Covariance Estimation and Geostatistical Simulation for ...earth.esa.int/fringe07/participants/225/pres_225_knospe.pdf · Covariance Estimation and Geostatistical Simulation for InSAR

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


• After removing orbit and topographic phase components, deformation phase remains

• However, some phase components do not vanish

φdiff = φdefo + φorb-err + φatmo + φtop-err + φnoise

Atmospheric signals can dominate deformation …28/11/2007 Steffen Knospe & Sigurjón Jónsson FRINGE 2007 2/20

Differential InSAR Measurements


28/11/2007 Steffen Knospe & Sigurjón Jónsson FRINGE 2007 3/20

Purpose • Error Analysis is important

to better understand dInSAR measurements and

• to get better results in geophysical inverse modelingand to get the model parameter uncertainty

• atmospheric signal is spatial autocorrelated with strong anisotropic anomalies

We show how to estimate covariance structuresand the simulation of anisotropic signal parts

How important is it to consider autocorrelated noise and anisotropy?



• Introduction• Atmospheric noise Structure Analysis

- 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



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



Data example• To study atmospheric phase delays, look at non-deforming

areas within interferograms or take …• ERS1/2 tandem interferograms

• directional phase anomalies



• 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



2D experimental variogram-maps•

• range-ratio

• direction



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 = −



Random Functions• replacement of experimental semi-variogram with the variogram

function type:Bessel-family

function type:Matérn-family



Geostatistical Simulation• we simulate Gaussian Random Fields

based on the covariance structures from our data examples



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 matrix as weights in 3 different scenarios– Uniform weight– Isotropic covariance– Anisotropic covariance

• STD values …characterizes quality and reliability of the inversion process


Geophysical Inversion Modeling

uniform weights

isotropic

anisotropic



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



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



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



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 …


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



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!



Sensitivity and Reliability Varying deformation Strength and Orientation (1 and 2)



Discussion• More realistic 2D error structure

– Nested models– Excluded for simplification – Geometric and Zonal anisotropy


• Two more examples from our data stack– Uniform weight– Isotropic C – Anisotropic C

• STD values …characterizes quality and reliability of the inversion process


Geophysical Inversion Modeling



Discussion• Wrong applied anisotropy

– Uniform weight– Isotropic C – Anisotropic C



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



Sensitivity

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Covariance Estimation and Geostatistical Simulation for ...earth.esa.int/fringe07/participants/225/pres_225_knospe.pdf · Covariance Estimation and Geostatistical Simulation for InSAR