SlideShare a Scribd company logo

Sat geo 03(1)

•
•
L
lenisepti

GPS

Engineering
1 of 294
Download to read offline
1 / 105 OBSERVATION TECHNIQUES IN SATELLITE GEODESY III Global Positioning System – GPS Observable and Data Processing – Wolfgang Keller Institute of Geodesy – University of Stuttgart May 1, 2007
GPS Observable 2 / 105
3 / 105 Three basic observables used with GPS system (in most cases): • pseudoranges from code measurements
3 / 105 Three basic observables used with GPS system (in most cases): • pseudoranges from code measurements • carrier phases or carrier phase differences
3 / 105 Three basic observables used with GPS system (in most cases): • pseudoranges from code measurements • carrier phases or carrier phase differences • differences in signal travel time from interferometric measurements.
Pseudoranges from Code 4 / 105 Fundamental observation equation for a single code-derived pseudorange: PRi = |Xi − XB| + cdtu = c · ∆i (1) with • Xi position of satellite i in CTS • XB position of receiver antenna B in CTS • dtu clock synchronization error between GPS time and receiver clock • ∆i observed signal travel time from satellite i to receiver B • c speed of light.
Pseudoranges from Code 4 / 105 Fundamental observation equation for a single code-derived pseudorange: PRi = |Xi − XB| + cdtu = c · ∆i (1) with • Xi position of satellite i in CTS • XB position of receiver antenna B in CTS • dtu clock synchronization error between GPS time and receiver clock • ∆i observed signal travel time from satellite i to receiver B • c speed of light. Derivation of coordinates of the receiver B needs at least four simultaneous pseudorange measurements.
Carrier Phases 5 / 105 Basic observation equation for a carrier phase measurement: ΦBi = 2π λ (|Xi − XB| + NBi λ + cdtu) (2) with • λ carrier wavelength • NBi unknown integer number of complete carrier cycles • Xi, XB, dtu, c as before
Carrier Phases 5 / 105 Basic observation equation for a carrier phase measurement: ΦBi = 2π λ (|Xi − XB| + NBi λ + cdtu) (2) with • λ carrier wavelength • NBi unknown integer number of complete carrier cycles • Xi, XB, dtu, c as before Main difficulty in the use of carrier phase observations: Determination of the unknown integer number NBi of cycle ambiguities. =⇒ Use of special sophisticated methods.
Carrier Phase Differences 6 / 105 Basic observation in most cases: usage of the difference of the phase observations of the signal of the same satellite i registered at two receivers A and B.
Carrier Phase Differences 6 / 105 Basic observation in most cases: usage of the difference of the phase observations of the signal of the same satellite i registered at two receivers A and B. Basic observation equation for this single phase difference: ∆ΦABi := ΦBi − ΦAi = 2π λ (|Xi − XB| − |Xi − XA| − (NBi − NAi )λ +c(dtuB − dtuA)). (3)
Interferometric Measurements 7 / 105 • usage of GPS signals without knowledge of the signal structure
Interferometric Measurements 7 / 105 • usage of GPS signals without knowledge of the signal structure • recording of signals with precise time marks at two different receivers A and B
Interferometric Measurements 7 / 105 • usage of GPS signals without knowledge of the signal structure • recording of signals with precise time marks at two different receivers A and B • correlate signals afterwards
Interferometric Measurements 7 / 105 • usage of GPS signals without knowledge of the signal structure • recording of signals with precise time marks at two different receivers A and B • correlate signals afterwards • fundamental observable: difference of the arrival times ∆τABi of the signal at the two receivers ∆τABi := |Xi − XB| − |Xi − XA| c (4)
Interferometric Measurements 7 / 105 • usage of GPS signals without knowledge of the signal structure • recording of signals with precise time marks at two different receivers A and B • correlate signals afterwards • fundamental observable: difference of the arrival times ∆τABi of the signal at the two receivers ∆τABi := |Xi − XB| − |Xi − XA| c (4) Quite similar to Very Long Baseline Interferometry (VLBI) which uses Quasars as radio sources.
Interferometric Measurements 7 / 105 • usage of GPS signals without knowledge of the signal structure • recording of signals with precise time marks at two different receivers A and B • correlate signals afterwards • fundamental observable: difference of the arrival times ∆τABi of the signal at the two receivers ∆τABi := |Xi − XB| − |Xi − XA| c (4) Quite similar to Very Long Baseline Interferometry (VLBI) which uses Quasars as radio sources. In Geodesy mostly: only code derived pseudoranges and carrier phase observations.
Parameter Estimation 8 / 105
Linear Combinations and Derived Observable 9 / 105 Carrier phases and code phases on both frequencies =⇒ pseudoranges
Linear Combinations and Derived Observable 9 / 105 Carrier phases and code phases on both frequencies =⇒ pseudoranges For elimination of errors in observables, formation of linear combinations of those observables
Linear Combinations and Derived Observable 9 / 105 Carrier phases and code phases on both frequencies =⇒ pseudoranges For elimination of errors in observables, formation of linear combinations of those observables Different kinds of linear combinations between observations ... • at different stations
Linear Combinations and Derived Observable 9 / 105 Carrier phases and code phases on both frequencies =⇒ pseudoranges For elimination of errors in observables, formation of linear combinations of those observables Different kinds of linear combinations between observations ... • at different stations • of different satellites
Linear Combinations and Derived Observable 9 / 105 Carrier phases and code phases on both frequencies =⇒ pseudoranges For elimination of errors in observables, formation of linear combinations of those observables Different kinds of linear combinations between observations ... • at different stations • of different satellites • at different epochs
Linear Combinations and Derived Observable 9 / 105 Carrier phases and code phases on both frequencies =⇒ pseudoranges For elimination of errors in observables, formation of linear combinations of those observables Different kinds of linear combinations between observations ... • at different stations • of different satellites • at different epochs • on different frequencies
10 / 105 Figure 1: satellite-receiver configuration for forming differences, Rc ab: pseudorange at receiver a and receiver b to the satellite c
11 / 105 Introduction of notations for differences: • between-receiver single differences ∆(•) := (•)receiver j − (•)receiver i (5)
11 / 105 Introduction of notations for differences: • between-receiver single differences ∆(•) := (•)receiver j − (•)receiver i (5) • between-satellite single differences ∇(•) := (•)satellite j − (•)satellite i (6)
11 / 105 Introduction of notations for differences: • between-receiver single differences ∆(•) := (•)receiver j − (•)receiver i (5) • between-satellite single differences ∇(•) := (•)satellite j − (•)satellite i (6) • between-epoch single differences δ(•) := (•)epoch 2 − (•)epoch 1 (7)
12 / 105 For code-phases between-receiver single differences read ∆PR p CDij = ∆R p ij + c(dtuj − dtui) + c(dtaj − dtai) + c(dtsp − dtsp) + ǫ. (8)
12 / 105 For code-phases between-receiver single differences read ∆PR p CDij = ∆R p ij + c(dtuj − dtui) + c(dtaj − dtai) + c(dtsp − dtsp) + ǫ. (8) Here, notations mean: • distance difference from satellite p to the two receivers i, j: ∆R p ij = |Xp − Xj| − |Xp − Xi|
12 / 105 For code-phases between-receiver single differences read ∆PR p CDij = ∆R p ij + c(dtuj − dtui) + c(dtaj − dtai) + c(dtsp − dtsp) + ǫ. (8) Here, notations mean: • distance difference from satellite p to the two receivers i, j: ∆R p ij = |Xp − Xj| − |Xp − Xi| • dtuk clock error of receiver k
12 / 105 For code-phases between-receiver single differences read ∆PR p CDij = ∆R p ij + c(dtuj − dtui) + c(dtaj − dtai) + c(dtsp − dtsp) + ǫ. (8) Here, notations mean: • distance difference from satellite p to the two receivers i, j: ∆R p ij = |Xp − Xj| − |Xp − Xi| • dtuk clock error of receiver k • dtak atmospheric delay of signal travelling from satellite p to receiver k
12 / 105 For code-phases between-receiver single differences read ∆PR p CDij = ∆R p ij + c(dtuj − dtui) + c(dtaj − dtai) + c(dtsp − dtsp) + ǫ. (8) Here, notations mean: • distance difference from satellite p to the two receivers i, j: ∆R p ij = |Xp − Xj| − |Xp − Xi| • dtuk clock error of receiver k • dtak atmospheric delay of signal travelling from satellite p to receiver k • dtsp clock error of the satellite p
13 / 105 • satellite clock error cancels out in the between-receiver single difference
13 / 105 • satellite clock error cancels out in the between-receiver single difference • single difference: only differential influence of atmospheric delay
13 / 105 • satellite clock error cancels out in the between-receiver single difference • single difference: only differential influence of atmospheric delay • for stations close together: dai ≈ daj =⇒ atmospheric delay cancels out.
13 / 105 • satellite clock error cancels out in the between-receiver single difference • single difference: only differential influence of atmospheric delay • for stations close together: dai ≈ daj =⇒ atmospheric delay cancels out. • the same is true for the differential effect of the orbital errors
13 / 105 • satellite clock error cancels out in the between-receiver single difference • single difference: only differential influence of atmospheric delay • for stations close together: dai ≈ daj =⇒ atmospheric delay cancels out. • the same is true for the differential effect of the orbital errors for carrier-phases between-receiver single differences read ∆PR p CRij = ∆R p ij + c(dtuj − dtui) + c(dtaj − dtai) +c(dtsp − dtsp) + λ(Ni − Nj) + ǫ. (9) Ni, Nj: unknown integer phase ambiguities in the undifferenced carrier phase pseudoranges from satellite p to receivers i, j
13 / 105 • satellite clock error cancels out in the between-receiver single difference • single difference: only differential influence of atmospheric delay • for stations close together: dai ≈ daj =⇒ atmospheric delay cancels out. • the same is true for the differential effect of the orbital errors for carrier-phases between-receiver single differences read ∆PR p CRij = ∆R p ij + c(dtuj − dtui) + c(dtaj − dtai) +c(dtsp − dtsp) + λ(Ni − Nj) + ǫ. (9) Ni, Nj: unknown integer phase ambiguities in the undifferenced carrier phase pseudoranges from satellite p to receivers i, j in single differences for phase pseudoranges the satellite clock error cancels out
13 / 105 • satellite clock error cancels out in the between-receiver single difference • single difference: only differential influence of atmospheric delay • for stations close together: dai ≈ daj =⇒ atmospheric delay cancels out. • the same is true for the differential effect of the orbital errors for carrier-phases between-receiver single differences read ∆PR p CRij = ∆R p ij + c(dtuj − dtui) + c(dtaj − dtai) +c(dtsp − dtsp) + λ(Ni − Nj) + ǫ. (9) Ni, Nj: unknown integer phase ambiguities in the undifferenced carrier phase pseudoranges from satellite p to receivers i, j in single differences for phase pseudoranges the satellite clock error cancels out atmospheric delay and orbital error influences the single difference only differentially
14 / 105 double differences: • usually formed between receivers and satellites
14 / 105 double differences: • usually formed between receivers and satellites • defined as the difference between two between-receiver single differences
14 / 105 double differences: • usually formed between receivers and satellites • defined as the difference between two between-receiver single differences double difference for code pseudoranges: ∇∆PR pq CDij := ∆PR q CDij − ∆PR p CDij = ∇∆R pq ij + c((dtuj − dtui) − (dtuj − dtui)) +c((dt q aj − dt q ai) − (dt p aj − dt p ai)) + ǫ
14 / 105 double differences: • usually formed between receivers and satellites • defined as the difference between two between-receiver single differences double difference for code pseudoranges: ∇∆PR pq CDij := ∆PR q CDij − ∆PR p CDij = ∇∆R pq ij + c((dtuj − dtui) − (dtuj − dtui)) +c((dt q aj − dt q ai) − (dt p aj − dt p ai)) + ǫ double difference for carrier phase pseudoranges: ∇∆PR pq CRij := ∆PR q CRij − ∆PR p CRij = ∇∆R pq ij + c((dtuj − dtui) − (dtuj − dtui)) +c((dt q aj − dt q ai) − (dt p aj − dt p ai)) +λ((N q j − N q i ) − (N p j − N p i )) + ǫ
15 / 105 double differences: • besides the satellite clock error also the receiver clock error drops out • for the differential effect of the atmospheric delay and the orbital error the same as for the single differences is true
15 / 105 double differences: • besides the satellite clock error also the receiver clock error drops out • for the differential effect of the atmospheric delay and the orbital error the same as for the single differences is true carrier phase pseudorange differences: • between different epochs the unknown phase ambiguity drops out • used as an auxiliary observation for determination of the phase ambiguities
15 / 105 double differences: • besides the satellite clock error also the receiver clock error drops out • for the differential effect of the atmospheric delay and the orbital error the same as for the single differences is true carrier phase pseudorange differences: • between different epochs the unknown phase ambiguity drops out • used as an auxiliary observation for determination of the phase ambiguities differences between different frequencies: • elimination of ionospheric delay for long baselines (where this effect doesn’t cancel out due to single or double differencing
16 / 105 • linear combination of phase observation: sum of carrier phase observations on both frequencies (previously multiplied by constant factors)
16 / 105 • linear combination of phase observation: sum of carrier phase observations on both frequencies (previously multiplied by constant factors) • phase can be given in radian or in metrical units: factors of the same linear combination differ for both representations
16 / 105 • linear combination of phase observation: sum of carrier phase observations on both frequencies (previously multiplied by constant factors) • phase can be given in radian or in metrical units: factors of the same linear combination differ for both representations • radian representation of a phase measurement: Φ, its metric representation: L.
16 / 105 • linear combination of phase observation: sum of carrier phase observations on both frequencies (previously multiplied by constant factors) • phase can be given in radian or in metrical units: factors of the same linear combination differ for both representations • radian representation of a phase measurement: Φ, its metric representation: L. phase linear combination of two carrier phases: Φn,m := nΦ1 + mΦ2, n, m ∈ Z (10)
16 / 105 • linear combination of phase observation: sum of carrier phase observations on both frequencies (previously multiplied by constant factors) • phase can be given in radian or in metrical units: factors of the same linear combination differ for both representations • radian representation of a phase measurement: Φ, its metric representation: L. phase linear combination of two carrier phases: Φn,m := nΦ1 + mΦ2, n, m ∈ Z (10) frequency of newly generated signal: ωn,m = nω1 + mω2 (11)
16 / 105 • linear combination of phase observation: sum of carrier phase observations on both frequencies (previously multiplied by constant factors) • phase can be given in radian or in metrical units: factors of the same linear combination differ for both representations • radian representation of a phase measurement: Φ, its metric representation: L. phase linear combination of two carrier phases: Φn,m := nΦ1 + mΦ2, n, m ∈ Z (10) frequency of newly generated signal: ωn,m = nω1 + mω2 (11) its wavelength: λn,m = c ωn,m . (12)
17 / 105 same linear combination in metrical units: Ln,m = xL1 + yL2 = x λ1 2π Φ1 + y λ2 2π Φ2. (13)
17 / 105 same linear combination in metrical units: Ln,m = xL1 + yL2 = x λ1 2π Φ1 + y λ2 2π Φ2. (13) relation between, n, m and x, y: x = nλn,m λ1 , y = mλn,m λ2 (14)
17 / 105 same linear combination in metrical units: Ln,m = xL1 + yL2 = x λ1 2π Φ1 + y λ2 2π Φ2. (13) relation between, n, m and x, y: x = nλn,m λ1 , y = mλn,m λ2 (14) Expression of linear combinations in radians or metric units: depending on the purpose.
17 / 105 same linear combination in metrical units: Ln,m = xL1 + yL2 = x λ1 2π Φ1 + y λ2 2π Φ2. (13) relation between, n, m and x, y: x = nλn,m λ1 , y = mλn,m λ2 (14) Expression of linear combinations in radians or metric units: depending on the purpose. Original intention of the introduction of two frequencies in the GPS systems: elimination of the ionospheric time delay.
17 / 105 same linear combination in metrical units: Ln,m = xL1 + yL2 = x λ1 2π Φ1 + y λ2 2π Φ2. (13) relation between, n, m and x, y: x = nλn,m λ1 , y = mλn,m λ2 (14) Expression of linear combinations in radians or metric units: depending on the purpose. Original intention of the introduction of two frequencies in the GPS systems: elimination of the ionospheric time delay. How influences the ionospheric time delay the artificial frequencies?
18 / 105 Linear approximation of ionospheric phase delay on a frequency ω: δΦ = − Cne ω2 , (15) C = 40.3 and ne (unknown): characterises electron density in the ionosphere.
18 / 105 Linear approximation of ionospheric phase delay on a frequency ω: δΦ = − Cne ω2 , (15) C = 40.3 and ne (unknown): characterises electron density in the ionosphere. Ionospheric phase delay on the artificial frequency ωn,m: δΦn,m = nδΦ1 + mδΦ2 = −Cne( n ω2 1 + m ω2 2 ) = − Cne ω2 1ω2 2 (nω2 2 + mω2 1) ≈ − CI ω1ω2 (nω2 + mω1). (16)
19 / 105 Influence of the phase delay on frequency ωn,m on the pseudorange on this frequency: δLn,m = λn,m 2π δΦn,m = − CIc 2πω1ω2 nω2 + mω1 nω1 + mω2 (17)
19 / 105 Influence of the phase delay on frequency ωn,m on the pseudorange on this frequency: δLn,m = λn,m 2π δΦn,m = − CIc 2πω1ω2 nω2 + mω1 nω1 + mω2 (17) Phase noise changes when artificial frequencies are built: σn,m = λn,m 2π σΦn,m = λn,m 2π n2 + m2σΦ (18)
19 / 105 Influence of the phase delay on frequency ωn,m on the pseudorange on this frequency: δLn,m = λn,m 2π δΦn,m = − CIc 2πω1ω2 nω2 + mω1 nω1 + mω2 (17) Phase noise changes when artificial frequencies are built: σn,m = λn,m 2π σΦn,m = λn,m 2π n2 + m2σΦ (18) After these preparations the most common frequency combinations can be considered.
wide-lane combination 20 / 105 defined by: L∆ := λ∆ 2π Φ1,−1 = c 2π(ω1 − ω2) ( 2π λ1 L1 − 2π λ2 L2) = ω1 ω1 − ω2 L1 − ω2 ω1 − ω2 L2 (19)
21 / 105 Ionospheric phase delay on the wide-lane: δΦ∆ = − CI ω1ω2 (ω2 − ω1) (20)
21 / 105 Ionospheric phase delay on the wide-lane: δΦ∆ = − CI ω1ω2 (ω2 − ω1) (20) relates to a pseudorange change of: δL∆ = − CIc ω1ω2 2πω2 − ω1 ω1 − ω2 = CIc 2πω1ω2 (21)
21 / 105 Ionospheric phase delay on the wide-lane: δΦ∆ = − CI ω1ω2 (ω2 − ω1) (20) relates to a pseudorange change of: δL∆ = − CIc ω1ω2 2πω2 − ω1 ω1 − ω2 = CIc 2πω1ω2 (21) Wavelength for wide-lane combination: λ∆ = 86 cm (∼ four times the original wavelength).
21 / 105 Ionospheric phase delay on the wide-lane: δΦ∆ = − CI ω1ω2 (ω2 − ω1) (20) relates to a pseudorange change of: δL∆ = − CIc ω1ω2 2πω2 − ω1 ω1 − ω2 = CIc 2πω1ω2 (21) Wavelength for wide-lane combination: λ∆ = 86 cm (∼ four times the original wavelength). The phase noise increases from about 3 mm at the L1, L2 frequencies to σ∆ = 19.4 mm.
narrow-lane combination 22 / 105 defined by: LΣ := λΣ 2π Φ1,1 = c 2π(ω1 + ω2) ( 2π λ1 L1 + 2π λ2 L2) = ω1 ω1 + ω2 L1 + ω2 ω1 + ω2 L2 (22)
23 / 105 Ionospheric phase delay on the narrow-lane: δΦΣ = − CI ω1ω2 (ω2 + ω1) (23)
23 / 105 Ionospheric phase delay on the narrow-lane: δΦΣ = − CI ω1ω2 (ω2 + ω1) (23) relates to a pseudorange change of: δLΣ = − CIc ω1ω2 ω2 + ω1 ω1 + ω2 = − CIc 2πω1ω2 . (24)
23 / 105 Ionospheric phase delay on the narrow-lane: δΦΣ = − CI ω1ω2 (ω2 + ω1) (23) relates to a pseudorange change of: δLΣ = − CIc ω1ω2 ω2 + ω1 ω1 + ω2 = − CIc 2πω1ω2 . (24) =⇒ Influence of the ionospheric delay has exact the same magnitude on the wide- and on the narrow lane, they only differ in the sign.
23 / 105 Ionospheric phase delay on the narrow-lane: δΦΣ = − CI ω1ω2 (ω2 + ω1) (23) relates to a pseudorange change of: δLΣ = − CIc ω1ω2 ω2 + ω1 ω1 + ω2 = − CIc 2πω1ω2 . (24) =⇒ Influence of the ionospheric delay has exact the same magnitude on the wide- and on the narrow lane, they only differ in the sign. =⇒ Elimination of the ionospheric range delay by computing the mean of the wide- and of the narrow lane combination. Resulting combination: ionosphere-free combination.
23 / 105 Ionospheric phase delay on the narrow-lane: δΦΣ = − CI ω1ω2 (ω2 + ω1) (23) relates to a pseudorange change of: δLΣ = − CIc ω1ω2 ω2 + ω1 ω1 + ω2 = − CIc 2πω1ω2 . (24) =⇒ Influence of the ionospheric delay has exact the same magnitude on the wide- and on the narrow lane, they only differ in the sign. =⇒ Elimination of the ionospheric range delay by computing the mean of the wide- and of the narrow lane combination. Resulting combination: ionosphere-free combination. Wavelength on the narrow-lane: λΣ = 10.7 mm
23 / 105 Ionospheric phase delay on the narrow-lane: δΦΣ = − CI ω1ω2 (ω2 + ω1) (23) relates to a pseudorange change of: δLΣ = − CIc ω1ω2 ω2 + ω1 ω1 + ω2 = − CIc 2πω1ω2 . (24) =⇒ Influence of the ionospheric delay has exact the same magnitude on the wide- and on the narrow lane, they only differ in the sign. =⇒ Elimination of the ionospheric range delay by computing the mean of the wide- and of the narrow lane combination. Resulting combination: ionosphere-free combination. Wavelength on the narrow-lane: λΣ = 10.7 mm Phase noise reduces to σΣ = 2.1 mm
ionosphere-free combination L3 24 / 105 defined by: L3 := L∆ + LΣ 2 = 1 2 ( ω1 ω1 − ω2 + ω1 ω1 + ω2 )L1 + ( ω2 ω1 + ω2 − ω2 ω1 − ω2 )L2 = 1 2 ω1(ω1 + ω2) + ω1(ω1 − ω2) ω2 1 − ω2 2 L1 + ω2(ω1 − ω2) − ω2(ω1 + ω2) ω2 1 − ω2 2 L2 = ω2 1 ω2 1 − ω2 2 L1 − ω2 2 ω2 1 − ω2 2 L2 (25)
ionosphere-free combination L3 24 / 105 defined by: L3 := L∆ + LΣ 2 = 1 2 ( ω1 ω1 − ω2 + ω1 ω1 + ω2 )L1 + ( ω2 ω1 + ω2 − ω2 ω1 − ω2 )L2 = 1 2 ω1(ω1 + ω2) + ω1(ω1 − ω2) ω2 1 − ω2 2 L1 + ω2(ω1 − ω2) − ω2(ω1 + ω2) ω2 1 − ω2 2 L2 = ω2 1 ω2 1 − ω2 2 L1 − ω2 2 ω2 1 − ω2 2 L2 (25) Due to the non-integer nature of the factors in the ionosphere free combination no wavelength can be assigned to L3.
geometry-free combination 25 / 105 defined by: L4 = LI := LΣ − L∆ = 2ω1ω2 ω2 1 − ω2 2 [L2 − L1] (26)
geometry-free combination 25 / 105 defined by: L4 = LI := LΣ − L∆ = 2ω1ω2 ω2 1 − ω2 2 [L2 − L1] (26) free from the influence of • orbital errors
geometry-free combination 25 / 105 defined by: L4 = LI := LΣ − L∆ = 2ω1ω2 ω2 1 − ω2 2 [L2 − L1] (26) free from the influence of • orbital errors • errors in the initial position of the receiver
geometry-free combination 25 / 105 defined by: L4 = LI := LΣ − L∆ = 2ω1ω2 ω2 1 − ω2 2 [L2 − L1] (26) free from the influence of • orbital errors • errors in the initial position of the receiver • clock errors
geometry-free combination 25 / 105 defined by: L4 = LI := LΣ − L∆ = 2ω1ω2 ω2 1 − ω2 2 [L2 − L1] (26) free from the influence of • orbital errors • errors in the initial position of the receiver • clock errors since these errors occur in the L1 and the L2 frequency and cancel out in this linear combination.
geometry-free combination 25 / 105 defined by: L4 = LI := LΣ − L∆ = 2ω1ω2 ω2 1 − ω2 2 [L2 − L1] (26) free from the influence of • orbital errors • errors in the initial position of the receiver • clock errors since these errors occur in the L1 and the L2 frequency and cancel out in this linear combination. Geometry-free combination contains twice the influence of the ionosphere =⇒ used for ionosphere modelling.
Navigation Solutions 26 / 105
27 / 105 • determination of the position of a GPS receiver (regardless to the positions of other receivers also observing the same satellites)
27 / 105 • determination of the position of a GPS receiver (regardless to the positions of other receivers also observing the same satellites) • practically only pseudoranges from code measurements are used
27 / 105 • determination of the position of a GPS receiver (regardless to the positions of other receivers also observing the same satellites) • practically only pseudoranges from code measurements are used At n epochs t1, . . . , tn the k satellites above the radio horizon are observed. These observations result in N = k · n observation equations. PRi(tj) = |Xi(tj) − xB| + cdtu,i, i = 1, . . . , k, j = 1, . . . , n. (27)
27 / 105 • determination of the position of a GPS receiver (regardless to the positions of other receivers also observing the same satellites) • practically only pseudoranges from code measurements are used At n epochs t1, . . . , tn the k satellites above the radio horizon are observed. These observations result in N = k · n observation equations. PRi(tj) = |Xi(tj) − xB| + cdtu,i, i = 1, . . . , k, j = 1, . . . , n. (27) cdtu,i contains all systematic errors such as • receiver clock error • satellite clock error • atmospheric delay • ionospheric delay
28 / 105 Observation equations (27) are nonlinear in the unknown receiver coordinates xB,m because of |Xi(tj) − xB| = 3 ∑ m=1 (Xi,m(tj) − xB,m)2
28 / 105 Observation equations (27) are nonlinear in the unknown receiver coordinates xB,m because of |Xi(tj) − xB| = 3 ∑ m=1 (Xi,m(tj) − xB,m)2 For linearization of the observation equations a prior information x (0) B about the position is required.
28 / 105 Observation equations (27) are nonlinear in the unknown receiver coordinates xB,m because of |Xi(tj) − xB| = 3 ∑ m=1 (Xi,m(tj) − xB,m)2 For linearization of the observation equations a prior information x (0) B about the position is required. Difference ∆x := xB − x (0) B is supposed to be small enough to neglect quadratic or higher order terms in ∆x
29 / 105 =⇒ linearized observation equations: PRi(tj) − |Xi(tj) − x (0) B | = ∂|Xi(tj) − xB| ∂xB · ∆x + cdtu,i (28) i = 1, . . . , k, j = 1, . . . , n
29 / 105 =⇒ linearized observation equations: PRi(tj) − |Xi(tj) − x (0) B | = ∂|Xi(tj) − xB| ∂xB · ∆x + cdtu,i (28) i = 1, . . . , k, j = 1, . . . , n N equations for 3 + k unknowns ∆x and dtu,1, . . . , dtu,k.
29 / 105 =⇒ linearized observation equations: PRi(tj) − |Xi(tj) − x (0) B | = ∂|Xi(tj) − xB| ∂xB · ∆x + cdtu,i (28) i = 1, . . . , k, j = 1, . . . , n N equations for 3 + k unknowns ∆x and dtu,1, . . . , dtu,k. ∂|Xi(tj) − xB| ∂xB = − Xi(tj) − x (0) B |Xi(tj) − x (0) B | holds, with the following definitions...
30 / 105 Y :=                         PR1(t1) − |X1(t1) − x (0) B | ... PRk(t1) − |Xk(t1) − x (0) B | PR1(t2) − |X1(t2) − x (0) B | ... PRk(t2) − |Xk(t2) − x (0) B | ... ... PR1(tn) − |X1(tn) − x (0) B | ... PRk(tn) − |Xk(tn) − x (0) B |                         ,
31 / 105 A :=                                − X1,1(t1)−x (0) B,1 |X1(t1)−x (0) B | − X1,2(t1)−x (0) B,2 |X1(t1)−x (0) B | − X1,3(t1)−x (0) B,3 |X1(t1)−x (0) B | c . . . 0 ... − Xk,1(t1)−x (0) B,1 |Xk(t1)−x (0) B | − Xk,2(t1)−x (0) B,2 |Xk(t1)−x (0) B | − Xk,3(t1)−x (0) B,3 |Xk(t1)−x (0) B | 0 . . . c − X1,1(t2)−x (0) B,1 |X1(t2)−x (0) B | − X1,2(t2)−x (0) B,2 |X1(t2)−x (0) B | − X1,3(t2)−x (0) B,3 |X1(t2)−x (0) B | c . . . 0 ... − Xk,1(t2)−x (0) B,1 |Xk(t2)−x (0) B | − Xk,2(t2)−x (0) B,2 |Xk(t2)−x (0) B | − Xk,3(t2)−x (0) B,3 |Xk(t2)−x (0) B | 0 . . . c ... ... − X1,1(tn)−x (0) B,1 |X1(tn)−x (0) B | − X1,2(tn)−x (0) B,2 |X1(tn)−x (0) B | − X1,3(tn)−x (0) B,3 |X1(tn)−x (0) B | c . . . 0 ... − Xk,1(tn)−x (0) B,1 |Xk(tn)−x (0) B | − Xk,2(tn)−x (0) B,2 |Xk(tn)−x (0) B | − Xk,3(tn)−x (0) B,3 |Xk(tn)−x (0) B | 0 . . . c                               
32 / 105 and ∆x := (x1,B − x (0) 1,B, x2,B − x (0) 2,B, x3,B − x (0) 3,B, dtu,1, . . . , dtu,k)⊤
32 / 105 and ∆x := (x1,B − x (0) 1,B, x2,B − x (0) 2,B, x3,B − x (0) 3,B, dtu,1, . . . , dtu,k)⊤ matrix notation of euqation (28): usual Gauss-Markov model of linear statistics E{Y} = A · ∆x, CYY = σ2 I
33 / 105 Usual least-squares estimation of position correction and error terms: ∆x = A⊤ A −1 A⊤ Y (29)
33 / 105 Usual least-squares estimation of position correction and error terms: ∆x = A⊤ A −1 A⊤ Y (29) with accuracy estimation: σ2 = | Y − A · ∆x 2 n · k − (3 + k) (30)
33 / 105 Usual least-squares estimation of position correction and error terms: ∆x = A⊤ A −1 A⊤ Y (29) with accuracy estimation: σ2 = | Y − A · ∆x 2 n · k − (3 + k) (30) and C∆x,∆x = σ2 A⊤ A −1 (31)
34 / 105 A few remarks have to be made: • at least 4 satellites at 2 epochs have to be observed: number n · k of observations has to exceed the number 3 + k of unknowns
34 / 105 A few remarks have to be made: • at least 4 satellites at 2 epochs have to be observed: number n · k of observations has to exceed the number 3 + k of unknowns • atmospheric and ionospheric delay are elevation dependent. =⇒ variation with time but variation is slow. For few observation epochs (typical for navigation solutions) delays can be considered constant =⇒ lumped error parameters dtu,i are only satellite, but not epoch dependent
34 / 105 A few remarks have to be made: • at least 4 satellites at 2 epochs have to be observed: number n · k of observations has to exceed the number 3 + k of unknowns • atmospheric and ionospheric delay are elevation dependent. =⇒ variation with time but variation is slow. For few observation epochs (typical for navigation solutions) delays can be considered constant =⇒ lumped error parameters dtu,i are only satellite, but not epoch dependent • systematic propagation- and clock errors modelled by nuisance parameters =⇒ remaining code phase errors can be considered uncorrelated
Satellite Geometry and Accuracy Measures 35 / 105 Accuracy of determined position for navigation solutions depends on two factors: 1. variance σ2 of a code-pseudorange observation 2. geometric configuration of observed satellites.
Satellite Geometry and Accuracy Measures 35 / 105 Accuracy of determined position for navigation solutions depends on two factors: 1. variance σ2 of a code-pseudorange observation 2. geometric configuration of observed satellites. Relation between standard deviation σ of a pseudorange observation and standard deviation of determined position σ∗: dilution of precision (DOP) σ∗ = DOP · σ. (32)
36 / 105 Usage of different DOP designations: • σH = HDOP · σ for horizontal positioning
36 / 105 Usage of different DOP designations: • σH = HDOP · σ for horizontal positioning • σV = VDOP · σ for vertical positioning
36 / 105 Usage of different DOP designations: • σH = HDOP · σ for horizontal positioning • σV = VDOP · σ for vertical positioning • σP = PDOP · σ for three-dimensional positioning
36 / 105 Usage of different DOP designations: • σH = HDOP · σ for horizontal positioning • σV = VDOP · σ for vertical positioning • σP = PDOP · σ for three-dimensional positioning • σT = TDOP · σ for timing
36 / 105 Usage of different DOP designations: • σH = HDOP · σ for horizontal positioning • σV = VDOP · σ for vertical positioning • σP = PDOP · σ for three-dimensional positioning • σT = TDOP · σ for timing Combined effect for three-dimensional positioning and timing: GDOP = PDOP2 + TDOP2. (33)
36 / 105 Usage of different DOP designations: • σH = HDOP · σ for horizontal positioning • σV = VDOP · σ for vertical positioning • σP = PDOP · σ for three-dimensional positioning • σT = TDOP · σ for timing Combined effect for three-dimensional positioning and timing: GDOP = PDOP2 + TDOP2. (33) Intuitive interpretation for PDOP: clock errors are supposed to be eliminated =⇒ one single epoch needed for positioning
37 / 105 Reduced observation equations: E{Y} = A · ∆x, CYY = σ2 I
37 / 105 Reduced observation equations: E{Y} = A · ∆x, CYY = σ2 I with Y :=    PR1(t1) − |X1(t1) − x (0) B | ... PRk(t1) − |Xk(t1) − x (0) B |   
37 / 105 Reduced observation equations: E{Y} = A · ∆x, CYY = σ2 I with Y :=    PR1(t1) − |X1(t1) − x (0) B | ... PRk(t1) − |Xk(t1) − x (0) B |    A :=        − X1,1(t1)−x (0) B,1 |X1(t1)−x (0) B | − X1,2(t1)−x (0) B,2 |X1(t1)−x (0) B | − X1,3(t1)−x (0) B,3 |X1(t1)−x (0) B | ... − Xk,1(t1)−x (0) B,1 |Xk(t1)−x (0) B | − Xk,2(t1)−x (0) B,2 |Xk(t1)−x (0) B | − Xk,3(t1)−x (0) B,3 |Xk(t1)−x (0) B |       
37 / 105 Reduced observation equations: E{Y} = A · ∆x, CYY = σ2 I with Y :=    PR1(t1) − |X1(t1) − x (0) B | ... PRk(t1) − |Xk(t1) − x (0) B |    A :=        − X1,1(t1)−x (0) B,1 |X1(t1)−x (0) B | − X1,2(t1)−x (0) B,2 |X1(t1)−x (0) B | − X1,3(t1)−x (0) B,3 |X1(t1)−x (0) B | ... − Xk,1(t1)−x (0) B,1 |Xk(t1)−x (0) B | − Xk,2(t1)−x (0) B,2 |Xk(t1)−x (0) B | − Xk,3(t1)−x (0) B,3 |Xk(t1)−x (0) B |        ∆x := (x1,B − x (0) 1,B, x2,B − x (0) 2,B, x3,B − x (0) 3,B)⊤
38 / 105 Estimated position corrections: ∆x = A⊤ A −1 A⊤ Y
38 / 105 Estimated position corrections: ∆x = A⊤ A −1 A⊤ Y Accuracy: C∆x,∆x = σ2 A⊤ A −1
38 / 105 Estimated position corrections: ∆x = A⊤ A −1 A⊤ Y Accuracy: C∆x,∆x = σ2 A⊤ A −1 Variance of estimated three-dimensional position (neglecting all cross-correlations): σ2 P = σ2 ∆x1 + σ2 ∆x2 + σ2 ∆x3 = σ2 trace A⊤ A −1
38 / 105 Estimated position corrections: ∆x = A⊤ A −1 A⊤ Y Accuracy: C∆x,∆x = σ2 A⊤ A −1 Variance of estimated three-dimensional position (neglecting all cross-correlations): σ2 P = σ2 ∆x1 + σ2 ∆x2 + σ2 ∆x3 = σ2 trace A⊤ A −1 =⇒ PDOP = trace A⊤A −1
39 / 105 Geometrical meaning of trace A⊤A −1 explained best in 2D: simultaneous observed pseudoranges to two satellites sufficient to determine the position. PR r1 r 2 PR α 1 2 Figure 2: geometrical satellite configuration
40 / 105 In this 2D one-epoch example the observation matrix consists of the direction unit vectors pointing to the satellites: A = −r1 −r2 (34)
40 / 105 In this 2D one-epoch example the observation matrix consists of the direction unit vectors pointing to the satellites: A = −r1 −r2 (34) Normal equation matrix: A⊤ A = r2 1,x + r2 2,x r1,xr1,y + r2,xr2,y r1,xr1,y + r2,xr2,y r2 1,y + r2 2,y (35)
40 / 105 In this 2D one-epoch example the observation matrix consists of the direction unit vectors pointing to the satellites: A = −r1 −r2 (34) Normal equation matrix: A⊤ A = r2 1,x + r2 2,x r1,xr1,y + r2,xr2,y r1,xr1,y + r2,xr2,y r2 1,y + r2 2,y (35) Determinant of the normal equation matrix: det A⊤ A = (r2 1,x + r2 2,x)(r2 1,y + r2 2,y) − (r1,xr1,y + r2,xr2,y)2 = (r1,xr2,y − r2,xr1,y)2 = sin2 α
41 / 105 Inverse of the normal equation matrix (computed by Schreibers rule): A⊤ A −1 = 1 sin2 α r2 1,y + r2 2,y −(r1,xr1,y + r2,xr2,y) −(r1,xr1,y + r2,xr2,y) r2 1,x + r2 2,x . (36)
41 / 105 Inverse of the normal equation matrix (computed by Schreibers rule): A⊤ A −1 = 1 sin2 α r2 1,y + r2 2,y −(r1,xr1,y + r2,xr2,y) −(r1,xr1,y + r2,xr2,y) r2 1,x + r2 2,x . (36) =⇒ PDOP: PDOP = trace A⊤A −1 (37) = 1 sin α r2 1,y + r2 2,y + r2 1,x + r2 2,x = √ 2 sin α
42 / 105 Area of the triangle spanned by the two unit vectors r1, r2: A = 1 2 sin α
42 / 105 Area of the triangle spanned by the two unit vectors r1, r2: A = 1 2 sin α =⇒ PDOP indirectly proportional to area spanned by the satellites: PDOP ∼ 1 A (38)
42 / 105 Area of the triangle spanned by the two unit vectors r1, r2: A = 1 2 sin α =⇒ PDOP indirectly proportional to area spanned by the satellites: PDOP ∼ 1 A (38) 2D −→ 3D: PDOP ∼ 1 V (39) V: volume of polyhedron spanned by the observer and the satellites.
43 / 105 P good PDOP P bad PDOP Figure 3: Good and bad satellite configuration
Baseline Solution 44 / 105
45 / 105 Navigation solution: limited in its accuracy (primarily relied on code phase measurements).
45 / 105 Navigation solution: limited in its accuracy (primarily relied on code phase measurements). For improvement of the accuracy two measurements: • use of carrier phases instead of code phases
45 / 105 Navigation solution: limited in its accuracy (primarily relied on code phase measurements). For improvement of the accuracy two measurements: • use of carrier phases instead of code phases • computation of baseline solutions instead of navigation, or single-point solutions
46 / 105 Basic idea of the baseline solution: • Coordinates of a baseline vector instead of coordinates of a point
46 / 105 Basic idea of the baseline solution: • Coordinates of a baseline vector instead of coordinates of a point • Baseline vector connects a known reference point with the point under consideration.
46 / 105 Basic idea of the baseline solution: • Coordinates of a baseline vector instead of coordinates of a point • Baseline vector connects a known reference point with the point under consideration. • Vector enters the observation model by forming differences of observations between a receiver at the reference point and a receiver at the current point.
46 / 105 Basic idea of the baseline solution: • Coordinates of a baseline vector instead of coordinates of a point • Baseline vector connects a known reference point with the point under consideration. • Vector enters the observation model by forming differences of observations between a receiver at the reference point and a receiver at the current point. =⇒ errors cancel out or are reduced in magnitude.
46 / 105 Basic idea of the baseline solution: • Coordinates of a baseline vector instead of coordinates of a point • Baseline vector connects a known reference point with the point under consideration. • Vector enters the observation model by forming differences of observations between a receiver at the reference point and a receiver at the current point. =⇒ errors cancel out or are reduced in magnitude. =⇒ remaining errors are small enough: determination of the integer phase ambiguities
46 / 105 Basic idea of the baseline solution: • Coordinates of a baseline vector instead of coordinates of a point • Baseline vector connects a known reference point with the point under consideration. • Vector enters the observation model by forming differences of observations between a receiver at the reference point and a receiver at the current point. =⇒ errors cancel out or are reduced in magnitude. =⇒ remaining errors are small enough: determination of the integer phase ambiguities =⇒ exploit the full accuracy potential of the observed carrier phases
46 / 105 Basic idea of the baseline solution: • Coordinates of a baseline vector instead of coordinates of a point • Baseline vector connects a known reference point with the point under consideration. • Vector enters the observation model by forming differences of observations between a receiver at the reference point and a receiver at the current point. =⇒ errors cancel out or are reduced in magnitude. =⇒ remaining errors are small enough: determination of the integer phase ambiguities =⇒ exploit the full accuracy potential of the observed carrier phases Baseline solutions can be computed on all frequency combinations. Special role of L3 combination: phase ambiguities lose their integer nature.
Single Differences Solution 47 / 105 Observation of k satellites at n epochs from two receivers – reference receiver r and rover receiver v.
Single Differences Solution 47 / 105 Observation of k satellites at n epochs from two receivers – reference receiver r and rover receiver v. Single differences observation equations (on an arbitrary frequency combination): ∆PR p CRrv(tj) = ∆R p rv(tj) + c(dtur − dtuv) + c(dtar − dtav)(tj) +λ(Nr − Nv) + ǫ (40) j = 1, . . . , n, p = 1, . . . , k.
48 / 105 ∆R p rv(tj) = 3 ∑ l=1 (X p l (tj) − xr,l)2 − 3 ∑ l=1 (X p l (tj) − xv,l)2 (41)
48 / 105 ∆R p rv(tj) = 3 ∑ l=1 (X p l (tj) − xr,l)2 − 3 ∑ l=1 (X p l (tj) − xv,l)2 (41) =⇒ single differences observation equations are nonlinear in the unknown coordinates xv of the rover receiver v.
48 / 105 ∆R p rv(tj) = 3 ∑ l=1 (X p l (tj) − xr,l)2 − 3 ∑ l=1 (X p l (tj) − xv,l)2 (41) =⇒ single differences observation equations are nonlinear in the unknown coordinates xv of the rover receiver v. (Coordinates of the reference receiver are assumed to be known.)
48 / 105 ∆R p rv(tj) = 3 ∑ l=1 (X p l (tj) − xr,l)2 − 3 ∑ l=1 (X p l (tj) − xv,l)2 (41) =⇒ single differences observation equations are nonlinear in the unknown coordinates xv of the rover receiver v. (Coordinates of the reference receiver are assumed to be known.) For the linearization a prior information x0 v about the position of the rover is required.
48 / 105 ∆R p rv(tj) = 3 ∑ l=1 (X p l (tj) − xr,l)2 − 3 ∑ l=1 (X p l (tj) − xv,l)2 (41) =⇒ single differences observation equations are nonlinear in the unknown coordinates xv of the rover receiver v. (Coordinates of the reference receiver are assumed to be known.) For the linearization a prior information x0 v about the position of the rover is required. =⇒ linearized single differences observation equations: ∆PR p CRrv(tj) − ∆R p,0 rv (tj) = ∂∆R p rv ∂xv (tj) · ∆xv + c(dtur − dtuv) +c(dtar − dtav)(tj) + λ(Nr − Nv) +ǫ. (42)
49 / 105 Computed carrier phase single difference: ∆R p,0 rv (tj) = 3 ∑ l=1 (X p l (tj) − xr,l)2 − 3 ∑ l=1 (X p l (tj) − x0 v,l)2 (43)
49 / 105 Computed carrier phase single difference: ∆R p,0 rv (tj) = 3 ∑ l=1 (X p l (tj) − xr,l)2 − 3 ∑ l=1 (X p l (tj) − x0 v,l)2 (43) Partial derivatives have the form: ∂∆R p rv ∂xv (tj) = Xp − x0 v |Xp − x0 v| (44)
50 / 105 Unknown position corrections: difference between the true but unknown position of the rover and the available prior information about this position: ∆xv = xv − x0 v (45)
50 / 105 Unknown position corrections: difference between the true but unknown position of the rover and the available prior information about this position: ∆xv = xv − x0 v (45) Among the remaining terms of the single differences observation equations two groups can be distinguished: • time independent terms like phase ambiguities, clock errors • time dependent atmospheric and ionospheric propagation-delay terms
51 / 105 Time independent terms remain in the observation equations as they are.
51 / 105 Time independent terms remain in the observation equations as they are. Modelling of atmospheric and ionospheric terms: c(dtar − dtav)(tj) ≈ M(tj, T, p, H) = 0 , short baselines m(tj, T, p, H) , long baselines m(tj, T, p, H): standard model for atmospheric and ionospheric propagation delay
51 / 105 Time independent terms remain in the observation equations as they are. Modelling of atmospheric and ionospheric terms: c(dtar − dtav)(tj) ≈ M(tj, T, p, H) = 0 , short baselines m(tj, T, p, H) , long baselines m(tj, T, p, H): standard model for atmospheric and ionospheric propagation delay re-arrangement of the terms: ∆PR p CRrv(tj) − ∆R p,0 rv (tj) − M(tj, T, p, H) = ∂∆R p rv ∂xv (tj) · ∆xv + c(dt p ur − dt p uv) + λ(N p r − N p v ) + ǫ
52 / 105 Individual terms are collected in vectors and in matrices: Y =                        ∆PR1 CRrv(t1) − ∆R1,0 rv (t1) − M(t1, T, p, H) ... ∆PRk CRrv(t1) − ∆Rk,0 rv (t1) − M(t1, T, p, H) ∆PR1 CRrv(t2) − ∆R1,0 rv (t2) − M(t2, T, p, H) ... ∆PRk CRrv(t2) − ∆Rk,0 rv (t2) − M(t2, T, p, H) ... ... ∆PR1 CRrv(tn) − ∆R1,0 rv (tn) − M(tn, T, p, H) ... ∆PRk CRrv(tn) − ∆Rk,0 rv (tn) − M(tn, T, p, H)                       
53 / 105 A =                               X1 1−x0 v,1 |X1−x0 v| (t1) X1 2−x0 v,2 |X1−x0 v| (t1) X1 3−x0 v,3 |X1−x0 v| (t1) c λ . . . 0 ... Xk 1−x0 v,1 |Xk−x0 v| (t1) Xk 2−x0 v,2 |Xk−x0 v| (t1) Xk 3−x0 v,3 |Xk−x0 v| (t1) c 0 . . . λ X1 1−x0 v,1 |X1−x0 v| (t2) X1 2−x0 v,2 |X1−x0 v| (t2) X1 3−x0 v,3 |X1−x0 v| (t2) c λ . . . 0 ... Xk 1−x0 v,1 |Xk−x0 v| (t2) Xk 2−x0 v,2 |Xk−x0 v| (t2) Xk 3−x0 v,3 |Xk−x0 v| (t2) c 0 . . . λ ... ... X1 1−x0 v,1 |X1−x0 v| (tn) X1 2−x0 v,2 |X1−x0 v| (tn) X1 3−x0 v,3 |X1−x0 v| (tn) c λ . . . 0 ... Xk 1−x0 v,1 |Xk−x0 v| (tn) Xk 2−x0 v,2 |Xk−x0 v| (tn) Xk 3−x0 v,3 |Xk−x0 v| (tn) c 0 . . . λ                              
54 / 105 ∆xv = (xv,1 − x0 v,1, xv,2 − x0 v,2, xv,3 − x0 v,3, dtur − dtuv, N1 r − N1 v , . . . , Nk r − Nk v)⊤ .
54 / 105 ∆xv = (xv,1 − x0 v,1, xv,2 − x0 v,2, xv,3 − x0 v,3, dtur − dtuv, N1 r − N1 v , . . . , Nk r − Nk v)⊤ . =⇒ Gauss-Markov form of the linearized single differences observation equations: E{Y} = A · ∆x. (46)
54 / 105 ∆xv = (xv,1 − x0 v,1, xv,2 − x0 v,2, xv,3 − x0 v,3, dtur − dtuv, N1 r − N1 v , . . . , Nk r − Nk v)⊤ . =⇒ Gauss-Markov form of the linearized single differences observation equations: E{Y} = A · ∆x. (46) Consider separately: covariance matrix CYY of the single difference observations.
54 / 105 ∆xv = (xv,1 − x0 v,1, xv,2 − x0 v,2, xv,3 − x0 v,3, dtur − dtuv, N1 r − N1 v , . . . , Nk r − Nk v)⊤ . =⇒ Gauss-Markov form of the linearized single differences observation equations: E{Y} = A · ∆x. (46) Consider separately: covariance matrix CYY of the single difference observations. Undifferenced phase observations are uncorrelated, but forming differences may generate correlations of the single difference observations.
55 / 105 Phases p, q of the satellites observed at the reference station r and at the rover station v at the epoch tj: Ψ(tj) := (Ψ p r (tj), Ψ p v(tj), Ψ q r (tj), Ψ q v(tj))⊤
55 / 105 Phases p, q of the satellites observed at the reference station r and at the rover station v at the epoch tj: Ψ(tj) := (Ψ p r (tj), Ψ p v(tj), Ψ q r (tj), Ψ q v(tj))⊤ Observations are uncorrelated: CΨΨ = σ2 I (47)
55 / 105 Phases p, q of the satellites observed at the reference station r and at the rover station v at the epoch tj: Ψ(tj) := (Ψ p r (tj), Ψ p v(tj), Ψ q r (tj), Ψ q v(tj))⊤ Observations are uncorrelated: CΨΨ = σ2 I (47) Formation of two single differences from these four undifferenced phase observations: ∆Ψ = Ψ p r (tj) − Ψ p v(tj) Ψ q r (tj) − Ψ q v(tj) . (48)
56 / 105 Establish a connection between ∆Ψ and Ψ: ∆Ψ = D · Ψ. D = 1 −1 0 0 0 0 1 −1
56 / 105 Establish a connection between ∆Ψ and Ψ: ∆Ψ = D · Ψ. D = 1 −1 0 0 0 0 1 −1 Laws of covariance propagation =⇒ covariance matrix of single differences: C∆Ψ∆Ψ = DCΨΨD⊤ = σ2 DD⊤ = σ2 2 0 0 2 = 2σ2 I.
57 / 105 =⇒ For the same epoch single differences to different satellites: • uncorrelated • twice the variance of the single difference observation.
57 / 105 =⇒ For the same epoch single differences to different satellites: • uncorrelated • twice the variance of the single difference observation. Single differences to the same satellite at different epochs: • uncorrelated • twice the variance of the undifferenced observations.
57 / 105 =⇒ For the same epoch single differences to different satellites: • uncorrelated • twice the variance of the single difference observation. Single differences to the same satellite at different epochs: • uncorrelated • twice the variance of the undifferenced observations. =⇒ Covariance-matrix of the observations: CYY = 2σ2 I. (49)
58 / 105 Least-squares estimation of the position correction and the error terms: ∆x = A⊤ A −1 A⊤ Y (50)
58 / 105 Least-squares estimation of the position correction and the error terms: ∆x = A⊤ A −1 A⊤ Y (50) Accuracy estimation: σ2 = | Y − A · ∆x 2 2 ∗ (n ∗ k − (4 + k)) (51)
58 / 105 Least-squares estimation of the position correction and the error terms: ∆x = A⊤ A −1 A⊤ Y (50) Accuracy estimation: σ2 = | Y − A · ∆x 2 2 ∗ (n ∗ k − (4 + k)) (51) and C∆x,∆x = 2σ2 A⊤ A −1 (52)
Double Differences Solution 59 / 105
60 / 105 k satellites have been observed at n epochs from two receivers – reference receiver r and rover receiver v.
60 / 105 k satellites have been observed at n epochs from two receivers – reference receiver r and rover receiver v. Double differences observation equations on an arbitrary frequency combination: ∇∆PR pq CRrv(tj) := ∆PR q CRrv(tj) − ∆PR p CRrv(tj) = ∇∆R pq rv (tj) +c((dt q ar(tj) − dt q av(tj)) −(dt p ar(tj) − dt p av(tj))) +λ((N q r − N q v ) − (N p r − N p v )) + ǫ. (53)
61 / 105 ∇∆R pq rv (tj) =   3 ∑ l=1 (x q l (tj) − xr,l)2 − 3 ∑ l=1 (x q l (tj) − xv,l)2   −   3 ∑ l=1 (x p l (tj) − xr,l)2 − 3 ∑ l=1 (x p l (tj) − xv,l)2   =⇒ Double differences observation equations are nonlinear in the unknown coordinates xv of the rover receiver v.
61 / 105 ∇∆R pq rv (tj) =   3 ∑ l=1 (x q l (tj) − xr,l)2 − 3 ∑ l=1 (x q l (tj) − xv,l)2   −   3 ∑ l=1 (x p l (tj) − xr,l)2 − 3 ∑ l=1 (x p l (tj) − xv,l)2   =⇒ Double differences observation equations are nonlinear in the unknown coordinates xv of the rover receiver v. (Coordinates of the reference receiver assumed to be known.)
61 / 105 ∇∆R pq rv (tj) =   3 ∑ l=1 (x q l (tj) − xr,l)2 − 3 ∑ l=1 (x q l (tj) − xv,l)2   −   3 ∑ l=1 (x p l (tj) − xr,l)2 − 3 ∑ l=1 (x p l (tj) − xv,l)2   =⇒ Double differences observation equations are nonlinear in the unknown coordinates xv of the rover receiver v. (Coordinates of the reference receiver assumed to be known.) For linearization: need of prior information x0 v about position of rover.
62 / 105 =⇒ Linearized single differences observation equations: ∇∆PR pq CRrv(tj) − ∇∆R pq,0 rv (tj) = ∂∇∆R pq rv ∂xv (tj) · ∆xv +c((dt q ar(tj) − dt q av(tj)) −(dt p ar(tj) − dt p av(tj))) +λ((N q r − N q v ) − (N p r − N p v )) +ǫ. (54)
63 / 105 Computed carrier phase double difference: ∇∆R pq,0 rv (tj) =   3 ∑ l=1 (X q l (tj) − xr,l)2 − 3 ∑ l=1 (X q l (tj) − x0 v,l)2  (55) −   3 ∑ l=1 (X p l (tj) − xr,l)2 − 3 ∑ l=1 (X p l (tj) − x0 v,l)2  
63 / 105 Computed carrier phase double difference: ∇∆R pq,0 rv (tj) =   3 ∑ l=1 (X q l (tj) − xr,l)2 − 3 ∑ l=1 (X q l (tj) − x0 v,l)2  (55) −   3 ∑ l=1 (X p l (tj) − xr,l)2 − 3 ∑ l=1 (X p l (tj) − x0 v,l)2   Partial derivatives: ∂∇∆R pq rv ∂xv (tj) = ( Xq − x0 v |Xq − x0 v| ) − ( Xp − x0 v |Xp − x0 v| ) (56)
64 / 105 Difference between true (but unknown) position of rover and available prior information about its position: Unknown position corrections: ∆xv = xv − x0 v
64 / 105 Difference between true (but unknown) position of rover and available prior information about its position: Unknown position corrections: ∆xv = xv − x0 v Differentiation of two groups among the remaining terms of the double differences observation equations: • time independent terms (e. g. phase ambiguities) • time dependent atmospheric and ionospheric propagation-delay terms
65 / 105 Time independent terms remain in the observation equations unchanged.
65 / 105 Time independent terms remain in the observation equations unchanged. Atmospheric and ionospheric terms: c(dt q ar − dt q av)(tj) − c(dt p ar − dt p av)(tj) ≈ Mpq (tj, T, p, H) M(tj, T, p, H): difference of two standard models for atmospheric and ionospheric propagation delay
65 / 105 Time independent terms remain in the observation equations unchanged. Atmospheric and ionospheric terms: c(dt q ar − dt q av)(tj) − c(dt p ar − dt p av)(tj) ≈ Mpq (tj, T, p, H) M(tj, T, p, H): difference of two standard models for atmospheric and ionospheric propagation delay ∇∆PR pq CRrv(tj) − ∇∆R pq,0 rv (tj) − Mpq (tj, T, p, H) = ∂∇∆R pq rv ∂xv (tj) · ∆xv + λ((N q r − N q v ) − (N p r − N p v )) + ǫ
66 / 105 Collection of individual terms in vectors and in matrices: Y =                 ∇∆PR12 CRrv(t1) − ∇∆R12,0 rv (t1) − M12(t1, T, p, H) ... ∇∆PR1k CRrv(t1) − ∇∆R1k,0 rv (t1) − M1k(t1, T, p, H) ... ... ∇∆PR12 CRrv(tn) − ∇∆R12,0 rv (tn) − M12(tn, T, p, H) ... ∇∆PR1k CRrv(tn) − ∇∆R1k,0 rv (tn) − M1k(tn, T, p, H)                
67 / 105 A =                                 −( X1 1 −x0 v,1 |X1−x0 v| − X2 1 −x0 v,1 |X2−x0 v| )(t1) −( X1 2−x0 v,2 |X1−x0 v| − X2 2−x0 v,2 |X2−x0 v| )(t1) −( X1 3−x0 v,3 |X1−x0 v| − X2 3−x0 v,3 |X2−x0 v| )(t1) λ . . . 0 . . . −( X1 1 −x0 v,1 |X1−x0 v| − Xk 1 −x0 v,1 |Xk−x0 v| )(t1) −( X1 2−x0 v,2 |X1−x0 v| − Xk 2−x0 v,2 |Xk−x0 v| )(t1) −( X1 3−x0 v,3 |X1−x0 v| − Xk 3−x0 v,3 |Xk−x0 v| )(t1) 0 . . . λ . . . . . . −( X1 1 −x0 v,1 |X1−x0 v| − X2 1 −x0 v,1 |X2−x0 v| )(tn) −( X1 2−x0 v,2 |X1−x0 v| − X2 2−x0 v,2 |X2−x0 v| )(tn) −( X1 3−x0 v,3 |X1−x0 v| − X2 3−x0 v,3 |X2−x0 v| )(tn) λ . . . 0 . . . −( X1 1 −x0 v,1 |X1−x0 v| − Xk 1 −x0 v,1 |Xk−x0 v| )(tn) −( X1 2−x0 v,2 |X1−x0 v| − Xk 2−x0 v,2 |Xk−x0 v| )(tn) −( X1 3−x0 v,3 |X1−x0 v| − Xk 3−x0 v,3 |Xk−x0 v| )(tn) 0 . . . λ                                
68 / 105 ∆xv = (xv,1 − x0 v,1, xv,2 − x0 v,2, xv,3 − x0 v,3, N12 , . . . , N1k )⊤ Npq: abbreviation of double differences of unknown integer phase ambiguities: Npq := (N q r − N q v ) − (N p r − N p v ).
68 / 105 ∆xv = (xv,1 − x0 v,1, xv,2 − x0 v,2, xv,3 − x0 v,3, N12 , . . . , N1k )⊤ Npq: abbreviation of double differences of unknown integer phase ambiguities: Npq := (N q r − N q v ) − (N p r − N p v ). =⇒ Gauss-Markov form of linearized double differences observation equations: E{Y} = A · ∆x.
68 / 105 ∆xv = (xv,1 − x0 v,1, xv,2 − x0 v,2, xv,3 − x0 v,3, N12 , . . . , N1k )⊤ Npq: abbreviation of double differences of unknown integer phase ambiguities: Npq := (N q r − N q v ) − (N p r − N p v ). =⇒ Gauss-Markov form of linearized double differences observation equations: E{Y} = A · ∆x. Consider separately: covariance matrix CYY of double difference observations.
69 / 105 Observation of k · n single differences ∆Ψi rv between reference receiver r and rover receiver v and k satellites at n epochs.
69 / 105 Observation of k · n single differences ∆Ψi rv between reference receiver r and rover receiver v and k satellites at n epochs. =⇒ m = (k − 1) double differences ∇∆Ψ pq rv (tj) can be formed.
69 / 105 Observation of k · n single differences ∆Ψi rv between reference receiver r and rover receiver v and k satellites at n epochs. =⇒ m = (k − 1) double differences ∇∆Ψ pq rv (tj) can be formed. Single differences to k satellites recorded at the epoch tj: ∆Ψ(tj) := (∆Ψ1 (tj), . . . , ∆Ψk (tj))⊤
69 / 105 Observation of k · n single differences ∆Ψi rv between reference receiver r and rover receiver v and k satellites at n epochs. =⇒ m = (k − 1) double differences ∇∆Ψ pq rv (tj) can be formed. Single differences to k satellites recorded at the epoch tj: ∆Ψ(tj) := (∆Ψ1 (tj), . . . , ∆Ψk (tj))⊤ =⇒ m double differences can be formed at this epoch: ∇∆Ψ(tj) =      ∆Ψ1(tj) − ∆Ψ2(tj) ∆Ψ1(tj) − ∆Ψ3(tj) ... ∆Ψ1(tj) − ∆Ψk(tj)      .
70 / 105 Single and double differences can be related to each other: ∇∆Ψ(tj) = D · ∆Ψ(tj). D =      1 −1 1 −1 ... 1 −1     
71 / 105 Laws of covariance propagation =⇒ covariance matrix of double differences: C∇∆Ψ∇∆Ψ = DC∆Ψ∆ΨD⊤ = 2σ2 DD⊤ = 2σ2      2 1 1 . . . . . . 1 1 2 1 1 . . . 1 ... 1 . . . . . . 1 1 2      .
71 / 105 Laws of covariance propagation =⇒ covariance matrix of double differences: C∇∆Ψ∇∆Ψ = DC∆Ψ∆ΨD⊤ = 2σ2 DD⊤ = 2σ2      2 1 1 . . . . . . 1 1 2 1 1 . . . 1 ... 1 . . . . . . 1 1 2      . =⇒ Double differences at the same epoch are strongly correlated.
71 / 105 Laws of covariance propagation =⇒ covariance matrix of double differences: C∇∆Ψ∇∆Ψ = DC∆Ψ∆ΨD⊤ = 2σ2 DD⊤ = 2σ2      2 1 1 . . . . . . 1 1 2 1 1 . . . 1 ... 1 . . . . . . 1 1 2      . =⇒ Double differences at the same epoch are strongly correlated. Correlation has to be taken into account in the parameter estimation process.
72 / 105 C−1 ∇∆Ψ∇∆Ψ = 1 2σ2(k + 1)      k −1 −1 . . . . . . −1 −1 k −1 −1 . . . −1 ... −1 . . . . . . −1 −1 k      =: ˜P (57)
72 / 105 C−1 ∇∆Ψ∇∆Ψ = 1 2σ2(k + 1)      k −1 −1 . . . . . . −1 −1 k −1 −1 . . . −1 ... −1 . . . . . . −1 −1 k      =: ˜P (57) Double differences at different epochs are uncorrelated =⇒ weight matrix P is a block diagonal matrix, with matrix ˜P as diagonal blocks: P =      ˜P ˜P ... ˜P      . (58)
73 / 105 Usual least-squares estimation of the position correction and the error terms: ∆x = A⊤ PA −1 A⊤ PY (59)
73 / 105 Usual least-squares estimation of the position correction and the error terms: ∆x = A⊤ PA −1 A⊤ PY (59) With accuracy estimation: σ2 = | Y − A · ∆x 2 n(k − 1) − 2 − k = | Y − A · ∆x 2 (n − 1)(k − 1) − 3 (60)
73 / 105 Usual least-squares estimation of the position correction and the error terms: ∆x = A⊤ PA −1 A⊤ PY (59) With accuracy estimation: σ2 = | Y − A · ∆x 2 n(k − 1) − 2 − k = | Y − A · ∆x 2 (n − 1)(k − 1) − 3 (60) and C∆x,∆x = σ2 A⊤ PA −1 . (61)
Cycle-slip Detection 74 / 105
75 / 105 Acquisition phase: receiver determines phase shift of received and receiver-generated carrier (up to an unknown integer number of complete cycles).
75 / 105 Acquisition phase: receiver determines phase shift of received and receiver-generated carrier (up to an unknown integer number of complete cycles). Thereafter Costa’s loop tracks the change in the phase shift due to the changing distance between receiver and satellite.
75 / 105 Acquisition phase: receiver determines phase shift of received and receiver-generated carrier (up to an unknown integer number of complete cycles). Thereafter Costa’s loop tracks the change in the phase shift due to the changing distance between receiver and satellite. Introduction of additional unknown: integer number N of unknown phase cycles.
75 / 105 Acquisition phase: receiver determines phase shift of received and receiver-generated carrier (up to an unknown integer number of complete cycles). Thereafter Costa’s loop tracks the change in the phase shift due to the changing distance between receiver and satellite. Introduction of additional unknown: integer number N of unknown phase cycles. If for one satellite the signal-to-noise ratio gets too low, Costa’s loop is not capable to keep track of the phase shift change and a new acquisition for this satellite has to be carried out.
75 / 105 Acquisition phase: receiver determines phase shift of received and receiver-generated carrier (up to an unknown integer number of complete cycles). Thereafter Costa’s loop tracks the change in the phase shift due to the changing distance between receiver and satellite. Introduction of additional unknown: integer number N of unknown phase cycles. If for one satellite the signal-to-noise ratio gets too low, Costa’s loop is not capable to keep track of the phase shift change and a new acquisition for this satellite has to be carried out. In the time elapsed during this acquisition the satellite-receiver distance has changed and the number of unknown cycles has also changed. This effect is called cycle-slip.
76 / 105 Figure 4: Occurrence of a cycle-slip
77 / 105 • Assumption that at time 0 the acquisition has returned the phase shift ϕ(0)
77 / 105 • Assumption that at time 0 the acquisition has returned the phase shift ϕ(0) • At this moment number of unknown entire cycles is N0
77 / 105 • Assumption that at time 0 the acquisition has returned the phase shift ϕ(0) • At this moment number of unknown entire cycles is N0 • Then the receiver tracks the satellite continuously until t
77 / 105 • Assumption that at time 0 the acquisition has returned the phase shift ϕ(0) • At this moment number of unknown entire cycles is N0 • Then the receiver tracks the satellite continuously until t • During this time the phase shift changes to ϕ(t)
77 / 105 • Assumption that at time 0 the acquisition has returned the phase shift ϕ(0) • At this moment number of unknown entire cycles is N0 • Then the receiver tracks the satellite continuously until t • During this time the phase shift changes to ϕ(t) • At t Costa’s loop looses track to the signal until t + dt
77 / 105 • Assumption that at time 0 the acquisition has returned the phase shift ϕ(0) • At this moment number of unknown entire cycles is N0 • Then the receiver tracks the satellite continuously until t • During this time the phase shift changes to ϕ(t) • At t Costa’s loop looses track to the signal until t + dt • At t + dt the new acquisition is completed and the new phase shift ϕ(t + dt) and the new phase ambiguity N1 is determined
77 / 105 • Assumption that at time 0 the acquisition has returned the phase shift ϕ(0) • At this moment number of unknown entire cycles is N0 • Then the receiver tracks the satellite continuously until t • During this time the phase shift changes to ϕ(t) • At t Costa’s loop looses track to the signal until t + dt • At t + dt the new acquisition is completed and the new phase shift ϕ(t + dt) and the new phase ambiguity N1 is determined • The difference N1 − N0 is the cycle-slip occurred at t
77 / 105 • Assumption that at time 0 the acquisition has returned the phase shift ϕ(0) • At this moment number of unknown entire cycles is N0 • Then the receiver tracks the satellite continuously until t • During this time the phase shift changes to ϕ(t) • At t Costa’s loop looses track to the signal until t + dt • At t + dt the new acquisition is completed and the new phase shift ϕ(t + dt) and the new phase ambiguity N1 is determined • The difference N1 − N0 is the cycle-slip occurred at t Important task of GPS data pre-processing: detection and correction of cycle slips.
77 / 105 • Assumption that at time 0 the acquisition has returned the phase shift ϕ(0) • At this moment number of unknown entire cycles is N0 • Then the receiver tracks the satellite continuously until t • During this time the phase shift changes to ϕ(t) • At t Costa’s loop looses track to the signal until t + dt • At t + dt the new acquisition is completed and the new phase shift ϕ(t + dt) and the new phase ambiguity N1 is determined • The difference N1 − N0 is the cycle-slip occurred at t Important task of GPS data pre-processing: detection and correction of cycle slips. Multiple techniques for this purpose. Here only three of them: • analysis of double differences • analysis of the ionospheric residuals • code- and carrier phase combination
Analysis of Double Differences 78 / 105 Comparation of phase double differences with double differences of the slant ranges between receivers and satellites.
Analysis of Double Differences 78 / 105 Comparation of phase double differences with double differences of the slant ranges between receivers and satellites. Double differences observation equation: ∇∆PR pq CRrv(tj) := ∆PR q CRrv(tj) − ∆PR p CRrv(tj) = ∇∆R pq rv (tj) +c((dt q ar(tj) − dt q av(tj)) −(dt p ar(tj) − dt p av(tj))) +λ((N q r − N q v ) − (N p r − N p v )) + ǫ
79 / 105 Test quantity: difference between phase double difference and double difference of the slant ranges ∇∆r(tj) := ∇∆PR pq CRrv(tj) − ∇∆R pq rv (tj) = c((dt q ar(tj) − dt q av(tj)) −(dt p ar(tj) − dt p av(tj))) +λ((N q r − N q v ) − (N p r − N p v )) + ǫ
79 / 105 Test quantity: difference between phase double difference and double difference of the slant ranges ∇∆r(tj) := ∇∆PR pq CRrv(tj) − ∇∆R pq rv (tj) = c((dt q ar(tj) − dt q av(tj)) −(dt p ar(tj) − dt p av(tj))) +λ((N q r − N q v ) − (N p r − N p v )) + ǫ No cycle slip: • time variation of ∇∆r only caused by changes of tropospheric and ionospheric delay • slow changes: time variation of ∇∆r is a smooth curve
79 / 105 Test quantity: difference between phase double difference and double difference of the slant ranges ∇∆r(tj) := ∇∆PR pq CRrv(tj) − ∇∆R pq rv (tj) = c((dt q ar(tj) − dt q av(tj)) −(dt p ar(tj) − dt p av(tj))) +λ((N q r − N q v ) − (N p r − N p v )) + ǫ No cycle slip: • time variation of ∇∆r only caused by changes of tropospheric and ionospheric delay • slow changes: time variation of ∇∆r is a smooth curve Cycle slip: • indication through sudden jumps in ∇∆r
80 / 105 Figure 5: Double differences residuals with occurring of a cycle slip
81 / 105 Jump in ∇∆r small compared to absolute values of ∇∆r
81 / 105 Jump in ∇∆r small compared to absolute values of ∇∆r =⇒ jump is difficult to detect by statistical methods
81 / 105 Jump in ∇∆r small compared to absolute values of ∇∆r =⇒ jump is difficult to detect by statistical methods =⇒ a polynomial p(t) of a low degree is fitted to ∇∆r and differences ∇∆r − p are screened Figure 6: Polynomial p fitted to the double differences residuals
82 / 105 Figure 7: Differences between the fitted polynomial p and the double differences residuals
83 / 105 Cycle slip indicated by a sudden change in the sign =⇒ easy detection by statistical tests
83 / 105 Cycle slip indicated by a sudden change in the sign =⇒ easy detection by statistical tests advantages: • applicable already for single frequency receivers • even small cycle slips can be detected
83 / 105 Cycle slip indicated by a sudden change in the sign =⇒ easy detection by statistical tests advantages: • applicable already for single frequency receivers • even small cycle slips can be detected disadvantages: • no possibility to decide for which satellite and which receiver the cycle slip has occurred • sensitive to sudden changes in the ionospheric electron concentration
Analysis of the Ionospheric Residuals 84 / 105 Analysation of the geometry-free linear combination: L4 = LI = LΣ − L∆ = ω1 ω1 + ω2 L1 + ω2 ω1 + ω2 L2 − ω1 ω1 − ω2 L1 + ω2 ω1 − ω2 L2 = 2ω1ω2 ω2 1 − ω2 2 [−L1 + L2] = 2ω1ω2 ω2 1 − ω2 2 [−(|XS − XB| + N1λ1 + cdtu + cdtI1) +(|XS − XB| + N2λ2 + cdtu + cdtI2)] = 2ω1ω2 ω2 1 − ω2 2 [−N1λ1 + N2λ2 + (−cdtI1 + cdtI2)]
85 / 105 No cycle slip =⇒ variation of L4 only caused by variation of the ionospheric delay
85 / 105 No cycle slip =⇒ variation of L4 only caused by variation of the ionospheric delay =⇒ variation of L4 has to be smooth
85 / 105 No cycle slip =⇒ variation of L4 only caused by variation of the ionospheric delay =⇒ variation of L4 has to be smooth Sudden changes in L4: indication of a cycle slip
85 / 105 No cycle slip =⇒ variation of L4 only caused by variation of the ionospheric delay =⇒ variation of L4 has to be smooth Sudden changes in L4: indication of a cycle slip advantages: • method works already with a single receiver, no baselines have to be performed
85 / 105 No cycle slip =⇒ variation of L4 only caused by variation of the ionospheric delay =⇒ variation of L4 has to be smooth Sudden changes in L4: indication of a cycle slip advantages: • method works already with a single receiver, no baselines have to be performed disadvantages: • at detection of a cycle slip: frequency on which this cycle slip had happened cannot be found
Analysis of the Code- Carrier Combination 86 / 105 Comparation of code and the carrier pseudorange on the same frequency: PRCR,CD(t) := PRCD − PRCR = |XS − XB| + cdtu + cdtI −(|XS − XB| + NBi λ + cdtu − cdtI) = −NBi λ + 2cdtI
Analysis of the Code- Carrier Combination 86 / 105 Comparation of code and the carrier pseudorange on the same frequency: PRCR,CD(t) := PRCD − PRCR = |XS − XB| + cdtu + cdtI −(|XS − XB| + NBi λ + cdtu − cdtI) = −NBi λ + 2cdtI =⇒ • ionospheric signal delay 2cdtI is very smooth • changes in ionospheric signal are constant • quantity PRCR,CD is constant
Analysis of the Code- Carrier Combination 86 / 105 Comparation of code and the carrier pseudorange on the same frequency: PRCR,CD(t) := PRCD − PRCR = |XS − XB| + cdtu + cdtI −(|XS − XB| + NBi λ + cdtu − cdtI) = −NBi λ + 2cdtI =⇒ • ionospheric signal delay 2cdtI is very smooth • changes in ionospheric signal are constant • quantity PRCR,CD is constant =⇒ Indication of a cycle-slip: sudden jumps in PRCR,CD.
87 / 105 • Magnitude of cycle slip identical to magnitude in the jump of PRCR,CD.
87 / 105 • Magnitude of cycle slip identical to magnitude in the jump of PRCR,CD. • P-Code pseudorange accuracy ∼ 6 dm =⇒ Accuracy of magnitude of a cycle slip: 3 cycles
87 / 105 • Magnitude of cycle slip identical to magnitude in the jump of PRCR,CD. • P-Code pseudorange accuracy ∼ 6 dm =⇒ Accuracy of magnitude of a cycle slip: 3 cycles • =⇒ Sufficient to input results of the code-carrier combination into analysis of ionospheric residuals
Phase Ambiguities Solution 88 / 105
89 / 105 Carrier phase measurements contain an unknown integer number N of cycles.
89 / 105 Carrier phase measurements contain an unknown integer number N of cycles. Number must be found =⇒ full accuracy potential of GPS carrier phase measurements
89 / 105 Carrier phase measurements contain an unknown integer number N of cycles. Number must be found =⇒ full accuracy potential of GPS carrier phase measurements Large number of methods for fixing these ambiguities. Three of them will be discussed: • geometric method • combination of code and carrier phase • search methods
The Geometric Method 90 / 105 Makes use of time differences of carrier phase observations.
The Geometric Method 90 / 105 Makes use of time differences of carrier phase observations. Assumption that at three epochs t1, t2, t3 carrier phase observation on one frequency to the same satellite are carried out: Φ(t1) = 2π λ (|XS(t1) − XB| + Nλ) (62) Φ(t2) = 2π λ (|XS(t2) − XB| + Nλ) (63) Φ(t3) = 2π λ (|XS(t3) − XB| + Nλ) (64)
91 / 105 Out of these three phase observations two time-differences can be formed: δΦ(t1) := Φ(t2) − Φ(t1) = 2π λ (|XS(t2) − XB| − |XS(t1) − XB|) (65) δΦ(t2) := Φ(t3) − Φ(t2) = 2π λ (|XS(t3) − XB| − |XS(t2) − XB|) (66)
91 / 105 Out of these three phase observations two time-differences can be formed: δΦ(t1) := Φ(t2) − Φ(t1) = 2π λ (|XS(t2) − XB| − |XS(t1) − XB|) (65) δΦ(t2) := Φ(t3) − Φ(t2) = 2π λ (|XS(t3) − XB| − |XS(t2) − XB|) (66) Equations (65) and (66): equations of hyperboloids with focal points in the known satellite positions XS(t1), XS(t2) and XS(t2), XS(t3).
92 / 105 =⇒Position of observer is on the intersection of the two hyperboloids. Figure 8: Geometric method of ambiguity resolution
93 / 105 Position of the observer: intersection of at least three hyperboloids.
93 / 105 Position of the observer: intersection of at least three hyperboloids. Larger number of intersecting hyperboloids =⇒ more precise estimation for the observer position ˆXB
93 / 105 Position of the observer: intersection of at least three hyperboloids. Larger number of intersecting hyperboloids =⇒ more precise estimation for the observer position ˆXB Process continues until the accuracy of the obtained estimation is better than λ/2.
93 / 105 Position of the observer: intersection of at least three hyperboloids. Larger number of intersecting hyperboloids =⇒ more precise estimation for the observer position ˆXB Process continues until the accuracy of the obtained estimation is better than λ/2. Then the estimation is inserted into an observation equation and solved for the unknown ambiguity: ˆN = 1 2π Φ(t) − 1 λ |XS(t) − ˆXB| (67)
93 / 105 Position of the observer: intersection of at least three hyperboloids. Larger number of intersecting hyperboloids =⇒ more precise estimation for the observer position ˆXB Process continues until the accuracy of the obtained estimation is better than λ/2. Then the estimation is inserted into an observation equation and solved for the unknown ambiguity: ˆN = 1 2π Φ(t) − 1 λ |XS(t) − ˆXB| (67) Obtained float solution ˆN is rounded to the nearest integer.
94 / 105 advantages: • simple and clear modelling • applicable also for single point positioning • single frequency receiver sufficient
94 / 105 advantages: • simple and clear modelling • applicable also for single point positioning • single frequency receiver sufficient disadvantages: • long arcs necessary • sensitive to unmodelled effects (ionosphere, troposphere, orbits, clocks) • no cycle slips allowed during ambiguity resolution
Combination of Code and Carrier Phase 95 / 105 Difference between pseudorange from carrier phase observations and pseudorange from code observation is used: PRCR,CD(t) := PRCD − PRCR = |XS − XB| + cdtu + cdtI −(|XS − XB| + NBλ + cdtu − cdtI) = −NBλ + 2cdtI (68)
96 / 105 Difference of the pseudoranges PRCR,CD(t) equals length Nλ of unknown integer number of carrier phase cycles – up to ionospheric error cdtI.
96 / 105 Difference of the pseudoranges PRCR,CD(t) equals length Nλ of unknown integer number of carrier phase cycles – up to ionospheric error cdtI. For short and medium length baselines elimination of ionospheric error by forming single differences: ∆PRCR,CDij := PRCR,CDi − PRCR,CDj = λ(NBi − NBj ) = λ∆Nij (69)
96 / 105 Difference of the pseudoranges PRCR,CD(t) equals length Nλ of unknown integer number of carrier phase cycles – up to ionospheric error cdtI. For short and medium length baselines elimination of ionospheric error by forming single differences: ∆PRCR,CDij := PRCR,CDi − PRCR,CDj = λ(NBi − NBj ) = λ∆Nij (69) Reduction of random errors contained in observations by computing the time average over some minutes
96 / 105 Difference of the pseudoranges PRCR,CD(t) equals length Nλ of unknown integer number of carrier phase cycles – up to ionospheric error cdtI. For short and medium length baselines elimination of ionospheric error by forming single differences: ∆PRCR,CDij := PRCR,CDi − PRCR,CDj = λ(NBi − NBj ) = λ∆Nij (69) Reduction of random errors contained in observations by computing the time average over some minutes Estimation of single difference ambiguity: ∆Nij = 1 λT T 0 ∆PRCR,CDij (t)dt (70)
97 / 105 ∆Nij: float solution, can be rounded to the nearest integer as soon as its standard deviation ≤ 0.5.
97 / 105 ∆Nij: float solution, can be rounded to the nearest integer as soon as its standard deviation ≤ 0.5. Because wavelength λ is in the denominator of equation (70) =⇒ easier ambiguity resolution for longer wavelength of λ
97 / 105 ∆Nij: float solution, can be rounded to the nearest integer as soon as its standard deviation ≤ 0.5. Because wavelength λ is in the denominator of equation (70) =⇒ easier ambiguity resolution for longer wavelength of λ =⇒ carrier code combination mostly used for wide lane frequency combination LΣ.
97 / 105 ∆Nij: float solution, can be rounded to the nearest integer as soon as its standard deviation ≤ 0.5. Because wavelength λ is in the denominator of equation (70) =⇒ easier ambiguity resolution for longer wavelength of λ =⇒ carrier code combination mostly used for wide lane frequency combination LΣ. Fixation of wide lane ambiguity if float solution for the single difference ambiguity has a standard deviation smaller than half a cycle.
97 / 105 ∆Nij: float solution, can be rounded to the nearest integer as soon as its standard deviation ≤ 0.5. Because wavelength λ is in the denominator of equation (70) =⇒ easier ambiguity resolution for longer wavelength of λ =⇒ carrier code combination mostly used for wide lane frequency combination LΣ. Fixation of wide lane ambiguity if float solution for the single difference ambiguity has a standard deviation smaller than half a cycle. For short and medium length baselines the ionospheric combination has to vanish identical: LI = LΣ − L∆ = 0 (71)
97 / 105 ∆Nij: float solution, can be rounded to the nearest integer as soon as its standard deviation ≤ 0.5. Because wavelength λ is in the denominator of equation (70) =⇒ easier ambiguity resolution for longer wavelength of λ =⇒ carrier code combination mostly used for wide lane frequency combination LΣ. Fixation of wide lane ambiguity if float solution for the single difference ambiguity has a standard deviation smaller than half a cycle. For short and medium length baselines the ionospheric combination has to vanish identical: LI = LΣ − L∆ = 0 (71) Wavelength of wide lane: ∼ 8 × wavelength of narrow lane
97 / 105 ∆Nij: float solution, can be rounded to the nearest integer as soon as its standard deviation ≤ 0.5. Because wavelength λ is in the denominator of equation (70) =⇒ easier ambiguity resolution for longer wavelength of λ =⇒ carrier code combination mostly used for wide lane frequency combination LΣ. Fixation of wide lane ambiguity if float solution for the single difference ambiguity has a standard deviation smaller than half a cycle. For short and medium length baselines the ionospheric combination has to vanish identical: LI = LΣ − L∆ = 0 (71) Wavelength of wide lane: ∼ 8 × wavelength of narrow lane =⇒ Initial estimation of narrow lane single difference ambiguity with an accuracy of about 8 cycles: condition (71).
98 / 105 advantages: • fast, • independent of geometry.
98 / 105 advantages: • fast, • independent of geometry. disadvantages: • dual frequency P-code receiver necessary, • only wide lane ambiguities can be resolved in short time.
Search Methods 99 / 105 Test of all possible integer values of the ambiguities and selection of the most plausible integer value.
Search Methods 99 / 105 Test of all possible integer values of the ambiguities and selection of the most plausible integer value. For more precision consider the adjustment problem for a single- or double differences baseline solution: l = A · x + v. (72)
Search Methods 99 / 105 Test of all possible integer values of the ambiguities and selection of the most plausible integer value. For more precision consider the adjustment problem for a single- or double differences baseline solution: l = A · x + v. (72) Division of unknown vector x in two parts: x = (x1, x2)
Search Methods 99 / 105 Test of all possible integer values of the ambiguities and selection of the most plausible integer value. For more precision consider the adjustment problem for a single- or double differences baseline solution: l = A · x + v. (72) Division of unknown vector x in two parts: x = (x1, x2) x1 containing all the unknown of non-integer nature: • receiver clock errors • parameters of atmospheric and ionospheric delay models • coordinates of the receivers x2: remaining integer ambiguities.
100 / 105 Also partition of adjustment problem: l = [A1, A2] · x1 x2 + v (73)
100 / 105 Also partition of adjustment problem: l = [A1, A2] · x1 x2 + v (73) Corresponding normal equations: N11 N12 N21 N22 · x1 x2 = b1 b2 (74) with Nij = A⊤ i Aj, bi = A⊤ i l (75)
101 / 105 Least squares solution: ˆx1 ˆx2 = Q11 Q12 Q21 Q22 · b1 b2 (76) with Q11 Q12 Q21 Q22 = N11 N12 N21 N22 −1 (77)
101 / 105 Least squares solution: ˆx1 ˆx2 = Q11 Q12 Q21 Q22 · b1 b2 (76) with Q11 Q12 Q21 Q22 = N11 N12 N21 N22 −1 (77) Derivation of standard deviation of estimated float ambiguities and standard deviation of difference between estimated float ambiguities from variance-covariance matrix Q: ˆσ2 = l − Aˆx 2 n − u (78) σNi = ˆσ Q22ii (79) σNi−Nj = ˆσ Q22ii − 2Q22ij + Q22jj (80)
102 / 105 Assuming a normal distribution of the observations =⇒ confidence regions for the unknown ambiguity parameters: P(Ni ∈ [ ˆNi − tασNi , ˆNi + tασNi ]) = 1 − α, (81) P(Ni − Nj ∈ [( ˆNi − ˆNj) − tασNi−Nj , ( ˆNi − ˆNj) + tασNi−Nj ]) = 1 − α, (82) tα: quantile of student distribution for confidence level α.
102 / 105 Assuming a normal distribution of the observations =⇒ confidence regions for the unknown ambiguity parameters: P(Ni ∈ [ ˆNi − tασNi , ˆNi + tασNi ]) = 1 − α, (81) P(Ni − Nj ∈ [( ˆNi − ˆNj) − tασNi−Nj , ( ˆNi − ˆNj) + tασNi−Nj ]) = 1 − α, (82) tα: quantile of student distribution for confidence level α. Intersection of all this confidence regions: region around the float solution ˆx2 = ( ˆN1, . . . , ˆNn)
102 / 105 Assuming a normal distribution of the observations =⇒ confidence regions for the unknown ambiguity parameters: P(Ni ∈ [ ˆNi − tασNi , ˆNi + tασNi ]) = 1 − α, (81) P(Ni − Nj ∈ [( ˆNi − ˆNj) − tασNi−Nj , ( ˆNi − ˆNj) + tασNi−Nj ]) = 1 − α, (82) tα: quantile of student distribution for confidence level α. Intersection of all this confidence regions: region around the float solution ˆx2 = ( ˆN1, . . . , ˆNn) Float solution contains true integer ambiguities with a probability of 1 − α.
102 / 105 Assuming a normal distribution of the observations =⇒ confidence regions for the unknown ambiguity parameters: P(Ni ∈ [ ˆNi − tασNi , ˆNi + tασNi ]) = 1 − α, (81) P(Ni − Nj ∈ [( ˆNi − ˆNj) − tασNi−Nj , ( ˆNi − ˆNj) + tασNi−Nj ]) = 1 − α, (82) tα: quantile of student distribution for confidence level α. Intersection of all this confidence regions: region around the float solution ˆx2 = ( ˆN1, . . . , ˆNn) Float solution contains true integer ambiguities with a probability of 1 − α. Trivial case: only two ambiguities N1, N2 (displayed in figure 9)
103 / 105 Figure 9: Confidence region for the integer ambiguities
104 / 105 In what follows this confidence region: C ⊂ Rn.
104 / 105 In what follows this confidence region: C ⊂ Rn. All vectors x2 ∈ Nn ∩ C: grid of possible integer ambiguity solutions x2,h, h = 1, . . . N. Figure 10: Candidates for integer solution
105 / 105 Introduction of each possible integer ambiguity solution to a subsequent adjustment. Chosen integer ambiguity solution: treated as known quantity. A1 · x1 = l − A2 · x2,h = lh (83) ˆx1 = (A⊤ 1 A1)−1 A⊤ 1 lh (84) ˆσ2 h = lh − A1 ˆx1 2 n − u1 , (85)
105 / 105 Introduction of each possible integer ambiguity solution to a subsequent adjustment. Chosen integer ambiguity solution: treated as known quantity. A1 · x1 = l − A2 · x2,h = lh (83) ˆx1 = (A⊤ 1 A1)−1 A⊤ 1 lh (84) ˆσ2 h = lh − A1 ˆx1 2 n − u1 , (85) Final solution: integer ambiguity solution candidate x2,h with smallest variation ˆσ2 h. Unless:
105 / 105 Introduction of each possible integer ambiguity solution to a subsequent adjustment. Chosen integer ambiguity solution: treated as known quantity. A1 · x1 = l − A2 · x2,h = lh (83) ˆx1 = (A⊤ 1 A1)−1 A⊤ 1 lh (84) ˆσ2 h = lh − A1 ˆx1 2 n − u1 , (85) Final solution: integer ambiguity solution candidate x2,h with smallest variation ˆσ2 h. Unless: 1. variance ˆσ2 h is not compatible with the variance of the L3 solution, or
105 / 105 Introduction of each possible integer ambiguity solution to a subsequent adjustment. Chosen integer ambiguity solution: treated as known quantity. A1 · x1 = l − A2 · x2,h = lh (83) ˆx1 = (A⊤ 1 A1)−1 A⊤ 1 lh (84) ˆσ2 h = lh − A1 ˆx1 2 n − u1 , (85) Final solution: integer ambiguity solution candidate x2,h with smallest variation ˆσ2 h. Unless: 1. variance ˆσ2 h is not compatible with the variance of the L3 solution, or 2. there is another integer solution candidate yielding almost identical variance.

Recommended

MO3.L09.2 - BISTATIC SAR BASED ON TERRASAR-X AND GROUND BASED RECEIVERS
MO3.L09.2 - BISTATIC SAR BASED ON TERRASAR-X AND GROUND BASED RECEIVERSMO3.L09.2 - BISTATIC SAR BASED ON TERRASAR-X AND GROUND BASED RECEIVERS
MO3.L09.2 - BISTATIC SAR BASED ON TERRASAR-X AND GROUND BASED RECEIVERS
grssieee
 
FIRST BISTATIC SPACEBORNE SAR EXPERIMENTS WITH TANDEM-X.ppt
FIRST BISTATIC SPACEBORNE SAR EXPERIMENTS WITH TANDEM-X.pptFIRST BISTATIC SPACEBORNE SAR EXPERIMENTS WITH TANDEM-X.ppt
FIRST BISTATIC SPACEBORNE SAR EXPERIMENTS WITH TANDEM-X.ppt
grssieee
 
MO4.L09 - POTENTIAL AND LIMITATIONS OF FORWARD-LOOKING BISTATIC SAR
MO4.L09 - POTENTIAL AND LIMITATIONS OF FORWARD-LOOKING BISTATIC SARMO4.L09 - POTENTIAL AND LIMITATIONS OF FORWARD-LOOKING BISTATIC SAR
MO4.L09 - POTENTIAL AND LIMITATIONS OF FORWARD-LOOKING BISTATIC SAR
grssieee
 
TGS NSA- Firestone 3D
TGS NSA- Firestone 3DTGS NSA- Firestone 3D
TGS NSA- Firestone 3D
TGS
 
Quadrature amplitude modulation qam transmitter
Quadrature amplitude modulation qam transmitterQuadrature amplitude modulation qam transmitter
Quadrature amplitude modulation qam transmitter
Asaad Drake
 
TU1.L09.3 - Fully Polarimetric TerraSAR-X Data: Data Quality and Scientific ...
TU1.L09.3 - Fully Polarimetric TerraSAR-X Data: Data Quality and Scientific ...TU1.L09.3 - Fully Polarimetric TerraSAR-X Data: Data Quality and Scientific ...
TU1.L09.3 - Fully Polarimetric TerraSAR-X Data: Data Quality and Scientific ...
grssieee
 
Introduction to velocity model building
Introduction to velocity model buildingIntroduction to velocity model building
Introduction to velocity model building
Grupo de Geofísica Computacional, UNICAMP
 
TGS NSA- Loyal 3D
TGS NSA- Loyal 3D TGS NSA- Loyal 3D
TGS NSA- Loyal 3D
TGS
 

Recommended

MO3.L09.2 - BISTATIC SAR BASED ON TERRASAR-X AND GROUND BASED RECEIVERS
MO3.L09.2 - BISTATIC SAR BASED ON TERRASAR-X AND GROUND BASED RECEIVERSMO3.L09.2 - BISTATIC SAR BASED ON TERRASAR-X AND GROUND BASED RECEIVERS
MO3.L09.2 - BISTATIC SAR BASED ON TERRASAR-X AND GROUND BASED RECEIVERS
grssieee
 
FIRST BISTATIC SPACEBORNE SAR EXPERIMENTS WITH TANDEM-X.ppt
FIRST BISTATIC SPACEBORNE SAR EXPERIMENTS WITH TANDEM-X.pptFIRST BISTATIC SPACEBORNE SAR EXPERIMENTS WITH TANDEM-X.ppt
FIRST BISTATIC SPACEBORNE SAR EXPERIMENTS WITH TANDEM-X.ppt
grssieee
 
MO4.L09 - POTENTIAL AND LIMITATIONS OF FORWARD-LOOKING BISTATIC SAR
MO4.L09 - POTENTIAL AND LIMITATIONS OF FORWARD-LOOKING BISTATIC SARMO4.L09 - POTENTIAL AND LIMITATIONS OF FORWARD-LOOKING BISTATIC SAR
MO4.L09 - POTENTIAL AND LIMITATIONS OF FORWARD-LOOKING BISTATIC SAR
grssieee
 
TGS NSA- Firestone 3D
TGS NSA- Firestone 3DTGS NSA- Firestone 3D
TGS NSA- Firestone 3D
TGS
 
Quadrature amplitude modulation qam transmitter
Quadrature amplitude modulation qam transmitterQuadrature amplitude modulation qam transmitter
Quadrature amplitude modulation qam transmitter
Asaad Drake
 
TU1.L09.3 - Fully Polarimetric TerraSAR-X Data: Data Quality and Scientific ...
TU1.L09.3 - Fully Polarimetric TerraSAR-X Data: Data Quality and Scientific ...TU1.L09.3 - Fully Polarimetric TerraSAR-X Data: Data Quality and Scientific ...
TU1.L09.3 - Fully Polarimetric TerraSAR-X Data: Data Quality and Scientific ...
grssieee
 
Introduction to velocity model building
Introduction to velocity model buildingIntroduction to velocity model building
Introduction to velocity model building
Grupo de Geofísica Computacional, UNICAMP
 
TGS NSA- Loyal 3D
TGS NSA- Loyal 3D TGS NSA- Loyal 3D
TGS NSA- Loyal 3D
TGS
 
IGARSS11_FFBP_CSAR_v3.ppt
IGARSS11_FFBP_CSAR_v3.pptIGARSS11_FFBP_CSAR_v3.ppt
IGARSS11_FFBP_CSAR_v3.ppt
grssieee
 
TGS NSA- Blanchard 3D
TGS NSA- Blanchard 3DTGS NSA- Blanchard 3D
TGS NSA- Blanchard 3D
TGS
 
TGS Arcis- East Coast- Nova Scotia Renaissance phase
TGS Arcis- East Coast- Nova Scotia Renaissance phaseTGS Arcis- East Coast- Nova Scotia Renaissance phase
TGS Arcis- East Coast- Nova Scotia Renaissance phase
TGS
 
Tim lucas-id2ox
Tim lucas-id2oxTim lucas-id2ox
Tim lucas-id2ox
Tim Lucas
 
Seismic QC & Filtering with Geostatistics
Seismic QC & Filtering with GeostatisticsSeismic QC & Filtering with Geostatistics
Seismic QC & Filtering with Geostatistics
Geovariances
 
IGARSS2011_TU1.T03.2_Zhan.ppt
IGARSS2011_TU1.T03.2_Zhan.pptIGARSS2011_TU1.T03.2_Zhan.ppt
IGARSS2011_TU1.T03.2_Zhan.ppt
grssieee
 
TGS NSA- Waterford
TGS NSA- WaterfordTGS NSA- Waterford
TGS NSA- Waterford
TGS
 
TGS NSA- Freeport 3D
TGS NSA- Freeport 3DTGS NSA- Freeport 3D
TGS NSA- Freeport 3D
TGS
 
Numerical Simulation: Flight Dynamic Stability Analysis Using Unstructured Ba...
Numerical Simulation: Flight Dynamic Stability Analysis Using Unstructured Ba...Numerical Simulation: Flight Dynamic Stability Analysis Using Unstructured Ba...
Numerical Simulation: Flight Dynamic Stability Analysis Using Unstructured Ba...
Masahiro Kanazaki
 
Vortex Dissipation Due to Airfoil-Vortex Interaction
Vortex Dissipation Due to Airfoil-Vortex InteractionVortex Dissipation Due to Airfoil-Vortex Interaction
Vortex Dissipation Due to Airfoil-Vortex Interaction
Masahiro Kanazaki
 
Electronic Circuit and Communication System
Electronic Circuit and Communication SystemElectronic Circuit and Communication System
Electronic Circuit and Communication System
Yong Heui Cho
 
Spacecraft RF Communications Course Sampler
Spacecraft RF Communications Course SamplerSpacecraft RF Communications Course Sampler
Spacecraft RF Communications Course Sampler
Jim Jenkins
 
TGS Arcis- East Coast- Labrador Sea 2D deep basin
TGS Arcis- East Coast- Labrador Sea 2D deep basinTGS Arcis- East Coast- Labrador Sea 2D deep basin
TGS Arcis- East Coast- Labrador Sea 2D deep basin
TGS
 
2.5 pda-capwap - gray
2.5 pda-capwap - gray2.5 pda-capwap - gray
2.5 pda-capwap - gray
cfpbolivia
 
TU1.T10.2.pptx
TU1.T10.2.pptxTU1.T10.2.pptx
TU1.T10.2.pptx
grssieee
 
Fast Factorized Backprojection Algorithm for UWB Bistatic.pdf
Fast Factorized Backprojection Algorithm for UWB Bistatic.pdfFast Factorized Backprojection Algorithm for UWB Bistatic.pdf
Fast Factorized Backprojection Algorithm for UWB Bistatic.pdf
grssieee
 
Iea59 optimiz tobita aaa
Iea59 optimiz tobita aaaIea59 optimiz tobita aaa
Iea59 optimiz tobita aaa
wixapps
 
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course SamplerFundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Jim Jenkins
 
Filter_Designs
Filter_DesignsFilter_Designs
Filter_Designs
"Katherine \"Nikki\"​" Quinn
 
Presentation
PresentationPresentation
Presentation
Christophe Demaziere
 
Errors and biases in gps
Errors and biases in gpsErrors and biases in gps
Errors and biases in gps
maneeb
 
Single and Dual Frequency Solution In GPS
Single and Dual Frequency Solution In GPSSingle and Dual Frequency Solution In GPS
Single and Dual Frequency Solution In GPS
Jithin Raj K
 

More Related Content

What's hot

IGARSS11_FFBP_CSAR_v3.ppt
IGARSS11_FFBP_CSAR_v3.pptIGARSS11_FFBP_CSAR_v3.ppt
IGARSS11_FFBP_CSAR_v3.ppt
grssieee
 
TGS NSA- Blanchard 3D
TGS NSA- Blanchard 3DTGS NSA- Blanchard 3D
TGS NSA- Blanchard 3D
TGS
 
TGS Arcis- East Coast- Nova Scotia Renaissance phase
TGS Arcis- East Coast- Nova Scotia Renaissance phaseTGS Arcis- East Coast- Nova Scotia Renaissance phase
TGS Arcis- East Coast- Nova Scotia Renaissance phase
TGS
 
IGARSS2011_TU1.T03.2_Zhan.ppt
IGARSS2011_TU1.T03.2_Zhan.pptIGARSS2011_TU1.T03.2_Zhan.ppt
IGARSS2011_TU1.T03.2_Zhan.ppt
grssieee
 
TGS NSA- Waterford
TGS NSA- WaterfordTGS NSA- Waterford
TGS NSA- Waterford
TGS
 
TGS NSA- Freeport 3D
TGS NSA- Freeport 3DTGS NSA- Freeport 3D
TGS NSA- Freeport 3D
TGS
 
TGS Arcis- East Coast- Labrador Sea 2D deep basin
TGS Arcis- East Coast- Labrador Sea 2D deep basinTGS Arcis- East Coast- Labrador Sea 2D deep basin
TGS Arcis- East Coast- Labrador Sea 2D deep basin
TGS
 
TU1.T10.2.pptx
TU1.T10.2.pptxTU1.T10.2.pptx
TU1.T10.2.pptx
grssieee
 
Fast Factorized Backprojection Algorithm for UWB Bistatic.pdf
Fast Factorized Backprojection Algorithm for UWB Bistatic.pdfFast Factorized Backprojection Algorithm for UWB Bistatic.pdf
Fast Factorized Backprojection Algorithm for UWB Bistatic.pdf
grssieee
 
Iea59 optimiz tobita aaa
Iea59 optimiz tobita aaaIea59 optimiz tobita aaa
Iea59 optimiz tobita aaa
wixapps
 

What's hot (20)

IGARSS11_FFBP_CSAR_v3.ppt
IGARSS11_FFBP_CSAR_v3.pptIGARSS11_FFBP_CSAR_v3.ppt
IGARSS11_FFBP_CSAR_v3.ppt
 
TGS NSA- Blanchard 3D
TGS NSA- Blanchard 3DTGS NSA- Blanchard 3D
TGS NSA- Blanchard 3D
 
TGS Arcis- East Coast- Nova Scotia Renaissance phase
TGS Arcis- East Coast- Nova Scotia Renaissance phaseTGS Arcis- East Coast- Nova Scotia Renaissance phase
TGS Arcis- East Coast- Nova Scotia Renaissance phase
 
Tim lucas-id2ox
Tim lucas-id2oxTim lucas-id2ox
Tim lucas-id2ox
 
Seismic QC & Filtering with Geostatistics
Seismic QC & Filtering with GeostatisticsSeismic QC & Filtering with Geostatistics
Seismic QC & Filtering with Geostatistics
 
IGARSS2011_TU1.T03.2_Zhan.ppt
IGARSS2011_TU1.T03.2_Zhan.pptIGARSS2011_TU1.T03.2_Zhan.ppt
IGARSS2011_TU1.T03.2_Zhan.ppt
 
TGS NSA- Waterford
TGS NSA- WaterfordTGS NSA- Waterford
TGS NSA- Waterford
 
TGS NSA- Freeport 3D
TGS NSA- Freeport 3DTGS NSA- Freeport 3D
TGS NSA- Freeport 3D
 
Numerical Simulation: Flight Dynamic Stability Analysis Using Unstructured Ba...
Numerical Simulation: Flight Dynamic Stability Analysis Using Unstructured Ba...Numerical Simulation: Flight Dynamic Stability Analysis Using Unstructured Ba...
Numerical Simulation: Flight Dynamic Stability Analysis Using Unstructured Ba...
 
Vortex Dissipation Due to Airfoil-Vortex Interaction
Vortex Dissipation Due to Airfoil-Vortex InteractionVortex Dissipation Due to Airfoil-Vortex Interaction
Vortex Dissipation Due to Airfoil-Vortex Interaction
 
Electronic Circuit and Communication System
Electronic Circuit and Communication SystemElectronic Circuit and Communication System
Electronic Circuit and Communication System
 
Spacecraft RF Communications Course Sampler
Spacecraft RF Communications Course SamplerSpacecraft RF Communications Course Sampler
Spacecraft RF Communications Course Sampler
 
TGS Arcis- East Coast- Labrador Sea 2D deep basin
TGS Arcis- East Coast- Labrador Sea 2D deep basinTGS Arcis- East Coast- Labrador Sea 2D deep basin
TGS Arcis- East Coast- Labrador Sea 2D deep basin
 
2.5 pda-capwap - gray
2.5 pda-capwap - gray2.5 pda-capwap - gray
2.5 pda-capwap - gray
 
TU1.T10.2.pptx
TU1.T10.2.pptxTU1.T10.2.pptx
TU1.T10.2.pptx
 
Fast Factorized Backprojection Algorithm for UWB Bistatic.pdf
Fast Factorized Backprojection Algorithm for UWB Bistatic.pdfFast Factorized Backprojection Algorithm for UWB Bistatic.pdf
Fast Factorized Backprojection Algorithm for UWB Bistatic.pdf
 
Iea59 optimiz tobita aaa
Iea59 optimiz tobita aaaIea59 optimiz tobita aaa
Iea59 optimiz tobita aaa
 
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course SamplerFundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
 
Filter_Designs
Filter_DesignsFilter_Designs
Filter_Designs
 
Presentation
PresentationPresentation
Presentation
 

Viewers also liked

iono_statmodel_final-meyer.ppt
iono_statmodel_final-meyer.pptiono_statmodel_final-meyer.ppt
iono_statmodel_final-meyer.ppt
grssieee
 
2-Bordoni_IGARSS11_APC.ppt
2-Bordoni_IGARSS11_APC.ppt2-Bordoni_IGARSS11_APC.ppt
2-Bordoni_IGARSS11_APC.ppt
grssieee
 
Introduction to gps and gnss
Introduction to gps and gnssIntroduction to gps and gnss
Introduction to gps and gnss
Vivek Srivastava
 
Global positioning system ppt
Global positioning system pptGlobal positioning system ppt
Global positioning system ppt
Swapnil Ramgirwar
 

Viewers also liked (18)

Errors and biases in gps
Errors and biases in gpsErrors and biases in gps
Errors and biases in gps
 
Single and Dual Frequency Solution In GPS
Single and Dual Frequency Solution In GPSSingle and Dual Frequency Solution In GPS
Single and Dual Frequency Solution In GPS
 
iono_statmodel_final-meyer.ppt
iono_statmodel_final-meyer.pptiono_statmodel_final-meyer.ppt
iono_statmodel_final-meyer.ppt
 
Caroline
CarolineCaroline
Caroline
 
DGdefranceschi
DGdefranceschiDGdefranceschi
DGdefranceschi
 
2-Bordoni_IGARSS11_APC.ppt
2-Bordoni_IGARSS11_APC.ppt2-Bordoni_IGARSS11_APC.ppt
2-Bordoni_IGARSS11_APC.ppt
 
Earthquake forecasting based on ionosphere statistical monitoring
Earthquake forecasting based on ionosphere statistical monitoringEarthquake forecasting based on ionosphere statistical monitoring
Earthquake forecasting based on ionosphere statistical monitoring
 
Master Thesis Final Presentation: Ionosphere monitoring in GBAS using Dual Fr...
Master Thesis Final Presentation: Ionosphere monitoring in GBAS using Dual Fr...Master Thesis Final Presentation: Ionosphere monitoring in GBAS using Dual Fr...
Master Thesis Final Presentation: Ionosphere monitoring in GBAS using Dual Fr...
 
GPS-errors-1
GPS-errors-1GPS-errors-1
GPS-errors-1
 
Gps measurements
Gps measurementsGps measurements
Gps measurements
 
Seminar on GPS by Haleem
Seminar on GPS by HaleemSeminar on GPS by Haleem
Seminar on GPS by Haleem
 
Introduction to gps and gnss
Introduction to gps and gnssIntroduction to gps and gnss
Introduction to gps and gnss
 
A seminar on GPS Technology
A seminar on GPS TechnologyA seminar on GPS Technology
A seminar on GPS Technology
 
GNSS
GNSSGNSS
GNSS
 
Gps signal structure
Gps signal structureGps signal structure
Gps signal structure
 
Global positioning system ppt
Global positioning system pptGlobal positioning system ppt
Global positioning system ppt
 
Global Positioning System
Global Positioning SystemGlobal Positioning System
Global Positioning System
 
GPS ppt.
GPS ppt. GPS ppt.
GPS ppt.
 

Similar to Sat geo 03(1)

Pilot Contamination Mitigation for Wideband Massive MIMO: Number of Cells Vs ...
Pilot Contamination Mitigation for Wideband Massive MIMO: Number of Cells Vs ...Pilot Contamination Mitigation for Wideband Massive MIMO: Number of Cells Vs ...
Pilot Contamination Mitigation for Wideband Massive MIMO: Number of Cells Vs ...
T. E. BOGALE
 
GPS cycle slips detection and repair through various signal combinations
GPS cycle slips detection and repair through various signal combinationsGPS cycle slips detection and repair through various signal combinations
GPS cycle slips detection and repair through various signal combinations
IJMER
 
GPS cycle slips detection and repair through various signal combinations
GPS cycle slips detection and repair through various signal combinationsGPS cycle slips detection and repair through various signal combinations
GPS cycle slips detection and repair through various signal combinations
IJMER
 

Similar to Sat geo 03(1) (20)

DGPS
DGPSDGPS
DGPS
 
Gps2
Gps2Gps2
Gps2
 
GPS in remote sensing,P K MANI
GPS in remote sensing,P K MANIGPS in remote sensing,P K MANI
GPS in remote sensing,P K MANI
 
Gps in remote sensing, pk mani
Gps in remote sensing, pk maniGps in remote sensing, pk mani
Gps in remote sensing, pk mani
 
Pilot Contamination Mitigation for Wideband Massive MIMO: Number of Cells Vs ...
Pilot Contamination Mitigation for Wideband Massive MIMO: Number of Cells Vs ...Pilot Contamination Mitigation for Wideband Massive MIMO: Number of Cells Vs ...
Pilot Contamination Mitigation for Wideband Massive MIMO: Number of Cells Vs ...
 
GPS cycle slips detection and repair through various signal combinations
GPS cycle slips detection and repair through various signal combinationsGPS cycle slips detection and repair through various signal combinations
GPS cycle slips detection and repair through various signal combinations
 
Fractal Antenna
Fractal AntennaFractal Antenna
Fractal Antenna
 
GPS cycle slips detection and repair through various signal combinations
GPS cycle slips detection and repair through various signal combinationsGPS cycle slips detection and repair through various signal combinations
GPS cycle slips detection and repair through various signal combinations
 
Traversing Notes |surveying II | Sudip khadka
Traversing Notes |surveying II | Sudip khadka Traversing Notes |surveying II | Sudip khadka
Traversing Notes |surveying II | Sudip khadka
 
Tracking and parameter estimation.pptx
Tracking and parameter estimation.pptxTracking and parameter estimation.pptx
Tracking and parameter estimation.pptx
 
The GPS/GNSS Signal (2)
The GPS/GNSS Signal (2)The GPS/GNSS Signal (2)
The GPS/GNSS Signal (2)
 
GNSS Observation (4)
GNSS Observation (4)GNSS Observation (4)
GNSS Observation (4)
 
OFDM Basics.ppt
OFDM Basics.pptOFDM Basics.ppt
OFDM Basics.ppt
 
1 survey system design_and_engg
1 survey system design_and_engg1 survey system design_and_engg
1 survey system design_and_engg
 
Anti collision monitoring
Anti collision monitoringAnti collision monitoring
Anti collision monitoring
 
MIRAS: the instrument aboard SMOS
MIRAS: the instrument aboard SMOSMIRAS: the instrument aboard SMOS
MIRAS: the instrument aboard SMOS
 
Multi-Carrier Transmission over Mobile Radio Channels.ppt
Multi-Carrier Transmission over Mobile Radio Channels.pptMulti-Carrier Transmission over Mobile Radio Channels.ppt
Multi-Carrier Transmission over Mobile Radio Channels.ppt
 
GPS-errors-2
GPS-errors-2GPS-errors-2
GPS-errors-2
 
Gps
GpsGps
Gps
 
EC6602 - AWP UNI-4
EC6602 - AWP UNI-4EC6602 - AWP UNI-4
EC6602 - AWP UNI-4
 

Recently uploaded

FULL ENJOY Call Girls In Mahipalpur Delhi Contact Us 8377877756
FULL ENJOY Call Girls In Mahipalpur Delhi Contact Us 8377877756FULL ENJOY Call Girls In Mahipalpur Delhi Contact Us 8377877756
FULL ENJOY Call Girls In Mahipalpur Delhi Contact Us 8377877756
dollysharma2066
 
CCS335 _ Neural Networks and Deep Learning Laboratory_Lab Complete Record
CCS335 _ Neural Networks and Deep Learning Laboratory_Lab Complete RecordCCS335 _ Neural Networks and Deep Learning Laboratory_Lab Complete Record
CCS335 _ Neural Networks and Deep Learning Laboratory_Lab Complete Record
Asst.prof M.Gokilavani
 
UNIT-V FMM.HYDRAULIC TURBINE - Construction and working
UNIT-V FMM.HYDRAULIC TURBINE - Construction and workingUNIT-V FMM.HYDRAULIC TURBINE - Construction and working
UNIT-V FMM.HYDRAULIC TURBINE - Construction and working
rknatarajan
 
The Most Attractive Pune Call Girls Manchar 8250192130 Will You Miss This Cha...
The Most Attractive Pune Call Girls Manchar 8250192130 Will You Miss This Cha...The Most Attractive Pune Call Girls Manchar 8250192130 Will You Miss This Cha...
The Most Attractive Pune Call Girls Manchar 8250192130 Will You Miss This Cha...
ranjana rawat
 
Structural Analysis and Design of Foundations: A Comprehensive Handbook for S...
Structural Analysis and Design of Foundations: A Comprehensive Handbook for S...Structural Analysis and Design of Foundations: A Comprehensive Handbook for S...
Structural Analysis and Design of Foundations: A Comprehensive Handbook for S...
Dr.Costas Sachpazis
 
Call for Papers - Educational Administration: Theory and Practice, E-ISSN: 21...
Call for Papers - Educational Administration: Theory and Practice, E-ISSN: 21...Call for Papers - Educational Administration: Theory and Practice, E-ISSN: 21...
Call for Papers - Educational Administration: Theory and Practice, E-ISSN: 21...
Christo Ananth
 
result management system report for college project
result management system report for college projectresult management system report for college project
result management system report for college project
Tonystark477637
 
Call Now ≽ 9953056974 ≼🔝 Call Girls In New Ashok Nagar ≼🔝 Delhi door step de...
Call Now ≽ 9953056974 ≼🔝 Call Girls In New Ashok Nagar ≼🔝 Delhi door step de...Call Now ≽ 9953056974 ≼🔝 Call Girls In New Ashok Nagar ≼🔝 Delhi door step de...
Call Now ≽ 9953056974 ≼🔝 Call Girls In New Ashok Nagar ≼🔝 Delhi door step de...
9953056974 Low Rate Call Girls In Saket, Delhi NCR
 
(INDIRA) Call Girl Bhosari Call Now 8617697112 Bhosari Escorts 24x7
(INDIRA) Call Girl Bhosari Call Now 8617697112 Bhosari Escorts 24x7(INDIRA) Call Girl Bhosari Call Now 8617697112 Bhosari Escorts 24x7
(INDIRA) Call Girl Bhosari Call Now 8617697112 Bhosari Escorts 24x7
Call Girls in Nagpur High Profile Call Girls
 
(INDIRA) Call Girl Aurangabad Call Now 8617697112 Aurangabad Escorts 24x7
(INDIRA) Call Girl Aurangabad Call Now 8617697112 Aurangabad Escorts 24x7(INDIRA) Call Girl Aurangabad Call Now 8617697112 Aurangabad Escorts 24x7
(INDIRA) Call Girl Aurangabad Call Now 8617697112 Aurangabad Escorts 24x7
Call Girls in Nagpur High Profile Call Girls
 

Recently uploaded (20)

FULL ENJOY Call Girls In Mahipalpur Delhi Contact Us 8377877756
FULL ENJOY Call Girls In Mahipalpur Delhi Contact Us 8377877756FULL ENJOY Call Girls In Mahipalpur Delhi Contact Us 8377877756
FULL ENJOY Call Girls In Mahipalpur Delhi Contact Us 8377877756
 
CCS335 _ Neural Networks and Deep Learning Laboratory_Lab Complete Record
CCS335 _ Neural Networks and Deep Learning Laboratory_Lab Complete RecordCCS335 _ Neural Networks and Deep Learning Laboratory_Lab Complete Record
CCS335 _ Neural Networks and Deep Learning Laboratory_Lab Complete Record
 
Water Industry Process Automation & Control Monthly - April 2024
Water Industry Process Automation & Control Monthly - April 2024Water Industry Process Automation & Control Monthly - April 2024
Water Industry Process Automation & Control Monthly - April 2024
 
Roadmap to Membership of RICS - Pathways and Routes
Roadmap to Membership of RICS - Pathways and RoutesRoadmap to Membership of RICS - Pathways and Routes
Roadmap to Membership of RICS - Pathways and Routes
 
UNIT-V FMM.HYDRAULIC TURBINE - Construction and working
UNIT-V FMM.HYDRAULIC TURBINE - Construction and workingUNIT-V FMM.HYDRAULIC TURBINE - Construction and working
UNIT-V FMM.HYDRAULIC TURBINE - Construction and working
 
The Most Attractive Pune Call Girls Manchar 8250192130 Will You Miss This Cha...
The Most Attractive Pune Call Girls Manchar 8250192130 Will You Miss This Cha...The Most Attractive Pune Call Girls Manchar 8250192130 Will You Miss This Cha...
The Most Attractive Pune Call Girls Manchar 8250192130 Will You Miss This Cha...
 
Structural Analysis and Design of Foundations: A Comprehensive Handbook for S...
Structural Analysis and Design of Foundations: A Comprehensive Handbook for S...Structural Analysis and Design of Foundations: A Comprehensive Handbook for S...
Structural Analysis and Design of Foundations: A Comprehensive Handbook for S...
 
UNIT-II FMM-Flow Through Circular Conduits
UNIT-II FMM-Flow Through Circular ConduitsUNIT-II FMM-Flow Through Circular Conduits
UNIT-II FMM-Flow Through Circular Conduits
 
Call for Papers - Educational Administration: Theory and Practice, E-ISSN: 21...
Call for Papers - Educational Administration: Theory and Practice, E-ISSN: 21...Call for Papers - Educational Administration: Theory and Practice, E-ISSN: 21...
Call for Papers - Educational Administration: Theory and Practice, E-ISSN: 21...
 
result management system report for college project
result management system report for college projectresult management system report for college project
result management system report for college project
 
Extrusion Processes and Their Limitations
Extrusion Processes and Their LimitationsExtrusion Processes and Their Limitations
Extrusion Processes and Their Limitations
 
Vivazz, Mieres Social Housing Design Spain
Vivazz, Mieres Social Housing Design SpainVivazz, Mieres Social Housing Design Spain
Vivazz, Mieres Social Housing Design Spain
 
Call Now ≽ 9953056974 ≼🔝 Call Girls In New Ashok Nagar ≼🔝 Delhi door step de...
Call Now ≽ 9953056974 ≼🔝 Call Girls In New Ashok Nagar ≼🔝 Delhi door step de...Call Now ≽ 9953056974 ≼🔝 Call Girls In New Ashok Nagar ≼🔝 Delhi door step de...
Call Now ≽ 9953056974 ≼🔝 Call Girls In New Ashok Nagar ≼🔝 Delhi door step de...
 
Coefficient of Thermal Expansion and their Importance.pptx
Coefficient of Thermal Expansion and their Importance.pptxCoefficient of Thermal Expansion and their Importance.pptx
Coefficient of Thermal Expansion and their Importance.pptx
 
NFPA 5000 2024 standard .
NFPA 5000 2024 standard .NFPA 5000 2024 standard .
NFPA 5000 2024 standard .
 
UNIT-III FMM. DIMENSIONAL ANALYSIS
UNIT-III FMM. DIMENSIONAL ANALYSISUNIT-III FMM. DIMENSIONAL ANALYSIS
UNIT-III FMM. DIMENSIONAL ANALYSIS
 
ONLINE FOOD ORDER SYSTEM PROJECT REPORT.pdf
ONLINE FOOD ORDER SYSTEM PROJECT REPORT.pdfONLINE FOOD ORDER SYSTEM PROJECT REPORT.pdf
ONLINE FOOD ORDER SYSTEM PROJECT REPORT.pdf
 
(INDIRA) Call Girl Bhosari Call Now 8617697112 Bhosari Escorts 24x7
(INDIRA) Call Girl Bhosari Call Now 8617697112 Bhosari Escorts 24x7(INDIRA) Call Girl Bhosari Call Now 8617697112 Bhosari Escorts 24x7
(INDIRA) Call Girl Bhosari Call Now 8617697112 Bhosari Escorts 24x7
 
Unit 1 - Soil Classification and Compaction.pdf
Unit 1 - Soil Classification and Compaction.pdfUnit 1 - Soil Classification and Compaction.pdf
Unit 1 - Soil Classification and Compaction.pdf
 
(INDIRA) Call Girl Aurangabad Call Now 8617697112 Aurangabad Escorts 24x7
(INDIRA) Call Girl Aurangabad Call Now 8617697112 Aurangabad Escorts 24x7(INDIRA) Call Girl Aurangabad Call Now 8617697112 Aurangabad Escorts 24x7
(INDIRA) Call Girl Aurangabad Call Now 8617697112 Aurangabad Escorts 24x7
 

Sat geo 03(1)

  • 1. 1 / 105 OBSERVATION TECHNIQUES IN SATELLITE GEODESY III Global Positioning System – GPS Observable and Data Processing – Wolfgang Keller Institute of Geodesy – University of Stuttgart May 1, 2007
  • 3. 3 / 105 Three basic observables used with GPS system (in most cases): • pseudoranges from code measurements
  • 4. 3 / 105 Three basic observables used with GPS system (in most cases): • pseudoranges from code measurements • carrier phases or carrier phase differences
  • 5. 3 / 105 Three basic observables used with GPS system (in most cases): • pseudoranges from code measurements • carrier phases or carrier phase differences • differences in signal travel time from interferometric measurements.
  • 6. Pseudoranges from Code 4 / 105 Fundamental observation equation for a single code-derived pseudorange: PRi = |Xi − XB| + cdtu = c · ∆i (1) with • Xi position of satellite i in CTS • XB position of receiver antenna B in CTS • dtu clock synchronization error between GPS time and receiver clock • ∆i observed signal travel time from satellite i to receiver B • c speed of light.
  • 7. Pseudoranges from Code 4 / 105 Fundamental observation equation for a single code-derived pseudorange: PRi = |Xi − XB| + cdtu = c · ∆i (1) with • Xi position of satellite i in CTS • XB position of receiver antenna B in CTS • dtu clock synchronization error between GPS time and receiver clock • ∆i observed signal travel time from satellite i to receiver B • c speed of light. Derivation of coordinates of the receiver B needs at least four simultaneous pseudorange measurements.
  • 8. Carrier Phases 5 / 105 Basic observation equation for a carrier phase measurement: ΦBi = 2π λ (|Xi − XB| + NBi λ + cdtu) (2) with • λ carrier wavelength • NBi unknown integer number of complete carrier cycles • Xi, XB, dtu, c as before
  • 9. Carrier Phases 5 / 105 Basic observation equation for a carrier phase measurement: ΦBi = 2π λ (|Xi − XB| + NBi λ + cdtu) (2) with • λ carrier wavelength • NBi unknown integer number of complete carrier cycles • Xi, XB, dtu, c as before Main difficulty in the use of carrier phase observations: Determination of the unknown integer number NBi of cycle ambiguities. =⇒ Use of special sophisticated methods.
  • 10. Carrier Phase Differences 6 / 105 Basic observation in most cases: usage of the difference of the phase observations of the signal of the same satellite i registered at two receivers A and B.
  • 11. Carrier Phase Differences 6 / 105 Basic observation in most cases: usage of the difference of the phase observations of the signal of the same satellite i registered at two receivers A and B. Basic observation equation for this single phase difference: ∆ΦABi := ΦBi − ΦAi = 2π λ (|Xi − XB| − |Xi − XA| − (NBi − NAi )λ +c(dtuB − dtuA)). (3)
  • 12. Interferometric Measurements 7 / 105 • usage of GPS signals without knowledge of the signal structure
  • 13. Interferometric Measurements 7 / 105 • usage of GPS signals without knowledge of the signal structure • recording of signals with precise time marks at two different receivers A and B
  • 14. Interferometric Measurements 7 / 105 • usage of GPS signals without knowledge of the signal structure • recording of signals with precise time marks at two different receivers A and B • correlate signals afterwards
  • 15. Interferometric Measurements 7 / 105 • usage of GPS signals without knowledge of the signal structure • recording of signals with precise time marks at two different receivers A and B • correlate signals afterwards • fundamental observable: difference of the arrival times ∆τABi of the signal at the two receivers ∆τABi := |Xi − XB| − |Xi − XA| c (4)
  • 16. Interferometric Measurements 7 / 105 • usage of GPS signals without knowledge of the signal structure • recording of signals with precise time marks at two different receivers A and B • correlate signals afterwards • fundamental observable: difference of the arrival times ∆τABi of the signal at the two receivers ∆τABi := |Xi − XB| − |Xi − XA| c (4) Quite similar to Very Long Baseline Interferometry (VLBI) which uses Quasars as radio sources.
  • 17. Interferometric Measurements 7 / 105 • usage of GPS signals without knowledge of the signal structure • recording of signals with precise time marks at two different receivers A and B • correlate signals afterwards • fundamental observable: difference of the arrival times ∆τABi of the signal at the two receivers ∆τABi := |Xi − XB| − |Xi − XA| c (4) Quite similar to Very Long Baseline Interferometry (VLBI) which uses Quasars as radio sources. In Geodesy mostly: only code derived pseudoranges and carrier phase observations.
  • 19. Linear Combinations and Derived Observable 9 / 105 Carrier phases and code phases on both frequencies =⇒ pseudoranges
  • 20. Linear Combinations and Derived Observable 9 / 105 Carrier phases and code phases on both frequencies =⇒ pseudoranges For elimination of errors in observables, formation of linear combinations of those observables
  • 21. Linear Combinations and Derived Observable 9 / 105 Carrier phases and code phases on both frequencies =⇒ pseudoranges For elimination of errors in observables, formation of linear combinations of those observables Different kinds of linear combinations between observations ... • at different stations
  • 22. Linear Combinations and Derived Observable 9 / 105 Carrier phases and code phases on both frequencies =⇒ pseudoranges For elimination of errors in observables, formation of linear combinations of those observables Different kinds of linear combinations between observations ... • at different stations • of different satellites
  • 23. Linear Combinations and Derived Observable 9 / 105 Carrier phases and code phases on both frequencies =⇒ pseudoranges For elimination of errors in observables, formation of linear combinations of those observables Different kinds of linear combinations between observations ... • at different stations • of different satellites • at different epochs
  • 24. Linear Combinations and Derived Observable 9 / 105 Carrier phases and code phases on both frequencies =⇒ pseudoranges For elimination of errors in observables, formation of linear combinations of those observables Different kinds of linear combinations between observations ... • at different stations • of different satellites • at different epochs • on different frequencies
  • 25. 10 / 105 Figure 1: satellite-receiver configuration for forming differences, Rc ab: pseudorange at receiver a and receiver b to the satellite c
  • 26. 11 / 105 Introduction of notations for differences: • between-receiver single differences ∆(•) := (•)receiver j − (•)receiver i (5)
  • 27. 11 / 105 Introduction of notations for differences: • between-receiver single differences ∆(•) := (•)receiver j − (•)receiver i (5) • between-satellite single differences ∇(•) := (•)satellite j − (•)satellite i (6)
  • 28. 11 / 105 Introduction of notations for differences: • between-receiver single differences ∆(•) := (•)receiver j − (•)receiver i (5) • between-satellite single differences ∇(•) := (•)satellite j − (•)satellite i (6) • between-epoch single differences δ(•) := (•)epoch 2 − (•)epoch 1 (7)
  • 29. 12 / 105 For code-phases between-receiver single differences read ∆PR p CDij = ∆R p ij + c(dtuj − dtui) + c(dtaj − dtai) + c(dtsp − dtsp) + ǫ. (8)
  • 30. 12 / 105 For code-phases between-receiver single differences read ∆PR p CDij = ∆R p ij + c(dtuj − dtui) + c(dtaj − dtai) + c(dtsp − dtsp) + ǫ. (8) Here, notations mean: • distance difference from satellite p to the two receivers i, j: ∆R p ij = |Xp − Xj| − |Xp − Xi|
  • 31. 12 / 105 For code-phases between-receiver single differences read ∆PR p CDij = ∆R p ij + c(dtuj − dtui) + c(dtaj − dtai) + c(dtsp − dtsp) + ǫ. (8) Here, notations mean: • distance difference from satellite p to the two receivers i, j: ∆R p ij = |Xp − Xj| − |Xp − Xi| • dtuk clock error of receiver k
  • 32. 12 / 105 For code-phases between-receiver single differences read ∆PR p CDij = ∆R p ij + c(dtuj − dtui) + c(dtaj − dtai) + c(dtsp − dtsp) + ǫ. (8) Here, notations mean: • distance difference from satellite p to the two receivers i, j: ∆R p ij = |Xp − Xj| − |Xp − Xi| • dtuk clock error of receiver k • dtak atmospheric delay of signal travelling from satellite p to receiver k
  • 33. 12 / 105 For code-phases between-receiver single differences read ∆PR p CDij = ∆R p ij + c(dtuj − dtui) + c(dtaj − dtai) + c(dtsp − dtsp) + ǫ. (8) Here, notations mean: • distance difference from satellite p to the two receivers i, j: ∆R p ij = |Xp − Xj| − |Xp − Xi| • dtuk clock error of receiver k • dtak atmospheric delay of signal travelling from satellite p to receiver k • dtsp clock error of the satellite p
  • 34. 13 / 105 • satellite clock error cancels out in the between-receiver single difference
  • 35. 13 / 105 • satellite clock error cancels out in the between-receiver single difference • single difference: only differential influence of atmospheric delay
  • 36. 13 / 105 • satellite clock error cancels out in the between-receiver single difference • single difference: only differential influence of atmospheric delay • for stations close together: dai ≈ daj =⇒ atmospheric delay cancels out.
  • 37. 13 / 105 • satellite clock error cancels out in the between-receiver single difference • single difference: only differential influence of atmospheric delay • for stations close together: dai ≈ daj =⇒ atmospheric delay cancels out. • the same is true for the differential effect of the orbital errors
  • 38. 13 / 105 • satellite clock error cancels out in the between-receiver single difference • single difference: only differential influence of atmospheric delay • for stations close together: dai ≈ daj =⇒ atmospheric delay cancels out. • the same is true for the differential effect of the orbital errors for carrier-phases between-receiver single differences read ∆PR p CRij = ∆R p ij + c(dtuj − dtui) + c(dtaj − dtai) +c(dtsp − dtsp) + λ(Ni − Nj) + ǫ. (9) Ni, Nj: unknown integer phase ambiguities in the undifferenced carrier phase pseudoranges from satellite p to receivers i, j
  • 39. 13 / 105 • satellite clock error cancels out in the between-receiver single difference • single difference: only differential influence of atmospheric delay • for stations close together: dai ≈ daj =⇒ atmospheric delay cancels out. • the same is true for the differential effect of the orbital errors for carrier-phases between-receiver single differences read ∆PR p CRij = ∆R p ij + c(dtuj − dtui) + c(dtaj − dtai) +c(dtsp − dtsp) + λ(Ni − Nj) + ǫ. (9) Ni, Nj: unknown integer phase ambiguities in the undifferenced carrier phase pseudoranges from satellite p to receivers i, j in single differences for phase pseudoranges the satellite clock error cancels out
  • 40. 13 / 105 • satellite clock error cancels out in the between-receiver single difference • single difference: only differential influence of atmospheric delay • for stations close together: dai ≈ daj =⇒ atmospheric delay cancels out. • the same is true for the differential effect of the orbital errors for carrier-phases between-receiver single differences read ∆PR p CRij = ∆R p ij + c(dtuj − dtui) + c(dtaj − dtai) +c(dtsp − dtsp) + λ(Ni − Nj) + ǫ. (9) Ni, Nj: unknown integer phase ambiguities in the undifferenced carrier phase pseudoranges from satellite p to receivers i, j in single differences for phase pseudoranges the satellite clock error cancels out atmospheric delay and orbital error influences the single difference only differentially
  • 41. 14 / 105 double differences: • usually formed between receivers and satellites
  • 42. 14 / 105 double differences: • usually formed between receivers and satellites • defined as the difference between two between-receiver single differences
  • 43. 14 / 105 double differences: • usually formed between receivers and satellites • defined as the difference between two between-receiver single differences double difference for code pseudoranges: ∇∆PR pq CDij := ∆PR q CDij − ∆PR p CDij = ∇∆R pq ij + c((dtuj − dtui) − (dtuj − dtui)) +c((dt q aj − dt q ai) − (dt p aj − dt p ai)) + ǫ
  • 44. 14 / 105 double differences: • usually formed between receivers and satellites • defined as the difference between two between-receiver single differences double difference for code pseudoranges: ∇∆PR pq CDij := ∆PR q CDij − ∆PR p CDij = ∇∆R pq ij + c((dtuj − dtui) − (dtuj − dtui)) +c((dt q aj − dt q ai) − (dt p aj − dt p ai)) + ǫ double difference for carrier phase pseudoranges: ∇∆PR pq CRij := ∆PR q CRij − ∆PR p CRij = ∇∆R pq ij + c((dtuj − dtui) − (dtuj − dtui)) +c((dt q aj − dt q ai) − (dt p aj − dt p ai)) +λ((N q j − N q i ) − (N p j − N p i )) + ǫ
  • 45. 15 / 105 double differences: • besides the satellite clock error also the receiver clock error drops out • for the differential effect of the atmospheric delay and the orbital error the same as for the single differences is true
  • 46. 15 / 105 double differences: • besides the satellite clock error also the receiver clock error drops out • for the differential effect of the atmospheric delay and the orbital error the same as for the single differences is true carrier phase pseudorange differences: • between different epochs the unknown phase ambiguity drops out • used as an auxiliary observation for determination of the phase ambiguities
  • 47. 15 / 105 double differences: • besides the satellite clock error also the receiver clock error drops out • for the differential effect of the atmospheric delay and the orbital error the same as for the single differences is true carrier phase pseudorange differences: • between different epochs the unknown phase ambiguity drops out • used as an auxiliary observation for determination of the phase ambiguities differences between different frequencies: • elimination of ionospheric delay for long baselines (where this effect doesn’t cancel out due to single or double differencing
  • 48. 16 / 105 • linear combination of phase observation: sum of carrier phase observations on both frequencies (previously multiplied by constant factors)
  • 49. 16 / 105 • linear combination of phase observation: sum of carrier phase observations on both frequencies (previously multiplied by constant factors) • phase can be given in radian or in metrical units: factors of the same linear combination differ for both representations
  • 50. 16 / 105 • linear combination of phase observation: sum of carrier phase observations on both frequencies (previously multiplied by constant factors) • phase can be given in radian or in metrical units: factors of the same linear combination differ for both representations • radian representation of a phase measurement: Φ, its metric representation: L.
  • 51. 16 / 105 • linear combination of phase observation: sum of carrier phase observations on both frequencies (previously multiplied by constant factors) • phase can be given in radian or in metrical units: factors of the same linear combination differ for both representations • radian representation of a phase measurement: Φ, its metric representation: L. phase linear combination of two carrier phases: Φn,m := nΦ1 + mΦ2, n, m ∈ Z (10)
  • 52. 16 / 105 • linear combination of phase observation: sum of carrier phase observations on both frequencies (previously multiplied by constant factors) • phase can be given in radian or in metrical units: factors of the same linear combination differ for both representations • radian representation of a phase measurement: Φ, its metric representation: L. phase linear combination of two carrier phases: Φn,m := nΦ1 + mΦ2, n, m ∈ Z (10) frequency of newly generated signal: ωn,m = nω1 + mω2 (11)
  • 53. 16 / 105 • linear combination of phase observation: sum of carrier phase observations on both frequencies (previously multiplied by constant factors) • phase can be given in radian or in metrical units: factors of the same linear combination differ for both representations • radian representation of a phase measurement: Φ, its metric representation: L. phase linear combination of two carrier phases: Φn,m := nΦ1 + mΦ2, n, m ∈ Z (10) frequency of newly generated signal: ωn,m = nω1 + mω2 (11) its wavelength: λn,m = c ωn,m . (12)
  • 54. 17 / 105 same linear combination in metrical units: Ln,m = xL1 + yL2 = x λ1 2π Φ1 + y λ2 2π Φ2. (13)
  • 55. 17 / 105 same linear combination in metrical units: Ln,m = xL1 + yL2 = x λ1 2π Φ1 + y λ2 2π Φ2. (13) relation between, n, m and x, y: x = nλn,m λ1 , y = mλn,m λ2 (14)
  • 56. 17 / 105 same linear combination in metrical units: Ln,m = xL1 + yL2 = x λ1 2π Φ1 + y λ2 2π Φ2. (13) relation between, n, m and x, y: x = nλn,m λ1 , y = mλn,m λ2 (14) Expression of linear combinations in radians or metric units: depending on the purpose.
  • 57. 17 / 105 same linear combination in metrical units: Ln,m = xL1 + yL2 = x λ1 2π Φ1 + y λ2 2π Φ2. (13) relation between, n, m and x, y: x = nλn,m λ1 , y = mλn,m λ2 (14) Expression of linear combinations in radians or metric units: depending on the purpose. Original intention of the introduction of two frequencies in the GPS systems: elimination of the ionospheric time delay.
  • 58. 17 / 105 same linear combination in metrical units: Ln,m = xL1 + yL2 = x λ1 2π Φ1 + y λ2 2π Φ2. (13) relation between, n, m and x, y: x = nλn,m λ1 , y = mλn,m λ2 (14) Expression of linear combinations in radians or metric units: depending on the purpose. Original intention of the introduction of two frequencies in the GPS systems: elimination of the ionospheric time delay. How influences the ionospheric time delay the artificial frequencies?
  • 59. 18 / 105 Linear approximation of ionospheric phase delay on a frequency ω: δΦ = − Cne ω2 , (15) C = 40.3 and ne (unknown): characterises electron density in the ionosphere.
  • 60. 18 / 105 Linear approximation of ionospheric phase delay on a frequency ω: δΦ = − Cne ω2 , (15) C = 40.3 and ne (unknown): characterises electron density in the ionosphere. Ionospheric phase delay on the artificial frequency ωn,m: δΦn,m = nδΦ1 + mδΦ2 = −Cne( n ω2 1 + m ω2 2 ) = − Cne ω2 1ω2 2 (nω2 2 + mω2 1) ≈ − CI ω1ω2 (nω2 + mω1). (16)
  • 61. 19 / 105 Influence of the phase delay on frequency ωn,m on the pseudorange on this frequency: δLn,m = λn,m 2π δΦn,m = − CIc 2πω1ω2 nω2 + mω1 nω1 + mω2 (17)
  • 62. 19 / 105 Influence of the phase delay on frequency ωn,m on the pseudorange on this frequency: δLn,m = λn,m 2π δΦn,m = − CIc 2πω1ω2 nω2 + mω1 nω1 + mω2 (17) Phase noise changes when artificial frequencies are built: σn,m = λn,m 2π σΦn,m = λn,m 2π n2 + m2σΦ (18)
  • 63. 19 / 105 Influence of the phase delay on frequency ωn,m on the pseudorange on this frequency: δLn,m = λn,m 2π δΦn,m = − CIc 2πω1ω2 nω2 + mω1 nω1 + mω2 (17) Phase noise changes when artificial frequencies are built: σn,m = λn,m 2π σΦn,m = λn,m 2π n2 + m2σΦ (18) After these preparations the most common frequency combinations can be considered.
  • 64. wide-lane combination 20 / 105 defined by: L∆ := λ∆ 2π Φ1,−1 = c 2π(ω1 − ω2) ( 2π λ1 L1 − 2π λ2 L2) = ω1 ω1 − ω2 L1 − ω2 ω1 − ω2 L2 (19)
  • 65. 21 / 105 Ionospheric phase delay on the wide-lane: δΦ∆ = − CI ω1ω2 (ω2 − ω1) (20)
  • 66. 21 / 105 Ionospheric phase delay on the wide-lane: δΦ∆ = − CI ω1ω2 (ω2 − ω1) (20) relates to a pseudorange change of: δL∆ = − CIc ω1ω2 2πω2 − ω1 ω1 − ω2 = CIc 2πω1ω2 (21)
  • 67. 21 / 105 Ionospheric phase delay on the wide-lane: δΦ∆ = − CI ω1ω2 (ω2 − ω1) (20) relates to a pseudorange change of: δL∆ = − CIc ω1ω2 2πω2 − ω1 ω1 − ω2 = CIc 2πω1ω2 (21) Wavelength for wide-lane combination: λ∆ = 86 cm (∼ four times the original wavelength).
  • 68. 21 / 105 Ionospheric phase delay on the wide-lane: δΦ∆ = − CI ω1ω2 (ω2 − ω1) (20) relates to a pseudorange change of: δL∆ = − CIc ω1ω2 2πω2 − ω1 ω1 − ω2 = CIc 2πω1ω2 (21) Wavelength for wide-lane combination: λ∆ = 86 cm (∼ four times the original wavelength). The phase noise increases from about 3 mm at the L1, L2 frequencies to σ∆ = 19.4 mm.
  • 69. narrow-lane combination 22 / 105 defined by: LΣ := λΣ 2π Φ1,1 = c 2π(ω1 + ω2) ( 2π λ1 L1 + 2π λ2 L2) = ω1 ω1 + ω2 L1 + ω2 ω1 + ω2 L2 (22)
  • 70. 23 / 105 Ionospheric phase delay on the narrow-lane: δΦΣ = − CI ω1ω2 (ω2 + ω1) (23)
  • 71. 23 / 105 Ionospheric phase delay on the narrow-lane: δΦΣ = − CI ω1ω2 (ω2 + ω1) (23) relates to a pseudorange change of: δLΣ = − CIc ω1ω2 ω2 + ω1 ω1 + ω2 = − CIc 2πω1ω2 . (24)
  • 72. 23 / 105 Ionospheric phase delay on the narrow-lane: δΦΣ = − CI ω1ω2 (ω2 + ω1) (23) relates to a pseudorange change of: δLΣ = − CIc ω1ω2 ω2 + ω1 ω1 + ω2 = − CIc 2πω1ω2 . (24) =⇒ Influence of the ionospheric delay has exact the same magnitude on the wide- and on the narrow lane, they only differ in the sign.
  • 73. 23 / 105 Ionospheric phase delay on the narrow-lane: δΦΣ = − CI ω1ω2 (ω2 + ω1) (23) relates to a pseudorange change of: δLΣ = − CIc ω1ω2 ω2 + ω1 ω1 + ω2 = − CIc 2πω1ω2 . (24) =⇒ Influence of the ionospheric delay has exact the same magnitude on the wide- and on the narrow lane, they only differ in the sign. =⇒ Elimination of the ionospheric range delay by computing the mean of the wide- and of the narrow lane combination. Resulting combination: ionosphere-free combination.
  • 74. 23 / 105 Ionospheric phase delay on the narrow-lane: δΦΣ = − CI ω1ω2 (ω2 + ω1) (23) relates to a pseudorange change of: δLΣ = − CIc ω1ω2 ω2 + ω1 ω1 + ω2 = − CIc 2πω1ω2 . (24) =⇒ Influence of the ionospheric delay has exact the same magnitude on the wide- and on the narrow lane, they only differ in the sign. =⇒ Elimination of the ionospheric range delay by computing the mean of the wide- and of the narrow lane combination. Resulting combination: ionosphere-free combination. Wavelength on the narrow-lane: λΣ = 10.7 mm
  • 75. 23 / 105 Ionospheric phase delay on the narrow-lane: δΦΣ = − CI ω1ω2 (ω2 + ω1) (23) relates to a pseudorange change of: δLΣ = − CIc ω1ω2 ω2 + ω1 ω1 + ω2 = − CIc 2πω1ω2 . (24) =⇒ Influence of the ionospheric delay has exact the same magnitude on the wide- and on the narrow lane, they only differ in the sign. =⇒ Elimination of the ionospheric range delay by computing the mean of the wide- and of the narrow lane combination. Resulting combination: ionosphere-free combination. Wavelength on the narrow-lane: λΣ = 10.7 mm Phase noise reduces to σΣ = 2.1 mm
  • 76. ionosphere-free combination L3 24 / 105 defined by: L3 := L∆ + LΣ 2 = 1 2 ( ω1 ω1 − ω2 + ω1 ω1 + ω2 )L1 + ( ω2 ω1 + ω2 − ω2 ω1 − ω2 )L2 = 1 2 ω1(ω1 + ω2) + ω1(ω1 − ω2) ω2 1 − ω2 2 L1 + ω2(ω1 − ω2) − ω2(ω1 + ω2) ω2 1 − ω2 2 L2 = ω2 1 ω2 1 − ω2 2 L1 − ω2 2 ω2 1 − ω2 2 L2 (25)
  • 77. ionosphere-free combination L3 24 / 105 defined by: L3 := L∆ + LΣ 2 = 1 2 ( ω1 ω1 − ω2 + ω1 ω1 + ω2 )L1 + ( ω2 ω1 + ω2 − ω2 ω1 − ω2 )L2 = 1 2 ω1(ω1 + ω2) + ω1(ω1 − ω2) ω2 1 − ω2 2 L1 + ω2(ω1 − ω2) − ω2(ω1 + ω2) ω2 1 − ω2 2 L2 = ω2 1 ω2 1 − ω2 2 L1 − ω2 2 ω2 1 − ω2 2 L2 (25) Due to the non-integer nature of the factors in the ionosphere free combination no wavelength can be assigned to L3.
  • 78. geometry-free combination 25 / 105 defined by: L4 = LI := LΣ − L∆ = 2ω1ω2 ω2 1 − ω2 2 [L2 − L1] (26)
  • 79. geometry-free combination 25 / 105 defined by: L4 = LI := LΣ − L∆ = 2ω1ω2 ω2 1 − ω2 2 [L2 − L1] (26) free from the influence of • orbital errors
  • 80. geometry-free combination 25 / 105 defined by: L4 = LI := LΣ − L∆ = 2ω1ω2 ω2 1 − ω2 2 [L2 − L1] (26) free from the influence of • orbital errors • errors in the initial position of the receiver
  • 81. geometry-free combination 25 / 105 defined by: L4 = LI := LΣ − L∆ = 2ω1ω2 ω2 1 − ω2 2 [L2 − L1] (26) free from the influence of • orbital errors • errors in the initial position of the receiver • clock errors
  • 82. geometry-free combination 25 / 105 defined by: L4 = LI := LΣ − L∆ = 2ω1ω2 ω2 1 − ω2 2 [L2 − L1] (26) free from the influence of • orbital errors • errors in the initial position of the receiver • clock errors since these errors occur in the L1 and the L2 frequency and cancel out in this linear combination.
  • 83. geometry-free combination 25 / 105 defined by: L4 = LI := LΣ − L∆ = 2ω1ω2 ω2 1 − ω2 2 [L2 − L1] (26) free from the influence of • orbital errors • errors in the initial position of the receiver • clock errors since these errors occur in the L1 and the L2 frequency and cancel out in this linear combination. Geometry-free combination contains twice the influence of the ionosphere =⇒ used for ionosphere modelling.
  • 85. 27 / 105 • determination of the position of a GPS receiver (regardless to the positions of other receivers also observing the same satellites)
  • 86. 27 / 105 • determination of the position of a GPS receiver (regardless to the positions of other receivers also observing the same satellites) • practically only pseudoranges from code measurements are used
  • 87. 27 / 105 • determination of the position of a GPS receiver (regardless to the positions of other receivers also observing the same satellites) • practically only pseudoranges from code measurements are used At n epochs t1, . . . , tn the k satellites above the radio horizon are observed. These observations result in N = k · n observation equations. PRi(tj) = |Xi(tj) − xB| + cdtu,i, i = 1, . . . , k, j = 1, . . . , n. (27)
  • 88. 27 / 105 • determination of the position of a GPS receiver (regardless to the positions of other receivers also observing the same satellites) • practically only pseudoranges from code measurements are used At n epochs t1, . . . , tn the k satellites above the radio horizon are observed. These observations result in N = k · n observation equations. PRi(tj) = |Xi(tj) − xB| + cdtu,i, i = 1, . . . , k, j = 1, . . . , n. (27) cdtu,i contains all systematic errors such as • receiver clock error • satellite clock error • atmospheric delay • ionospheric delay
  • 89. 28 / 105 Observation equations (27) are nonlinear in the unknown receiver coordinates xB,m because of |Xi(tj) − xB| = 3 ∑ m=1 (Xi,m(tj) − xB,m)2
  • 90. 28 / 105 Observation equations (27) are nonlinear in the unknown receiver coordinates xB,m because of |Xi(tj) − xB| = 3 ∑ m=1 (Xi,m(tj) − xB,m)2 For linearization of the observation equations a prior information x (0) B about the position is required.
  • 91. 28 / 105 Observation equations (27) are nonlinear in the unknown receiver coordinates xB,m because of |Xi(tj) − xB| = 3 ∑ m=1 (Xi,m(tj) − xB,m)2 For linearization of the observation equations a prior information x (0) B about the position is required. Difference ∆x := xB − x (0) B is supposed to be small enough to neglect quadratic or higher order terms in ∆x
  • 92. 29 / 105 =⇒ linearized observation equations: PRi(tj) − |Xi(tj) − x (0) B | = ∂|Xi(tj) − xB| ∂xB · ∆x + cdtu,i (28) i = 1, . . . , k, j = 1, . . . , n
  • 93. 29 / 105 =⇒ linearized observation equations: PRi(tj) − |Xi(tj) − x (0) B | = ∂|Xi(tj) − xB| ∂xB · ∆x + cdtu,i (28) i = 1, . . . , k, j = 1, . . . , n N equations for 3 + k unknowns ∆x and dtu,1, . . . , dtu,k.
  • 94. 29 / 105 =⇒ linearized observation equations: PRi(tj) − |Xi(tj) − x (0) B | = ∂|Xi(tj) − xB| ∂xB · ∆x + cdtu,i (28) i = 1, . . . , k, j = 1, . . . , n N equations for 3 + k unknowns ∆x and dtu,1, . . . , dtu,k. ∂|Xi(tj) − xB| ∂xB = − Xi(tj) − x (0) B |Xi(tj) − x (0) B | holds, with the following definitions...
  • 95. 30 / 105 Y :=                         PR1(t1) − |X1(t1) − x (0) B | ... PRk(t1) − |Xk(t1) − x (0) B | PR1(t2) − |X1(t2) − x (0) B | ... PRk(t2) − |Xk(t2) − x (0) B | ... ... PR1(tn) − |X1(tn) − x (0) B | ... PRk(tn) − |Xk(tn) − x (0) B |                         ,
  • 96. 31 / 105 A :=                                − X1,1(t1)−x (0) B,1 |X1(t1)−x (0) B | − X1,2(t1)−x (0) B,2 |X1(t1)−x (0) B | − X1,3(t1)−x (0) B,3 |X1(t1)−x (0) B | c . . . 0 ... − Xk,1(t1)−x (0) B,1 |Xk(t1)−x (0) B | − Xk,2(t1)−x (0) B,2 |Xk(t1)−x (0) B | − Xk,3(t1)−x (0) B,3 |Xk(t1)−x (0) B | 0 . . . c − X1,1(t2)−x (0) B,1 |X1(t2)−x (0) B | − X1,2(t2)−x (0) B,2 |X1(t2)−x (0) B | − X1,3(t2)−x (0) B,3 |X1(t2)−x (0) B | c . . . 0 ... − Xk,1(t2)−x (0) B,1 |Xk(t2)−x (0) B | − Xk,2(t2)−x (0) B,2 |Xk(t2)−x (0) B | − Xk,3(t2)−x (0) B,3 |Xk(t2)−x (0) B | 0 . . . c ... ... − X1,1(tn)−x (0) B,1 |X1(tn)−x (0) B | − X1,2(tn)−x (0) B,2 |X1(tn)−x (0) B | − X1,3(tn)−x (0) B,3 |X1(tn)−x (0) B | c . . . 0 ... − Xk,1(tn)−x (0) B,1 |Xk(tn)−x (0) B | − Xk,2(tn)−x (0) B,2 |Xk(tn)−x (0) B | − Xk,3(tn)−x (0) B,3 |Xk(tn)−x (0) B | 0 . . . c                               
  • 97. 32 / 105 and ∆x := (x1,B − x (0) 1,B, x2,B − x (0) 2,B, x3,B − x (0) 3,B, dtu,1, . . . , dtu,k)⊤
  • 98. 32 / 105 and ∆x := (x1,B − x (0) 1,B, x2,B − x (0) 2,B, x3,B − x (0) 3,B, dtu,1, . . . , dtu,k)⊤ matrix notation of euqation (28): usual Gauss-Markov model of linear statistics E{Y} = A · ∆x, CYY = σ2 I
  • 99. 33 / 105 Usual least-squares estimation of position correction and error terms: ∆x = A⊤ A −1 A⊤ Y (29)
  • 100. 33 / 105 Usual least-squares estimation of position correction and error terms: ∆x = A⊤ A −1 A⊤ Y (29) with accuracy estimation: σ2 = | Y − A · ∆x 2 n · k − (3 + k) (30)
  • 101. 33 / 105 Usual least-squares estimation of position correction and error terms: ∆x = A⊤ A −1 A⊤ Y (29) with accuracy estimation: σ2 = | Y − A · ∆x 2 n · k − (3 + k) (30) and C∆x,∆x = σ2 A⊤ A −1 (31)
  • 102. 34 / 105 A few remarks have to be made: • at least 4 satellites at 2 epochs have to be observed: number n · k of observations has to exceed the number 3 + k of unknowns
  • 103. 34 / 105 A few remarks have to be made: • at least 4 satellites at 2 epochs have to be observed: number n · k of observations has to exceed the number 3 + k of unknowns • atmospheric and ionospheric delay are elevation dependent. =⇒ variation with time but variation is slow. For few observation epochs (typical for navigation solutions) delays can be considered constant =⇒ lumped error parameters dtu,i are only satellite, but not epoch dependent
  • 104. 34 / 105 A few remarks have to be made: • at least 4 satellites at 2 epochs have to be observed: number n · k of observations has to exceed the number 3 + k of unknowns • atmospheric and ionospheric delay are elevation dependent. =⇒ variation with time but variation is slow. For few observation epochs (typical for navigation solutions) delays can be considered constant =⇒ lumped error parameters dtu,i are only satellite, but not epoch dependent • systematic propagation- and clock errors modelled by nuisance parameters =⇒ remaining code phase errors can be considered uncorrelated
  • 105. Satellite Geometry and Accuracy Measures 35 / 105 Accuracy of determined position for navigation solutions depends on two factors: 1. variance σ2 of a code-pseudorange observation 2. geometric configuration of observed satellites.
  • 106. Satellite Geometry and Accuracy Measures 35 / 105 Accuracy of determined position for navigation solutions depends on two factors: 1. variance σ2 of a code-pseudorange observation 2. geometric configuration of observed satellites. Relation between standard deviation σ of a pseudorange observation and standard deviation of determined position σ∗: dilution of precision (DOP) σ∗ = DOP · σ. (32)
  • 107. 36 / 105 Usage of different DOP designations: • σH = HDOP · σ for horizontal positioning
  • 108. 36 / 105 Usage of different DOP designations: • σH = HDOP · σ for horizontal positioning • σV = VDOP · σ for vertical positioning
  • 109. 36 / 105 Usage of different DOP designations: • σH = HDOP · σ for horizontal positioning • σV = VDOP · σ for vertical positioning • σP = PDOP · σ for three-dimensional positioning
  • 110. 36 / 105 Usage of different DOP designations: • σH = HDOP · σ for horizontal positioning • σV = VDOP · σ for vertical positioning • σP = PDOP · σ for three-dimensional positioning • σT = TDOP · σ for timing
  • 111. 36 / 105 Usage of different DOP designations: • σH = HDOP · σ for horizontal positioning • σV = VDOP · σ for vertical positioning • σP = PDOP · σ for three-dimensional positioning • σT = TDOP · σ for timing Combined effect for three-dimensional positioning and timing: GDOP = PDOP2 + TDOP2. (33)
  • 112. 36 / 105 Usage of different DOP designations: • σH = HDOP · σ for horizontal positioning • σV = VDOP · σ for vertical positioning • σP = PDOP · σ for three-dimensional positioning • σT = TDOP · σ for timing Combined effect for three-dimensional positioning and timing: GDOP = PDOP2 + TDOP2. (33) Intuitive interpretation for PDOP: clock errors are supposed to be eliminated =⇒ one single epoch needed for positioning
  • 113. 37 / 105 Reduced observation equations: E{Y} = A · ∆x, CYY = σ2 I
  • 114. 37 / 105 Reduced observation equations: E{Y} = A · ∆x, CYY = σ2 I with Y :=    PR1(t1) − |X1(t1) − x (0) B | ... PRk(t1) − |Xk(t1) − x (0) B |   
  • 115. 37 / 105 Reduced observation equations: E{Y} = A · ∆x, CYY = σ2 I with Y :=    PR1(t1) − |X1(t1) − x (0) B | ... PRk(t1) − |Xk(t1) − x (0) B |    A :=        − X1,1(t1)−x (0) B,1 |X1(t1)−x (0) B | − X1,2(t1)−x (0) B,2 |X1(t1)−x (0) B | − X1,3(t1)−x (0) B,3 |X1(t1)−x (0) B | ... − Xk,1(t1)−x (0) B,1 |Xk(t1)−x (0) B | − Xk,2(t1)−x (0) B,2 |Xk(t1)−x (0) B | − Xk,3(t1)−x (0) B,3 |Xk(t1)−x (0) B |       
  • 116. 37 / 105 Reduced observation equations: E{Y} = A · ∆x, CYY = σ2 I with Y :=    PR1(t1) − |X1(t1) − x (0) B | ... PRk(t1) − |Xk(t1) − x (0) B |    A :=        − X1,1(t1)−x (0) B,1 |X1(t1)−x (0) B | − X1,2(t1)−x (0) B,2 |X1(t1)−x (0) B | − X1,3(t1)−x (0) B,3 |X1(t1)−x (0) B | ... − Xk,1(t1)−x (0) B,1 |Xk(t1)−x (0) B | − Xk,2(t1)−x (0) B,2 |Xk(t1)−x (0) B | − Xk,3(t1)−x (0) B,3 |Xk(t1)−x (0) B |        ∆x := (x1,B − x (0) 1,B, x2,B − x (0) 2,B, x3,B − x (0) 3,B)⊤
  • 117. 38 / 105 Estimated position corrections: ∆x = A⊤ A −1 A⊤ Y
  • 118. 38 / 105 Estimated position corrections: ∆x = A⊤ A −1 A⊤ Y Accuracy: C∆x,∆x = σ2 A⊤ A −1
  • 119. 38 / 105 Estimated position corrections: ∆x = A⊤ A −1 A⊤ Y Accuracy: C∆x,∆x = σ2 A⊤ A −1 Variance of estimated three-dimensional position (neglecting all cross-correlations): σ2 P = σ2 ∆x1 + σ2 ∆x2 + σ2 ∆x3 = σ2 trace A⊤ A −1
  • 120. 38 / 105 Estimated position corrections: ∆x = A⊤ A −1 A⊤ Y Accuracy: C∆x,∆x = σ2 A⊤ A −1 Variance of estimated three-dimensional position (neglecting all cross-correlations): σ2 P = σ2 ∆x1 + σ2 ∆x2 + σ2 ∆x3 = σ2 trace A⊤ A −1 =⇒ PDOP = trace A⊤A −1
  • 121. 39 / 105 Geometrical meaning of trace A⊤A −1 explained best in 2D: simultaneous observed pseudoranges to two satellites sufficient to determine the position. PR r1 r 2 PR α 1 2 Figure 2: geometrical satellite configuration
  • 122. 40 / 105 In this 2D one-epoch example the observation matrix consists of the direction unit vectors pointing to the satellites: A = −r1 −r2 (34)
  • 123. 40 / 105 In this 2D one-epoch example the observation matrix consists of the direction unit vectors pointing to the satellites: A = −r1 −r2 (34) Normal equation matrix: A⊤ A = r2 1,x + r2 2,x r1,xr1,y + r2,xr2,y r1,xr1,y + r2,xr2,y r2 1,y + r2 2,y (35)
  • 124. 40 / 105 In this 2D one-epoch example the observation matrix consists of the direction unit vectors pointing to the satellites: A = −r1 −r2 (34) Normal equation matrix: A⊤ A = r2 1,x + r2 2,x r1,xr1,y + r2,xr2,y r1,xr1,y + r2,xr2,y r2 1,y + r2 2,y (35) Determinant of the normal equation matrix: det A⊤ A = (r2 1,x + r2 2,x)(r2 1,y + r2 2,y) − (r1,xr1,y + r2,xr2,y)2 = (r1,xr2,y − r2,xr1,y)2 = sin2 α
  • 125. 41 / 105 Inverse of the normal equation matrix (computed by Schreibers rule): A⊤ A −1 = 1 sin2 α r2 1,y + r2 2,y −(r1,xr1,y + r2,xr2,y) −(r1,xr1,y + r2,xr2,y) r2 1,x + r2 2,x . (36)
  • 126. 41 / 105 Inverse of the normal equation matrix (computed by Schreibers rule): A⊤ A −1 = 1 sin2 α r2 1,y + r2 2,y −(r1,xr1,y + r2,xr2,y) −(r1,xr1,y + r2,xr2,y) r2 1,x + r2 2,x . (36) =⇒ PDOP: PDOP = trace A⊤A −1 (37) = 1 sin α r2 1,y + r2 2,y + r2 1,x + r2 2,x = √ 2 sin α
  • 127. 42 / 105 Area of the triangle spanned by the two unit vectors r1, r2: A = 1 2 sin α
  • 128. 42 / 105 Area of the triangle spanned by the two unit vectors r1, r2: A = 1 2 sin α =⇒ PDOP indirectly proportional to area spanned by the satellites: PDOP ∼ 1 A (38)
  • 129. 42 / 105 Area of the triangle spanned by the two unit vectors r1, r2: A = 1 2 sin α =⇒ PDOP indirectly proportional to area spanned by the satellites: PDOP ∼ 1 A (38) 2D −→ 3D: PDOP ∼ 1 V (39) V: volume of polyhedron spanned by the observer and the satellites.
  • 130. 43 / 105 P good PDOP P bad PDOP Figure 3: Good and bad satellite configuration
  • 132. 45 / 105 Navigation solution: limited in its accuracy (primarily relied on code phase measurements).
  • 133. 45 / 105 Navigation solution: limited in its accuracy (primarily relied on code phase measurements). For improvement of the accuracy two measurements: • use of carrier phases instead of code phases
  • 134. 45 / 105 Navigation solution: limited in its accuracy (primarily relied on code phase measurements). For improvement of the accuracy two measurements: • use of carrier phases instead of code phases • computation of baseline solutions instead of navigation, or single-point solutions
  • 135. 46 / 105 Basic idea of the baseline solution: • Coordinates of a baseline vector instead of coordinates of a point
  • 136. 46 / 105 Basic idea of the baseline solution: • Coordinates of a baseline vector instead of coordinates of a point • Baseline vector connects a known reference point with the point under consideration.
  • 137. 46 / 105 Basic idea of the baseline solution: • Coordinates of a baseline vector instead of coordinates of a point • Baseline vector connects a known reference point with the point under consideration. • Vector enters the observation model by forming differences of observations between a receiver at the reference point and a receiver at the current point.
  • 138. 46 / 105 Basic idea of the baseline solution: • Coordinates of a baseline vector instead of coordinates of a point • Baseline vector connects a known reference point with the point under consideration. • Vector enters the observation model by forming differences of observations between a receiver at the reference point and a receiver at the current point. =⇒ errors cancel out or are reduced in magnitude.
  • 139. 46 / 105 Basic idea of the baseline solution: • Coordinates of a baseline vector instead of coordinates of a point • Baseline vector connects a known reference point with the point under consideration. • Vector enters the observation model by forming differences of observations between a receiver at the reference point and a receiver at the current point. =⇒ errors cancel out or are reduced in magnitude. =⇒ remaining errors are small enough: determination of the integer phase ambiguities
  • 140. 46 / 105 Basic idea of the baseline solution: • Coordinates of a baseline vector instead of coordinates of a point • Baseline vector connects a known reference point with the point under consideration. • Vector enters the observation model by forming differences of observations between a receiver at the reference point and a receiver at the current point. =⇒ errors cancel out or are reduced in magnitude. =⇒ remaining errors are small enough: determination of the integer phase ambiguities =⇒ exploit the full accuracy potential of the observed carrier phases
  • 141. 46 / 105 Basic idea of the baseline solution: • Coordinates of a baseline vector instead of coordinates of a point • Baseline vector connects a known reference point with the point under consideration. • Vector enters the observation model by forming differences of observations between a receiver at the reference point and a receiver at the current point. =⇒ errors cancel out or are reduced in magnitude. =⇒ remaining errors are small enough: determination of the integer phase ambiguities =⇒ exploit the full accuracy potential of the observed carrier phases Baseline solutions can be computed on all frequency combinations. Special role of L3 combination: phase ambiguities lose their integer nature.
  • 142. Single Differences Solution 47 / 105 Observation of k satellites at n epochs from two receivers – reference receiver r and rover receiver v.
  • 143. Single Differences Solution 47 / 105 Observation of k satellites at n epochs from two receivers – reference receiver r and rover receiver v. Single differences observation equations (on an arbitrary frequency combination): ∆PR p CRrv(tj) = ∆R p rv(tj) + c(dtur − dtuv) + c(dtar − dtav)(tj) +λ(Nr − Nv) + ǫ (40) j = 1, . . . , n, p = 1, . . . , k.
  • 144. 48 / 105 ∆R p rv(tj) = 3 ∑ l=1 (X p l (tj) − xr,l)2 − 3 ∑ l=1 (X p l (tj) − xv,l)2 (41)
  • 145. 48 / 105 ∆R p rv(tj) = 3 ∑ l=1 (X p l (tj) − xr,l)2 − 3 ∑ l=1 (X p l (tj) − xv,l)2 (41) =⇒ single differences observation equations are nonlinear in the unknown coordinates xv of the rover receiver v.
  • 146. 48 / 105 ∆R p rv(tj) = 3 ∑ l=1 (X p l (tj) − xr,l)2 − 3 ∑ l=1 (X p l (tj) − xv,l)2 (41) =⇒ single differences observation equations are nonlinear in the unknown coordinates xv of the rover receiver v. (Coordinates of the reference receiver are assumed to be known.)
  • 147. 48 / 105 ∆R p rv(tj) = 3 ∑ l=1 (X p l (tj) − xr,l)2 − 3 ∑ l=1 (X p l (tj) − xv,l)2 (41) =⇒ single differences observation equations are nonlinear in the unknown coordinates xv of the rover receiver v. (Coordinates of the reference receiver are assumed to be known.) For the linearization a prior information x0 v about the position of the rover is required.
  • 148. 48 / 105 ∆R p rv(tj) = 3 ∑ l=1 (X p l (tj) − xr,l)2 − 3 ∑ l=1 (X p l (tj) − xv,l)2 (41) =⇒ single differences observation equations are nonlinear in the unknown coordinates xv of the rover receiver v. (Coordinates of the reference receiver are assumed to be known.) For the linearization a prior information x0 v about the position of the rover is required. =⇒ linearized single differences observation equations: ∆PR p CRrv(tj) − ∆R p,0 rv (tj) = ∂∆R p rv ∂xv (tj) · ∆xv + c(dtur − dtuv) +c(dtar − dtav)(tj) + λ(Nr − Nv) +ǫ. (42)
  • 149. 49 / 105 Computed carrier phase single difference: ∆R p,0 rv (tj) = 3 ∑ l=1 (X p l (tj) − xr,l)2 − 3 ∑ l=1 (X p l (tj) − x0 v,l)2 (43)
  • 150. 49 / 105 Computed carrier phase single difference: ∆R p,0 rv (tj) = 3 ∑ l=1 (X p l (tj) − xr,l)2 − 3 ∑ l=1 (X p l (tj) − x0 v,l)2 (43) Partial derivatives have the form: ∂∆R p rv ∂xv (tj) = Xp − x0 v |Xp − x0 v| (44)
  • 151. 50 / 105 Unknown position corrections: difference between the true but unknown position of the rover and the available prior information about this position: ∆xv = xv − x0 v (45)
  • 152. 50 / 105 Unknown position corrections: difference between the true but unknown position of the rover and the available prior information about this position: ∆xv = xv − x0 v (45) Among the remaining terms of the single differences observation equations two groups can be distinguished: • time independent terms like phase ambiguities, clock errors • time dependent atmospheric and ionospheric propagation-delay terms
  • 153. 51 / 105 Time independent terms remain in the observation equations as they are.
  • 154. 51 / 105 Time independent terms remain in the observation equations as they are. Modelling of atmospheric and ionospheric terms: c(dtar − dtav)(tj) ≈ M(tj, T, p, H) = 0 , short baselines m(tj, T, p, H) , long baselines m(tj, T, p, H): standard model for atmospheric and ionospheric propagation delay
  • 155. 51 / 105 Time independent terms remain in the observation equations as they are. Modelling of atmospheric and ionospheric terms: c(dtar − dtav)(tj) ≈ M(tj, T, p, H) = 0 , short baselines m(tj, T, p, H) , long baselines m(tj, T, p, H): standard model for atmospheric and ionospheric propagation delay re-arrangement of the terms: ∆PR p CRrv(tj) − ∆R p,0 rv (tj) − M(tj, T, p, H) = ∂∆R p rv ∂xv (tj) · ∆xv + c(dt p ur − dt p uv) + λ(N p r − N p v ) + ǫ
  • 156. 52 / 105 Individual terms are collected in vectors and in matrices: Y =                        ∆PR1 CRrv(t1) − ∆R1,0 rv (t1) − M(t1, T, p, H) ... ∆PRk CRrv(t1) − ∆Rk,0 rv (t1) − M(t1, T, p, H) ∆PR1 CRrv(t2) − ∆R1,0 rv (t2) − M(t2, T, p, H) ... ∆PRk CRrv(t2) − ∆Rk,0 rv (t2) − M(t2, T, p, H) ... ... ∆PR1 CRrv(tn) − ∆R1,0 rv (tn) − M(tn, T, p, H) ... ∆PRk CRrv(tn) − ∆Rk,0 rv (tn) − M(tn, T, p, H)                       
  • 157. 53 / 105 A =                               X1 1−x0 v,1 |X1−x0 v| (t1) X1 2−x0 v,2 |X1−x0 v| (t1) X1 3−x0 v,3 |X1−x0 v| (t1) c λ . . . 0 ... Xk 1−x0 v,1 |Xk−x0 v| (t1) Xk 2−x0 v,2 |Xk−x0 v| (t1) Xk 3−x0 v,3 |Xk−x0 v| (t1) c 0 . . . λ X1 1−x0 v,1 |X1−x0 v| (t2) X1 2−x0 v,2 |X1−x0 v| (t2) X1 3−x0 v,3 |X1−x0 v| (t2) c λ . . . 0 ... Xk 1−x0 v,1 |Xk−x0 v| (t2) Xk 2−x0 v,2 |Xk−x0 v| (t2) Xk 3−x0 v,3 |Xk−x0 v| (t2) c 0 . . . λ ... ... X1 1−x0 v,1 |X1−x0 v| (tn) X1 2−x0 v,2 |X1−x0 v| (tn) X1 3−x0 v,3 |X1−x0 v| (tn) c λ . . . 0 ... Xk 1−x0 v,1 |Xk−x0 v| (tn) Xk 2−x0 v,2 |Xk−x0 v| (tn) Xk 3−x0 v,3 |Xk−x0 v| (tn) c 0 . . . λ                              
  • 158. 54 / 105 ∆xv = (xv,1 − x0 v,1, xv,2 − x0 v,2, xv,3 − x0 v,3, dtur − dtuv, N1 r − N1 v , . . . , Nk r − Nk v)⊤ .
  • 159. 54 / 105 ∆xv = (xv,1 − x0 v,1, xv,2 − x0 v,2, xv,3 − x0 v,3, dtur − dtuv, N1 r − N1 v , . . . , Nk r − Nk v)⊤ . =⇒ Gauss-Markov form of the linearized single differences observation equations: E{Y} = A · ∆x. (46)
  • 160. 54 / 105 ∆xv = (xv,1 − x0 v,1, xv,2 − x0 v,2, xv,3 − x0 v,3, dtur − dtuv, N1 r − N1 v , . . . , Nk r − Nk v)⊤ . =⇒ Gauss-Markov form of the linearized single differences observation equations: E{Y} = A · ∆x. (46) Consider separately: covariance matrix CYY of the single difference observations.
  • 161. 54 / 105 ∆xv = (xv,1 − x0 v,1, xv,2 − x0 v,2, xv,3 − x0 v,3, dtur − dtuv, N1 r − N1 v , . . . , Nk r − Nk v)⊤ . =⇒ Gauss-Markov form of the linearized single differences observation equations: E{Y} = A · ∆x. (46) Consider separately: covariance matrix CYY of the single difference observations. Undifferenced phase observations are uncorrelated, but forming differences may generate correlations of the single difference observations.
  • 162. 55 / 105 Phases p, q of the satellites observed at the reference station r and at the rover station v at the epoch tj: Ψ(tj) := (Ψ p r (tj), Ψ p v(tj), Ψ q r (tj), Ψ q v(tj))⊤
  • 163. 55 / 105 Phases p, q of the satellites observed at the reference station r and at the rover station v at the epoch tj: Ψ(tj) := (Ψ p r (tj), Ψ p v(tj), Ψ q r (tj), Ψ q v(tj))⊤ Observations are uncorrelated: CΨΨ = σ2 I (47)
  • 164. 55 / 105 Phases p, q of the satellites observed at the reference station r and at the rover station v at the epoch tj: Ψ(tj) := (Ψ p r (tj), Ψ p v(tj), Ψ q r (tj), Ψ q v(tj))⊤ Observations are uncorrelated: CΨΨ = σ2 I (47) Formation of two single differences from these four undifferenced phase observations: ∆Ψ = Ψ p r (tj) − Ψ p v(tj) Ψ q r (tj) − Ψ q v(tj) . (48)
  • 165. 56 / 105 Establish a connection between ∆Ψ and Ψ: ∆Ψ = D · Ψ. D = 1 −1 0 0 0 0 1 −1
  • 166. 56 / 105 Establish a connection between ∆Ψ and Ψ: ∆Ψ = D · Ψ. D = 1 −1 0 0 0 0 1 −1 Laws of covariance propagation =⇒ covariance matrix of single differences: C∆Ψ∆Ψ = DCΨΨD⊤ = σ2 DD⊤ = σ2 2 0 0 2 = 2σ2 I.
  • 167. 57 / 105 =⇒ For the same epoch single differences to different satellites: • uncorrelated • twice the variance of the single difference observation.
  • 168. 57 / 105 =⇒ For the same epoch single differences to different satellites: • uncorrelated • twice the variance of the single difference observation. Single differences to the same satellite at different epochs: • uncorrelated • twice the variance of the undifferenced observations.
  • 169. 57 / 105 =⇒ For the same epoch single differences to different satellites: • uncorrelated • twice the variance of the single difference observation. Single differences to the same satellite at different epochs: • uncorrelated • twice the variance of the undifferenced observations. =⇒ Covariance-matrix of the observations: CYY = 2σ2 I. (49)
  • 170. 58 / 105 Least-squares estimation of the position correction and the error terms: ∆x = A⊤ A −1 A⊤ Y (50)
  • 171. 58 / 105 Least-squares estimation of the position correction and the error terms: ∆x = A⊤ A −1 A⊤ Y (50) Accuracy estimation: σ2 = | Y − A · ∆x 2 2 ∗ (n ∗ k − (4 + k)) (51)
  • 172. 58 / 105 Least-squares estimation of the position correction and the error terms: ∆x = A⊤ A −1 A⊤ Y (50) Accuracy estimation: σ2 = | Y − A · ∆x 2 2 ∗ (n ∗ k − (4 + k)) (51) and C∆x,∆x = 2σ2 A⊤ A −1 (52)
  • 174. 60 / 105 k satellites have been observed at n epochs from two receivers – reference receiver r and rover receiver v.
  • 175. 60 / 105 k satellites have been observed at n epochs from two receivers – reference receiver r and rover receiver v. Double differences observation equations on an arbitrary frequency combination: ∇∆PR pq CRrv(tj) := ∆PR q CRrv(tj) − ∆PR p CRrv(tj) = ∇∆R pq rv (tj) +c((dt q ar(tj) − dt q av(tj)) −(dt p ar(tj) − dt p av(tj))) +λ((N q r − N q v ) − (N p r − N p v )) + ǫ. (53)
  • 176. 61 / 105 ∇∆R pq rv (tj) =   3 ∑ l=1 (x q l (tj) − xr,l)2 − 3 ∑ l=1 (x q l (tj) − xv,l)2   −   3 ∑ l=1 (x p l (tj) − xr,l)2 − 3 ∑ l=1 (x p l (tj) − xv,l)2   =⇒ Double differences observation equations are nonlinear in the unknown coordinates xv of the rover receiver v.
  • 177. 61 / 105 ∇∆R pq rv (tj) =   3 ∑ l=1 (x q l (tj) − xr,l)2 − 3 ∑ l=1 (x q l (tj) − xv,l)2   −   3 ∑ l=1 (x p l (tj) − xr,l)2 − 3 ∑ l=1 (x p l (tj) − xv,l)2   =⇒ Double differences observation equations are nonlinear in the unknown coordinates xv of the rover receiver v. (Coordinates of the reference receiver assumed to be known.)
  • 178. 61 / 105 ∇∆R pq rv (tj) =   3 ∑ l=1 (x q l (tj) − xr,l)2 − 3 ∑ l=1 (x q l (tj) − xv,l)2   −   3 ∑ l=1 (x p l (tj) − xr,l)2 − 3 ∑ l=1 (x p l (tj) − xv,l)2   =⇒ Double differences observation equations are nonlinear in the unknown coordinates xv of the rover receiver v. (Coordinates of the reference receiver assumed to be known.) For linearization: need of prior information x0 v about position of rover.
  • 179. 62 / 105 =⇒ Linearized single differences observation equations: ∇∆PR pq CRrv(tj) − ∇∆R pq,0 rv (tj) = ∂∇∆R pq rv ∂xv (tj) · ∆xv +c((dt q ar(tj) − dt q av(tj)) −(dt p ar(tj) − dt p av(tj))) +λ((N q r − N q v ) − (N p r − N p v )) +ǫ. (54)
  • 180. 63 / 105 Computed carrier phase double difference: ∇∆R pq,0 rv (tj) =   3 ∑ l=1 (X q l (tj) − xr,l)2 − 3 ∑ l=1 (X q l (tj) − x0 v,l)2  (55) −   3 ∑ l=1 (X p l (tj) − xr,l)2 − 3 ∑ l=1 (X p l (tj) − x0 v,l)2  
  • 181. 63 / 105 Computed carrier phase double difference: ∇∆R pq,0 rv (tj) =   3 ∑ l=1 (X q l (tj) − xr,l)2 − 3 ∑ l=1 (X q l (tj) − x0 v,l)2  (55) −   3 ∑ l=1 (X p l (tj) − xr,l)2 − 3 ∑ l=1 (X p l (tj) − x0 v,l)2   Partial derivatives: ∂∇∆R pq rv ∂xv (tj) = ( Xq − x0 v |Xq − x0 v| ) − ( Xp − x0 v |Xp − x0 v| ) (56)
  • 182. 64 / 105 Difference between true (but unknown) position of rover and available prior information about its position: Unknown position corrections: ∆xv = xv − x0 v
  • 183. 64 / 105 Difference between true (but unknown) position of rover and available prior information about its position: Unknown position corrections: ∆xv = xv − x0 v Differentiation of two groups among the remaining terms of the double differences observation equations: • time independent terms (e. g. phase ambiguities) • time dependent atmospheric and ionospheric propagation-delay terms
  • 184. 65 / 105 Time independent terms remain in the observation equations unchanged.
  • 185. 65 / 105 Time independent terms remain in the observation equations unchanged. Atmospheric and ionospheric terms: c(dt q ar − dt q av)(tj) − c(dt p ar − dt p av)(tj) ≈ Mpq (tj, T, p, H) M(tj, T, p, H): difference of two standard models for atmospheric and ionospheric propagation delay
  • 186. 65 / 105 Time independent terms remain in the observation equations unchanged. Atmospheric and ionospheric terms: c(dt q ar − dt q av)(tj) − c(dt p ar − dt p av)(tj) ≈ Mpq (tj, T, p, H) M(tj, T, p, H): difference of two standard models for atmospheric and ionospheric propagation delay ∇∆PR pq CRrv(tj) − ∇∆R pq,0 rv (tj) − Mpq (tj, T, p, H) = ∂∇∆R pq rv ∂xv (tj) · ∆xv + λ((N q r − N q v ) − (N p r − N p v )) + ǫ
  • 187. 66 / 105 Collection of individual terms in vectors and in matrices: Y =                 ∇∆PR12 CRrv(t1) − ∇∆R12,0 rv (t1) − M12(t1, T, p, H) ... ∇∆PR1k CRrv(t1) − ∇∆R1k,0 rv (t1) − M1k(t1, T, p, H) ... ... ∇∆PR12 CRrv(tn) − ∇∆R12,0 rv (tn) − M12(tn, T, p, H) ... ∇∆PR1k CRrv(tn) − ∇∆R1k,0 rv (tn) − M1k(tn, T, p, H)                
  • 188. 67 / 105 A =                                 −( X1 1 −x0 v,1 |X1−x0 v| − X2 1 −x0 v,1 |X2−x0 v| )(t1) −( X1 2−x0 v,2 |X1−x0 v| − X2 2−x0 v,2 |X2−x0 v| )(t1) −( X1 3−x0 v,3 |X1−x0 v| − X2 3−x0 v,3 |X2−x0 v| )(t1) λ . . . 0 . . . −( X1 1 −x0 v,1 |X1−x0 v| − Xk 1 −x0 v,1 |Xk−x0 v| )(t1) −( X1 2−x0 v,2 |X1−x0 v| − Xk 2−x0 v,2 |Xk−x0 v| )(t1) −( X1 3−x0 v,3 |X1−x0 v| − Xk 3−x0 v,3 |Xk−x0 v| )(t1) 0 . . . λ . . . . . . −( X1 1 −x0 v,1 |X1−x0 v| − X2 1 −x0 v,1 |X2−x0 v| )(tn) −( X1 2−x0 v,2 |X1−x0 v| − X2 2−x0 v,2 |X2−x0 v| )(tn) −( X1 3−x0 v,3 |X1−x0 v| − X2 3−x0 v,3 |X2−x0 v| )(tn) λ . . . 0 . . . −( X1 1 −x0 v,1 |X1−x0 v| − Xk 1 −x0 v,1 |Xk−x0 v| )(tn) −( X1 2−x0 v,2 |X1−x0 v| − Xk 2−x0 v,2 |Xk−x0 v| )(tn) −( X1 3−x0 v,3 |X1−x0 v| − Xk 3−x0 v,3 |Xk−x0 v| )(tn) 0 . . . λ                                
  • 189. 68 / 105 ∆xv = (xv,1 − x0 v,1, xv,2 − x0 v,2, xv,3 − x0 v,3, N12 , . . . , N1k )⊤ Npq: abbreviation of double differences of unknown integer phase ambiguities: Npq := (N q r − N q v ) − (N p r − N p v ).
  • 190. 68 / 105 ∆xv = (xv,1 − x0 v,1, xv,2 − x0 v,2, xv,3 − x0 v,3, N12 , . . . , N1k )⊤ Npq: abbreviation of double differences of unknown integer phase ambiguities: Npq := (N q r − N q v ) − (N p r − N p v ). =⇒ Gauss-Markov form of linearized double differences observation equations: E{Y} = A · ∆x.
  • 191. 68 / 105 ∆xv = (xv,1 − x0 v,1, xv,2 − x0 v,2, xv,3 − x0 v,3, N12 , . . . , N1k )⊤ Npq: abbreviation of double differences of unknown integer phase ambiguities: Npq := (N q r − N q v ) − (N p r − N p v ). =⇒ Gauss-Markov form of linearized double differences observation equations: E{Y} = A · ∆x. Consider separately: covariance matrix CYY of double difference observations.
  • 192. 69 / 105 Observation of k · n single differences ∆Ψi rv between reference receiver r and rover receiver v and k satellites at n epochs.
  • 193. 69 / 105 Observation of k · n single differences ∆Ψi rv between reference receiver r and rover receiver v and k satellites at n epochs. =⇒ m = (k − 1) double differences ∇∆Ψ pq rv (tj) can be formed.
  • 194. 69 / 105 Observation of k · n single differences ∆Ψi rv between reference receiver r and rover receiver v and k satellites at n epochs. =⇒ m = (k − 1) double differences ∇∆Ψ pq rv (tj) can be formed. Single differences to k satellites recorded at the epoch tj: ∆Ψ(tj) := (∆Ψ1 (tj), . . . , ∆Ψk (tj))⊤
  • 195. 69 / 105 Observation of k · n single differences ∆Ψi rv between reference receiver r and rover receiver v and k satellites at n epochs. =⇒ m = (k − 1) double differences ∇∆Ψ pq rv (tj) can be formed. Single differences to k satellites recorded at the epoch tj: ∆Ψ(tj) := (∆Ψ1 (tj), . . . , ∆Ψk (tj))⊤ =⇒ m double differences can be formed at this epoch: ∇∆Ψ(tj) =      ∆Ψ1(tj) − ∆Ψ2(tj) ∆Ψ1(tj) − ∆Ψ3(tj) ... ∆Ψ1(tj) − ∆Ψk(tj)      .
  • 196. 70 / 105 Single and double differences can be related to each other: ∇∆Ψ(tj) = D · ∆Ψ(tj). D =      1 −1 1 −1 ... 1 −1     
  • 197. 71 / 105 Laws of covariance propagation =⇒ covariance matrix of double differences: C∇∆Ψ∇∆Ψ = DC∆Ψ∆ΨD⊤ = 2σ2 DD⊤ = 2σ2      2 1 1 . . . . . . 1 1 2 1 1 . . . 1 ... 1 . . . . . . 1 1 2      .
  • 198. 71 / 105 Laws of covariance propagation =⇒ covariance matrix of double differences: C∇∆Ψ∇∆Ψ = DC∆Ψ∆ΨD⊤ = 2σ2 DD⊤ = 2σ2      2 1 1 . . . . . . 1 1 2 1 1 . . . 1 ... 1 . . . . . . 1 1 2      . =⇒ Double differences at the same epoch are strongly correlated.
  • 199. 71 / 105 Laws of covariance propagation =⇒ covariance matrix of double differences: C∇∆Ψ∇∆Ψ = DC∆Ψ∆ΨD⊤ = 2σ2 DD⊤ = 2σ2      2 1 1 . . . . . . 1 1 2 1 1 . . . 1 ... 1 . . . . . . 1 1 2      . =⇒ Double differences at the same epoch are strongly correlated. Correlation has to be taken into account in the parameter estimation process.
  • 200. 72 / 105 C−1 ∇∆Ψ∇∆Ψ = 1 2σ2(k + 1)      k −1 −1 . . . . . . −1 −1 k −1 −1 . . . −1 ... −1 . . . . . . −1 −1 k      =: ˜P (57)
  • 201. 72 / 105 C−1 ∇∆Ψ∇∆Ψ = 1 2σ2(k + 1)      k −1 −1 . . . . . . −1 −1 k −1 −1 . . . −1 ... −1 . . . . . . −1 −1 k      =: ˜P (57) Double differences at different epochs are uncorrelated =⇒ weight matrix P is a block diagonal matrix, with matrix ˜P as diagonal blocks: P =      ˜P ˜P ... ˜P      . (58)
  • 202. 73 / 105 Usual least-squares estimation of the position correction and the error terms: ∆x = A⊤ PA −1 A⊤ PY (59)
  • 203. 73 / 105 Usual least-squares estimation of the position correction and the error terms: ∆x = A⊤ PA −1 A⊤ PY (59) With accuracy estimation: σ2 = | Y − A · ∆x 2 n(k − 1) − 2 − k = | Y − A · ∆x 2 (n − 1)(k − 1) − 3 (60)
  • 204. 73 / 105 Usual least-squares estimation of the position correction and the error terms: ∆x = A⊤ PA −1 A⊤ PY (59) With accuracy estimation: σ2 = | Y − A · ∆x 2 n(k − 1) − 2 − k = | Y − A · ∆x 2 (n − 1)(k − 1) − 3 (60) and C∆x,∆x = σ2 A⊤ PA −1 . (61)
  • 206. 75 / 105 Acquisition phase: receiver determines phase shift of received and receiver-generated carrier (up to an unknown integer number of complete cycles).
  • 207. 75 / 105 Acquisition phase: receiver determines phase shift of received and receiver-generated carrier (up to an unknown integer number of complete cycles). Thereafter Costa’s loop tracks the change in the phase shift due to the changing distance between receiver and satellite.
  • 208. 75 / 105 Acquisition phase: receiver determines phase shift of received and receiver-generated carrier (up to an unknown integer number of complete cycles). Thereafter Costa’s loop tracks the change in the phase shift due to the changing distance between receiver and satellite. Introduction of additional unknown: integer number N of unknown phase cycles.
  • 209. 75 / 105 Acquisition phase: receiver determines phase shift of received and receiver-generated carrier (up to an unknown integer number of complete cycles). Thereafter Costa’s loop tracks the change in the phase shift due to the changing distance between receiver and satellite. Introduction of additional unknown: integer number N of unknown phase cycles. If for one satellite the signal-to-noise ratio gets too low, Costa’s loop is not capable to keep track of the phase shift change and a new acquisition for this satellite has to be carried out.
  • 210. 75 / 105 Acquisition phase: receiver determines phase shift of received and receiver-generated carrier (up to an unknown integer number of complete cycles). Thereafter Costa’s loop tracks the change in the phase shift due to the changing distance between receiver and satellite. Introduction of additional unknown: integer number N of unknown phase cycles. If for one satellite the signal-to-noise ratio gets too low, Costa’s loop is not capable to keep track of the phase shift change and a new acquisition for this satellite has to be carried out. In the time elapsed during this acquisition the satellite-receiver distance has changed and the number of unknown cycles has also changed. This effect is called cycle-slip.
  • 211. 76 / 105 Figure 4: Occurrence of a cycle-slip
  • 212. 77 / 105 • Assumption that at time 0 the acquisition has returned the phase shift ϕ(0)
  • 213. 77 / 105 • Assumption that at time 0 the acquisition has returned the phase shift ϕ(0) • At this moment number of unknown entire cycles is N0
  • 214. 77 / 105 • Assumption that at time 0 the acquisition has returned the phase shift ϕ(0) • At this moment number of unknown entire cycles is N0 • Then the receiver tracks the satellite continuously until t
  • 215. 77 / 105 • Assumption that at time 0 the acquisition has returned the phase shift ϕ(0) • At this moment number of unknown entire cycles is N0 • Then the receiver tracks the satellite continuously until t • During this time the phase shift changes to ϕ(t)
  • 216. 77 / 105 • Assumption that at time 0 the acquisition has returned the phase shift ϕ(0) • At this moment number of unknown entire cycles is N0 • Then the receiver tracks the satellite continuously until t • During this time the phase shift changes to ϕ(t) • At t Costa’s loop looses track to the signal until t + dt
  • 217. 77 / 105 • Assumption that at time 0 the acquisition has returned the phase shift ϕ(0) • At this moment number of unknown entire cycles is N0 • Then the receiver tracks the satellite continuously until t • During this time the phase shift changes to ϕ(t) • At t Costa’s loop looses track to the signal until t + dt • At t + dt the new acquisition is completed and the new phase shift ϕ(t + dt) and the new phase ambiguity N1 is determined
  • 218. 77 / 105 • Assumption that at time 0 the acquisition has returned the phase shift ϕ(0) • At this moment number of unknown entire cycles is N0 • Then the receiver tracks the satellite continuously until t • During this time the phase shift changes to ϕ(t) • At t Costa’s loop looses track to the signal until t + dt • At t + dt the new acquisition is completed and the new phase shift ϕ(t + dt) and the new phase ambiguity N1 is determined • The difference N1 − N0 is the cycle-slip occurred at t
  • 219. 77 / 105 • Assumption that at time 0 the acquisition has returned the phase shift ϕ(0) • At this moment number of unknown entire cycles is N0 • Then the receiver tracks the satellite continuously until t • During this time the phase shift changes to ϕ(t) • At t Costa’s loop looses track to the signal until t + dt • At t + dt the new acquisition is completed and the new phase shift ϕ(t + dt) and the new phase ambiguity N1 is determined • The difference N1 − N0 is the cycle-slip occurred at t Important task of GPS data pre-processing: detection and correction of cycle slips.
  • 220. 77 / 105 • Assumption that at time 0 the acquisition has returned the phase shift ϕ(0) • At this moment number of unknown entire cycles is N0 • Then the receiver tracks the satellite continuously until t • During this time the phase shift changes to ϕ(t) • At t Costa’s loop looses track to the signal until t + dt • At t + dt the new acquisition is completed and the new phase shift ϕ(t + dt) and the new phase ambiguity N1 is determined • The difference N1 − N0 is the cycle-slip occurred at t Important task of GPS data pre-processing: detection and correction of cycle slips. Multiple techniques for this purpose. Here only three of them: • analysis of double differences • analysis of the ionospheric residuals • code- and carrier phase combination
  • 221. Analysis of Double Differences 78 / 105 Comparation of phase double differences with double differences of the slant ranges between receivers and satellites.
  • 222. Analysis of Double Differences 78 / 105 Comparation of phase double differences with double differences of the slant ranges between receivers and satellites. Double differences observation equation: ∇∆PR pq CRrv(tj) := ∆PR q CRrv(tj) − ∆PR p CRrv(tj) = ∇∆R pq rv (tj) +c((dt q ar(tj) − dt q av(tj)) −(dt p ar(tj) − dt p av(tj))) +λ((N q r − N q v ) − (N p r − N p v )) + ǫ
  • 223. 79 / 105 Test quantity: difference between phase double difference and double difference of the slant ranges ∇∆r(tj) := ∇∆PR pq CRrv(tj) − ∇∆R pq rv (tj) = c((dt q ar(tj) − dt q av(tj)) −(dt p ar(tj) − dt p av(tj))) +λ((N q r − N q v ) − (N p r − N p v )) + ǫ
  • 224. 79 / 105 Test quantity: difference between phase double difference and double difference of the slant ranges ∇∆r(tj) := ∇∆PR pq CRrv(tj) − ∇∆R pq rv (tj) = c((dt q ar(tj) − dt q av(tj)) −(dt p ar(tj) − dt p av(tj))) +λ((N q r − N q v ) − (N p r − N p v )) + ǫ No cycle slip: • time variation of ∇∆r only caused by changes of tropospheric and ionospheric delay • slow changes: time variation of ∇∆r is a smooth curve
  • 225. 79 / 105 Test quantity: difference between phase double difference and double difference of the slant ranges ∇∆r(tj) := ∇∆PR pq CRrv(tj) − ∇∆R pq rv (tj) = c((dt q ar(tj) − dt q av(tj)) −(dt p ar(tj) − dt p av(tj))) +λ((N q r − N q v ) − (N p r − N p v )) + ǫ No cycle slip: • time variation of ∇∆r only caused by changes of tropospheric and ionospheric delay • slow changes: time variation of ∇∆r is a smooth curve Cycle slip: • indication through sudden jumps in ∇∆r
  • 226. 80 / 105 Figure 5: Double differences residuals with occurring of a cycle slip
  • 227. 81 / 105 Jump in ∇∆r small compared to absolute values of ∇∆r
  • 228. 81 / 105 Jump in ∇∆r small compared to absolute values of ∇∆r =⇒ jump is difficult to detect by statistical methods
  • 229. 81 / 105 Jump in ∇∆r small compared to absolute values of ∇∆r =⇒ jump is difficult to detect by statistical methods =⇒ a polynomial p(t) of a low degree is fitted to ∇∆r and differences ∇∆r − p are screened Figure 6: Polynomial p fitted to the double differences residuals
  • 230. 82 / 105 Figure 7: Differences between the fitted polynomial p and the double differences residuals
  • 231. 83 / 105 Cycle slip indicated by a sudden change in the sign =⇒ easy detection by statistical tests
  • 232. 83 / 105 Cycle slip indicated by a sudden change in the sign =⇒ easy detection by statistical tests advantages: • applicable already for single frequency receivers • even small cycle slips can be detected
  • 233. 83 / 105 Cycle slip indicated by a sudden change in the sign =⇒ easy detection by statistical tests advantages: • applicable already for single frequency receivers • even small cycle slips can be detected disadvantages: • no possibility to decide for which satellite and which receiver the cycle slip has occurred • sensitive to sudden changes in the ionospheric electron concentration
  • 234. Analysis of the Ionospheric Residuals 84 / 105 Analysation of the geometry-free linear combination: L4 = LI = LΣ − L∆ = ω1 ω1 + ω2 L1 + ω2 ω1 + ω2 L2 − ω1 ω1 − ω2 L1 + ω2 ω1 − ω2 L2 = 2ω1ω2 ω2 1 − ω2 2 [−L1 + L2] = 2ω1ω2 ω2 1 − ω2 2 [−(|XS − XB| + N1λ1 + cdtu + cdtI1) +(|XS − XB| + N2λ2 + cdtu + cdtI2)] = 2ω1ω2 ω2 1 − ω2 2 [−N1λ1 + N2λ2 + (−cdtI1 + cdtI2)]
  • 235. 85 / 105 No cycle slip =⇒ variation of L4 only caused by variation of the ionospheric delay
  • 236. 85 / 105 No cycle slip =⇒ variation of L4 only caused by variation of the ionospheric delay =⇒ variation of L4 has to be smooth
  • 237. 85 / 105 No cycle slip =⇒ variation of L4 only caused by variation of the ionospheric delay =⇒ variation of L4 has to be smooth Sudden changes in L4: indication of a cycle slip
  • 238. 85 / 105 No cycle slip =⇒ variation of L4 only caused by variation of the ionospheric delay =⇒ variation of L4 has to be smooth Sudden changes in L4: indication of a cycle slip advantages: • method works already with a single receiver, no baselines have to be performed
  • 239. 85 / 105 No cycle slip =⇒ variation of L4 only caused by variation of the ionospheric delay =⇒ variation of L4 has to be smooth Sudden changes in L4: indication of a cycle slip advantages: • method works already with a single receiver, no baselines have to be performed disadvantages: • at detection of a cycle slip: frequency on which this cycle slip had happened cannot be found
  • 240. Analysis of the Code- Carrier Combination 86 / 105 Comparation of code and the carrier pseudorange on the same frequency: PRCR,CD(t) := PRCD − PRCR = |XS − XB| + cdtu + cdtI −(|XS − XB| + NBi λ + cdtu − cdtI) = −NBi λ + 2cdtI
  • 241. Analysis of the Code- Carrier Combination 86 / 105 Comparation of code and the carrier pseudorange on the same frequency: PRCR,CD(t) := PRCD − PRCR = |XS − XB| + cdtu + cdtI −(|XS − XB| + NBi λ + cdtu − cdtI) = −NBi λ + 2cdtI =⇒ • ionospheric signal delay 2cdtI is very smooth • changes in ionospheric signal are constant • quantity PRCR,CD is constant
  • 242. Analysis of the Code- Carrier Combination 86 / 105 Comparation of code and the carrier pseudorange on the same frequency: PRCR,CD(t) := PRCD − PRCR = |XS − XB| + cdtu + cdtI −(|XS − XB| + NBi λ + cdtu − cdtI) = −NBi λ + 2cdtI =⇒ • ionospheric signal delay 2cdtI is very smooth • changes in ionospheric signal are constant • quantity PRCR,CD is constant =⇒ Indication of a cycle-slip: sudden jumps in PRCR,CD.
  • 243. 87 / 105 • Magnitude of cycle slip identical to magnitude in the jump of PRCR,CD.
  • 244. 87 / 105 • Magnitude of cycle slip identical to magnitude in the jump of PRCR,CD. • P-Code pseudorange accuracy ∼ 6 dm =⇒ Accuracy of magnitude of a cycle slip: 3 cycles
  • 245. 87 / 105 • Magnitude of cycle slip identical to magnitude in the jump of PRCR,CD. • P-Code pseudorange accuracy ∼ 6 dm =⇒ Accuracy of magnitude of a cycle slip: 3 cycles • =⇒ Sufficient to input results of the code-carrier combination into analysis of ionospheric residuals
  • 247. 89 / 105 Carrier phase measurements contain an unknown integer number N of cycles.
  • 248. 89 / 105 Carrier phase measurements contain an unknown integer number N of cycles. Number must be found =⇒ full accuracy potential of GPS carrier phase measurements
  • 249. 89 / 105 Carrier phase measurements contain an unknown integer number N of cycles. Number must be found =⇒ full accuracy potential of GPS carrier phase measurements Large number of methods for fixing these ambiguities. Three of them will be discussed: • geometric method • combination of code and carrier phase • search methods
  • 250. The Geometric Method 90 / 105 Makes use of time differences of carrier phase observations.
  • 251. The Geometric Method 90 / 105 Makes use of time differences of carrier phase observations. Assumption that at three epochs t1, t2, t3 carrier phase observation on one frequency to the same satellite are carried out: Φ(t1) = 2π λ (|XS(t1) − XB| + Nλ) (62) Φ(t2) = 2π λ (|XS(t2) − XB| + Nλ) (63) Φ(t3) = 2π λ (|XS(t3) − XB| + Nλ) (64)
  • 252. 91 / 105 Out of these three phase observations two time-differences can be formed: δΦ(t1) := Φ(t2) − Φ(t1) = 2π λ (|XS(t2) − XB| − |XS(t1) − XB|) (65) δΦ(t2) := Φ(t3) − Φ(t2) = 2π λ (|XS(t3) − XB| − |XS(t2) − XB|) (66)
  • 253. 91 / 105 Out of these three phase observations two time-differences can be formed: δΦ(t1) := Φ(t2) − Φ(t1) = 2π λ (|XS(t2) − XB| − |XS(t1) − XB|) (65) δΦ(t2) := Φ(t3) − Φ(t2) = 2π λ (|XS(t3) − XB| − |XS(t2) − XB|) (66) Equations (65) and (66): equations of hyperboloids with focal points in the known satellite positions XS(t1), XS(t2) and XS(t2), XS(t3).
  • 254. 92 / 105 =⇒Position of observer is on the intersection of the two hyperboloids. Figure 8: Geometric method of ambiguity resolution
  • 255. 93 / 105 Position of the observer: intersection of at least three hyperboloids.
  • 256. 93 / 105 Position of the observer: intersection of at least three hyperboloids. Larger number of intersecting hyperboloids =⇒ more precise estimation for the observer position ˆXB
  • 257. 93 / 105 Position of the observer: intersection of at least three hyperboloids. Larger number of intersecting hyperboloids =⇒ more precise estimation for the observer position ˆXB Process continues until the accuracy of the obtained estimation is better than λ/2.
  • 258. 93 / 105 Position of the observer: intersection of at least three hyperboloids. Larger number of intersecting hyperboloids =⇒ more precise estimation for the observer position ˆXB Process continues until the accuracy of the obtained estimation is better than λ/2. Then the estimation is inserted into an observation equation and solved for the unknown ambiguity: ˆN = 1 2π Φ(t) − 1 λ |XS(t) − ˆXB| (67)
  • 259. 93 / 105 Position of the observer: intersection of at least three hyperboloids. Larger number of intersecting hyperboloids =⇒ more precise estimation for the observer position ˆXB Process continues until the accuracy of the obtained estimation is better than λ/2. Then the estimation is inserted into an observation equation and solved for the unknown ambiguity: ˆN = 1 2π Φ(t) − 1 λ |XS(t) − ˆXB| (67) Obtained float solution ˆN is rounded to the nearest integer.
  • 260. 94 / 105 advantages: • simple and clear modelling • applicable also for single point positioning • single frequency receiver sufficient
  • 261. 94 / 105 advantages: • simple and clear modelling • applicable also for single point positioning • single frequency receiver sufficient disadvantages: • long arcs necessary • sensitive to unmodelled effects (ionosphere, troposphere, orbits, clocks) • no cycle slips allowed during ambiguity resolution
  • 262. Combination of Code and Carrier Phase 95 / 105 Difference between pseudorange from carrier phase observations and pseudorange from code observation is used: PRCR,CD(t) := PRCD − PRCR = |XS − XB| + cdtu + cdtI −(|XS − XB| + NBλ + cdtu − cdtI) = −NBλ + 2cdtI (68)
  • 263. 96 / 105 Difference of the pseudoranges PRCR,CD(t) equals length Nλ of unknown integer number of carrier phase cycles – up to ionospheric error cdtI.
  • 264. 96 / 105 Difference of the pseudoranges PRCR,CD(t) equals length Nλ of unknown integer number of carrier phase cycles – up to ionospheric error cdtI. For short and medium length baselines elimination of ionospheric error by forming single differences: ∆PRCR,CDij := PRCR,CDi − PRCR,CDj = λ(NBi − NBj ) = λ∆Nij (69)
  • 265. 96 / 105 Difference of the pseudoranges PRCR,CD(t) equals length Nλ of unknown integer number of carrier phase cycles – up to ionospheric error cdtI. For short and medium length baselines elimination of ionospheric error by forming single differences: ∆PRCR,CDij := PRCR,CDi − PRCR,CDj = λ(NBi − NBj ) = λ∆Nij (69) Reduction of random errors contained in observations by computing the time average over some minutes
  • 266. 96 / 105 Difference of the pseudoranges PRCR,CD(t) equals length Nλ of unknown integer number of carrier phase cycles – up to ionospheric error cdtI. For short and medium length baselines elimination of ionospheric error by forming single differences: ∆PRCR,CDij := PRCR,CDi − PRCR,CDj = λ(NBi − NBj ) = λ∆Nij (69) Reduction of random errors contained in observations by computing the time average over some minutes Estimation of single difference ambiguity: ∆Nij = 1 λT T 0 ∆PRCR,CDij (t)dt (70)
  • 267. 97 / 105 ∆Nij: float solution, can be rounded to the nearest integer as soon as its standard deviation ≤ 0.5.
  • 268. 97 / 105 ∆Nij: float solution, can be rounded to the nearest integer as soon as its standard deviation ≤ 0.5. Because wavelength λ is in the denominator of equation (70) =⇒ easier ambiguity resolution for longer wavelength of λ
  • 269. 97 / 105 ∆Nij: float solution, can be rounded to the nearest integer as soon as its standard deviation ≤ 0.5. Because wavelength λ is in the denominator of equation (70) =⇒ easier ambiguity resolution for longer wavelength of λ =⇒ carrier code combination mostly used for wide lane frequency combination LΣ.
  • 270. 97 / 105 ∆Nij: float solution, can be rounded to the nearest integer as soon as its standard deviation ≤ 0.5. Because wavelength λ is in the denominator of equation (70) =⇒ easier ambiguity resolution for longer wavelength of λ =⇒ carrier code combination mostly used for wide lane frequency combination LΣ. Fixation of wide lane ambiguity if float solution for the single difference ambiguity has a standard deviation smaller than half a cycle.
  • 271. 97 / 105 ∆Nij: float solution, can be rounded to the nearest integer as soon as its standard deviation ≤ 0.5. Because wavelength λ is in the denominator of equation (70) =⇒ easier ambiguity resolution for longer wavelength of λ =⇒ carrier code combination mostly used for wide lane frequency combination LΣ. Fixation of wide lane ambiguity if float solution for the single difference ambiguity has a standard deviation smaller than half a cycle. For short and medium length baselines the ionospheric combination has to vanish identical: LI = LΣ − L∆ = 0 (71)
  • 272. 97 / 105 ∆Nij: float solution, can be rounded to the nearest integer as soon as its standard deviation ≤ 0.5. Because wavelength λ is in the denominator of equation (70) =⇒ easier ambiguity resolution for longer wavelength of λ =⇒ carrier code combination mostly used for wide lane frequency combination LΣ. Fixation of wide lane ambiguity if float solution for the single difference ambiguity has a standard deviation smaller than half a cycle. For short and medium length baselines the ionospheric combination has to vanish identical: LI = LΣ − L∆ = 0 (71) Wavelength of wide lane: ∼ 8 × wavelength of narrow lane
  • 273. 97 / 105 ∆Nij: float solution, can be rounded to the nearest integer as soon as its standard deviation ≤ 0.5. Because wavelength λ is in the denominator of equation (70) =⇒ easier ambiguity resolution for longer wavelength of λ =⇒ carrier code combination mostly used for wide lane frequency combination LΣ. Fixation of wide lane ambiguity if float solution for the single difference ambiguity has a standard deviation smaller than half a cycle. For short and medium length baselines the ionospheric combination has to vanish identical: LI = LΣ − L∆ = 0 (71) Wavelength of wide lane: ∼ 8 × wavelength of narrow lane =⇒ Initial estimation of narrow lane single difference ambiguity with an accuracy of about 8 cycles: condition (71).
  • 274. 98 / 105 advantages: • fast, • independent of geometry.
  • 275. 98 / 105 advantages: • fast, • independent of geometry. disadvantages: • dual frequency P-code receiver necessary, • only wide lane ambiguities can be resolved in short time.
  • 276. Search Methods 99 / 105 Test of all possible integer values of the ambiguities and selection of the most plausible integer value.
  • 277. Search Methods 99 / 105 Test of all possible integer values of the ambiguities and selection of the most plausible integer value. For more precision consider the adjustment problem for a single- or double differences baseline solution: l = A · x + v. (72)
  • 278. Search Methods 99 / 105 Test of all possible integer values of the ambiguities and selection of the most plausible integer value. For more precision consider the adjustment problem for a single- or double differences baseline solution: l = A · x + v. (72) Division of unknown vector x in two parts: x = (x1, x2)
  • 279. Search Methods 99 / 105 Test of all possible integer values of the ambiguities and selection of the most plausible integer value. For more precision consider the adjustment problem for a single- or double differences baseline solution: l = A · x + v. (72) Division of unknown vector x in two parts: x = (x1, x2) x1 containing all the unknown of non-integer nature: • receiver clock errors • parameters of atmospheric and ionospheric delay models • coordinates of the receivers x2: remaining integer ambiguities.
  • 280. 100 / 105 Also partition of adjustment problem: l = [A1, A2] · x1 x2 + v (73)
  • 281. 100 / 105 Also partition of adjustment problem: l = [A1, A2] · x1 x2 + v (73) Corresponding normal equations: N11 N12 N21 N22 · x1 x2 = b1 b2 (74) with Nij = A⊤ i Aj, bi = A⊤ i l (75)
  • 282. 101 / 105 Least squares solution: ˆx1 ˆx2 = Q11 Q12 Q21 Q22 · b1 b2 (76) with Q11 Q12 Q21 Q22 = N11 N12 N21 N22 −1 (77)
  • 283. 101 / 105 Least squares solution: ˆx1 ˆx2 = Q11 Q12 Q21 Q22 · b1 b2 (76) with Q11 Q12 Q21 Q22 = N11 N12 N21 N22 −1 (77) Derivation of standard deviation of estimated float ambiguities and standard deviation of difference between estimated float ambiguities from variance-covariance matrix Q: ˆσ2 = l − Aˆx 2 n − u (78) σNi = ˆσ Q22ii (79) σNi−Nj = ˆσ Q22ii − 2Q22ij + Q22jj (80)
  • 284. 102 / 105 Assuming a normal distribution of the observations =⇒ confidence regions for the unknown ambiguity parameters: P(Ni ∈ [ ˆNi − tασNi , ˆNi + tασNi ]) = 1 − α, (81) P(Ni − Nj ∈ [( ˆNi − ˆNj) − tασNi−Nj , ( ˆNi − ˆNj) + tασNi−Nj ]) = 1 − α, (82) tα: quantile of student distribution for confidence level α.
  • 285. 102 / 105 Assuming a normal distribution of the observations =⇒ confidence regions for the unknown ambiguity parameters: P(Ni ∈ [ ˆNi − tασNi , ˆNi + tασNi ]) = 1 − α, (81) P(Ni − Nj ∈ [( ˆNi − ˆNj) − tασNi−Nj , ( ˆNi − ˆNj) + tασNi−Nj ]) = 1 − α, (82) tα: quantile of student distribution for confidence level α. Intersection of all this confidence regions: region around the float solution ˆx2 = ( ˆN1, . . . , ˆNn)
  • 286. 102 / 105 Assuming a normal distribution of the observations =⇒ confidence regions for the unknown ambiguity parameters: P(Ni ∈ [ ˆNi − tασNi , ˆNi + tασNi ]) = 1 − α, (81) P(Ni − Nj ∈ [( ˆNi − ˆNj) − tασNi−Nj , ( ˆNi − ˆNj) + tασNi−Nj ]) = 1 − α, (82) tα: quantile of student distribution for confidence level α. Intersection of all this confidence regions: region around the float solution ˆx2 = ( ˆN1, . . . , ˆNn) Float solution contains true integer ambiguities with a probability of 1 − α.
  • 287. 102 / 105 Assuming a normal distribution of the observations =⇒ confidence regions for the unknown ambiguity parameters: P(Ni ∈ [ ˆNi − tασNi , ˆNi + tασNi ]) = 1 − α, (81) P(Ni − Nj ∈ [( ˆNi − ˆNj) − tασNi−Nj , ( ˆNi − ˆNj) + tασNi−Nj ]) = 1 − α, (82) tα: quantile of student distribution for confidence level α. Intersection of all this confidence regions: region around the float solution ˆx2 = ( ˆN1, . . . , ˆNn) Float solution contains true integer ambiguities with a probability of 1 − α. Trivial case: only two ambiguities N1, N2 (displayed in figure 9)
  • 288. 103 / 105 Figure 9: Confidence region for the integer ambiguities
  • 289. 104 / 105 In what follows this confidence region: C ⊂ Rn.
  • 290. 104 / 105 In what follows this confidence region: C ⊂ Rn. All vectors x2 ∈ Nn ∩ C: grid of possible integer ambiguity solutions x2,h, h = 1, . . . N. Figure 10: Candidates for integer solution
  • 291. 105 / 105 Introduction of each possible integer ambiguity solution to a subsequent adjustment. Chosen integer ambiguity solution: treated as known quantity. A1 · x1 = l − A2 · x2,h = lh (83) ˆx1 = (A⊤ 1 A1)−1 A⊤ 1 lh (84) ˆσ2 h = lh − A1 ˆx1 2 n − u1 , (85)
  • 292. 105 / 105 Introduction of each possible integer ambiguity solution to a subsequent adjustment. Chosen integer ambiguity solution: treated as known quantity. A1 · x1 = l − A2 · x2,h = lh (83) ˆx1 = (A⊤ 1 A1)−1 A⊤ 1 lh (84) ˆσ2 h = lh − A1 ˆx1 2 n − u1 , (85) Final solution: integer ambiguity solution candidate x2,h with smallest variation ˆσ2 h. Unless:
  • 293. 105 / 105 Introduction of each possible integer ambiguity solution to a subsequent adjustment. Chosen integer ambiguity solution: treated as known quantity. A1 · x1 = l − A2 · x2,h = lh (83) ˆx1 = (A⊤ 1 A1)−1 A⊤ 1 lh (84) ˆσ2 h = lh − A1 ˆx1 2 n − u1 , (85) Final solution: integer ambiguity solution candidate x2,h with smallest variation ˆσ2 h. Unless: 1. variance ˆσ2 h is not compatible with the variance of the L3 solution, or
  • 294. 105 / 105 Introduction of each possible integer ambiguity solution to a subsequent adjustment. Chosen integer ambiguity solution: treated as known quantity. A1 · x1 = l − A2 · x2,h = lh (83) ˆx1 = (A⊤ 1 A1)−1 A⊤ 1 lh (84) ˆσ2 h = lh − A1 ˆx1 2 n − u1 , (85) Final solution: integer ambiguity solution candidate x2,h with smallest variation ˆσ2 h. Unless: 1. variance ˆσ2 h is not compatible with the variance of the L3 solution, or 2. there is another integer solution candidate yielding almost identical variance.