1 Principles of the Global Positioning System Lecture 05 YUAN Linguo Email: lgyuan@home.swjtu.edu.cn Dept. of Surveying Engineering, Southwest Jiaotong University Principles of the Global Positioning System 2005-3-25 2 Outline Review: ? Examined basics of GPS signal structure and how signal is tracked ? Looked at methods used to acquire satellites and start tracking Today we look at: ? Basic GPS observables ? Biases and noise ? Examine RINEX format and look at some “raw” data 2 Principles of the Global Positioning System 2005-3-25 3 Review ：： GPS Signal Summary Table Component Frequency [MHz] Ratio of fundamental frequency f o Wavelength [cm] Fundamental frequency f o 10.23 1 2932.6 L1 Carrier 1,575.42 154?f o 19.04 L2 Carrier 1,227.60 120?f o 24.45 L5 Carrier 1,176.45 115?f o 25.5 P-code 10.23 1 2932.6 C/A code 1.023 f o /10 29326 W-code 0.5115 f o /20 58651 Navigation message 50?10 -6 fo/204,600 N/A Principles of the Global Positioning System 2005-3-25 4 Basic GPS Observables Code pseudoranges ? precise/protected P1, P2 codes ? - available only to the military users ? clear/acquisition C/A code - available to the civilian users Phase pseuodranges ? L1, L2 phases, used mainly in geodesy and surveying Doppler data 3 Principles of the Global Positioning System 2005-3-25 5 Code pseudoranges ? When a GPS receiver measures the time offset it needs to apply to its replica of the code to reach maximum correlation with received signal, what is it measuring? ? It is measuring the time difference between when a signal was transmitted (based on satellite clock) and when it was received (based on receiver clock). ? If the satellite and receiver clocks were synchronized, this would be a measure of range ? Since they are not synchronized, it is called “pseudornage” Principles of the Global Positioning System 2005-3-25 6 Code pseudoranges Pseudorange: cttR S R ??= )( Where R is the pseudorange between receiver R and satellite S; t R is the receiver clock time, t S is the satellite transmit time; and c is the speed of light This expression can be related to the true range by introducing corrections to the clock times SSS RRR tttt ?+=?+= ττ τ R and τ S are true times; Δ t R and Δ t S are clock corrections 4 Principles of the Global Positioning System 2005-3-25 7 Code pseudoranges Substituting into the equation of the pseudorange yields ρ R S is true range, and the ionospheric and atmospheric terms are introduced because the propagation velocity is not c. [ ] N N delay cAtmospheri delay Ionspheric )( )()( S R S R S R S R S R S R AIcttR cttR ++????+= ????+?= ρ ττ Principles of the Global Positioning System 2005-3-25 8 Millisecond problem in C/A- code ?The C/A-code repeats every millisecond which corresponds to 300km in range. Since the satellites are distance of about 20,000km from the earth, C/A-code pseudoranges are ambiguous. How to resolve this problem? ?Introduce approximate (within some few hundred kilometers) position coordinates of the receiver in initial satellite acquisition. ?The maximum radial velocity for GPS satellites in the case of a stationary receiver is ≈ 0.9km/s, and the travel time of the satellite signal is about 0.07s. The correction term in Eq., thus, amounts to some 60m. ρ tttttttt SSSS R S ?+=?+== )()())(,(),( ρρρρρ  5 Principles of the Global Positioning System 2005-3-25 9 Notes of code pseudoranges ? The equation for the pseudorange uses the true range and corrections applied for propagation delays because the propagation velocity is not the in-vacuum value, c, 2.99792458x10 8 m/s ? To convert times to distance c is used and then corrections applied for the actual velocity not equaling c. In RINEX data files, pseudorange is given in distance units. ? The true range is related to the positions of the ground receiver and satellite. ? Also need to account for noise in measurements. P-code pseudoranges can be as good as 20 cm or less, while the L1 C/A code range noise level reaches even a meter or more . Principles of the Global Positioning System 2005-3-25 10 Phase pseudoranges ? 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 ? contains the unknown integer ambiguity, Nown , i.e., the number of phase cycles at the starting epoch that remains constant as long as the tracking is continuous ? phase cycle slip or loss of lock introduces a new ambiguity unknown. ? typical noise of phase measurements is generally of the order of a few millimeters or less 6 Principles of the Global Positioning System 2005-3-25 11 Phase pseudoranges ? 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 ?(t0) to zero, and taking into account the signal travel time tr corresponding to the satellite-receiver distance ρ, we get dt d f ? = ∫ = t t dtf 0 ? () ? ? ? ? ? ? ?=?= c tfttf tr ρ ? Principles of the Global Positioning System 2005-3-25 12 ? s (t) phase of received carrier with frequency f s ? R (t) phase of reconstructed carrier with frequency f R tffdtfdtf c fttt dtfdtf where tft c ftft R s RR sss R ss R RRcR sss c s ccRo cRRR s c sss )()()()( and errorsclock are and )( )( ,0,0 ,0, ,0 ,0 ?+?+?=?= ?=?= ?= ??= ρ ??? ?? ?? ?? ? ρ ? Phase pseudoranges 7 Principles of the Global Positioning System 2005-3-25 13 [m]in range phase is which )()( )()( 1 )( :h wavelengta is where c f denoting and phasebeat fractional initial thegIntroducin unknown isreceiver theand satellitebetween cycles of Nnumber integer initial the,epoch t initial at the measured is phase ofpart fractional only the since )()( :simplified becan equation thenegligible is 1.5GHz)f and y,instabilit oscillator theis 10/ (assuming 101.5 oforder at the differencefrequency theassuming 00 00 0 0 12 -3 R s R s R s R ss R R ss R Ndtdtc Ndtdt c t dtdtf c ft fdf Hz ??λλρ ?? λ ρ λ ? λ λ ? ρ ? ?++?+?=Φ ?++?+?= = ?+?= == ? ? Principles of the Global Positioning System 2005-3-25 14 ? R S (t)＝ ? R S (t 0 )+ N R S (t－ t 0 ) ， Φ R S (t)= ? R S (t)+N R S (t 0 ) or ? R S (t) = Φ R S (t) -N R S (t 0 t 1 earth T R t 0 t 2 ? R S (t 0 ) ? R S (t 1 ) ? R S (t 2 ) N R S (t 0 ) N R S (t 0 ) N R S (t 0 ) Geometrical interpretation of phase range 8 Principles of the Global Positioning System 2005-3-25 15 Precision of phase measurements ? Nominally phase can be measured to 1% of wavelength (~2mm L1 and ~2.4 mm L2) ? Again effected by multipath, ionospheric delays (~30m), atmospheric delays (3-30m). ? Since phase is more precise than range, more effects need to be carefully accounted for with phase. ? Precise and consistent definition of time of events is one the most critical areas ? In general, phase can be treated like range measurement with unknown offset due to cycles and offsets of oscillator phases. Principles of the Global Positioning System 2005-3-25 16 P I f T c dt dt b M e P I f T c dt dt b M e i k i k i k i k i k ii k i k i k i k i k i k i k ii k i k , ,,, , ,,, () () 1 211 2 322 1 2 2 2 =+++ ? ++ + =+++ ? ++ + ρ ρ ( ) ( ) Φ Φ i k i k i k i k i k i kk ii k i k i k i k i k i k i k i k i k ii k i k I f TNcdtdt m I f TNcdtdtb m ,, , , , () () 1 1 1101 2 2 21202 1 2 0 2 2 0 =?++ + ? + ? + + =?++ + ? ++ ? + + ρλ λ? ε ρλ λ? ε ()()()[ ] 222 0, i k i k i kk i ZZYYXXsqrt ?+?+?=ρ 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 Basic GPS Observables 9 Principles of the Global Positioning System 2005-3-25 17 PP i k i k ,, , 12 ? pseudoranges measured between station i and satellite k on L1 and L2 ΦΦ i k i k ,, , 12 ?phase ranges measured between station i and satellite k on L1 and L2 ? ? 0 0 k i , ?initial fractional phases at the transmitter and the receiver, respectively NN i k i k ,, , 12 ? ambiguities associated with L and L , respectively 12 λ 1 ≈ 19 cm and λ 2 ≈ 24 cm are wavelengths of L 1 and L 2 phases ρ i k - geometric distance between the satellite k and receiver i, I f I f i k i k 1 2 2 2 , - ionospheric refraction on L1 and L2, respectively T i k - the tropospheric refraction term Basic GPS Observables (cont.) Principles of the Global Positioning System 2005-3-25 18 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 ee i k i k i k i k ,, , ,12 1 2 , , , - measurement noise for pseudoranges and phases on L1 and L2 ε ε 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 b ii k i k ,,,112 - interchannel bias between and ΦΦ bb P P ii i k i k i k i k ,, , , , , , 23 1 1 1 2 ? biases between and , and ΦΦ MMmm i k i k i k i k ,,,,1212 ? multipath on phases and ranges Basic GPS Observables (cont.) 10 Principles of the Global Positioning System 2005-3-25 19 ? 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). Principles of the Global Positioning System 2005-3-25 20 R 1 = ρ + c?dt +Ι / f 1 2 + T + e R1 R 2 = ρ + c?dt +Ι / f 2 2 + T + e R2 λ 1 Φ 1 = ρ ? Ι / f 1 2 + T + λ 1 Ν 1 + ε Φ1 λ 2 Φ 2 = ρ ? Ι / f 2 2 + T + λ 2 Ν 2 + ε Φ2 Ν 1 , Ν 2 - integer ambiguities R ? pseudorange I / f 2 - ionospheric effect Φ ? phase T - tropospheric effect ρ?geometric range e R1, e R2, ε Φ1, ε Φ2 ? white noise λ ? wavelength Basic GPS observables (simplified form) 11 Principles of the Global Positioning System 2005-3-25 21 ? Observed Doppler shift scaled to range rate; time derivative of the phase or pseudorange observation equation satellitejreceiveridtdt t tc j i j i j i j i j i ==? ? ?+=Φ ,),(error clock combined theof derivative a is     ρλ θρ cosv j i ?= Instantaneous radial velocity between the satellite j and the receiver i, and v is satellite tangential velocity The raw Doppler shift Principles of the Global Positioning System 2005-3-25 22 GPS Errors Bias errors - can be removed from the direct observables, or at least significantly reduced, by using empirical models (eg., tropospheric models), or by differencing direct observables - satellite orbital errors (imperfect orbit modeling), - station position errors - propagation media errors and receiver errors White noise 12 Principles of the Global Positioning System 2005-3-25 23 GPS Errors Bias errors ? Satellite and receiver clock errors ? Satellite orbit errors ? Atmospheric effects (ionosphere, troposphere) ? Multipath: signal reflected from surfaces near the receiver ? Selective Availability (SA) ? - epsilon process: falsifying the navigation broadcast data ? - delta process: dithering or systematic destabilizing of the satellite clock frequency ? Anti-spoofing (AS): limits the number of unauthorized users and the level of accuracy for nonmilitary applications ? Antenna phase center Principles of the Global Positioning System 2005-3-25 24 Linear phase combinations General remarks 12 11 2 2 i i 11 2 2 11 2 2 The linear combination of two phase and is defined by =n n and f t f=n f t n f t c Therefore, f=n f n f and f To compute the linear combination, proper noise levels must be taken into account. ?? ?? ? ? λ += + += 13 Principles of the Global Positioning System 2005-3-25 25 Linear combination with integer numbers L1+L2 L1 L2 L1-L2 L1 L2 L1+L2 L1-L2 The narrow lane observable The wide lane observable The corresponding wavelength are 10.7cm 86.2cm The lane signals are applied for ambiguity resolution.The advantage λ λ Φ=Φ+Φ Φ=Φ?Φ = = of a linear combination with integer numbers is that the integer nature of the ambiguitiesis preserved. Principles of the Global Positioning System 2005-3-25 26 Linear combination with real numbers L2 L2, L1 L1 L2 L1 L1 L1, L2 L1 L2 L2 Geometric residual f f This quantity is the kernel in a combination used to reduce ionospheric effects Ionospheric residual f f This quantity is used that the integer natu Φ=Φ?Φ Φ=Φ?Φ re of cycle slip detection 14 Principles of the Global Positioning System 2005-3-25 27 GPS Data file formats ? Receivers use there own propriety (binary) formats but programs convert these to standard format called Receiver Independent Exchange Format (RINEX) ? teqc available at http://www.unavco.ucar.edu/data_support/software/teqc/teqc.html is one of the most common ? The link to the RINEX format is: ftp://igscb.jpl.nasa.gov/igscb/data/format/rinex2.txt Principles of the Global Positioning System 2005-3-25 28 Rinex header 2 OBSERVATION DATA G (GPS) RINEX VERSION / TYPE ASHTORIN 04 - JAN - 03 22:56 PGM / RUN BY / DATE COMMENT 0015 MARKER NAME MARKER NUMBER OBSERVER / AGENCY ASHTECH UZ-12 ZC00 0A13 REC # / TYPE / VERS ANT # / TYPE -1332774.6000 5325356.9700 3237371.2900 APPROX POSITION XYZ 0.1132 0.0000 0.0000 ANTENNA: DELTA H/E/N 1 1 WAVELENGTH FACT L1/2 7 L1 L2 C1 P1 P2 D1 D2 # / TYPES OF OBSERV 10.0000 INTERVAL LEAP SECONDS 2003 1 1 1 52 10.000000 GPS TIME OF FIRST OBS 2003 1 1 7 32 0.000000 GPS TIME OF LAST OBS END OF HEADER 15 Principles of the Global Positioning System 2005-3-25 29 RINEX Data block 03 1 1 1 52 10.0000000 0 6G01G02G03G20G25G13 0.000000001 10185587.54311 7948396.28051 21348512.858 21348513.6115 21348509.7485 2211.739 1723.433 10290568.39211 8028016.96051 20908561.352 20908561.5665 20908558.0755 -88.269 -68.781 10137141.95811 7921677.35351 22080166.190 22080166.4985 22080163.9795 2985.965 2326.726 10376475.59911 8090767.50551 22834982.907 22834983.5025 22834980.0605 -1798.220 -1401.210 10349756.47411 8068684.52551 22116508.688 22116510.1875 22116506.7085 -1259.935 -981.768 10111007.33711 7906165.31451 24517873.958 24517875.5665 24517871.0025 3439.280 2679.958 03 1 1 1 52 20.0000000 0 6G01G02G03G20G25G13 0.000000006 10163499.790 1 7931185.04841 21344309.738 21344310.4274 21344306.5554 2206.774 1719.564 10291477.555 1 8028725.40041 20908734.284 20908734.6324 20908731.0804 -92.527 -72.099 ………………………………………………… ? Phase in cycles, range in meters Principles of the Global Positioning System 2005-3-25 30 Summary ?Looked at definitions of data types ?Looked at data and its characteristics. ?Next class, we finish observables and will examine: ? GPS major error sources and they characteristics