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Basics in Geostatistics 1Geostatistical structure analysis:

The variogram

Hans Wackernagel

MINES ParisTech

NERSC • April 2013

http://hans.wackernagel.free.fr

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.

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


The Variogram

The vector x =

(x1x2

): coordinates of a point in 2D.

Let h be the vector separating two points:

l

l

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:

ll

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lll l

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l l l

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l ll

l lll

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l l

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l l

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l

l

h

(z(x+h) − z(x))

2

2

The Experimental Variogram

Averages within distance (and angle) classes hk arecomputed:

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lll l

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l l l

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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


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 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


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

0.2

0.4

0.6

0.8

1.0

DISTANCE

VA

RIO

GR

AM

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

DISTANCE

VA

RIO

GR

AM

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

DISTANCE

VA

RIO

GR

AM

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

RIO

GR

AM

p=1.5

p=1

p=0.5

Observe the different behavior at the origin!

Nested variogram

Nested variogramand corresponding random function model


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

nu

gg

et

lon

g

ra

ng

e

sh

ort

ra

ng

e

(h)γ

h

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


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.

80.

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.


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

http://rgeos.free.fr

http://www.r-project.org/

http://www.rstudio.com/ide/

http://www.geovariances.com

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