Generation of slant tropospheric delay time series based ... · von Karman spectrum (→...
Transcript of Generation of slant tropospheric delay time series based ... · von Karman spectrum (→...
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ErdmessungInstitut für 1 / 17
Generation of slant tropospheric delay time series
based on turbulence theory
Markus VennebuschSteffen Schon
Institut fur Erdmessung, Leibniz Universitat Hannover, Germany
1. September 2009
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M. Vennebusch, S. SchönIAG 2009, Buenos Aires, 1. September 2009Erdmessung
Institut für
Motivation
Introduction
Motivation
Stochastic modelSlant delaysimulation
Analysis objectives
Simulations
Impact of parametervariations
Summary &Conclusions
2 / 17
Slant tropospheric delay:
■ long-periodic variations:
◆ caused by e.g. daily - hourly variations of temperature,pressure, partial pressure of water vapor, ...
◆ mean behaviour described by deterministic models(e.g., Hopfield, Saastamoinen, ... & mapping functions)
■ short-periodic variations (periods of [min] to [sec]):
◆ caused by:turbulent flow inatmospheric boundary layer
◆ water vapor variations⇒ index of refractivity variations
◆ behaviour described stochastically
(⇒ Turbulence theory)
Wallace, Hobbs (2006): Atmospheric Science
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M. Vennebusch, S. SchönIAG 2009, Buenos Aires, 1. September 2009Erdmessung
Institut für
Motivation
Introduction
Motivation
Stochastic modelSlant delaysimulation
Analysis objectives
Simulations
Impact of parametervariations
Summary &Conclusions
3 / 17
Turbulence theory / ’Wave propagation in turbulent media’:
■ Spectrum ofturbulence kinetic energy:
■ von Karman spectrum (→ non-stationary process):
Φn(κ) =0.033 C2
n
(κ2 + κ2
0)
11
6
∝ κ−11/3, 0 < κ < κS
C2
nstructure constant of refractivity
κ0 = 2π/L0 wavenumber corresponding to outer scale length L0
⇒ Stochastic model of GNSS phase observations(can be regarded as stochastic model of slant delays)
Wallace, Hobbs (2006): Atmospheric Science
![Page 4: Generation of slant tropospheric delay time series based ... · von Karman spectrum (→ non-stationary process): ... Test of processing strategy Basis for analysis of real GNSS data](https://reader036.fdocuments.in/reader036/viewer/2022081410/60945fee1abbc50edf3f96df/html5/thumbnails/4.jpg)
M. Vennebusch, S. SchönIAG 2009, Buenos Aires, 1. September 2009Erdmessung
Institut für
Stochastic model
Introduction
Motivation
Stochastic modelSlant delaysimulation
Analysis objectives
Simulations
Impact of parametervariations
Summary &Conclusions
4 / 17
Turbulence theory-based covariance (Schon/Brunner, JGeod 2007 82(1), pp. 47-57):
〈ϕiA(tA), ϕj
B(tB)〉 = 〈τ1
A(tA), τ2
B(tB)〉 =
=12
5
0.033
Γ(
5
6
)
√π3κ
−
2
3
02−
1
3
sin εiA sin εj
B
C2
n
×H
∫
0
H∫
0
(κ0d)1
3 K−
1
3
(κ0d) dz1 dz2
ϕiA(tA) Phase observation (station A, satellite i, epoch tA)C2
n Structure constant (characterises strength of turbulence)L0 Outer scale length (κ0 = 2π/L0)H Integration heightεi
A Elevation of satellite i at station Ad separation between integration pointsK modified Bessel functionΓ Gamma function
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M. Vennebusch, S. SchönIAG 2009, Buenos Aires, 1. September 2009Erdmessung
Institut für
Generation of slant delay variations
Introduction
Motivation
Stochastic modelSlant delaysimulation
Analysis objectives
Simulations
Impact of parametervariations
Summary &Conclusions
5 / 17
Simulation of slant delay variations using EVD of Σ:
y = G√
Λx withG eigenvectors of Σ
Λ diagonal matrix, eigenvalues λi of Σ
x gaussian random vector with N(0, 1)→ variations of slant delays (5 realisations)
Stochastic properties:
■ Temporal structure function / Variogram:
Dn(τ) = 〈[n(t + τ) − n(t)]2〉 ∝ τ5
3
log( )τ
slope 5/3
log(
stru
ctur
e fu
nctio
n)
2 x Variance
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M. Vennebusch, S. SchönIAG 2009, Buenos Aires, 1. September 2009Erdmessung
Institut für
Analysis objectives
Introduction
Motivation
Stochastic modelSlant delaysimulation
Analysis objectives
Simulations
Impact of parametervariations
Summary &Conclusions
6 / 17
Analysis objectives:Investigation of impact of parameter variations on:
■ Variance-covariance matrices and correlation matrices■ Stochastic behaviour of simulated variations of slant delays
Purpose / Motivation:
■ Dominant model parameters? (→ must be precisely known)■ Test of processing strategy■ Basis for analysis of real GNSS data in future
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M. Vennebusch, S. SchönIAG 2009, Buenos Aires, 1. September 2009Erdmessung
Institut für
Simulations
Introduction
SimulationsScenarios ¶meter sets
Impact of parametervariations
Summary &Conclusions
7 / 17
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M. Vennebusch, S. SchönIAG 2009, Buenos Aires, 1. September 2009Erdmessung
Institut für
Simulation scenarios and parameter sets
Introduction
SimulationsScenarios ¶meter sets
Impact of parametervariations
Summary &Conclusions
8 / 17
Scenarios:
EW
S
N
Scenario 1 (low elevation)
EW
S
N
Scenario 2 (zenith)
EW
S
N
Scenario 3 (rising satellite)
100 observations 100 observations 1000 observations10 [sec] sampling 10 [sec] sampling 10 [sec] sampling
Elevation: 10 - 17 [deg] Elevation: 90 [deg] Elevation: 10 - 90 [deg]Azimuth: 200 - 202 [deg] Azimuth: 0 [deg] Azimuth: 200 - 280 [deg]
(Fixed satellite) (from real satellite)
Turbulence parameter variations:
C2
n [m−2/3] L0 [m] v [m/s] H [m] αv [deg] Comment:
0.3 × 10−14 3000 8 2000 0 Reference set
5.76 × 10−14 6000 15 1000 90
9.0 × 10−14 180
270
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ErdmessungInstitut für
Simulations - zenith scenario
9 / 17
Variance−covariance matrix
20 40 60 80 100
20
40
60
80
100 0
2
4
6
8
10
x 10−8
0 10 20 30 40 500
0.5
1
1.5
2
2.5x 10
−6
Anti−diagonal element
(Co−
)Var
ianc
e [m
2 ]
VCM anti−diagonals Correlation matrix
20 40 60 80 100
20
40
60
80
100 0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 500
0.2
0.4
0.6
0.8
1
Anti−diagonal element
Cor
rela
tion
[−]
Correlation matrix anti−diagonals
0/90 deg 20/90 deg 40/90 deg 60/90 deg 80/90 deg 100/90 deg−0.005
−0.0025
0
0.0025
0.005
epoch (@ 10 [sec] sampling) / elevation [deg]
Sla
nt d
elay
[m]
Simulated slant delay time series
average variance= 122e−9 [m]
1 2 3 4 5 6 7 8 9 10 2010
−9
10−8
10−7
10−6
10−5
10−4
log(Time shift τ) [10 sec]
log(
DS
D(τ
)) [m
2 ]
(Temporal) structure function (loglog plot, axis offset=1.3981e−08)
5/3−PL reference2/3−PL reference
Scenario: zenith, parameter set: 5 (= average turbulence parameters):
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ErdmessungInstitut für
Simulations - low elevation
10 / 17
Variance−covariance matrix
20 40 60 80 100
20
40
60
80
100 0
1
2
3
4
5
6
7
x 10−7
0 10 20 30 40 500
0.5
1
1.5
2
2.5x 10
−6
Anti−diagonal element
(Co−
)Var
ianc
e [m
2 ]
VCM anti−diagonals Correlation matrix
20 40 60 80 100
20
40
60
80
100 0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 500
0.2
0.4
0.6
0.8
1
Anti−diagonal element
Cor
rela
tion
[−]
Correlation matrix anti−diagonals
0/10.0 deg 20/11.4 deg 40/12.8 deg 60/14.2 deg 80/15.6 deg 100/17.0 deg−0.005
−0.0025
0
0.0025
0.005
epoch (@ 10 [sec] sampling) / elevation [deg]
Sla
nt d
elay
[m]
Simulated slant delay time series
average variance= 460e−9 [m]
1 2 3 4 5 6 7 8 9 10 2010
−9
10−8
10−7
10−6
10−5
10−4
log(Time shift τ) [10 sec]
log(
DS
D(τ
)) [m
2 ]
(Temporal) structure function (loglog plot, axis offset=1.4725e−08)
5/3−PL reference2/3−PL reference
Scenario: low elevation, parameter set: 5 (= average turbulence parameters):
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ErdmessungInstitut für
Simulations - rising satellite
11 / 17
Variance−covariance matrix
200 400 600 800 1000
200
400
600
800
1000 0
1
2
3
4
5
6
7
x 10−7
0 10 20 30 40 500
0.5
1
1.5
2
2.5x 10
−6
Anti−diagonal element
(Co−
)Var
ianc
e [m
2 ]
VCM anti−diagonals Correlation matrix
200 400 600 800 1000
200
400
600
800
1000 0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 500
0.2
0.4
0.6
0.8
1
Anti−diagonal element
Cor
rela
tion
[−]
Correlation matrix anti−diagonals
0/10 deg 200/24 deg 400/40 deg 600/56 deg 800/72 deg 1000/90 deg−0.005
−0.0025
0
0.0025
0.005
epoch (@ 10 [sec] sampling) / elevation [deg]
Sla
nt d
elay
[m]
Simulated slant delay time series
average variance= 216e−9 [m]
1 2 3 4 5 6 7 8 9 10 2010
−9
10−8
10−7
10−6
10−5
10−4
log(Time shift τ) [10 sec]
log(
DS
D(τ
)) [m
2 ]
(Temporal) structure function (loglog plot, axis offset=1.3393e−08)
5/3−PL reference2/3−PL reference
Scenario: rising satellite, parameter set: 5 (= average turbulence parameters):
![Page 12: Generation of slant tropospheric delay time series based ... · von Karman spectrum (→ non-stationary process): ... Test of processing strategy Basis for analysis of real GNSS data](https://reader036.fdocuments.in/reader036/viewer/2022081410/60945fee1abbc50edf3f96df/html5/thumbnails/12.jpg)
M. Vennebusch, S. SchönIAG 2009, Buenos Aires, 1. September 2009Erdmessung
Institut für
Impact of parameter variations
Introduction
Simulations
Impact of parametervariations
Summary &Conclusions
12 / 17
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ErdmessungInstitut für
Impact of parameter variations
13 / 17
Impact of L0 variations on mean correlations and average SD-variance:
0.0
0.2
0.4
0.6
0.8
1.0
0 10 20 30 40 50
Cor
rela
tion
[-]
Anti-diagonal element
Low elevation scenario: Impact of L0 variations on correlations
L0 = 3000 [m]L0 = 6000 [m]
0.0
0.2
0.4
0.6
0.8
1.0
0 10 20 30 40 50
Cor
rela
tion
[-]
Anti-diagonal element
Zenith scenario: Impact of L0 variations on correlations
L0 = 3000 [m]L0 = 6000 [m]
0
200
400
600
800
1000
L0 = 3000 [m] L0 = 6000 [m]
Ave
rage
var
ianc
e [e
-9 m
2 ]
Low elevation scenario: Impact of L0 variations on average SD-variance
0
200
400
600
800
1000
L0 = 3000 [m] L0 = 6000 [m]
Ave
rage
var
ianc
e [e
-9 m
2 ]
Zenith scenario: Impact of L0 variations on average SD-variance
⇒ increasing L0 → stronger turbulence, longer correlation time
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ErdmessungInstitut für
Impact of parameter variations
14 / 17
Impact of H variations on mean correlations and average SD-variance:
0.0
0.2
0.4
0.6
0.8
1.0
0 10 20 30 40 50
Cor
rela
tion
[-]
Anti-diagonal element
Low elevation scenario: Impact of H variations on correlations
H = 2000 [m]H = 1000 [m]
0.0
0.2
0.4
0.6
0.8
1.0
0 10 20 30 40 50
Cor
rela
tion
[-]
Anti-diagonal element
Zenith scenario: Impact of H variations on correlations
H = 2000 [m]H = 1000 [m]
0
200
400
600
800
1000
H = 2000 [m] H = 1000 [m]
Ave
rage
var
ianc
e [e
-9 m
2 ]
Low elevation scenario: Impact of H variations on average SD-variance
0
200
400
600
800
1000
H = 2000 [m] H = 1000 [m]
Ave
rage
var
ianc
e [e
-9 m
2 ]
Zenith scenario: Impact of H variations on average SD-variance
⇒ increasing H → stronger turbulence, small increase of correlations
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ErdmessungInstitut für
Impact of parameter variations
15 / 17
Impact of wind speed variations on mean correlations and average SD-variance:
0.0
0.2
0.4
0.6
0.8
1.0
0 10 20 30 40 50
Cor
rela
tion
[-]
Anti-diagonal element
Low elevation scenario: Impact of wind speed variations on correlations
v = 8 [m/s]v = 15 [m/s]
0.0
0.2
0.4
0.6
0.8
1.0
0 10 20 30 40 50
Cor
rela
tion
[-]
Anti-diagonal element
Zenith scenario: Impact of wind speed variations on correlations
v = 8 [m/s]v = 15 [m/s]
0
200
400
600
800
1000
v = 8 [m/s] v = 15 [m/s]
Ave
rage
var
ianc
e [e
-9 m
2 ]
Low elevation scenario: Impact of wind speed variations on average SD-variance
0
200
400
600
800
1000
v = 8 [m/s] v = 15 [m/s]
Ave
rage
var
ianc
e [e
-9 m
2 ]
Zenith scenario: Impact of wind speed variations on average SD-variance
⇒ depending on geometry: increasing wind speed → decorrelating effect
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ErdmessungInstitut für
Impact of parameter variations
16 / 17
Impact of wind direction variations on mean correlations and average SD-variance:
0.0
0.2
0.4
0.6
0.8
1.0
0 10 20 30 40 50
Cor
rela
tion
[-]
Anti-diagonal element
Low elevation scenario: Impact of wind direction variations on correlations
az = 0 [deg]az = 90 [deg]
az = 180 [deg]az = 270 [deg]
0.0
0.2
0.4
0.6
0.8
1.0
0 10 20 30 40 50
Cor
rela
tion
[-]
Anti-diagonal element
Zenith scenario: Impact of wind direction variations on correlations
az = 0 [deg]az = 90 [deg]
az = 180 [deg]az = 270 [deg]
0
200
400
600
800
1000
az = 0 [deg] az = 90 [deg] az = 180 [deg] az = 270 [deg]
Ave
rage
var
ianc
e [e
-9 m
2 ]
Low elevation scenario: Impact of wind direction variations on average SD-variance
0
200
400
600
800
1000
az = 0 [deg] az = 90 [deg] az = 180 [deg] az = 270 [deg]
Ave
rage
var
ianc
e [e
-9 m
2 ]
Zenith scenario: Impact of wind direction variations on average SD-variance
⇒ depending on geometry: decorrelating effect for orthogonal wind
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M. Vennebusch, S. SchönIAG 2009, Buenos Aires, 1. September 2009Erdmessung
Institut für
Summary & Conclusions
Introduction
Simulations
Impact of parametervariations
Summary &Conclusions
17 / 17
Summary & Conclusions:
■ generation of variance-covariance matricesof slant delays possible
■ generation of slant delay variations possible
◆ typical variations: ± 1 - 3 [mm]◆ correlation lengths: ≈ 200 [sec]
■ simulated slant delay variations as expected:
◆ 5/3 power law for all simulated time series◆ higher variations for low elevations
■ superposition of geometric effects and atmosphericturbulence needs further investigation
■ now: analysis of real GNSS data
Acknowledgements:
This project is funded by Deutsche Forschungsgemeinschaft (SCHO 1314/1-1).