Basics in Geostatistics 1
Geostatistical structure analysis:
The variogram Hans Wackernagel MINES ParisTech
NERSC • April 2013
http://hans.wackernagel.free.fr
Basic concepts
Geostatistics
Basic concepts
Geostatistics
Hans Wackernagel Wackernagel (MINES ParisTech) ParisTech)
Basics in Geostatistics 1
NERSC
•
April 2013
Geostatistics
Geostatistics is
an application of the theory of Regionalized Variables to the the prob proble lem m of pred predic icti ting ng sp spat atia iall phenomena. Georges Matheron (1930-2000) Note: the regionalized variable (reality) is viewed as a realization of a random function, which is a collection of random variables.
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Basic concepts
Geostatistics
Hans Wackernagel Wackernagel (MINES ParisTech) ParisTech)
Basics in Geostatistics 1
NERSC
•
April 2013
Geostatistics
Geostatistics is
an application of the theory of Regionalized Variables to the the prob proble lem m of pred predic icti ting ng sp spat atia iall phenomena. Georges Matheron (1930-2000) Note: the regionalized variable (reality) is viewed as a realization of a random function, which is a collection of random variables.
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Geostatistics
Geostatistics is
an application of the theory of Regionalized Variables to the the prob proble lem m of pred predic icti ting ng sp spat atia iall phenomena. Georges Matheron (1930-2000) Note: the regionalized variable (reality) is viewed as a realization of 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 th e ’90ies.
Geostatistics
Geostatistics is
an application of the theory of Regionalized Variables to the problem of predicting spatial phenomena. Georges Matheron (1930-2000) Note: the regionalized variable (reality) is viewed as a realization of a random function, which is a collection of random variables.
Geostatistics
Geostatistics is
an application of the theory of Regionalized Variables to the problem of predicting spatial phenomena. Georges Matheron (1930-2000) Note: the regionalized variable (reality) is viewed as a realization of 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 th e ’90ies.
Concepts Variogram: function describing the spatial correlation of a phenomenon.
Concepts Variogram: function describing the spatial correlation of a phenomenon.
Concepts Variogram: function describing the spatial correlation of a phenomenon. Kriging: linear regression method for estimating values at any location of a region.
Daniel G. Krige (1919-2013)
Concepts Variogram: function describing the spatial correlation of a phenomenon. Kriging: linear regression method for estimating values at any location of a region.
Daniel G. Krige (1919-2013)
Concepts Variogram: function describing the spatial correlation of a phenomenon. Kriging: linear regression method for estimating values at any location of a region.
Daniel G. Krige (1919-2013)
Concepts Variogram: function describing the spatial correlation of a phenomenon. Kriging: linear regression method for estimating values at any location of a region.
Daniel G. Krige (1919-2013)
Conditional simulation: simulation of an ensemble of realizations 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:
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 process without drift. Both types of series can be characterized with a variogram. Structure analysis
Variogram
Structure analysis
Variogram
Hans Wackernagel (MINES ParisTech)
Basics in Geostatistics 1
NERSC
The Variogram
x 1 : coordinates of a point in 2D. x 2 Let h be the vector separating two points:
The vector x =
D
xβ h
xα
•
April 2013
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The Variogram
x 1 : coordinates of a point in 2D. x 2 Let h be the vector separating two points:
The vector x =
D
xβ h
xα
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: (z(x+h) − z(x)) 2
2
The Variogram Cloud Variogram values are plotted against distance in space: (z(x+h) − z(x)) 2
2
h
The Experimental Variogram Averages within distance (and angle) classes hk are computed: γ (h ) k
The Experimental Variogram Averages within distance (and angle) classes hk are computed: γ (h ) k
h1 h2 h3 h4 h5 h6 h7 h8 h9
The Theoretical Variogram
A theoretical model is fitted: γ (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)
Properties - zero at the origin - positive values - even function
=
1 E 2
2
Z (x+h) − Z (x)
γ (0)
= 0 γ (h) ≥ 0 γ (h) = γ (−h)
The variogram shape near the origin is linked to the smoothness of the phenomenon:
The theoretical Variogram Variogram: average of squared increments for a spacing h, γ (h)
=
Properties - zero at the origin - positive values - even function
1 E 2
2
Z (x+h) − Z (x)
γ (0)
= 0 γ (h) ≥ 0 γ (h) = γ (−h)
The variogram shape near the origin is linked to the smoothness of the phenomenon: Regionalized variable smooth rough speckled
Behavior of γ (h) at origin ←→ continuous and differentiable ←→ not differentiable ←→ discontinuous
Structure analysis
The empirical variogram
Structure analysis
The empirical variogram
Hans Wackernagel (MINES ParisTech)
Basics in Geostatistics 1
NERSC
•
April 2013
Empirical variogram
Variogram: average of squared increments for a class hk ,
γ
1 (hk ) = 2 N(hk )
( Z (xα ) − Z (xβ ))2
xα −xβ ∈hk
where N(hk ) is the number of lags h = xα −xβ within the distance (and angle) class hk .
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Empirical variogram
Variogram: average of squared increments for a class hk ,
γ
1 (hk ) = 2 N(hk )
( Z (xα ) − Z (xβ ))2
xα −xβ ∈hk
where N(hk ) is the number of lags h = xα −xβ within the distance (and angle) class hk .
Example 1D Transect :
Example 1D Transect :
Example 1D Transect :
(1) =
(2) =
γ γ
1 (02 + 22 + 02 + 12 + 32 + 42 + 22 + 42 + 02 ) = 2.78 2×9 1 (22 + 22 + 12 + 22 + 12 + 62 + 62 + 42 ) = 6 .38 2×8
Example 1D Transect :
(1) =
(2) =
γ γ
1 (02 + 22 + 02 + 12 + 32 + 42 + 22 + 42 + 02 ) = 2.78 2×9 1 (22 + 22 + 12 + 22 + 12 + 62 + 62 + 42 ) = 6 .38 2×8
Example 1D Transect :
(1) =
(2) =
γ γ
1 (02 + 22 + 02 + 12 + 32 + 42 + 22 + 42 + 02 ) = 2.78 2×9 1 (22 + 22 + 12 + 22 + 12 + 62 + 62 + 42 ) = 6 .38 2×8
Example 1D Transect :
(1) =
(2) =
γ γ
1 (02 + 22 + 02 + 12 + 32 + 42 + 22 + 42 + 02 ) = 2.78 2×9 1 (22 + 22 + 12 + 22 + 12 + 62 + 62 + 42 ) = 6 .38 2×8
Example 2D
Example 2D
The directional variograms overlay: the variogram is isotropic.
Variogram: anisotropy Computing the variogram for two pairs of directions.
Variogram: anisotropy Computing the variogram for two pairs of directions.
The anisotropy becomes apparent when computing the pair of directions 45 and 135 degrees.
Variogram map: SST Skagerrak, 30 June 2005, 2am
Variogram map: SST Skagerrak, 30 June 2005, 2am
The variogram exhibits a more complex anisotropy: different shapes according to direction. . Structure analysis
Variogram model
Structure analysis
Variogram model
Hans Wackernagel (MINES ParisTech)
Basics in Geostatistics 1
NERSC
•
April 2013
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Variogram calculation and fitting 1) Sample map
2) Experimental variogram
Variogram Cloud (small datasets)
3) Theoretical variogram
Variogram calculation and fitting 1) Sample map
Variogram Cloud (small datasets)
2) Experimental variogram
3) Theoretical variogram
Nugget-effect variogram The nugget-effect is equivalent to white noise
0 . 1 8 . 0 M A R G O I R A V
6 . 0 4 . 0 2 . 0 0 . 0
Nugget-effect variogram The nugget-effect is equivalent to white noise
0 . 1 8 . 0 M A R G O I R A V
6 . 0 4 . 0 2 . 0 0 . 0
0
2
4
6
8
10
DISTANCE
No spatial structure
Discontinuity at the origin
Three bounded variogram models The smoothness of the (simulated) surfaces is linked to the shape at the origin of γ (h)
Rough
Smooth
Spherical model 0 .
Cubic model 0 .
Rough
Exponential model 0 .
Three bounded variogram models The smoothness of the (simulated) surfaces is linked to the shape at the origin of γ (h)
Rough
Smooth
Spherical model
M A R G O I R A V
Cubic model
0 . 1
0 . 1
8 . 0
8 . 0
M A R G O I R A V
6 . 0 4 . 0 2 . 0
Rough
0 . 0
Exponential model 0 . 1
M A R G O I R A V
6 . 0 4 . 0 2 . 0 0 . 0
0
2
4
6
8
10
8 . 0 6 . 0 4 . 0 2 . 0 0 . 0
0
2
DISTANCE
4
6
8
10
0
DISTANCE
Linear at origin
Unbounded variogram variogram models
0
4
p=1.5 3
p=1 M A R G O I R A V
2
1
6
Linear
Power model family
= |h| p ,
4
DISTANCE
Parabolic
γ (h)
2
p=0.5
8
10
Power model family Unbounded variogram variogram models
γ (h)
= |h| p ,
0
4
p=1.5 3
p=1 M A R G O I R A V
2
p=0.5
1
0
−10
−5
0
5
10
DISTANCE
Observe the different behavior at the origin! Nested variogram
Nested variogram and corresponding random function model
Nested variogram
Nested variogram and corresponding random function model
Hans Wackernagel (MINES ParisTech)
Basics in Geostatistics 1
NERSC
•
April 2013
Nested Variogram Model Variogram 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. γ (h)
1.0
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Nested Variogram Model Variogram 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. γ (h)
1.0
0.5
t e g g u n 0.0 0.
1.
2.
e g n a r
e g n a r
t r o h s
g n o l
3.
4.
5.
6.
7.
8.
9.
10.
11.
h
Nested scales
We can define a random function model that goes with the nested variogram: Z (x) =
Y 0 (x)
+ Y 1 (x) + Y 2 (x)
micro-scale
small scale
large scale
This statistical model can be used to extract a specific component Y (x) from the data.
Nested scales
We can define a random function model that goes with the nested variogram: Z (x) =
Y 0 (x)
+ Y 1 (x) + Y 2 (x)
micro-scale
small scale
large scale
This statistical model can be used to extract a specific component Y (x) from the data.
Filtering
Case study: human fertility in France
Filtering
Case study: human fertility in France
Hans Wackernagel (MINES ParisTech)
Basics in Geostatistics 1
NERSC
•
April 2013
Fertility in France Mean annual number of births per 1000 women over the ’90ies 150
100 0 9 ’ y t i l i t r e f l a u n n a n a e M 50
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Fertility in France Mean annual number of births per 1000 women over the ’90ies 150
100 0 9 ’ y t i l i t r e f l a u n n a n a e M 50
0 100
500
5000 10000 25000 Nb of women per "commune"
50000
5e+05
FERT500 class
FERT500: index for communes with 100 to 500 women.
Scales identified on the variogram Three functions are fitted: nugget-effect, short- and long-range sphericals 110. 100.
Directional variograms
0 0 5 T R E F
90.
:
60.
m a r g o i r a V
50.
80. 70.
40. 30. 20. 10.
M1 D1
Scales identified on the variogram Three functions are fitted: nugget-effect, short- and long-range sphericals 110. 100.
Directional variograms show isotropy.
0 0 5 T R E F
90.
:
60.
m a r g o i r a V
50.
M1 D1
80. 70.
40. 30. 20. 10. 0.
0.
100.
200.
300.
Distance (km)
short range
long range
The variogram characterizes three scales: micro-, small- and large-scale variation.
Filtering large-scale component Micro- and small-scale components are removed
400.
Filtering large-scale component Micro- and small-scale components are removed
Fertility tends to be particularly high in the eastern Bretagne and above average in the Auvergne.
Conclusion
Summary
We have seen that: the variogram model characterizes the variability at different scales, a random function model with several components can be associated to the structures identified on the variogram, these components can be extracted by kriging and mapped. We will see next how to formulate different kriging algorithms.
Conclusion
Summary
We have seen that: the variogram model characterizes the variability at different scales, a random function model with several components can be associated to the structures identified on the variogram, these components can be extracted by kriging and mapped. 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.
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 with Applications. Springer-Verlag, Berlin, 3rd edition, 2003.
Software
Public domain
The free (though not open source) geostatistical software package 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 by installing additionnally: http://www.rstudio.com/ide/ Commercial The window and driven software Isatis is available