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• Basics in Geostatistics 2Geostatistical interpolation/estimation:

Kriging methods

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 2 NERSC April 2013 2 / 35

• Concepts

Geostatistical model

The experimental variogram is used to analyzethe spatial structure of the data from aregionalized variable z(x).It is fitted with a nested variogram model, thusproviding the structure function of a randomfunction.The regionalized variable (reality) is viewed asone realization of the random function Z(x),which is a collection of random variables.

Kriging: a linear regression method for estimatingpoint values (or spatial averages) at any locationof a region.

Conditional simulation: simulation of an ensemble ofrealizations of a random function,conditional upon data for non-linear estimation.

• Concepts

Geostatistical model

The experimental variogram is used to analyzethe spatial structure of the data from aregionalized variable z(x).It is fitted with a nested variogram model, thusproviding the structure function of a randomfunction.The regionalized variable (reality) is viewed asone realization of the random function Z(x),which is a collection of random variables.

Kriging: a linear regression method for estimatingpoint values (or spatial averages) at any locationof a region.

Conditional simulation: simulation of an ensemble ofrealizations of a random function,conditional upon data for non-linear estimation.

• Concepts

Geostatistical model

The experimental variogram is used to analyzethe spatial structure of the data from aregionalized variable z(x).It is fitted with a nested variogram model, thusproviding the structure function of a randomfunction.The regionalized variable (reality) is viewed asone realization of the random function Z(x),which is a collection of random variables.

Kriging: a linear regression method for estimatingpoint values (or spatial averages) at any locationof a region.

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 m and covariance function C(h)

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.

• Kriging of the mean

Kriging of the mean of a random function

Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 2 NERSC April 2013 5 / 35

• Spatially Correlated Data

Sample locations x in a spatial domain:

With spatial correlation we need to consider that:

each sample point plays a different role in estimating themean of the spatial domain,

distances to neighboring points play a role.

How should samples thus be weighted in an optimal way?

• Estimation of the Mean Value

Using the arithmetic mean:

M? =1n

n=1

Z(x)

all samples get the same weight:1n

We rather need an estimator:

M? =n

=1

w Z(x)

with weights w reflecting the spatial correlation.

• Stationary random function

We assume translation-invariance of the mean:

x D : E[Z(x)

]= m

and of the covariance:

x,x D with xx = h : cov(Z(x),Z(x)) = C(xx) = C(h)

• Unbiased estimator

The estimation error in our statistical model:

M? estimated value

m true value

should be zero on average:

E[M? m

]= 0

No bias: the estimator M? does not on average yield avalue that is different from m.

• No biasBias is avoided using weights of unit sum:

n=1

w = 1

Consider:

E[M? m

]= E

[ n=1

w Z(x)m]

=n

=1

w E[Z(x)

]

m

m

= mn

=1

w 1

m = 0

• Variance of the estimation error

The variance of the estimation error is:

2E = var(M? m) = E

[(M? m)2

] E

[M? m

]2

0

= E[M?2 2mM? +m2

]=

n=1

n=1

ww E[Z(x)Z(x)

]

2mn

=1

w E[Z(x)

]

m

+m2

2E =n

=1

n=1

ww C(x x)

• Minimal estimation variance

Aim: weights w that produce a minimal estimation variance:

minimize 2E subject ton

=1

w = 1

The objective function has n+1 parameters:

(w1, . . . ,wn, ) = var(M? m)

criterion

2( n=1

w 1)

condition

where is a Lagrange multiplier.

Setting partial derivatives to zero, we obtain n+1 equations:

: (w1, . . . ,wn, )w

= 0,(w1, . . . ,wn, )

= 0

• Kriging equations

The method of Lagrange yields the equation system forthe optimal weights wKM of the estimation of the mean:

n=1

wKM C(x x) KM = 0 for = 1, . . . ,n

n=1

wKM = 1

The variance at the minimum:

2KM = KM

is equal to the Lagrange multiplier.

• Special case: no spatial correlation

When the covariance model is only nugget-effect:

C(x x) ={2 if x = x0 if x 6= x

the kriging of the mean system simplifies to:wKM

2 = KM for = 1, . . . ,nn

=1

wKM = 1

The solution weights are all equal: wKM =1n

M? = 1n

n=1

Z(x) with variance KM = 2KM =1n

2

• Special case: no spatial correlation

When the covariance model is only nugget-effect:

C(x x) ={2 if x = x0 if x 6= x

the kriging of the mean system simplifies to:wKM

2 = KM for = 1, . . . ,nn

=1

wKM = 1

The solution weights are all equal: wKM =1n

M? = 1n

n=1

Z(x) with variance KM = 2KM =1n

2

• Ordinary Kriging

Ordinary Kriging

Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 2 NERSC April 2013 15 / 35

• Estimation at an unsampled location

Sample locations x (blue dots) in a spatial domain D:

x0

Aim: estimate Z? at an unsampled location x0.

• Ordinary kriging

The estimate Z? is a weighted average of data values Z(x):

Z?(x0) =n

=1

w Z(x) withn

=1

w = 1

The weights wOK of the Best Linear Unbiased Estimator (BLUE)are solution of the system:

n=1

wOK (xx) + OK = (xx0) for = 1, . . . ,n

n=1

wOK = 1

Minimal variance: 2OK = OK +n

=1

wOK (xx0)

• Ordinary kriging

The estimate Z? is a weighted average of data values Z(x):

Z?(x0) =n

=1

w Z(x) withn

=1

w = 1

The weights wOK of the Best Linear Unbiased Estimator (BLUE)are solution of the system:

n=1

wOK (xx) + OK = (xx0) for = 1, . . . ,n

n=1

wOK = 1

Minimal variance: 2OK = OK +n

=1

wOK (xx0)

• Kriging weights

The behavior of kriging weights

Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 2 NERSC April 2013 18 / 35

• Geometric anisotropySpherical variogram, weights sum up to 100%

Isotropic model:

L

25%

25%

25%

25%

Anisotropic (ranges horizontal 1.5L, vertical .75L):

L

17.6%

32.4%

17.6%

32.4%

• Geometric anisotropySpherical variogram, weights sum up to 100%

Isotropic model:

L

25%

25%

25%

25%

Anisotropic (ranges horizontal 1.5L, vertical .75L):

L

17.6%

32.4%

17.6%

32.4%

• The screen effectSpherical variogram with range 2L

Left sample is at 1L and right at 2L from target 2OK = 1.14

A B

34.4%65.6%

Introducing an extra sample 2OK = 0.87

A

49.1%

C

48.2%

B

2.7%

Adding the sample C screens off the sample B.

• The screen effectSpherical variogram with range 2L

Left sample is at 1L and right at 2L from target 2OK = 1.14

A B

34.4%65.6%

Introducing an extra sample 2OK = 0.87

A

49.1%

C

48.2%

B

2.7%

Adding the sample C screens off the sample B.

• Case-study

Filtering noisy images by cokriging

Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 2 NERSC April 2013 21 / 35

• Application: metallography

Trace elements are usually masked by instrumental noise.

Data: 512512 image of phosphorus (P) trace elements.Images for chrome (Cr) and (Ni) are less noisy.

Geostatistical filtering is used to remove the noise.

• Structural analysis

Image of phosphorus Nested variogram(h) = 384 nug(h) + 75 exp(h) + 13 |h|

• Filtering the nugget-effect

Raw image of phosphorus Filtered image

• Multivariate data

P Cr Ni

• Multivariate structural analysisDirect and cross variograms

Matrix variogram model: G(h) = B0 nug(h) + B1 exp(h) + B1 |h|

• Filtering the nugget-effectPhosphorus

Filtered by kriging Filtered by cokriging

• Case study

Space-time filteringSguret & Huchon, JGR 1990

Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 2 NERSC April 2013 28 / 35

• Earth magnetism

Magnetic anomalies are essential to study earth history.

Magnetism is influenced by several external factors like:

solar wind explaining daily fluctuations(period: 24 hours)rotation of the moon around the earth(period: 28 days)solar pe