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

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

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


Institut für

Motivation

Introduction

Motivation


Analysis objectives

Simulations



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


Institut für

Stochastic model

Introduction

Motivation


Analysis objectives

Simulations



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


Institut für

Generation of slant delay variations

Introduction

Motivation


Analysis objectives

Simulations



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


Institut für

Analysis objectives

Introduction

Motivation


Analysis objectives

Simulations



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


Institut für

Simulations

Introduction

SimulationsScenarios &parameter sets



7 / 17


Institut für

Simulation scenarios and parameter sets

Introduction

SimulationsScenarios &parameter sets



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

ErdmessungInstitut für

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


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):


Simulations - low elevation

10 / 17


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


(Co−

)Var

ianc

e [m

2 ]


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


Cor

rela

tion

[−]


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


Sla

nt d

elay

[m]



1 2 3 4 5 6 7 8 9 10 2010

−9

10−8

10−7

10−6

10−5

10−4


log(

DS

D(τ

)) [m

2 ]



Scenario: low elevation, parameter set: 5 (= average turbulence parameters):


Simulations - rising satellite

11 / 17


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


(Co−

)Var

ianc

e [m

2 ]


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


Cor

rela

tion

[−]


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


Sla

nt d

elay

[m]



1 2 3 4 5 6 7 8 9 10 2010

−9

10−8

10−7

10−6

10−5

10−4


log(

DS

D(τ

)) [m

2 ]



Scenario: rising satellite, parameter set: 5 (= average turbulence parameters):


Institut für

Impact of parameter variations

Introduction

Simulations



12 / 17



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

[-]


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



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

[-]


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

[-]


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



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

[-]


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

[-]


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



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

[-]


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

[-]


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


Institut für

Summary & Conclusions

Introduction

Simulations



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).

Documents

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