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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Generation of slant tropospheric delay time series
based on turbulence theory
Markus VennebuschSteffen Schon
Institut fur Erdmessung, Leibniz Universitat Hannover, Germany
1. September 2009
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
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
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
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
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
M. Vennebusch, S. SchönIAG 2009, Buenos Aires, 1. September 2009Erdmessung
Institut für
Simulations
Introduction
SimulationsScenarios ¶meter sets
Impact of parametervariations
Summary &Conclusions
7 / 17
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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Simulations - zenith scenario
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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):
ErdmessungInstitut für
Simulations - low elevation
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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):
ErdmessungInstitut für
Simulations - rising satellite
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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):
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
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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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
ErdmessungInstitut für
Impact of parameter variations
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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
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
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).