1. Civil and Environmental Engineering and Geodetic Science Part IV TYPES OF GPS OBSERVABLE AND METHODS OF THEIR PROCESSING GS608

2. Civil and Environmental Engineering and Geodetic Science GPS Basic GPS Observables Pseudoranges precise/protected P1, P2 codes (Y-code under AS) - available only to the military users clear/acquisition C/A code - available to the civilian users Carrier phases L1, L2 phases, used mainly in geodesy and surveying Range-rate (Doppler)

3. Civil and Environmental Engineering and Geodetic Science Pseudoranges - geometric range between the transmitter and the receiver, distorted by the lack of synchronization between satellite and receiver clocks, and the propagation media time difference recovered from the measured time difference between the instant of transmission and the epoch of reception. good as 20 cm P-code pseudoranges can be as good as 20 cm or less, while the L 1 C/A code range noise level reaches even a meter or more Basic GPS Observables

4. Civil and Environmental Engineering and Geodetic Science Carrier phase - a difference between the phases of a carrier signal received from a spacecraft and a reference signal generated by the receiver’s internal oscillator unknown integer ambiguity, N contains the unknown integer ambiguity, N, i.e., the number of phase cycles at the starting epoch that remains constant as long as the tracking is continuous cycle slip loss of lock phase cycle slip or loss of lock introduces a new ambiguity unknown. noise typical noise of phase measurements is generally of the order of a few millimeters or less Basic GPS observables

5. Civil and Environmental Engineering and Geodetic Science Ambiguity: the initial bias in a carrier-phase observation of an arbitrary number of cycles between the satellite and the receiver; the uncertainty of the number of complete cycles a receiver is attempting to count. The initial phase measurement made when a GPS receiver first locks onto a satellite signal is ambiguous by an integer number of cycles since the receiver has no way of knowing when the carrier wave left the satellite. This ambiguity remains constant as long as the receiver remains locked onto the satellite signal and is resolved when the carrier-phase data are processed. If wavelength is known, the distance to a satellite can be computed once the total number of cycles is established via carrier-phase processing.

6. Civil and Environmental Engineering and Geodetic Science Doppler Effect on GPS observable The Doppler equation for electromagnetic wave, where f r and f s are received and transmitted frequencies In case of moving emitter or moving receiver the receiver frequency is Doppler shifted The difference between the receiver and emitted frequencies is proportional to the radial velocity v r of the emitter with respect to the receiver

7. Civil and Environmental Engineering and Geodetic Science Doppler Effect on GPS observable For GPS satellites orbiting with the mean velocity of 3.9 km/s, assuming stationary receiver, neglecting Earth rotation, the maximum radial velocity 0.9 km/s is at horizon and is zero at the epoch of closest approach For 1.5 GHz frequency the Doppler shift is 4.5·10 3 Hz we get: 4.5 cycles phase change after 1 millisecond, or change in the range by 90 cm

8. Civil and Environmental Engineering and Geodetic Science Phase Observable Instantaneous circular frequency f is a derivative of the phase with respect to time By integrating frequency between two time epochs the signal’s phase results Assuming constant frequency, setting the initial phase  (t 0 ) to zero, and taking into account the signal travel time t tr corresponding to the satellite-receiver distance , we get

9. Civil and Environmental Engineering and Geodetic Science Pseudorange Observable - geometric range to the satellite t r, t s – time of signal reception at the receiver and the signal transmit at by the satellite (both are subject to time errors, i.e., offsets from the true GPS time) dt r,dt s – receiver and transmitter (satellite) clock corrections (errors) c – speed of light e – random errors (white noise)

10. Civil and Environmental Engineering and Geodetic Science Taking into account all error sources (and also simplifying some terms), we arrive at the final observation equations of the following form (for pseudorange and phase observable)

11. Civil and Environmental Engineering and Geodetic Science Basic GPS Observable 1/4 and The primary unknowns are Xi, Yi, Zi – coordinates of the user (receiver) 1,2 stand for frequency on L1 and L2, respectively i –denotes the receiver, while k denotes the satellite

12. Civil and Environmental Engineering and Geodetic Science Basic GPS Observable 2/4 1  19 cm and 2  24 cm are wavelengths of L 1 and L 2 phases Using our earlier notation for the ionospheric correction we have: i.e., in our earlier notation

13. Civil and Environmental Engineering and Geodetic Science Basic GPS Observables 3/4 dt i - the i-th receiver clock error dt k - the k-th transmitter (satellite) clock error f 1, f 2 - carrier frequencies c - the vacuum speed of light b i,1, b i,2, b i,3 - interchannel bias terms for receiver i that represent the possible time non-synchronization of the four measurements multipath on phases and ranges

14. Civil and Environmental Engineering and Geodetic Science The above equations are non-linear and require linearization (Taylor series expansion) in order to be solved for the unknown receiver positions and (possibly) for other nuisance unknowns, such as receiver clock correction Since we normally have more observations than the unknowns, we have a redundancy in the observation system, which must consequently be solved by the Least Squares Adjustment technique Secondary (nuisance) parameters, or unknowns in the above equations are satellite and clock errors, troposperic and ionospheric errors, multipath, interchannel biases and integer ambiguities. These are usually removed by differential GPS processing or by a proper empirical model (for example troposphere), and processing of a dual frequency signal (ionosphere).

15. Civil and Environmental Engineering and Geodetic Science Basic GPS Observable 4/4 Assume that ionospheric effect is removed from the equation by applying the model provided by the navigation message Assume that tropospheric effect is removed from the equation by estimating the dry+wet effect based on the tropospheric model (e.g., by Saastamoinen, Goad and Goodman, Chao, Lanyi) Satellite clock correction is also applied based on the navigation message Multipath and interchannel bias are neglected The resulting range equation :  corrected observable Four unknowns: 3 receiver coordinates and receiver clock correction

16. Civil and Environmental Engineering and Geodetic Science Instantaneous Doppler Observed Doppler shift scaled to range rate; time derivative of the phase or pseudorange observation equation Instantaneous radial velocity between the satellite j and the receiver i, and v is satellite tangential velocity, see a slide “Doppler effect on GPS observable” (corresponds to in the notation used in figure 6.3)

17. Civil and Environmental Engineering and Geodetic Science Instantaneous Doppler Used primarily to support velocity estimation Can be used for point positioning Are instantaneous position vector of the satellite, and the unknown receiver position vector; correspond to r s and r p in the notation used in Figure 6.3 dot denotes time derivative

18. Civil and Environmental Engineering and Geodetic Science Integrated Doppler Observable The frequency difference between the nominal (sent) signal and the locally generated replica f g can be used to recover pseudorange difference through so-called integrated Doppler count (more accurate than instantaneous Doppler): Observed: N jk Where  ik and  ij are the distances from the receiver i to the position of the satellite at epochs k and j.

19. Civil and Environmental Engineering and Geodetic Science

20. R 1 =  c  dt +  f 1 2 + T + e R1 R 2 =  c  dt  f 2 2 + T + e R2   1 =  f 1 2 + T +        2 =  f 2 2 + T +           - integer ambiguities R  pseudorange I / f 2 - ionospheric effect  phase T - tropospheric effect   geometric range e R1, e R2,      white noise  wavelength Basic GPS observables (simplified form)

21. Civil and Environmental Engineering and Geodetic Science GPS Positioning (point positioning with pseudoranges) tt signal transmitted signal received range,  = c  t    

22. Civil and Environmental Engineering and Geodetic Science

23. Point Positioning with Pseudoranges Assume that ionospheric effect is removed from the equation by applying the model provided by the navigation message Assume that tropospheric effect is removed from the equation by estimating the dry+wet effect based on the tropospheric model (e.g., by Saastamoinen, Goad and Goodman, Chao, Lanyi) Satellite clock correction is also applied based on the navigation message Multipath and interchannel bias are neglected The resulting equation : corrected observable 

24. Civil and Environmental Engineering and Geodetic Science Point Positioning with Pseudoranges Linearized observation equation Geometric distance obtained from known satellite coordinates (broadcast ephemeris) and approximated station coordinates Objective: drive (“observed – computed” term) to zero by iterating the solution from the sufficient number of satellites (see next slide)

25. Civil and Environmental Engineering and Geodetic Science Point Positioning with Pseudoranges Minimum of four independent observations to four satellites k, l, m, n is needed to solve for station i coordinates and the receiver clock correction Iterations: reset station coordinates, compute better approximation of the geometric range Solve again until left hand side of the above system is driven to zero

26. Civil and Environmental Engineering and Geodetic Science In the case of multiple epochs of observation (or more than 4 satellites)  Least Squares Adjustment problem! Number of unknowns: 3 coordinates + n receiver clock error terms, each corresponding to a separate epoch of observation 1 to n

27. Civil and Environmental Engineering and Geodetic Science Dilution of Precision (DOP) Accuracy of GPS positioning depends on: the accuracy of the range observables the geometric configuration of the satellites used (reflected in the design matrix A) the relation between the measurement error,  obs, and the positioning error:  pos = DOP  obs DOP is called dilution of precision for 3D positioning, PDOP (position dilution of precision), is defined as a square root of a sum of the diagonal elements of the normal matrix (A T A) -1 (corresponding to x, y and z unknowns) In differential GPS we use RDOP (relative DOP) term

28. Civil and Environmental Engineering and Geodetic Science Dilution of Precision PDOP is interpreted as the reciprocal value of the volume of tetrahedron that is formed from the satellite and user positions Receiver Good PDOP (usually < 7) Bad PDOP Position error  p =  r PDOP, where  r is the observation error (or standard deviation)

29. Civil and Environmental Engineering and Geodetic Science Dilution of Precision The observation standard deviation, denoted as  r or  obs is the number that best describes the quality of the pseudorange (or phase) observation, thus is is about 0.2 – 1.0 m for P-code range and reaches a few meters for the C/A-code pseudorange. Thus, DOP is a geometric factor that amplifies the single range observation error to show the factual positioning accuracy obtained from multiple observations It is very important to use the right numbers for  r to properly describe the factual quality of of your measurements. However, most of the time, these values are pre-defined within the GPS processing software (remember that Geomatics Office never prompted you about the observation error (or standard deviation)) and user has no way to manipulate that. This values are derived as average for a particular class of receivers (and it works well for most applications!)

30. Civil and Environmental Engineering and Geodetic Science Dilution of Precision DOP concept is of most interest to navigation. If a four channel receiver is used, the best four-satellite configuration will be used automatically based on the lowest DOP (however, most of modern receivers have more than 4 channels) This is also an important issue for differential GPS, as both stations must use the same satellites (actually with the current full constellation the common observability is not a problematic issue, even for very long baselines) DOP is not that crucial for surveying results, where multiple (redundant) satellites are used, and where the Least Squares Adjustment is used to arrive at the most optimal solution However, DOP is very important in the surveying planning and control (especially for kinematic and fast static modes), where the best observability window can be selected based on the highest number of satellites and the best geometry (lowest DOP); check the Quick Plan option under Utilities menu in Geomatics Office

31. Civil and Environmental Engineering and Geodetic Science Differential GPS (DGPS) DGPS is applied in geodesy and surveying (for the highest accuracy, cm-level) as well as in GIS-type of data collection (sub meter or less accuracy required) Data collected simultaneously by two stations (one with known location) can be processed in a differential mode, by differing respective observables from both stations The user can set up his own base (reference) station for DGPS or use differential services provided by, for example, Coast Guard, which provides differential correction to reduce the pseudorange error in the user’s observable

32. Civil and Environmental Engineering and Geodetic Science Differential GPS (DGPS) So, DGPS can be performed by collecting data (phase and/or range) by two simultaneously tracking receivers, where one of them is placed on the known location These data are then processed together in a single adjustment to provide high-accuracy positioning information Or, one can use DGPS services that provide correction terms, which account for error sources due to atmosphere and SA (when activated) in pseudorange measurement; this correction is applied by the receiver to the observed pseudorange, which is subsequently used for navigation/positioning

33. Civil and Environmental Engineering and Geodetic Science By differencing observables with respect to simultaneously tracking receivers, satellites and time epochs, a significant reduction of errors affecting the observables due to: satellite and receiver clock biases, atmospheric as well as SA effects (for short baselines), inter-channel biases is achieved DGPS: Objectives and Benefits

34. Civil and Environmental Engineering and Geodetic Science Differential GPS Selective Availability (SA), if it is on Satellite clock and orbit errors Atmospheric effects (for short baselines) Using data from two receivers observing the same satellite simultaneously removes (or significantly decreases) common errors, including: Base station with known location Unknown position Single difference mode

35. Civil and Environmental Engineering and Geodetic Science Differential GPS Receiver clock errors Atmospheric effects (ionosphere, troposphere) Receiver interchannel bias Using two satellites in the differencing process, further removes common errors such as: Base station with known location Unknown position Double difference mode

36. Civil and Environmental Engineering and Geodetic Science

37. Consider two stations i and j observing L1 pseudorange to the same two GPS satellites k and l:

38. Civil and Environmental Engineering and Geodetic Science DGPS Concept The single-differenced (SD) measurement is obtained by differencing two observables of the satellite k, tracked simultaneously by two stations i and j: It significantly reduces the atmospheric errors and removes the satellite clock and orbital errors; differential effects are still there (like iono, tropo and multipath, and the difference between the clock errors between the receivers) In the actual data processing some differential errors (tropo) can be neglected for short baselines, while remaining differential ionospheric, differential clock error, and interchannel biases might be estimated (if possible)

39. Civil and Environmental Engineering and Geodetic Science DGPS Concept By differencing one-way observables from two receivers, i and j, observing two satellites, k and l, or simply by differencing two single differences to satellites k and l, one arrives at the double-differenced (DD) measurement: In the actual data processing the differential tropospheric, ionospheric and multipath errors are neglected; the only unknowns are the station coordinates Double difference Two single differences

40. Civil and Environmental Engineering and Geodetic Science Note: the SD and DD equations were derived here for pseudorange observable, only as an example, because pseudorange equation is simpler (and shorter) than phase equation. SD and DD are most often used with phase observations Pseudorange observations are most often (but not only) used in navigation and point-positioning mode Or DGPS services are used to obtain the pseudorange correction (see the future notes for more info on DGPS services) in order to achieve sub-meter accuracy from pseudorange observations (which is otherwise in the order of a few meters)

41. Civil and Environmental Engineering and Geodetic Science Differential Phase Observations Double difference Two single differences Single difference ambiguity

42. Civil and Environmental Engineering and Geodetic Science Differential Phase Observations Double differenced (DD) mode is the most popular for phase data processing In DD the unknowns are station coordinates and the integer ambiguities In DD the differential atmospheric and multipath effects are very small and are neglected The achievable accuracy is cm-level for short baselines (below 10- 15 km); for longer distances, DD ionospheric-free combination is used (see the future notes for reference!) Single differencing is also frequently used, however, the problem there is non-integer ambiguity term (see previous slide), which does not provide such strong constraints into the solution as the integer ambiguity for DD

43. Civil and Environmental Engineering and Geodetic Science Triple Difference Observable Differencing two double differences, separated by the time interval dt provides triple-differenced measurement, that in case of phase observables effectively cancels the phase ambiguity biases, N 1 and N 2 In both equations, for short baselines, the differential effects are neglected and the station coordinates are the only unknowns

44. Civil and Environmental Engineering and Geodetic Science Note: Observed phases (in cycles) are converted to so-called phase ranges (in meters) by multiplying the raw phase by the respective wavelength of L1 or L2 signals  Thus, the units in the above equations are meters!  Positioning with phase ranges is much more accurate as compared to pseudoranges, but more complicated since integer ambiguities (such as DD ambiguities) must be fixed before the preciase positioning can be achieved  So called float solution (with ambiguities approximated by real numbers) is less accurate that the fixed solution  Triple difference (TD) equation does not contain ambiguities, but its noise level is higher as compared to SD or DD, so it is not recommended if the highest accuracy is expected

45. Civil and Environmental Engineering and Geodetic Science St. 1 St. 2 2 (base) 3 4 1 Positioning with phase observations: A Concept

46. Civil and Environmental Engineering and Geodetic Science Positioning with phase observations: A Concept Three double difference (based on four satellites) is a minimum to do DGPS with phase ranges after ambiguities have been fixed to their integer values Minimum of five simultaneously observed satellites is needed to resolve ambiguities Thus, ambiguities must be resolved first, then positioning step can be performed Ambiguities stay fixed and unchanged until cycle slip (CS) happens

47. Civil and Environmental Engineering and Geodetic Science Cycle Slips Sudden jump in the carrier phase observable by an integer number of cycles All observations after CS are shifted by the same integer amount Due to signal blockage (trees, buildings, bridges) Receiver malfunction (due to severe ionospheric distortion, multipath or high dynamics that pushes the signal beyond the receiver’s bandwidth) Interference Jamming (intentional interference) Consequently, the new ambiguities must be found

48. Civil and Environmental Engineering and Geodetic Science

49. Some useful linear combinations Created usually from double-differenced (DD) phase observations, derived as a linear combination of the phase observations on L1 and L2 frequencies Ion-free combination - eliminates ionospheric effects Widelane – its long wavelength of 86.2 cm supports fast ambiguity resolution

50. Civil and Environmental Engineering and Geodetic Science Useful linear combinations Ion-free combination The conditions applied to derive this linear combination are: sum of ionospheric effects on both frequencies multiplied by constants (to be determined) must be zero sum of the constants is 1, or one constant is set to 1 Used over long baselines (over 15 km), where DD differential ionospheric effect becomes significant

51. Civil and Environmental Engineering and Geodetic Science Ionosphere-free combination ionosphere-free phase measurement complication: ambiguity term is non-integer ! similarly, ionosphere-free pseudorange can be obtained

52. Civil and Environmental Engineering and Geodetic Science Useful linear combinations widelanewidelane where is in cycles the corresponding wavelength [meter] Simplifies ambiguity resolution, as for the long wavelength it is much easier as opposed to L1 or L2 phase observations Complication:  ionospheric effects are amplified by a factor of 77/60 (i.e., f 1 /f 2 ),  higher noise

53. Civil and Environmental Engineering and Geodetic Science Differential Global Positioning System (DGPS) services provide differential corrections to a GPS receiver in order to improve the accuracy of the navigation solution. DGPS corrections originate from a reference station at a known location. The receivers in these reference stations can estimate errors in the GPS because, unlike the general population of GPS receivers, they have an accurate knowledge of their position. As a result of applying DGPS corrections, the horizontal accuracy of the system can be improved from 10-15 m (100m under SA) (95% of the time) to better than 1m (95% of the time). Differential GPS (DGPS) Services

54. Civil and Environmental Engineering and Geodetic Science DGPS Services: A Concept There exists a reference station (or a network of stations) with a known location that can determine the range corrections (due to atmospheric, orbital and clock errors), and transmit them to the users equipped with proper radio modem. The DGPS reference station transmits pseudorange correction information for each satellite in view on a separate radio frequency carrier in real time. DGPS is normally limited to about 100 km separation between stations. Improves positioning with ranges by 100 times (to sub-meter level)

55. Civil and Environmental Engineering and Geodetic Science DGPS Services Starfix II OMNI-STAR (John E. Chance & Assoc, Inc.) U.S. Coast Guard Federal Aviation Administration GLOBAL SURVEYOR™ II NATIONAL, Natural Resources Canada Differential Global Positioning System (DGPS) Service, AMSA, Australia

56. Civil and Environmental Engineering and Geodetic Science Wide Area Differential GPS (WADGPS) Differential GPS operation over a wider region that employs a set of monitor stations spread out geographically, with a central control or monitor station. WADGPS uses geostationary satellites to transmit the corrections in real time (5-10 sec delay) to the remote users. For example: OMNISTAR, Differential Corrections Inc., WAAS (FAA-developed Wide Area Augmentation System)

57. Civil and Environmental Engineering and Geodetic Science Atmospheric layer A Schematic of the WAAS

58. Civil and Environmental Engineering and Geodetic Science The WAAS improves the accuracy, integrity, and availability of the basic GPS signals A WAAS-capable receiver can give you a position accuracy of better than three meters, 95 percent of the time This system should allow GPS to be used as a primary means of navigation for enroute travel and non-precision approaches in the U.S., as well as for Category I approaches to selected airports throughout the nation The wide area of coverage for this system includes the entire United States and some outlying areas such as Canada and Mexico. The Wide Area Augmentation System is currently under development and test prior to FAA certification for safety-of-flight applications. WAAS

59. Civil and Environmental Engineering and Geodetic Science Total correction estimation is accomplished by the use of one or more GPS "Base Stations" that measure the errors in the GPS pseudo-ranges and generate corrections. A "real-time" DGPS involves some type of wireless transmission system. VHF systems for short ranges (FM Broadcast) low frequency transmitters for medium ranges (Beacons) geostationary satellites (OmniSTAR) for coverage of entire continents. A GPS base station tracks all GPS satellites that are in view at its location. Given the precise surveyed location of the base station antenna, and the location in space of all GPS satellites at any time from the ephemeris data that is broadcast from all GPS satellites an expected range to each satellite can be computed for any time The difference between that computed range and the measured range is the range error.WADGPS

60. Civil and Environmental Engineering and Geodetic Science If that information can quickly be transmitted to other nearby users, they can use those values as corrections to their own measured GPS ranges to the same satellites. The range and range rate correction are generated The range correction is an absolute value, in meters, for a given satellite at a given time of day. The range-rate term is the rate that correction is changing, in meters per second. That allows GPS users to continue to use the "correction, plus the rate- of-change" for some period of time while waiting for a new message. In practice, OmniSTAR TM would allow about 12 seconds in the "age of correction" before the error from that term would cause a one-meter position error. OmniSTAR TM transmits a new correction message every two and a half seconds, so even if an occasional message is missed, the user's "age of data" is still well below 12 seconds. WADGPS

61. Civil and Environmental Engineering and Geodetic Science

62. OmniSTAR's unique "Virtual Base Station" technology generates corrections optimized for the user's location. OmniSTAR receivers output both high quality RTCM-SC104 (Radio Technical Commission for Maritime Services) Version 2 corrections and differentially corrected Lat/Long in NMEA format (National Marine Electronics Association).

63. Civil and Environmental Engineering and Geodetic Science

64. OmniSTAR receiver

65. Civil and Environmental Engineering and Geodetic Science Radio Modems