Comparing VALERI sampling schemes to better represent
high spatial resolution satellite pixel from ground
measurements: How to characterize an ESU.
Sébastien Garrigues*, Denis Allard, Marie Weiss, Frédéric Baret
26 june 2002
*sebastien.garrigues@avignon.inra.fr
A/ Objectives
The aim of this study is to compare different sampling schemes (represented by the following
figures) according to their spatial representativity of a 20m resolution pixel. The local
sampling is made of 12 individual measurements and corresponds to an Elementary Sampling
Unit (ESU) used in the VALERI methodology (Baret et al, 2003)
20m
distribution a


A1
20m
distribution b
The b distribution: the sample are distributed along a cross pattern (20 m size).
The a distribution: 8 points are located on the square edges and four within the square.
The a distribution can differ according to the sample configuration within the pixel.
Three cases are tested depending on the location positions of the 4 central points.
A2
A3
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B/ Method
1/ Extension Variance definition
a/ Geostatistics theory
Spatial variation between the individual measurement (z(x)) of the ESU is described
by the experimental semivariogram γe(h) (commonly called experimental variogram) which is
the mean squared deviation of a variable at locations separated by a given lag distance:
N(h)
(1)
 (h) 1 * z(xi) z(xi  h)
e
2*N(h)
i 1
z(xi) is the measurement value at location i, z(xi+h) is the measurement value at lag h
from x and N(h) is the number of pairs of points separated by the distance h. Two main
parameters characterize the variogram:
 The total sill (²) corresponds to the true variance of the data.
 The range (r) is the lag at which the variogram reaches the sill. Up to this distance
data are spatially autocorrelated, beyond this distance samples are spatially
independent .
As the regionalized variable z(x) is difficult to model with a simple deterministic
function, it is seen as an outcome of a random function Z(x). In this context and assuming a
second-order stationarity hypothesis for Z(x), the theoretical variogram describing the spatial
dependence and spatial variation of the random function is introduced:
(h)0.5*EZ(x)Z(xh)²
In practice a theoretical model is fitted on the experimental variogram. Two kinds of
basic variogram functions are used (figure1):
 spherical model:

3

(h) ²1.5 h   0.5* h   for h  r
r
r


²

(2)
for h  r
exponential model:
(h) ²1exp 3h  

 r  
with r the pratical range (distance that corresponds to 95% of the sill)
2
sill(²)=1
range:250m
pratical range (95% sill):250m
1
variogram
0.8
exponential model
spherical model
0.6
0.4
0.2
0
0
100
200
300
400
500
600
distance (m)
figure1: Spheric and exponential covariance
The covariance function described the data spatial correlation and is obtained from the
variogram function:
C(h)C(0)(h)
So it will be assumed that data within the pixel domain and the ESU domain are
spatially correlated and the spatial structure is described by a given covariance model.
b/ Extension Variance definition
We consider a “V domain” that corresponds to the SPOT pixel (20m*20m) and a a”v’
domain” that corresponds to the ESU. The spatial structure within the V and v’ domain is
described by a given covariance function C(h). Zv is the Z(x) spatial average value over V
and Zmean is the arithmetic mean of the 12 point values of v’.
Zv(1/V) z(x)dx
V
12
Zmean 1 Z(xi)
12 i 1
The extension variance V to v’ is:
e²var( ZvZmean)
Zmean is considered as a non biased estimator of Zv, thus e² is the variance estimator error.
Therefore, e² indicates the sampling strategy (v’) capability to represent the Z(x) spatial
variance within the pixel V.
The Z(x) mean spatial mean variance over V is given by:
Var(Zv) 1
V²
 C(x x')dxdx C(V,V)
VV
The Z(x) arithmetic mean variance over v’ is given by:
Var(Zmean) 1
n²
C(xi  xj)C(v',v')
i
j
We also have :
C(V,v') 2  C(x xi)dx
n i xV
Thus the extension variance is given by :
eC(V,V)2*C(V,v')C(v',v')
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To compute the extension variance the integral are approached by the square discretisation
(1m).
2/ Extension Variance computing
The extension variance is computed for a spherical covariance and an exponential covariance
with a sill value of 1 ( to normalize the result) and different ranges.
e² is the variance estimator error when Zv is estimated by Zmean. Since the data are spatially
correlated e² depends on the ESU spatial distribution. The lowest e²is, the best the sampling
strategy (ESU v’ domain) represents the Z(x) spatial variance within the pixel V.
C/ Results
a)
b)
Figure 2: Extension Variance versus range for spherical model (a) and exponential model (b)
These results show :
-
-
difference between strategies: There is no great difference, the A3( equirepartition)
strategy seems to be the best and the A2 strategy the worst.
Beyond 100m scale the extension variance is very low but it is not negligeable. Indeed
if we consider for example the128m range case the extension variance for a spherical
covariance (B configuration case) is 0.0053 that corresponds to an error of
2* ² =0.15 with 95% confidence interval (assuming a normal law).
difference between ranges: The extension variance decreases with range increasing
since data are spatially correlated at larger distance and become more continuous and
4
-
more regular. When the range tends to zero, the error tends to the spatial mean
estimate error by twelve independent samples (1/12=0.0833).
Figure3 presents a zoom at small range. The different strategies show different
behaviors when the range is less than 20 m. The B strategy (crosses pattern) has the
lowest extension variance for small ranges. This is no the case of VALERI sites since
the ranges are higher than 20m .
a)
b)
Figure 3: Zoom at short range:(a): extension variance versus range; (b) extension variance
difference between sampling shemes versus range
Conclusions
These results show general low extension variance, so therefore all these strategies are
relevant sampling strategies for a wide range of spatial structures. The choice between
strategies will be done according to fields constraints.
References
H. Wackernagel : “Multivariate geostatistics, an introduction with applications”, Springer
Verlag, 1995, 257pp
Baret et al: “VALERI: a network of sites and a methodology for the validation of land
satellite products”, 2003, in preparation
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