The variogram

Basics in Geostatistics 1

Geostatistical structure analysis:

The variogram Hans Wackernagel MINES ParisTech

NERSC • April 2013

http://hans.wackernagel.free.fr

Basic concepts

Geostatistics

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Basics in Geostatistics 1

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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 the ’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 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 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)

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: 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

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The Variogram   x1 The vector x = : coordinates of a point in 2D. x2 Let h be the vector separating two points: 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 ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●● ● ●● ● ● ●● ● ●●

h

The Experimental Variogram Averages within distance (and angle) classes hk are computed: γ (hk) ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ●● ● ● ● ●● ● ● ●● ● ● ● ● ● ●●● ● ● ●● ● ●● ● ● ●● ● ●●

h1 h2 h3 h4 h5 h6 h7 h8 h9

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) =

2 i 1 h E Z(x+h) − Z(x) 2

Properties - zero at the origin - positive values - even function

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

Hans Wackernagel (MINES ParisTech)

Basics in Geostatistics 1

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Empirical variogram

Variogram: average of squared increments for a class hk , X 1 γ ? (hk ) = (Z(xα ) − Z(xβ ))2 2 N(hk ) 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 :

γ ? (1) = γ ? (2) =

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 :

γ ? (1) = γ ? (2) =

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

The directional variograms overlay: the variogram is isotropic.

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

The variogram exhibits a more complex anisotropy: different shapes according to direction. .

Structure analysis

Variogram model

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Variogram calculation and fitting 1) Sample map

2) Experimental variogram

Variogram Cloud (small datasets)

3) Theoretical variogram

Nugget-effect variogram

0.6 0.4 0.0

0.2

VARIOGRAM

0.8

1.0

The nugget-effect is equivalent to white noise

●

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

0

2

4

6

8

10

DISTANCE

Linear at origin

0.8 0.6 0.4

VARIOGRAM

0.0

0.2

1.0 0.8 0.6 0.4

VARIOGRAM

0.0

0.2

0.8 0.6 0.4 0.2 0.0

VARIOGRAM

Exponential model 1.0

Cubic model

1.0

Spherical model

Rough

0

2

4

6

DISTANCE

Parabolic

8

10

0

2

4

6

DISTANCE

Linear

8

10

Power model family Unbounded variogram variogram models

γ(h) = |h|p , 4

0