1 Characterization of Spatial Heterogeneity for Scaling Non Linear Processes S. Garrigues 1, D. Allard 2, F. Baret 1. 1 INRA-CSE, Avignon, France 2 INRA-Biométrie, Avignon, France

2 Different spatial and temporal scales 1. Background Vegetation monitoring at global scale (primary production, carbon cycle...) Transfer Function BV = f(R i ) Biophysical variable (LAI, FAPAR) Need of high time frequency data Non linear process Vegetation ground scene (different biome types) Reflectance Image ( {R i, i=1..n} ) Sensor function Technological constraints: Coarse spatial resolution sensor

3 2. Problematic Image spatial structure depends on vegetation type Counami: Tropical forest Alpilles: Cropland Puechabon: Woody savana Nezer: Pine forest Alpilles: Cropland Puechabon: Woody savana Nezer: Pine forest Counami: Tropical forest 20m SPOT NDVI image

4 2. Problematic 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) Spatial Resolution « Heterogeneous site » (Alpilles Cropland ) 2. Problematic 20m (SPOT) Image spatial structure depends on sensor spatial resolution 60m (  SPECTRA) 300m (  MERIS) 500m (  MODIS) 1000m (  VGT)

6 Heterogeneous pixel AB 2. Problematic Spatial heterogeneity and non linear process Spatial heterogeneity and non linear process Non linear transfer function between NDVI and LAI : LAI=f(NDVI) LAI B NDVI B NDVI A LAI A Bias: e=LAI apparent -LAI actual biais NDVI LAI apparent Apparent LAI LAI actual Actual LAI :

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 scale Utilisation of high spatial resolution (SPOT 20m) to characterize ground spatial structure at field spatial frequency.

8 Stochastic 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 The variogram: a structure function  Definition: spatial variance distribution of the regionalized variable z(x)  Sample variogram:  Theoretical variogram Sill (  ²) True Variance Range (r) Up to this distance data are spatially correlated  Variogram regionalization model of the image: nested structure 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 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= Counami,r1=57m, r2=676m, sill=

11 3. Spatial heterogeneity characterization Spatial heterogeneity typology Integral range is a yardstick that summarizes variogram on the image Variance Integral range

12 4. Spatial heterogeneity regularization with decreasing spatial resolution Spatial structure regularization is a function of sensor spatial characteristics Puechabon site Sensor regularization Image structureGround structure Point Spread Function Regularized variogram Ground variogram

13 Spatial heterogeneity quantification 4. Spatial heterogeneity regularization with decreasing spatial resolution  Sample dispersion variance Pixel x,Z(x) Pixel vi, Z(vi) o Our model of data regularization o Quantify spatial heterogeneity (spatial variance) with spatial resolution  Theoretical dispersion variance: Cropland Woody Savana Pine forest Tropical Forest

14 Biased LAI Sample dispersion variance 5 Bias correction model Univariate Model Non linearity degreeHeterogeneity degree

15 Cropland 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) 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) example

18 6. Multivariate spatial heterogeneity characterization Multivariate description of spatial heterogeneity Coregionalization variogram model  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) NDVI A LAI h LAI b LAI R NDVI B NDVI P A B C D R^R^ RVRV Raffy Method – Univariate case

21 Heterogeneous pixel AB 2. Problematic Spatial heterogeneity and non linear process Spatial heterogeneity and non linear process Non linear transfer function between NDVI and LAI : LAI=f(NDVI) LAI B NDVI B NDVI A LAI A Bias: e=LAI apparent -LAI actual biais NDVI LAI apparent Apparent LAI LAI actual Actual LAI :