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1
Characterization of Spatial Heterogeneity for Scaling Non Linear Processes
S. Garrigues1, D. Allard2, F. Baret1.1INRA-CSE, Avignon, France2INRA-Biométrie, Avignon, France
2
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
3
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
4
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
5
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)
6
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((
7
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.
8
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
9
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)
10
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
11
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
12
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 ,',
13
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
14
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
14
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
15
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
16
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
17
Resolution=1000m
5. Bias correction model
Cropland site (Alpilles) exampleCropland site (Alpilles) example
18
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)
19
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
20
LAI=f(NDVI)
NDVIA
LAIh
LAIb
LAIR
NDVIBNDVIP
A
B
C
D
R^
RV
Raffy Method – Univariate caseRaffy Method – Univariate case
21
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((