Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy 24/04/2007 Steffen Knospe & Sigurjón Jónsson ENVISAT 2007 1/20 Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy Steffen Knospe & Sigurjón Jónsson Institute of Geophysics, ETH Zurich
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Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
24/04/2007 Steffen Knospe & Sigurjón Jónsson ENVISAT 2007 1/20
Covariance Estimation and Geostatistical Simulation
for InSAR Observations in Presence of Strong Atmospheric Anisotropy
Steffen Knospe & Sigurjón JónssonInstitute of Geophysics, ETH Zurich
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
• After removing orbit and topographic phase components, deformation phase remains
- data example- covariance estimation – structure analysis- variogram model functions
• Simulation of noise structures• Source Parameter Inversion
from dInSAR deformation measurements• Results and open questions
Contents
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
28/11/2007 Steffen Knospe & Sigurjón Jónsson FRINGE 2007 5/20
Atmospheric signal• Methods to reduce atmospheric signal have been described
(atmospheric modeling, use of GPS and MERIS data, APS in PSI, etc.) …
• not applicable in every case …(especially for single interferogram studies)
• We use a stochastic model
to describe atmospheric signal characteristics and
to respect it in dInSAR data modeling
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
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Data example• To study atmospheric phase delays, look at non-deforming
areas within interferograms or take …• ERS1/2 tandem interferograms
• directional phase anomalies
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
28/11/2007 Steffen Knospe & Sigurjón Jónsson FRINGE 2007 7/20
• Experimental omnidirectional semi-variograms
Omnidirectional experimental variograms
( ) ( ) ( ) ( )( )( ) 2
1
12
N h
i ii
h z x z x hN h
γ=
= − +∑ sillshape
nugget range
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
28/11/2007 Steffen Knospe & Sigurjón Jónsson FRINGE 2007 8/20
2D experimental variogram-maps•
• range-ratio
• direction
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
28/11/2007 Steffen Knospe & Sigurjón Jónsson FRINGE 2007 9/20
Random Functions• we search for a Structure Function …
employing the theory of Random Functions • assuming, there exists a theoretical variogram
of a homogeneous isotropic Random Function with the same characteristics
• with the Random Function this spatial structure inherently obtains a physical meaning
• analytic negative-definite function, which guarantees positive variances for any linear combination of sample values
• the auto-covariance function C(h)is derived from the variogram γ(h)with variance C(0) under 2nd order stationarity
( ) ( ) ( )0h C C hγ = −
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
28/11/2007 Steffen Knospe & Sigurjón Jónsson FRINGE 2007 10/20
Random Functions• replacement of experimental semi-variogram with the variogram
function type:Bessel-family
function type:Matérn-family
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
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Geostatistical Simulation• we simulate Gaussian Random Fields
based on the covariance structures from our data examples
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
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Geophysical Inversion ModelingDoes it make a difference if anisotropy is included
in the data covariance matrix or not?
To answer, we carry out source parameter inversion from a deformation signal affected by autocorrelated noise
1.Deformation signal simulation using forward model calculation with a opening sill as deformation source(refilling of a magma chamber)
2.Add a realization of anisotropic, autocorrelated error signal
3.Invert for the source strength (sill opening)
4.Repeat 1000 times for different random realizations of the error signal
5.Use 3 different data covariance matrices as weights in the inversion algorithm in 3 different scenarios
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
• Covariance matrix as weights in 3 different scenarios– Uniform weight– Isotropic covariance– Anisotropic covariance
• STD values …characterizes quality and reliability of the inversion process
28/11/2007 Steffen Knospe & Sigurjón Jónsson FRINGE 2007 13/20
Geophysical Inversion Modeling
uniform weights
isotropic
anisotropic
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
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Sensitivity and Reliability How sensitive are these results? Does it depend on the deformation signal? Is there a maximum in the improvements?
To answer, we change parameter in the simulation of deformation signal and do the inversion multiple times.
1.We change sill depthto simulate a deformation signal with varying strength
2.We change sill striketo simulate a deformation signal with varying orientation
3.We change sill length to simulate a deformation signal with a varying shape (anisotropy-ratio)
The gain - ratio of standard deviation for inversion scenario 2 and 3
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
28/11/2007 Steffen Knospe & Sigurjón Jónsson FRINGE 2007 15/20
Sensitivity and Reliability Varying deformation Strength and Orientation (1 and 2)
maximum deformation: 0.009m to 0.13m (sill 8.5km x 1.25km, opening 0.2m)
maximum gain = 18.6
at:
• maximum deformation = 0.021m(sill depth = 5.75m)
compare: maximum error = 0.019m
• at orientation = 85° from North
compared to 78° for error anisotropygain = STDscenario2 / STDscenario3
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
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Sensitivity and Reliability 3. Varying Anisotropy-Ratiomaximum deformation: 0.003m to 0.13m (sill width 1.25km, opening 0.2m)(vs. maximum error = 0.019m)
sill length: 2.5km to 15.0kmmaximum gain at sill length = 8.5km (ratio: 6.8 : 1)
compared to error anisotropy-ratio: 6.7 : 1precision gain – sill length = 8.5 km precision gain – sill length = 2.5 km precision gain – sill length = 15 km
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
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Results• respecting spatial autocorrelation improves the
quality of inversion results (even if we neglect anisotropy)
• we dramatically improve results … taking anisotropy into account, the more the
stronger anisotropic effects are… the greater the similarity between deformation and error signal
It is! … important to consider autocorrelated noise and to respect anisotropy …
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
Acknowledgements Many thanks to:
• ESAfor providing me a post-doc research fellowship in the External fellowship program
• GAMMA Remote Sensingfor providing data and InSAR processing software
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
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Open Questions
• Atmosphere vs. Deformation– Strength of atmospheric noise and deformation signal characteristics
(variance) may be different, the spatial extent and the structure (inter alia smoothness and shape) are likely to be the same.
– The deformation phase component itself is spatial autocorrelated,
– We down-weight data with their error content (covariance matrix) to get more reliable results with inversion models in presence of correlated noise
– However, similarity of error (atmospheric delay) and signal (deformation)forces a down-weighting of the deformation based contribution in the data!
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
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Sensitivity and Reliability Varying deformation Strength and Orientation (1 and 2)
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
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Discussion• More realistic 2D error structure
– Nested models– Excluded for simplification – Geometric and Zonal anisotropy
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
• Two more examples from our data stack– Uniform weight– Isotropic C – Anisotropic C
• STD values …characterizes quality and reliability of the inversion process
28/11/2007 Steffen Knospe & Sigurjón Jónsson FRINGE 2007 22/20
Geophysical Inversion Modeling
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
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Discussion• Wrong applied anisotropy
– Uniform weight– Isotropic C – Anisotropic C
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
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Sensitivity and Reliability 2. Varying Anisotropy-Ratiomaximum deformation: 0.001m to 0.11m (sill opening 0.2m, width 1km)(vs. maximum error = 0.007m)
maximum gain = 1.41, orientation = 90° from North (error anisotropy: 82°)at sill length = 3.0km (error anisotropy-ratio: 3:1)
at maximum deformation = 0.055m (sill depth = 1.75m)precision gain – sill length = 3 km precision gain – sill length = 1.5 km precision gain – sill length = 15 km
Covariance Estimation and Geostatistical Simulation for InSAR Observations in Presence of Strong Atmospheric Anisotropy
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