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Characterization of Spatial Heterogeneity for Scaling Non Linear Processes

S. Garrigues1, D. Allard2, F. Baret1.1INRA-CSE, Avignon, France2INRA-Biométrie, Avignon, France

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Different spatial and temporal scales

1. Background

Vegetation monitoring at global scale (primary production, carbon cycle...)

Transfer Function

BV = f(Ri)

Biophysical variable (LAI, FAPAR)

Need of high time frequency data

Non linear process

Vegetation ground scene (different biome types)

Reflectance Image ({Ri, i=1..n})

Sensor

function

Technological constraints:Coarse spatial resolution sensor

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2. ProblematicImage spatial structure depends on vegetation type Image spatial structure depends on vegetation type

Counami: Tropical forest

Alpilles: CroplandPuechabon: Woody savana

Nezer: Pine forest

Alpilles: CroplandPuechabon: Woody savana

Nezer: Pine forest Counami: Tropical forest

20m SPOT NDVI image

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2. ProblematicImage spatial structure depends on vegetation type Image spatial structure depends on vegetation type

Alpilles:Cropland

Puechabon:Woody Savana

Nezer:Pine Forest

Counami:Woody Savana

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The sensor integrates the signal over the pixel; intra-pixel variance lost

Spatial heterogeneity depends on the spatial resolution

« Homogeneous »(Guyana Forest)

Sp

ati

al R

esolu

tion

« Heterogeneous site »

(Alpilles Cropland )

2. Problematic

20m (SPOT)

Image spatial structure depends on sensor spatial Image spatial structure depends on sensor spatial resolution resolution

60m ( SPECTRA)

300m ( MERIS)

500m ( MODIS)

1000m ( VGT)

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Heterogeneous pixel

A B

2. Problematic

Spatial heterogeneity and non linear processSpatial heterogeneity and non linear processNon linear transfer function between NDVI and LAI:

LAI=f(NDVI)

LAIB

NDVIBNDVIA

LAIABias: e=LAIapparent-LAIactual

biais

NDVI

LAIapparent

Apparent LAI

2NDVIBNDVIAfNDVIfLAIapparent

LAIactual

Actual LAI :

2LAIBLAI ALAIactual

lai

s

KNDVINDVINDVINDVI

LAI

))/()log((

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2. Problematic

Spatial heterogeneity definition: •quantitative information characterizing the ground spatial structure•spatial variance distribution of the variable considered, within the coarse resolution pixel Our aim: using spatial heterogeneity as an a priori information to

correct biophysical estimation biais, i.e. to scale up the transfer function at coarser spatial resolution

Spatial structure (i.e. spatial heterogeneity) depends on:- surface property variation - sensor regularization

- spatial characteristics: spatial resolution, support geometry (PSF), viewing angle…- spectral characteristic, atmospheric effects- image extent

Working scale: the field scaleUtilisation of high spatial resolution (SPOT 20m) to characterize ground spatial structure at field spatial frequency.

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Stochastic framework for image exploitationStochastic framework for image exploitation

The image is a realization of a random process (random function model) with the following characteristics:

•Ergodicity: one realization of the random process allows to infer the statistical properties of the random function.•Stationarity of the two first moments:

- the mean image value is constant over the image- the correlation between two pixel values depends only on the distance

between them.

Data support: SPOT pixel considered as punctual- No accounting for SPOT regularization (PSF)- No accounting for SPOT pixel radiometric uncertainties (measurement errors)

Variable studied : NDVI

3. Spatial heterogeneity characterization

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3. Spatial heterogeneity characterization

The variogram: a structure function The variogram: a structure function

Definition:• spatial variance distribution of the regionalized variable z(x)

)(

1

2

)()(*)(*2

1)(hN

iehxizxizhN

h Sample variogram:

Theoretical variogram

²)()(*5.0)( hxZxZEh

Sill ( ²)

True Variance

Range (r)

Up to this distance data are spatially correlated

Variogram regionalization model of the image: nested structure

)(1

hgbn

Standard variogram structure characterizes a spatial variation of the image

Range1 (r1)Range2 (r2)

Sill(²)

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3. Spatial heterogeneity characterization

Spatial structure characterization by the variogramSpatial structure characterization by the variogram

The variogram describes the ground spatial structure of different vegetation types.

Alpilles,r1=264m, r2=1148m, sill=0.042

Puechabon r1=260m, r2=1806m, sill=0.012

Nezer,r1=222m, r2=1533m, sill=0.0037

Counami,r1=57m, r2=676m, sill=0.00086

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3. Spatial heterogeneity characterization

Spatial heterogeneity typology Spatial heterogeneity typology

Integral range is a yardstick that summarizes variogram on the image

²²

)²)(1()(*)

²1(

RR

dhhdhhCA

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Var

ian

ce

Integral range

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4. Spatial heterogeneity regularization with decreasing spatial resolution

Spatial structure regularization is a function of sensor Spatial structure regularization is a function of sensor spatial characteristicsspatial characteristics

Puechabon site

Sensor regularization

Image structure Ground structure

)](*[ xZpZ v

Point Spread Function

Regularized variogram

Regularized variogram Ground variogram

vvvxvxv ,',

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Spatial heterogeneity quantificationSpatial heterogeneity quantification

4. Spatial heterogeneity regularization with decreasing spatial resolution

Sample dispersion variance

0.2

0.3

0.4

0.5

0.6

0.7

Image NDVIM

R

2 4 6 8 10 12 14

2

4

6

8

10

12

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Pixel x,Z(x)

Pixel vi, Z(vi)

ni x

vZxz ivnvxS

..1

2

)()(11),²(

o Our model of data regularization

o Quantify spatial heterogeneity (spatial variance) with spatial resolution

),( vv

),()),²((),²( vvVxSEVx

Theoretical dispersion variance:

Cropland

Woody Savana

Pine forest

Tropical Forest

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2)(2

)(''))((')()( NDVINDVI

NDVIfNDVINDVINDVIfNDVIfNDVIf

x

actual xNDVIfn

LAI ))((1

x

actualapparent xNDVIfn

NDVIfLAILAIe ))((1)(Biased LAI

x

NDVINDVIn

NDVIfvxe

2)(

1*

2

)(''),(

Sample dispersion variance ),²( xS

5 Bias correction model

Univariate Model Univariate Model

),(*2

)(''))(( vv

NDVIfveE

NDVI

Non linearity degree Heterogeneity degree

)(NDVIfLAI apparent

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Cropland site (Alpilles) exampleCropland site (Alpilles) example

5. Bias correction model

Resolution=500m

Model problem:• Non stationnarity pixel: sample dispersion variance (pixel spatial heterogeneity) is lower than theoretical variance dispersion predicted by variogram model

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Cropland site (Alpilles) exampleCropland site (Alpilles) example

5. Bias correction model

Resolution=500m

Model problem:• Non stationnarity pixel: the sample dispersion variance of the pixel is lower than the theoretical variance dispersion predicted by the variogram model

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Resolution=1000m

5. Bias correction model

Cropland site (Alpilles) exampleCropland site (Alpilles) example

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6. Multivariate spatial heterogeneity characterization

Multivariate description of spatial heterogeneityMultivariate description of spatial heterogeneity

Coregionalization variogram model

)(0

0)(*

)(0

0)(* 2

2

22

22

1

1

11

11

11

11

h

h

h

h

gg

bbbb

gg

bbbb

jij

jii

jij

jii

jij

iii

multi-spectral spatial heterogeneity description:

- more information on physical signal- using variance-covariance dispersion matrix to correct bias

Problems: disturbing factors (atmosphere) influence the spatial structure

Alpilles (Cropland)

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5. Conclusions and prospects

Using variograms to describe spatial heterogeneity :• it describes the spatial structure of different landscapes • it allows to model data regularization

Bias correction model :• based on variogram models and accounts for the non linearity of the transfer function• allows accounting for actual PSF (sensor spatial characteristics, registration for data fusion)

Problems:• How to adjust variogram models for bias correction?•Temporal stationnarity of the variogram models? •Transfer function diversity: development of a multivariate model

Accounting for image spatial information for quantitative remote sensing is an important concern

• Use of SPECTRA data to adjust variogram models and investig•ate their temporal stationnarity•Optimizing the PSF design of future missions

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LAI=f(NDVI)

NDVIA

LAIh

LAIb

LAIR

NDVIBNDVIP

A

B

C

D

R^

RV

Raffy Method – Univariate caseRaffy Method – Univariate case

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Heterogeneous pixel

A B

2. Problematic

Spatial heterogeneity and non linear processSpatial heterogeneity and non linear processNon linear transfer function between NDVI and LAI:

LAI=f(NDVI)

LAIB

NDVIBNDVIA

LAIABias: e=LAIapparent-LAIactual

biais

NDVI

LAIapparent

Apparent LAI

2NDVIBNDVIAfNDVIfLAIapparent

LAIactual

Actual LAI :

2LAIBLAI ALAIactual

lai

s

KNDVINDVINDVINDVI

LAI

))/()log((