates (reference station) • Solve for n-1 station coordinates • Form n-1 independent baselines

  41. Form the independent baselines • Each baseline has to include a new station • Based on shortest distance • Based on one reference station • User selection

  42. Network solution • Principle is the same as single baseline • Repair cycle slips: baseline by baseline • Algorithm: Least Squares • More parameters • Difficult to manipulate data in programming • Different number of satellite observed • Data missing • …………

  43. How to Form Double Differencing • For each station i, form undifferecing observation equations • Linearize the equation based on the given approximate coordinates of the station • l +v = aidxi+bidyi+cidzi+diNiK

  44. How to Form Double Differencing • The estimated parameters for the whole network: • X=(dx1, dy1, dz1, ….dxn, dyn, dzn, N11,N12,..N1m, N21,..N2m, …Nn1,…Nnm)T • Put ai, bi, ci, di at the corresponding location of A matrix • For all stations at the same epoch • The undifference equation: • AX=L+v, with the S(v) =Is0

  45. How to Form Double Differencing • Form DD multiplier matrix D • For example: • 3 stations • 3 satellites observed by each station • Undifference equations: 9 • Two independent baselines • Each based line 2 independent DD equations • Total 4 independent DD equations

  46. How to Form Double Differencing • Option 1: • Station: 1-2, 1-3 • Satellite: 1-2, 1-3

  47. How to Form Double Differencing • Option 2: • Station: 1-2, 2-3 • Satellite: 1-2, 2-3

  48. How to Form Double Differencing • Form Undifference Equations: • AX=L+v, with the S(v+) =Is0 • Form DD multiplier matrix D • Based on selected baselines and satellite pairs • The Final DD Observation Equation • DAX=DL+Dv, S(Dv)=DTDs0

  49. Reference Station and Reference Satellite • The Final DD Observation Equation • DAX=DL+Dv, S(Dv)=DTDs0 (1) • The equation does not have a solution • Need Constraints: • Reference station ref: dXref=0, s=0 (a small number) (2) • Reference satellite k: Nrk = N0, s=0 (a small number) (3) • Solve for X using (1), (2), (3) (Least Squares) => float solution • Try to fix ambiguity => fix solution

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