on the use of anisotropic covariance models of atmospheric...
TRANSCRIPT
On the use of anisotropic covariance models of atmospheric DInSAR
contributions
A. Refice, A. Belmonte, F. Bovenga, G. Pasquariello
ISSIA-CNR, Bari, ItalyE-mail: [email protected]
SAR data provided
by
ESA through
C1P.5367
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
Outline
•
Introduction: evidences of anisotropy•
APS estimation and reconstruction
•
The PSI context•
Observations and insights from simulations
•
Conclusions
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
Introduction
•
Atmospheric phase screen (APS) modeling is a long- standing problem in DInSAR
processing
•
In many applications, APS is a nuisance: it has to be estimated
and removed
from the interferogram before
quantitative interpretations of DInSAR
phase•
Accurate modeling
seems necessary to remove it as
best as possible•
Isotropy is often a simplification
•
Recent works advocate use of anisotropic
stochastic models for APS analysis
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
Evidence
of
anisotropy
Tandem interferograms•Flat
topography
-> no stratification•1 day
interval
-> no deformation
ONLY turbulent
mixing (troposphere)
pixels (range)
pixe
ls (a
zim
uth)
Tandem 02-03/06/1996
100 200 300 400 500 600 700 800 900 1000
100
200
300
400
500
600
700
800
900
1000
pixels (range)
pixe
ls (a
zim
uth)
Tandem 05-06/11/1995
100 200 300 400 500 600 700 800 900 1000
100
200
300
400
500
600
700
800
900
1000
pixels (range)
pixe
ls (a
zim
uth)
Tandem 10-11/12/1995
100 200 300 400 500 600 700 800 900 1000
100
200
300
400
500
600
700
800
900
1000
pixels (range)
pixe
ls (a
zim
uth)
Tandem 18-19/02/1996
100 200 300 400 500 600 700 800 900 1000
100
200
300
400
500
600
700
800
900
1000
pixels (range)
pixe
ls (a
zim
uth)
Tandem 23-24/07/1995
100 200 300 400 500 600 700 800 900 1000
100
200
300
400
500
600
700
800
900
1000
pixels (range)
pixe
ls (a
zim
uth)
Tandem 24-25/03/1996
100 200 300 400 500 600 700 800 900 1000
100
200
300
400
500
600
700
800
900
1000
pixels (range)
pixe
ls (a
zim
uth)
Tandem 28-29/04/1996
100 200 300 400 500 600 700 800 900 1000
100
200
300
400
500
600
700
800
900
1000
π
−π
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Estimating
1D and 2D variograms
-500
0
500 -500
0
500
0
10
20
y lag (samples)
bidimensional variogram
x lag (samples)
varia
nce
-500
0
500 -500
0
500
0
5
10
y lag (samples)
bidimensional variogram
x lag (samples)
varia
nce
Unwrapped phase 1D variograms 2D variograms
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A bit of
mathematics…•
Given a regionalized variable•
Intrinsic stationarity:
•
Second-order stationarity (bounded variogram):
•
Variogram estimator (method of moments):
•
1-D
•
2-D
( ) ( ) ( )[ ]2rzhrzEhV −+=
( ) ( ) ( )hVChC −= 0
( ) ( ) ( )[ ]∑ ∈−′−′=
hNrrh
rzrzN
hV 21
( ) ( ) ( )[ ]( )∑ ∈−′
−′=hNrr
h
rzrzN
hV 21
( )rz
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
Atmospheric
phase
screen
modeling
•
APS contributions have been described as self-similar (fractal)
processes
•
Stemming from the Kolmogorov
turbulence theory, a physically-based multi-fractal model
has been developed
(Hanssen, 2001)•
The multi-fractal paradigm solves some problems connected to the use of power-laws in variogram modeling –
e.g.
stationarity…•
Other models are used in geostatistics
to describe random
fields, with more desirable properties (e.g. differentiability, long-range stability, etc.)
•
In view of this, “simpler”
models are often used to describe the APS in operational contexts
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
Variogram models
•
Matérn
model ( ) ( ) ⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛Κ⋅⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅
Γ−⋅+=
−
Lh
LhSNLSNhV αα
αα α
αα 2221,,,1
0 cos sinsin cos0 1
u u
v v
h hh h
δ θ θθ θδ
′ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤= = ⋅ ⋅⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥′ −⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦h
( ) ( ) ( ) ( ); ;h V h V V V′→ → = ⋅ ⋅h h h S R h
•
Extension
to
geometric
anisotropy:
( ) ( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛⋅⎟
⎠⎞
⎜⎝⎛⋅+Γ−⋅+=
−
LhJ
LhSNLSNhV α
αα αα 121,,,
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−⋅+=
2
exp,,,LhSNLSNhV α
•
Bessel
model
•
Gauss model
(α = 2)
hv
huθ
hv
hu
hv
huθ
δ = hu / hv
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
The PSI context
•
In Persistent Scatterers (PSI) applications, APS must be predicted from a limited number
of pixels in each
interferogram over the entire raster grid•
Usually, stacks of several tens of interferograms
are
processed•
APS is a random field stochastic modeling and prediction (the kriging paradigm) applies naturally
•
Accurate modeling seems important for reconstruction
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
The PSI context
(cont.)
•
Modeling of stochastic fields is something of an art•
Experienced geostatisticians
dispose of a number of methods to
enhance modeling•
Nested models, Parameter profiling, …
•
Effective tools for APS prediction come from physically-based atmospheric models (MM5, ECMWF, …)
•
Recently, data from other sensors are being incorporated : GPS, MODIS, MERIS, etc.
HOWEVER…•
Ancillary data are not always available
•
Often, within PSI, the APS field is used as a “dustbin”
for other unmodeled
contributions
•
In PSI contexts, we should consider an operational
framework Emphasis is on automated and robust estimation methods
can user intervention be
reduced?
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
Estimating
variograms
from
limited
samples
(1)
Coherence
thresholding: sampling
at 15%
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
Estimating
variograms
from
limited
samples
(1)
Coherence
thresholding: sampling
at 10%
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
Estimating
variograms
from
limited
samples
(1)
Coherence
thresholding: sampling
at 5%
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
Estimating
variograms
from
limited
samples
(1)
Coherence
thresholding: sampling
at 3%
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
Estimating
variograms
from
limited
samples
(1)
Coherence
thresholding: sampling
at 1%
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
Estimating
variograms
from
limited
samples
(2)
Coherence
thresholding: sampling
at 15%
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
Estimating
variograms
from
limited
samples
(2)
Coherence
thresholding: sampling
at 10%
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
Estimating
variograms
from
limited
samples
(2)
Coherence
thresholding: sampling
at 5%
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
Estimating
variograms
from
limited
samples
(2)
Coherence
thresholding: sampling
at 3%
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
Estimating
variograms
from
limited
samples
(2)
Coherence
thresholding: sampling
at 1%
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
Questions
•
Obviously model estimation performances degrade with decreasing sampling densities
•
More complicated models involve higher degrees of uncertainty (“dimensionality curse”)
•
Therefore, it is not trivial to ask how much
do more sophisticated models actually help
in APS prediction from limited samples
•
N.B.: •
For ERS/ENVISAT
full-resolution data (5x20 m2
on ground)•
1% sampling ~ 100 samples per km2
•
Max observed PS densities ~ 1-2%
•
For higher resolution sensors (e.g. TerraSAR-X
or COSMO/SkyMed stripmap
data, 3x3 m2)•
1% sampling ~ 1000 samples per km2
•
Max observed PS densities > 3-4%
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
SimulationsRandom
surfaceSimulation
(spectral
methods)
LMS model
ParameterEstimation
(all
pts.)
δ = 1δ = 2Anisotropic
Fieldsδ = 10
Pts. Sampling10-5-3-2-1-0.5 %
(W)LMS parameterestimation
Ordinary
kriging
Reference
modelparameters
Estimated
modelparameters
Reconstructedsurfaces
Comparison(RMSE)
θ random,uniform
∈
[− π,π]
•1D model•2D model
w/ guess
values
estimated
from
all
pts. (“2D TRUE”)•2D model
w/ isotropic
guess
values, δ=1, θ=0 (“2D ISO”)
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
Example
1 –
Matérn
model Sampled
at 5%
Sampled
at 2%
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Example
2 –
Matérn
model Sampled
at 10%
Sampled
at 5%
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
Average
reconstruction
resultsMatérn
model –
LMS fit Matérn
model –
WLMS fit Gauss model –
WLMS fit
‘True’
anisotropic fit ---
2D model, (W)LMS with guess δ and θ values = reference values from all pts.Anisotropic fit ---
2D model, (W)LMS with guess values δ = 1, θ = 0.Isotropic fit ---
1D model
Bessel model –
WLMS fit
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
Interpretations•
2D models are more “complicated”
than corresponding 1D
models•
2D fitting is algorithmically less stable than 1D (local minima)
•
Estimating 2d experimental variograms
requires more sampling points than for 1D (number of required bins increases)
•
Therefore, for a certain sampling density, estimates using simpler, 1D models are more robust than complex 2D models.
•
For the reconstruction purpose, often robustness
seems to be more important than accuracy
•
Note: anisotropy parameters δ and θ are “different”
from other model parameters for what concerns weighting schemes •
δ and θ are better estimated “away from the origin”,
•
the rest of the model parameters, instead, have influence close to the origin [Stein, 1999].
Consiglio Nazionale delle Ricerche Istituto di Studio sui Sistemi Intelligenti per l’Automazione (ISSIA)
Conclusions
•
We have made some observations about the advocated use of anisotropic models for APS modeling in the specific PSI context
•
There is a trade-off between model complexity
and robustness in estimation and prediction of APS fields through kriging
•
Conditions
(low sampling densities) and requirements
(little user intervention) in PSI processing are demanding
•
Anisotropic APS models seem to be useful for:•
sufficient sampling densities, or if
•
ancillary information is available
•
Future work:•
Further explore alternative estimation techniques (ML/REML, etc.)
•
Better constrain requirements
based on sampling densities•
Quantify expected phase noise reduction