雨自文大電园地 Principles of the Global Positioning System Lecture 05 YUAN Linguo Email:Igyuan@home.switu.edu.cn Dept of Surveying Engineering, Southwest Jiaotong University Outline Review u Examined basics of GPS signal structure and how gnal is tracked a Looked at methods used to acquire satellites and start g Today we look at o Basic gps observables Biases and nois o Examine rinex format and look at some raw"' data wv Principles of the Global Positioning System
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
Review GPS Signal Summary Table odone Frequency Ratio of fundamental Wavelength [cm] 10.23 29326 LI Carrier 1, 575.42 154f6 19.04 L 2 Carrier 1.227.60 120f0 2445 L5 Carrier.176.45 l15f6 5 P-code 10.23 29326 C/A code 29326 W-code.5115 f/20 58651 Navigation 50-10 f。204,600 N/A Principles of the Global Positioning System 2005325(3 Basic GPs observables Code pseudoranges a precise/protected Pl, P2 codes available only to the military users a clear/acquisition C/A code available to the civilian users Phase pseuodranges a Ll, L2 phases, used mainly in geodesy and surveying Doppler data Gv Principles of the Global Positioning System 2005325(4
2 Principles of the Global Positioning System 2005-3-25 3 Review ：GPS Signal Summary Table GPS Signal Summary Table Component Frequency [MHz] Ratio of fundamental frequency fo Wavelength [cm] Fundamental frequency fo 10.23 1 2932.6 L1 Carrier 1,575.42 154⋅fo 19.04 L2 Carrier 1,227.60 120⋅fo 24.45 L5 Carrier 1,176.45 115⋅fo 25.5 P-code 10.23 1 2932.6 C/A code 1.023 fo/10 29326 W-code 0.5115 fo/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 GPS Code pseudoranges pseudoranges precise/protected P1, P2 codes - available only to the military users clear/acquisition C/A code - available to the civilian users Phase pseuodranges pseuodranges L1, L2 phases, used mainly in geodesy and surveying Doppler data Doppler data
Code pseudoranges e 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 wy Principles of the Global Positioning System 2005-325(5) Code pseudoranges Pseudorange R=(t C Where R is the pseudorange between receiver R and satellite S; tR is the receiver clock time. t is the satellite transmit time and c is the This expression can be related to the true range by introducing t=rR+△tt=r3+△ t Rand r are true times; Atg and 4N are clock corrections wEI Principles of the Global Positioning System 20053-25(6
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: R t t c S R = ( − )⋅ Where R is the pseudorange between receiver R and satellite S; tR 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 S S S R R R t =τ + ∆t t =τ + ∆t τR and τS are true times; ΔtR and Δt S are clock corrections
Code pseudoranges Substituting into the equation of the pseudorange ields R=(x2-x)+(△M-△)e R=p2+(△2-△r3)c+IA+A lonspheric Atmospheric PR IS true range, and the ionospheric and atmospheric terms are introduced because the propagation velocity is not c wy Principles of the Global Positioning System 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 p=p(r3,tn)=p(,、(3+△)=p(3)+p(r3)△ The maximum radial velocity for GPS satellites in the case of a stationary receiver is p 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 NEE Principles of the Global Positioning System 8
4 Principles of the Global Positioning System 2005-3-25 7 Code pseudoranges Substituting into the equation of the pseudorange yields ρR S is true range, and the ionospheric and atmospheric terms are introduced because the propagation velocity is not c. [ ] N N delay Atmospheric delay Ionspheric ( ) ( ) ( ) S R S R S R S R S R S R R t t c I A R t t c = + ∆ − ∆ ⋅ + + = − + ∆ − ∆ ⋅ ρ τ τ Principles of the Global Positioning System 2005-3-25 8 Millisecond problem in C/Acode 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 t t t t t t t t S S S S R S ρ = ρ( , ) = ρ( ,( + ∆ )) = ρ( ) + ρ( )∆
Notes of code pseudoranges corrections ap plied for propagation delays becauange and The equation for the pseudorange uses the true se the propagation velocity is not the in-vacuum value, c, 299792458x108m/s e 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 e Also need to account for noise in measurements p-code pseudoranges can be as good as 20 cm or less, while the LI C/A code range noise level reaches even a meter or more Principles of the Global Positioning System 2005-3-25(9) hase pseudoranges Carrier phase- a difference between the phases of a carrier signal received from a spacecraft and a reference signal generated by the receivers internal oscillator e 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 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 EX Principles of the Global Positioning System 5
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, 2.99792458x108 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 good as 20 cm or less, while the L1 C/A code range noise level reaches even a meter or more 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, N 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 phase cycle slip cycle slip or loss of lock loss of lock introduces a new ambiguity unknown. typical noise of phase measurements is generally of the order of a few millimeters or less
Phase pseudoranges Instantaneous circular frequency f is a derivative of the phase t to time d o By integrating frequency between two time epochs the ignal's phase results g=∫fd 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 p, we get =f(-n)=f|t C 2005-3-25 hase pseudoranges PRIt phase of reconstructed carrier with frequency R (t)=f°t-f PRoe and oc are clock errors o.=-f dt and frat ()=q(1)-g2(t) P f+f'dt'-frdtr+(fs-fr Gv Principles of the Global Positioning System 20053-25(12
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 ϕ(t0) to zero, and taking into account the signal travel time tr corresponding to the satellite-receiver distance ρ, we get dt d f ϕ = ∫ = t t f dt 0 ϕ ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = − = − c f t t f t tr ρ ϕ Principles of the Global Positioning System 2005-3-25 12 ϕs (t) phase of received carrier with frequency f s ϕR(t) phase of reconstructed carrier with frequency fR f dt f dt f f t c t t t f f dt f dt where t f t c t f t f R s R R s s s R s s R R c R R s s s c s Ro c c R R R c s c s s s ( ) ( ) ( ) ( ) and and are clock errors ( ) ( ) 0 , 0 , , 0 , 0 , 0 , = − = − + − + − = − = − = − = − − ρ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ρ ϕ Phase pseudoranges
assuming the frequency difference at the order of 1.5.10-H= (assuming df/f =10- is the oscillator instabilit y, and f=1.5GHz is negligible the equation can be simplified PR()=-f=+f(dt-dt r since only the fractional part of phase is measured at the initial epoch t o, the initial integer number N of cycles between satellite and the receiver is unknown Introducin g the initial fractional beat phase Po and denoting f=-where is a wavelengt h (dr-dI)+N+( -PoR) ap=-p+c(dt'-dtg)+aN+1( -or) which is phase range in [m] ev Principles of the Global Positioning System 13 Geometrical interpretation of phase range pr(t)=(R (to)+ NR(t-to (t-R(t+NR(to)or r(t)=R(t)-NR(to PRS(t Nr(to) earth NEV Principles of the Global Positioning System 2005325(10 7
7 Principles of the Global Positioning System 2005-3-25 13 ( ) ( ) which is phase range in [m] ( ) ( ) 1 ( ) where is a wavelengt h : c f Introducin g the initial fractional beat phase and denoting between satellite and the receiver is unknown initial epoch t , the initial integer number N of cycles since only the fractional part of phase is measured at the ( ) ( ) is negligible the equation can be simplified : (assuming / 10 is the oscillator instabilit y, and f 1.5GHz) assuming the frequency difference at the order of 1.5 10 0 0 0 0 0 0 12 -3 R s R s R s R s s R R s s R c dt dt N dt dt N c t f dt dt c t f df f Hz ρ λ λ ϕ ϕ ϕ ϕ λ ρ λ ϕ λ λ ϕ ρ ϕ Φ = − + − + + − = − + − + + − = = − + − = = ⋅ − Principles of the Global Positioning System 2005-3-25 14 ϕR S(t)＝ ϕR S (t0)+ NR S (t－t0) ， ΦR S (t)= ϕR S (t)+NR S (t0) or ϕR S (t) = ΦR S (t) -NR S (t0 t1 earth TR t0 t2 ϕR S (t0) ϕR S (t1) ϕR S (t2) NR S (t0) NR S (t0) NR S (t0) Geometrical interpretation of phase range
Precision of phase measurements Nominally phase can be measured to 1%of wavelength (2mm LI 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 e 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 Basic GPs observables c(dt, -dr)+b,2+M+e P.2=p4++T+c(dt;-dt)+b,3+M2+e2 8M+-d)+(-9)+m+ =川-7++1M2+c(-d)+b+入2(9-9)+m+Ea P o =sarl(x*x, +(r-r)+(zt-zPI The primary unknowns are Xi, Yi, Zi-coordinates of the user(receiver) 1, 2 stand for frequency on LI and L2, respectively 1-denotes the receiver. while k denotes the satellite EX Principles of the Global Positioning System 20053-25(16
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 i i k i k i k i k i k i k i k i i k i k , , ,, , , ,, ( ) ( ) 1 2 11 2 3 22 1 2 2 2 =+ ++ − ++ + =+ ++ − ++ + ρ ρ ( ) ( ) Φ Φ i k i k i k i k i k i k k i i k i k i k i k i k i k i k i k i k i i k i k I f T N c dt dt m I f T N c dt dt b m , , ,, , , , ,, ( ) ( ) 1 1 1 10 1 1 2 2 2 1 20 2 2 1 2 0 2 2 0 =−++ + − + − + + =−++ + − ++ − + + ρ λ λϕ ϕ ε ρ λ λϕ ϕ ε [( )( )( ) ] 2 2 2 ,0 i k i k i k k ρi = sqrt X − X + Y −Y + Z − Z 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 Basic GPS Observables
Basic GPS Observables(cont P,n, P2-pseudoranges measured between station i and satellite k on LI and L d., a-phase ranges measured between station i and satellite k on LI and L2 Po, P. -initial fractional phases at the transmitter and the receiver, respectively 4: -ambiguities associated with L, and L2, respectively M,=19 cm and n2= 24 cm are wavelengths of L, and L, phases p.geometric distance between the satellite k and receiver i, ionospheric refraction on LI and L2, respecti T-the tropospheric refraction term ey Principles of the Global Positioning System 2005-3- Basic GPS Observables(cont d -the i-th receiver clock error dr- the k-th transmitter(satellite)clock error fr, f,- carrier fre c- the vacuum speed of light e e,2 E E,- measurement noise for pseudoranges and phases on LI and L2 M, ,M,2,m, , m 2 -multipath on phases and ranges bil, bz, b3-interchannel bias terms for receiver i that represent the sible time non-synchronization of the four measurements b,- interchannel bias betweenΦandΦ , 2, 6, -biases between q. and Pl, dpi and P 2 Gv Principles of the Global Positioning System 18)
9 Principles of the Global Positioning System 2005-3-25 17 P P i k i k , , ,1 2 − pseudoranges measured between station i and satellite k on L1 and L2 Φ Φ i k i k , , , 1 2 −phase ranges measured between station i and satellite k on L1 and L2 ϕ 0 ϕ 0 k i , −initial fractional phases at the transmitter and the receiver, respectively N N i k i k , , , 1 2 −ambiguities associated with L and L , respectively 1 2 λ1 ≈ 19 cm and λ2 ≈ 24 cm are wavelengths of L1 and L2 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 Ti k - the tropospheric refraction term Basic GPS Observables (cont.) Basic GPS Observables (cont.) Principles of the Global Positioning System 2005-3-25 18 dti - the i-th receiver clock error dtk - the k-th transmitter (satellite) clock error f1, f2 - carrier frequencies c - the vacuum speed of light e e i k i k i k i k ,, , , 12 1 2 , , , - measurement noise for pseudoranges and phases on L1 and L2 ε ε bi,1, bi,2 , bi,3 - interchannel bias terms for receiver i that represent the possible time non-synchronization of the four measurements bi i k i k , ,, 1 12 - interchannel bias between and Φ Φ bb P P ii i k i k i k i k ,, , , , , , 23 1 1 1 2 − biases between and , and Φ Φ M M mm i k i k i k i k , , ,, , ,, 1 212 − multipath on phases and ranges Basic GPS Observables (cont.) Basic GPS Observables (cont.)
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 econdary(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 Ev Principles of the Global Positioning System 2005-3- 25 Basic GPS observables(simplified form) RI=p+cdt +I/f2+T+ eRI R2=p+cdt+/f22+T+eR2 入①1=p-I/f12+T+1N1+Ea1 n,2=p-1/5+T+nN,+Ea NI, N2-integer ambiguities T-tropospheric effect p-geometric range eri eR2 Egl.Ea2-white noise 入- wavelength Gv Principles of the Global Positioning System 10
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 R1 = ρ + c⋅dt +Ι / f1 2 + T + eR1 R2 = ρ + c⋅dt +Ι / f2 2 + T + eR2 λ1Φ1 = ρ − Ι / f1 2 + T + λ1Ν1 + εΦ1 λ2Φ2 = ρ − Ι / f2 2 + T + λ2Ν2 + εΦ2 Ν1 , Ν2 - integer ambiguities R − pseudorange I / f2 - ionospheric effect Φ − phase T - tropospheric effect ρ − geometric range eR1, eR2, εΦ1, εΦ2 − white noise λ − wavelength Basic GPS observables (simplified form) Basic GPS observables (simplified form)
