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Basics in Geostatistics 1Geostatistical structure analysis:
The variogram
Hans Wackernagel
MINES ParisTech
NERSC • April 2013
http://hans.wackernagel.free.fr
Basic concepts
Geostatistics
Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 1 NERSC • April 2013 2 / 32
Geostatistics
Geostatistics is an application ofthe theory of Regionalized Variablesto the problem of predicting spatialphenomena.
Georges Matheron (1930-2000)
Note: the regionalized variable (reality) is viewed as a realizationof a random function, which is a collection of random variables.
Geostatistics has been applied to:
geology and mining since the ’50ies,natural phenomena since the ’70ies.
It (re-)integrated mainstream statistics in the ’90ies.
Geostatistics
Geostatistics is an application ofthe theory of Regionalized Variablesto the problem of predicting spatialphenomena.
Georges Matheron (1930-2000)
Note: the regionalized variable (reality) is viewed as a realizationof a random function, which is a collection of random variables.
Geostatistics has been applied to:
geology and mining since the ’50ies,natural phenomena since the ’70ies.
It (re-)integrated mainstream statistics in the ’90ies.
Concepts
Variogram: function describing the spatial correlation of aphenomenon.
Kriging: linear regression method for estimating values atany location of a region.
Daniel G. Krige (1919-2013)
Conditional simulation: simulation of an ensemble ofrealizations of a random function,conditional upon data — for non-linear estimation.
Concepts
Variogram: function describing the spatial correlation of aphenomenon.
Kriging: linear regression method for estimating values atany location of a region.
Daniel G. Krige (1919-2013)
Conditional simulation: simulation of an ensemble ofrealizations of a random function,conditional upon data — for non-linear estimation.
Concepts
Variogram: function describing the spatial correlation of aphenomenon.
Kriging: linear regression method for estimating values atany location of a region.
Daniel G. Krige (1919-2013)
Conditional simulation: simulation of an ensemble ofrealizations of a random function,conditional upon data — for non-linear estimation.
Stationarity
For the top series:
stationary mean and variance make sense
For the bottom series:
mean and variance are not stationary,
actually the realization of a non-stationary processwithout drift.
Both types of series can be characterized with a variogram.
Structure analysis
Variogram
Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 1 NERSC • April 2013 6 / 32
The Variogram
The vector x =
(x1x2
): coordinates of a point in 2D.
Let h be the vector separating two points:
l
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h
D
x
x
β
α
We compare sample values z at a pair of points with:(z(x+ h)− z(x)
)2
2
The Variogram Cloud
Variogram values are plotted against distance in space:
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(z(x+h) − z(x))
2
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The Experimental Variogram
Averages within distance (and angle) classes hk arecomputed:
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h hh h h h h hh1 2 3 4 5 6 7 8 9
γk(h )
The Theoretical Variogram
A theoretical model is fitted:
γ
h
(h)
The theoretical Variogram
Variogram: average of squared increments for a spacing h,
γ(h) =12E[ (
Z(x+h)− Z(x))2 ]
Properties- zero at the origin γ(0) = 0- positive values γ(h) ≥ 0- even function γ(h) = γ(−h)
The variogram shape near the origin is linked to thesmoothness of the phenomenon:
Regionalized variable Behavior of γ(h) at origin
smooth ←→ continuous and differentiablerough ←→ not differentiable
speckled ←→ discontinuous
Structure analysis
The empirical variogram
Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 1 NERSC • April 2013 12 / 32
Empirical variogram
Variogram: average of squared increments for a class hk,
γ?(hk) =1
2N(hk)
∑xα−xβ∈hk
(Z(xα)− Z(xβ))2
where N(hk) is the number of lags h = xα−xβ withinthe distance (and angle) class hk.
Example 1D
Transect :
γ?(1) =1
2× 9(02 + 22 + 02 + 12 + 32 + 42 + 22 + 42 + 02) = 2.78
γ?(2) =1
2× 8(22 + 22 + 12 + 22 + 12 + 62 + 62 + 42) = 6.38
Example 1D
Transect :
γ?(1) =1
2× 9(02 + 22 + 02 + 12 + 32 + 42 + 22 + 42 + 02) = 2.78
γ?(2) =1
2× 8(22 + 22 + 12 + 22 + 12 + 62 + 62 + 42) = 6.38
Example 1D
Transect :
γ?(1) =1
2× 9(02 + 22 + 02 + 12 + 32 + 42 + 22 + 42 + 02) = 2.78
γ?(2) =1
2× 8(22 + 22 + 12 + 22 + 12 + 62 + 62 + 42) = 6.38
Example 2D
The directional variograms overlay: the variogram is isotropic.
Variogram: anisotropyComputing the variogram for two pairs of directions.
The anisotropy becomes apparent when computing the pair ofdirections 45 and 135 degrees.
Variogram map: SSTSkagerrak, 30 June 2005, 2am
The variogram exhibits a more complex anisotropy:
different shapes according to direction.
.
Structure analysis
Variogram model
Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 1 NERSC • April 2013 18 / 32
Variogram calculation and fitting1) Sample map Variogram Cloud
(small datasets)
2) Experimental variogram 3) Theoretical variogram
Nugget-effect variogramThe nugget-effect is equivalent to white noise
0 2 4 6 8 100.
00.
20.
40.
60.
81.
0
DISTANCE
VA
RIO
GR
AM
●
No spatial structure Discontinuity at the origin
Three bounded variogram modelsThe smoothness of the (simulated) surfaces is linked to
the shape at the origin of γ(h)
Rough Smooth Rough
Spherical model Cubic model Exponential model
0 2 4 6 8 10
0.0
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DISTANCE
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0 2 4 6 8 10
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0 2 4 6 8 10
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DISTANCE
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Linear at origin Parabolic Linear
Power model familyUnbounded variogram variogram models
γ(h) = |h|p, 0 < p ≤ 2
−10 −5 0 5 10
01
23
4
DISTANCE
VA
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p=1.5
p=1
p=0.5
Observe the different behavior at the origin!
Nested variogram
Nested variogramand corresponding random function model
Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 1 NERSC • April 2013 23 / 32
Nested Variogram ModelVariogram functions can be added to form a nested variogram
Example
A nugget-effect and two spherical structures:
γ(h) = b0 nug(h) + b1 sph(h,a1) + b2 sph(h,a2)
where:• b0, b1, b2 represent the variances at different scales,• a1, a2 are the parameters for short and long range.
0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.0.0
0.5
1.0
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(h)γ
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Nested scales
We can define a random function model that goes with thenested variogram:
Z(x) = Y0(x)︸ ︷︷ ︸micro-scale
+ Y1(x)︸ ︷︷ ︸small scale
+ Y2(x)︸ ︷︷ ︸large scale
This statistical model can be used to extract a specificcomponent Y(x) from the data.
Filtering
Case study: human fertility in France
Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 1 NERSC • April 2013 26 / 32
Fertility in FranceMean annual number of births per 1000 women over the ’90ies
Nb of women per "commune"
Mean a
nnual f
ert
ility
’90
0
50
100
150
100 500 5000 10000 25000 50000 5e+05
FERT500 class
FERT500: index for communes with 100 to 500 women.
Scales identified on the variogramThree functions are fitted: nugget-effect, short- and long-range sphericals
Directionalvariogramsshow isotropy.
D1M1
0. 100. 200. 300. 400.
Distance (km)
0.
10.
20.
30.
40.
50.
60.
70.
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90.
100.
110.
Variogram : FERT500
long rangeshort range
The variogram characterizes three scales:
micro-, small- and large-scale variation.
Filtering large-scale componentMicro- and small-scale components are removed
Fertility tends to be particularly high in the eastern Bretagneand above average in the Auvergne.
Conclusion
Summary
We have seen that:
the variogram model characterizes the variability atdifferent scales,
a random function model with several components can beassociated to the structures identified on the variogram,
these components can be extracted by kriging andmapped.
We will see next how to formulate different kriging algorithms.
Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 1 NERSC • April 2013 30 / 32
References
JP Chilès and P Delfiner.Geostatistics: Modeling Spatial Uncertainty.Wiley, New York, 2nd edition, 2012.
P. Diggle, M. Fuentes, A.E. Gelfand, and P. Guttorp, editors.Handbook of Spatial Statistics.Chapman Hall, 2010.
C Lantuéjoul.Geostatistical Simulation: Models and Algorithms.Springer-Verlag, Berlin, 2002.
H Wackernagel.Multivariate Geostatistics: an Introduction withApplications.Springer-Verlag, Berlin, 3rd edition, 2003.
Software
Public domain
The free (though not open source) geostatistical softwarepackage RgeoS is available for use in R at:http://rgeos.free.fr
R is free and available at http://www.r-project.org/
R can be used in a matlab-like graphical environement byinstalling additionnally: http://www.rstudio.com/ide/
Commercial
The window and menu driven software Isatis is availablefrom: http://www.geovariances.com