On the use of anisotropic covariance models of atmospheric...

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

mailto:[email protected]

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


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


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


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


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


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


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


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


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?


Estimating

variograms

from

limited

samples

(1)

Coherence

thresholding: sampling

at 15%


Estimating

variograms

from

limited

samples

(1)

Coherence


at 10%


Estimating

variograms

from

limited

samples

(1)

Coherence


at 5%


Estimating

variograms

from

limited

samples

(1)

Coherence


at 3%


Estimating

variograms

from

limited

samples

(1)

Coherence


at 1%


Estimating

variograms

from

limited

samples

(2)

Coherence


at 15%


Estimating

variograms

from

limited

samples

(2)

Coherence


at 10%


Estimating

variograms

from

limited

samples

(2)

Coherence


at 5%


Estimating

variograms

from

limited

samples

(2)

Coherence


at 3%


Estimating

variograms

from

limited

samples

(2)

Coherence


at 1%


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%


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”)


Example

1 –

Matérn

model Sampled

at 5%

Sampled

at 2%


Example

2 –

Matérn

model Sampled

at 10%

Sampled

at 5%


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


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].


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

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