Lecture 1 GNSS measurements and their combinations

Transcription

1 Lecture 1 GNSS measurements and their combinations Contact: jaume.sanz@upc.edu Web site: 1

2 Authorship statement The authorship of this material and the Intellectual Property Rights are owned by J. Sanz Subirana and J.M. Juan Zornoza. These slides can be obtained either from the server or Any partial reproduction should be previously authorized by the authors, clearly referring to the slides used. This authorship statement must be kept intact and unchanged at all times. 5 March

3 Contents 1. Review of GNSS measurements. 2. Linear combinations of measurements. 3. Carrier cycle-slips detection. 4. Carrier smoothing of code pseudorange. 5. Code Multipath. 3

4 Contents 1. Review of GNSS measurements. 2. Linear combinations of measurements. 3. Carrier cycle-slips detection. 4. Carrier smoothing of code pseudorange. 5. Code Multipath. 4

5 GPS SIGNAL STRUCTURE Two carriers in L-band: L 1 =154 fo= MHz L 2 =120 fo= MHz where fo=10.23 MHz C/A-code for civilian users [X C (t)] P-code only for military and authorized users [X P (t)] Navigation message with satellite ephemeris and clock corrections [D(t)] P(Y) MHz L 2 P(Y) C/A MHz L 1 S () t = a X () t D ()sin( t ωt+ φ ) + a X () t D ()cos( t ωt+ φ ) ( k) ( k) ( k) ( k) ( k) L P P 1 L C C 1 L S () t = bx () t D ()sin( t ω t+ φ ) ( k) ( k) ( k) L P P 2 L 2 2 5

6 GPS Code Pseudorange Measurements S () t = a X () t D ()sin( t ωt+ ϕ ) + a X () t D ()cos( t ωt+ ϕ ) ( k) ( k) ( k) ( k) ( k) L P P 1 L C C 1 L S () t = bx () t D ()sin( t ω t+ ϕ ) ( k) ( k) ( k) L P P 2 L 2 2 sat P( T) = c T = c trec ( T) t ( T T) From hereafter we will call: C 1 pseudorange computed from X C (t) binary code (on frequency 1) P 1 pseudorange computed from X P (t) binary code (on frequency 1) P 2 pseudorange computed from X P (t) binary code (on frequency 2) T binary code X P (t) C 1,P 1, P 2 6

7 GPS Carrier Phase Measurements S () t = a X () t D ()sin( t ωt+ ϕ ) + a X () t D ()cos( t ωt+ ϕ ) ( k) ( k) ( k) ( k) ( k) L P P 1 L C C 1 L S () t = bx () t D ()sin( t ω t+ ϕ ) ( k) ( k) ( k) L P P 2 L 2 2 Carrier beat phase: sat φ ( T) = φ ( T) φ ( T T ) L Lrec L c = T + N λ From hereafter we will call: L 1 =λ 1 φ L1 measur. computed from the carrier phase on frequency 1 L 2 =λ 2 φ L2 measur. computed from the carrier phase on frequency 2 C 1 pseudorange computed from X C (t) binary code (on frequency 1) P 1 pseudorange computed from X P (t) binary code (on frequency 1) P 2 pseudorange computed from X P (t) binary code (on frequency 2) Unknown ambiguity Carrier phase C 1,P 1, P 2 L 1, L 2 7

8 Carrier and Code pseudorange measurements P1 P1 P1 = c T= c [t rec (T)-t sat (T- T)] P 1 ρ + clock offset Km P 1 is basically the geometric range (ρ) between satellite and receiver, plus the relative clock offset. The range varies in time due to the satellite motion relative to the receiver. P 1 is an absolute measurement (unambiguous) 8

9 Phase and Code pseudorange measurements L ρ+ clock offset + λn Relative measurement (shifted by the unknown ambiguity λn ) Each time that the receiver lose the phase lock, the unknown ambiguity changes by an integer number of λ L ( T) = c T +λ N

10 Code and Carrier Phase measurements Ambiguity Code (unambiguous but noisier) Carrier Phase (ambiguous but precise) 10

11 GPS measurements: Code and Carrier Phase Antispoofing (A/S): The code P is encrypted to Y. Only the GPS code C signal at frequency L1 is available. C 1 P 1 (Y1): encrypted P 2 (Y2): encrypted L 1 L 2 Wavelength (chip-length) cm σ noise (1% of λ) [*] Code measurements 300 m 30 m 30 m cm 3 m 30 cm 30 cm Phase measurements 2 mm 2 mm Main characteristics Unambiguous but noisier Precise but ambiguous [*] the codes can be smoothed with the phases in order to reduce noise (i.e, C 1 smoothed with L 1 50 cm noise) 11

12 RINEX FILES 12

13 GNSS Format Descriptions GNSS data files follow a well defined set of standards formats: RINEX, ANTEX, SINEX Understanding a format description is a tough task. These standards are explained in a very easy and friendly way through a set of html files. Described formats: Observation RINEX Navigation RINEX RINEX CLOCKS SP3 Version C ANTEX Open GNSS Formats with Firefox internet browser More details at: gage/upc Research group of Astronomy & Geomatics Technical University of Catalonia Tutorial associated to the GNSS Data Processing book 13 J. Sanz Subirana, J.M. Juan Zornoza, M. Hernández-Pajares

14 RINEX measurement file HEADER MEASUREMENTS 14

15 RINEX measurement file 15

16 RINEX measurement file Measurement time (receive time tags) Number of tracked satellites Epoch flag 0: OK One satellite per row 16

17 RINEX measurement file Synthetic P2 (A/S=on) S/N indicatorsloss of lock indicator 17

18 Pseudorange modeling P= c T= c [t rec (T)-t sat (T- T)] = ρ sat + c ( dt dt sat ) + P sat δ rec rec rec δ Geometric range = Trop + Ion + K + K + sat sat sat rec rec rec Ionospheric delay Tropospheric delay Clock offsets Instrumental delays ε δ noise 18

19 19

20 20

21 Exercise: Exercise 1: a) Using the file coco o, generate the txt file coco txt (with data ordered in columns). b) Plot code and phase measurements for satellite PRN28 and discuss the results. Resolution: a) glab_linux -input:cfg meas.cfg -input:obs coco o b) See next plots: 21

22 An example of program to read the RINEX: glab RINEX file glab txt file sta Doy sec PRN L1 L2 C1/ P2 cambiar The RINEX file is converted to a columnar format to easily plot its contents and to analyze the measurements (the public domain free tool gnuplot is used in the book to make the plots). 22

23 Code measurements P1 The geometry ρ is the dominant term in the plot. The pattern in the figures is due to the variation of ρ P ρ c ( dt dt ) Trop Ion K K = ε sat sat sat sat sat sat 1sta sta sta sta 1sta 1sta P1

24 Code measurements P2 Similar plot for code measurements at f 2. Notice that Ionosphere (Ion) and Instrumental delays (K) depend on frequency. P ρ c ( dt dt ) Trop Ion K K = ε sat sat sat sat sat sat 2sta sta sta sta 2sta 2sta

25 Code and Phase measurements Ionosphere delays code and advances phase measurements sat C1 sta ; Code measurements: C1,P1,P2 P = ρ + c ( dt dt ) + Trop + Ion + K + K + ε sat sat sat sat sat sat 1sta sta sta sta 1sta 1sta 1 1 P = ρ + c ( dt dt ) + Trop + Ion + K + K + ε sat sat sat sat sat sat 2sta sta sta sta 2sta 2sta 2 2 Phase measurements: L1,L2 L = ρ + c ( dt dt ) + Trop Ion + b + b + λ N + λw + ν sat sat sat sat sat sat 1sta sta sta sta 1sta 1sta L = ρ + c ( dt dt ) + Trop Ion + b + b + λ N + λ w + ν sat sat sat sat sat sat 2sta sta sta sta 2sta 2sta Frequency dependent phase Ambiguities N 1, N 2 are integers Wind Up 25

26 Carrier Phase measurements The geometry ρ is the dominant term in the plot. The pattern in the figures is due to the variation of ρ. The curves are broken when the receiver loss the lock (cycle-slip). When a cycle-slip happens, the phase measurement L changes by un unknown integer number of cycles (N) L sat sat c ( dt dt sat ) Trop sat Ion sat b b sat N w = ρ λ + λ + ν 1sta sta sta sta 1sta 1sta

27 Carrier Phase measurements The geometry ρ is the dominant term in the plot. The pattern in the figures is due to the variation of ρ. The curves are broken when the receiver loss the lock (cycle-slip). When a cycle-slip happens, the phase measurement L changes by un unknown integer number of cycles (N) = ρ + ( ) λ + λ + ν sat Master of Science sat in GNSS sat sat sat sat L c dt dt Trop Ion b b N w 2sta sta sta sta 2sta 2sta

28 Contents 1. Review of GNSS measurements. 2. Linear combinations of measurements. 3. Carrier cycle-slips detection. 4. Carrier smoothing of code pseudorange. 5. Code Multipath. 28

29 Linear Combinations of measurements: Geometry-free (or Ionospheric) combination. Ionosphere-Free combination. Wide-lane and Narrow-lane combinations. 29

30 1. Geometry-free (or ionospheric) combination sat C1 sta P 2 -P 1 ; L 1 -L 2 P I = P 2 P 1 =Iono+ctt L I = L 1 L 2 = Iono+ctt+Ambig Ambiguity Code measurements: C 1,P 1,P 2 P = ρ + c ( dt dt ) + Trop + Ion + K + K + ε sat sat sat sat sat sat 1sta sta sta sta 1sta 1sta 1 1 P = ρ + c ( dt dt ) + Trop + Ion + K + K + ε sat sat sat sat sat sat 2sta sta sta sta 2sta 2sta 2 2 Carrier measurements: L1,L2 L = ρ + c ( dt dt ) + Trop Ion + b + b + λ N + λw + ν sat sat sat sat sat sat 1sta sta sta sta 1sta 1sta L = ρ + c ( dt dt ) + Trop Ion + b + b + λ N + λ w + ν sat sat sat sat sat sat 2sta sta sta sta 2sta 2sta Carrier Ambiguities 30

31 1. Geometry-free (or ionospheric) combination GPS observables P I = P 2 P 1 =Iono+ctt L I = L 1 L 2 = Iono+ctt+Ambig P ij = c t= c [t rec (T R )-t ems (T S )] Ambiguity The pattern corresponds to the ionospheric refraction (Ion), because the other terms (K) are constant. Notice that code measurements are noisier. 31

32 Ionospheric effects The ionospheric refraction depends on: Geographic location Time of day Time with respect to solar cycle (11y) 32

33 Ionospheric effects The ionospheric delay (Ion) is proportional to the electron density integrated along the ray path (STEC) Ion = 40.3 STEC 2 f STEC Ionosphere = Ambiguity r[ GPSreceiver ] N e r[ GPStransmitter ] ( r, t) dr 33

34 2. Ionosphere-free Ionospheric-Free Combination (Pc,Lc) The ionospheric refraction depends on the inverse of the squared frequency and can be removed up to 99.9% combining f1 and f2 signals: Pc f P f P = Lc = f1 f2 f1 f2 sat sat sat sat Pc = ρ + c ( dt dt ) + Trop + ε sta sta sta sta c Lc = ρ + c ( dt dt ) + Trop + λ w + b + b + λ ( N N ) + ν sat sat sat sat sat sat sat λw sat sta sta sta sta N sta c, sta c N 1sta λ W sta c The ionospheric refraction has been removed in Lc and Pc λ N = 10.7 cm, λ W =86.2cm f Ion L = 40.3 STEC 2 f f L 2 2 Note: K sat cancels in Pc and K sta included in dt sta N = N N W

35 Comments: Two-frequency receivers are needed to apply the ionosphere-free combination. If a one-frequency receiver is used, a ionospheric model must be applied to remove the ionospheric refraction. The GPS navigation message provides the parameters of the Klobuchar model which accounts for more than 50% (RMS) of the ionospheric delay. 35

36 3.- Narrow-lane Ionospheric-Free (P N ) and Wide-lane Combination Combination (L W ) The wide-lane combination L W provides a signal with a large wavelength (λ W =86.2cm ~ 4*λ 1 ). This makes it very useful for detecting cycle-slips through the Melbourne-Wübbena combination: L W P N P N = f P f f P f L W = The same sign f L f The ambiguities N W are INTEGER Numbers! P = ρ + c ( dt dt ) + Trop + Ion + K + K + ε sat sat sat sat sat sat N sta sta sta sta w w w N f L f L = ρ + c ( dt dt ) + Trop + Ion + b + b + λ N + ν sat sat sat sat sat sat sat W sta sta sta sta w w w w w, sta w sta sta sta sta N = N N W wind-up J. Sanz & J.M. Juan

37 Exercises: 1) Consider the wide-lane combination of carrier phase measurements L W =, where L W is given in length units (i.e. L i = λ i φ i ). Show that the corresponding wavelength is: Hint: fl f fl f L W = λ W φ W ; φ W = φ 1 φ 2 λ = 1 2 2) Assuming L 1, L 2 uncorrelated measurements with equal noise σ L, show that: 2 γ f 1 σl = σ ; W L γ12 = γ 1 f 12 2 W f c Slides associated to the GNSS Data Processing book J. Sanz Subirana, J.M. Juan Zornoza, M. Hernández-Pajares 37 f

38 Contents 1. Review of GNSS measurements. 2. Linear combinations of measurements. 3. Carrier cycle-slips detection. 4. Carrier smoothing of code pseudorange. 5. Code Multipath. 38

39 There is a cycleslip of only one cycle (~20cm) How to detect it? Detecting cycle-slips This cycle-slip involves millions of cycles it is easy to detect!! 39

40 Exercise: a) Using the file 95oct18casa r0.rnx, generate the txt file 95oct18casa.a (with data ordered in columns). b) Insert a cycle-slip of one wavelength (19cm) in L1 measurement at t=5000 s (and no cycle-slip in L2). c) Plot the measurements L1, L1-P1, LC-PC, Lw-PN and L1-L2 and discuss which combination/s should be used to detect the cycle-slip. Resolution: a) glab_linux -input:cfg meas.cfg -input:obs 95oct18casa_r0.rnx b) cat 95oct18casa.a gawk {if ($4==18) print $3,$5,$6,$7,$8} > s18.org cat s18.org gawk {if ($1>=5000) $2=$2+0.19; printf %s %f %f %f %f \n, $1,$2,$3,$4,$5} > s18.cl c) See next plots: 40

41 The geometry ρ is the dominant term in the plot. The variation of ρ in 1 sec may be hundreds of meters, many times greater than the cycle-slip (19 cm) the variation of ρ shadows the cycle-slip! 1 unit = 19 cm (L1 cycles) L1 (without the cycle-slip) L1 (with the cycle-slip) ρ A jump of λ=19 cm (one cycle in L1) has been introduced in L1 at t=5000s L c ( dt dt ) Trop Ion b b N w = ρ λ + λ + ν sat sat sat sat sat sat sat sat 1sta sta sta sta 1sta 1sta 1 1 1sta 1 sta 411

42 The geometry and clock offsets have been removed. The trend is due to the Ionosphere. The P1 code noise shadows the cycle-slip, and without the reference (in blue), the time where the cycle-slip happens could not be identified. 1 unit = 19cm (L1 cycles) L1-P1 (with the cycle-slip) L P = 2Ion + ctt + ambig + ε sat sat sat 1sta 1sta 1sta P = ρ + c ( dt dt ) + Trop + Ion + K + K + ε A jump of λ=19 cm (one cycle in L1) has been introduced in L1 at t=5000s sat sat sat sat sat sat 1sta sta sta sta 1sta 1sta 1 1 L c ( dt dt ) Trop Ion b b N w = ρ λ + λ + ν sat sat sat sat sat sat sat sat 1sta sta sta sta 1sta 1sta 1 1 1sta 1 sta 421

43 The geometry and clock offsets have been removed. The trend is due to the Ionosphere. The P1 code noise shadows the cycle-slip, and without the reference (in blue), the time where the cycle-slip happens could not be identified. 1 unit = 19cm (L1 cycles) L1-P1 (with the cycle-slip) L P = 2Ion + ctt + ambig + ε sat sat sat 1sta 1sta 1sta P = ρ + c ( dt dt ) + Trop + Ion + K + K + ε A jump of λ=19 cm (one cycle in L1) has been introduced in L1 at t=5000s sat sat sat sat sat sat 1sta sta sta sta 1sta 1sta 1 1 L c ( dt dt ) Trop Ion b b N w = ρ λ + λ + ν sat sat sat sat sat sat sat sat 1sta sta sta sta 1sta 1sta 1 1 1sta 1 sta 431

44 The geometry, clock offsets and iono have been removed. There is a constant pattern plus noise. The P C code noise also shadows the cycle-slip, and without the reference (in blue), the time where the cycle-slip happens could not be identified. A jump of λ=19 cm (one cycle in L1) has been introduced in L1 at t=5000s 1 unit = 10.7 cm sat sat sat sat Pc = ρ + c ( dt dt ) + Trop + ε sta sta sta sta c LC-PC (without the cycle-slip) LC-PC (with the cycle-slip) Lc Pc = ctt + ambig + ε λ Lc = ρ + c ( dt dt ) + Trop + λ w + b + b + λ ( N N ) + ν satmaster of sat Science in GNSS sat sat sat sat J. Sanz W & J.M. Juan sat sta sta sta sta N sta c, sta c N 1sta λ2 W sta c sat sta sat sta 44

45 The geometry, clock offsets and iono have been removed. There is a constant pattern plus noise. The P I code noise also shadows the cycle-slip, and without the reference (in blue), the time where the cycle-slip happens could not be identified. A jump of λ=19 cm (one cycle in L1) has been introduced in L1 at t=5000s 1 unit = 5.4 cm P = Ion + K + K + ε LI-PI (without the cycle-slip) LI-PI (with the cycle-slip) L P = ctt + ambig + ε sat I sta sat sat I sta I I sta I I sat I sta L = Ion + b + b + N N + w + λ λ ( λ λ ) ν sat sat sat sat sat I sta I I I 1 1sta 2 2sta 1 2 sta 45 I sta

46 The geometry, clock offsets and iono have been removed. There is a constant pattern plus noise. The PN code noise is under one cycle of Lw. Thence, the cycle-slip is clearly detected A jump of λ=19 cm (one cycle in L1) has been introduced in L1 at t=5000s 1 unit unit = 86.2cm 86.2 cm (Lw cycles) Lw-PN (without the cycle-slip) Lw-PN (with the cycle-slip) Melbourne Wübbena Combination L P = ctt + ambig + ε sat W sta sat N sta P = ρ + c ( dt dt ) + Trop + Ion + K + K + ε sat sat sat sat sat sat N sta sta sta sta w w w N L c ( dt dt ) Trop Ion b b N sta sta = ρ λ + ν sat sat sat sat sat sat sat W sta sta sta sta w w w w wsta 46w sta sta

47 The geometry and clock offsets have been removed. The trend is due to the Iono. The L I carrier noise is few mm, and the variation of the ionosphere in 1 second is lower than λ 1 =19 cm Thence, the cycle-slip is detected. A jump of λ=19 cm (one cycle in L1) has been introduced in L1 at t=5000s 1 unit = 5.4 cm LI (without the cycle-slip) LI (with the cycle-slip) Geometry-Free Combination L L = Ion Ion + ctt + ambig + ε sat sat sat sat 1, sta 2, sta 1, sta 2, sta ( λ λ )w << 1 2 L = ρ + c ( dt dt ) + Trop Ion + b + b + λ N + λw + ν sat sat sat sat sat sat sat sat 1sta sta sta sta 1sta 1sta 1 1 1sta 1 sta 1 = ρ λ + λ + ν L c ( dt dt ) Trop Ion b b N w ε ~mm sat sat sat sat sat sat sat sat 2sta sta sta sta 2sta 2sta 2 2 2sta 2 sta 47 2

48 Summary L1 LI-PI L1-P1 LC-PC 48

49 The cycle-slips are detected by the Ionospheric combination (LI=L1-L2) and the Melbourne Wübbena (MW=Lw-PN) LI Geometry Free Combination L1 Lw-PN Melbourne Wübbena Combination Two independent combinations, LI and Lw, allow to detect two independent cycle-slips (in L1 and L2 phase measur.). L2 Notice that, from L1, L2 is not possible to detect small cycle-slips 49

50 Summary of Cycle-slip detectors COMBI NATION MEAS Combination Noise (σ) λ σ/λ L 1 -P 1-2 Ion+K+λ 1 N 1 σ L1-P1 σ P1 =30 cm λ 1 =19.0 cm 1.58 L c -P c k c +λ N R c σ LC-PC σ PC =2.98 σ P = 89 cm λ N = 10.7 cm 8.32 L I -P I k I +λ 1 N 1 -λ 2 N 2 σ LI-PI σ PI = 2 σ P =42 cm λ 2 -λ 1 =5.4 cm 7.78 L W -P N λ W N w σ LW-PN σ PN = σ P / 2 =21 cm λ W =86.2 cm 0.25 L I Ion+k I +λ 1 N 1 -λ 2 N 2 σ LI = 2 σ L1 =3 mm λ 2 -λ 1 =5.4 cm

51 Contents 1. Review of GNSS measurements. 2. Linear combinations of measurements. 3. Carrier cycle-slips detection. 3.1 Cycle-slip Detection Algorithms 4. Carrier smoothing of code pseudorange. 5. Code Multipath. 51

52 Cycle-slip detector based on carrier phase data: The Geometry-free combination L L sk L sk I = 1(; ) 2(; ) The detection is based on fitting a psk (; ) L (; s k) p(; s k) > threshold I L(; sk) I second order polynomial over a sliding window of N I samples. The predicted value is compared with the observed one to detect cycle-slip. psk (; ) [ L(; sk N),, L(; sk 1) ] I I I L(; sk) I

53 L(; sk) I Carrier measurements psk (; ) Fitted Polynomial Under not disturbed ionospheric conditions, the geometry-free combination performs as a very precise and smooth test signal, driven by the ionospheric refraction. Although, for instance, the jump produced by a simultaneous one-cycle slip in both signals is smaller in this combination than in the original signals (λ 2 -λ 1 =5.4cm), it can provide reliable detection even for small jumps

54 Cycle-slip detector based on code and carrier phase data: The Melbourne-Wübbena combination B = L P = λ N + ε W W N W W The detection is based on real-time computation of mean (m BW ) and sigma (S BW ) values of the measurement test data Bw. A cycle-slip is declared when the measurement differs form the mean value by a predefined number of standard deviations (S BW )

55 Moving average Threshold Moving sigma Nevertheless, in spite of these benefits, the performance is worse than in the previous carrier-phase-only based detector and it is used as a secondary test. The Melbourne-Wübbena combination has a double benefit: The enlargement of the ambiguity spacing, thanks to the larger wavelength λ W =86.2cm. The noise is reduced by the narrow-lane combination of code measurement Cycle-slip detection

56 Exercises: 1) Show that N = 9 and N = produces jumps of few millimetres in the geometry-free combination. 2) Show that no jump happens in the geometry-free combination when N1/ N2 = 77 / 60. In particular when N 1 =77 and N 2 =60 the jump in the wide-lane combination is: 17λ W = 15m Hint: Consider the following relationships (from [RD-1]): 56

57 Example of Single frequency Cycle-slip detector ( P 1) ( L 1 ) dsk (; ) = L(; sk) Psk (; ) 1 1 The detection is based on real-time computation of mean and sigma values of the differences (d=l 1 -P 1 ) of the code pseudorange and carrier over a sliding window of N samples (e.g. N=100). A cycle-slip is declared when a measurement differs from the mean bias value over a predefined threshold. Missed detection

58 More details, exercises and examples of software code implementation of these detectors can be found in [RD-1] and [RD-2]. This detector is affected by the code pseudorange noise and multipath as well as the divergence of the ionosphere. Higher sampling rate improves detection performance, but shortest jumps can still escape from this detector. On the other hand, a minimum number of samples is needed for filter initialization in order to ensure a reliable value of sigma for the detection threshold

59 Contents 1. Review of GNSS measurements. 2. Linear combinations of measurements. 3. Carrier cycle-slips detection. 4. Carrier smoothing of code pseudorange. 5. Code Multipath. 59

60 Carrier smoothing of code pseudorange The noisy (but unambiguous) code pseudorange can be smoothed with the precise (but ambiguous) carrier. A simple algorithm is given next: Hatch filter: ˆ 1 n 1 Pk ( ) = Pk ( ) + n n Pk ( 1) + Lk ( ) Lk ( 1) where Pˆ (1) = P(1) and n= k; k < N n= N; k N ( ˆ ) This algorithm can be interpreted as a real-time alignment of the carrier phase with the code measurement: ˆ 1 n 1 Pk ( ) = Pk ( ) + Pk ( 1) + Lk ( ) Lk ( 1) = Lk ( ) + P L k n n ( ˆ ) ( )

61 This algorithm can be interpreted as a real-time alignment of the carrier phase with the code measurement: P L P L ( ˆ ) ( ) ˆ 1 n 1 Pk ( ) = Pk ( ) + Pk ( 1) + Lk ( ) Lk ( 1) = Lk ( ) + P L k n n 61

62 P ˆP L P L ˆ 1 n 1 Pk ( ) = Pk ( ) + Pk ( 1) + Lk ( ) Lk ( 1) = Lk ( ) + P L k n n ( ˆ ) ( ) 62

63 P ˆP L P L ˆ 1 n 1 Pk ( ) = Pk ( ) + Pk ( 1) + Lk ( ) Lk ( 1) = Lk ( ) + P L k n n ( ˆ ) ( ) 63

64 Code-carrier divergence: SF smoother Time varying ionosphere induces a bias in the single frequency (SF) smoothed code when it is averaged in the smoothing filter (Hatch filter). Let: P1 = ρ+ I1+ ε1 L = ρ I + B + ς thence, P L = 2I B + ε Substituting P L 1 1 Where ρ includes all non dispersive terms (geometric range, clock offsets, troposphere) and I 1 represents the frequency dependent terms (ionosphere and DCBs). B 1 is the carrier ambiguity, which is constant along continuous carrier phase arcs and ε1, ς1 account for code and carrier multipath and thermal noise. 2 I : Code-carrier divergence 1 in Hatch filter equation Pk ˆ( ) = Lk ( ) + P L = ρ ( k ) I ( k ) + B+ 2 I B = where, being the ambiguity term a constant bias, thence B1 B1, and cancels in the previous expression. ( k ) ( k) ( k) ( ) = ρ( k) + I1( k) + 2 I1 I ( ) 1( k) k bias I B 1 Pˆ 1 = ρ+ I1+ bias I + υ1 υ 1 where is the noise term after smoothing. 64

65 N=3600 s Iono

66 66

67 Halloween storm Data File: amc o_1hz STEC N=100 (i.e. filter smoothing time constant τ=100 sec). 67

68 Halloween storm Image from STEC [*] Ionospheric Threat Parameterization for Local Area Global-Positioning- System-Based Aircraft Landing Systems, Datta-Barua et al, Journal of Aircraft Vol. 47, No. 4, July August 2010, DOI: /

69 Carrier-smoothed pseudorange: DFree Divergence-Free (Dfree) smoother: With two frequency carrier measurements a combination of carriers with the same ionospheric delay (the same sign) as the code can be generated: P = ρ+ I + ε L = ρ+ I + B + ς 1, DF 1 1, DF 1, DF Thence, ( ) L = L + 2 α L L = ρ+ I + B + ς 1, DF , DF 1, DF With this new combination we have: P L = B + ε 1 1, DF 1, DF 1 No Code-carrier divergence! ˆ 1, DF ρ I1 υ12 P = + + f 1 α = = = γ f1 f2 77 γ = 60 This smoothed code is immune to temporal gradients (unlike the SF smoother), being the same ionospheric delay as in the original raw code (i.e. I 1 ). Nevertheless, as it is still affected by the ionosphere, its spatial decorrelation must be taken into account in differential positioning. 2

70 70

71 Carrier-smoothed pseudorange: IFree Ionosphere-Free (Ifree) smoother: Using both code and carrier dual-frequency measurements, it is possible to remove the frequency dependent effects using the ionosphere-free combination of code and carriers (PC and LC). Thence: P L C = ρ+ ε P C L L Thence, C = ρ+ B + υ PC LC = BC + ε P C C C γ P P γ P P ; L L L = = L γ 1 γ IFree C IFree C Pˆ Pˆ = ρ+ υ IFree C IFree 77 γ = 60 2 γ + 1 σpc = σp 3σ 1 P1 γ 1 This smoothed is based on the ionosphere-free combination of measurements, and therefore it is unaffected by either the spatial and temporal inospheric gradients, but has the disadvantage that the noise is amplified by a factor 3 (using the legacy GPS signals). 2

72 Vertical range: [-5 : 5] Vertical range: [-15:15] 72

73 C1, L1 PC, LC N=100 N=100 STEC Exercise: Justify that the ionosphere-free combination (PC) is (obviously) not affected by the code-carrier divergence, but it is 3 times noisier.

74 C1, L1 PC, LC N=360 N=360 N=100 N=100

75 C1, L1 PC, LC N=3600 N=3600 N=360 N=360

76 Halloween storm Data File: amc o_1hz STEC N=100 (i.e. filter smoothing time constant τ=100 sec). 76

77 Contents 1. Review of GNSS measurements. 2. Linear combinations of measurements. 3. Carrier cycle-slips detection. 4. Carrier smoothing of code pseudorange. 5. Code Multipath. 77

78 Multipath One or more reflected signals reach the antenna in addition to the direct signal. Reflective objects can be earth surface (ground and water), buildings, trees, hills, etc. It affects both code and carrier phase measurements, and it is more important at low elevation angles. Butterfly shape Code: up to 1.5 chip-length up to 450m for C1 [theoretically] Typically: less than 2-3 m. Phase: up to λ/4 up to 5 cm for L1 and L2 [theoretically] Typically: less than 1 cm 78

79 Exercise Plot code and phase geometry-free combination for satellite PRN 15 of file 97jan09coco_r0.rnx and discuss the results. Butterfly shape: High multipath for low elevation rays (when satellite rises and sets) 79

80 M = Pc Lc Pc

81 M = Pc Lc Pc

82 MMW = PN LW 82

83 MMW = PN LW 83

84 After one year, the directions of the Sun and Aries coincide again, but the number of laps relative to the Sun (solar days) is one less than those relative to Aries (sidereal days). 24h m s = 3 56 Aries Point direction Thus, a sidereal day is shorter than a solar day for about 3 m 56 s 84

85 Receiver noise and multipath GPS standalone (C1 code) 10,000 85

86 Receiver noise and multipath Same environment! GPS standalone (C1 code)

87 References [RD-1] J. Sanz Subirana, J.M. Juan Zornoza, M. Hernández-Pajares, GNSS Data processing. Volume 1: Fundamentals and Algorithms. ESA TM- 23/1. ESA Communications, [RD-2] J. Sanz Subirana, J.M. Juan Zornoza, M. Hernández-Pajares, GNSS Data processing. Volume 2: Laboratory Exercises. ESA TM-23/2. ESA Communications, [RD-3] Pratap Misra, Per Enge. Global Positioning System. Signals, Measurements, and Performance. Ganga-Jamuna Press, [RD-4] B. Hofmann-Wellenhof et al. GPS, Theory and Practice. Springer-Verlag. Wien, New York,

88 Thank you!