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Geostatistics Course

Dr. Arifudin Idrus

Department of Geological Engineering

Gadjah Mada UniversityE-mail: arifidrus@ugm.ac.id

KRIGING

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KRIGING

CONSTRAINED OPTIMIZATION PROBLEM

∑i=1

Wj .σ  ij− µ=   σ oi i= 1, …, N

LAGRANGE MULTIPLIER

∑i=1

 N 

Wi= 1 UNBIASED CONDITION

SIMULTANT equation N+1

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KRIGING EQUATION IN MATRIX:

A . x = B

 x = A-1 B

A =

σ 11 σ 12 ….. σ 1 n -

1

σ 21 σ 22 ….. σ 2 n -

1

Σn1 σ n2 ….. σ n n -1

1 1 ….. 1 0

 x =

W1

W2

Wn

µ

B =

σ oi

σ o2

σ on

1

KRIGING VARIANCE

σ k 2=  σ o2    µ−∑i=  1

Wi .σ  oi

Sampel variance (sill) for point kriging

Blok variance for blok kriging

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KRIGING :

Using Statistical distance concept’ 

Location and configuration of data is taken

into account

‘SCREENING EFFECT’ IS:

Sample situated close to block tend to reducethe role of the samples which located far

behind the sample.

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“SCEENING EFFECT”

Sample configuration 1 to 7

Having the same distance to estimated block

IDW : W1 W2 W3 W71

7……

KRIGING :

W4 + W5 + W6 + W71

4

W1 W2 W31

4

W7 ≈ 0

‘SCREENED BY 4

●

●●●

●

●

●7

6

5

4

3

2

1

≈ ≈ ≈ ≈

≈ ≈ ≈

≈

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Example:

100m

 1 0 0 m

10 0 m 

200m

200m

20m

GRID-1Zo*

Xo

3

2

1

Z1= 0.25%

Z3= 0.32%

Z2= 0.56%

100m

1 0 0 m 

 1 0 0 m

1

2

3

173mX

o

GRID-2

FROM VARIOGRAM:

•   γ(h) = 0.01 h untuk h ≤ 400m

•   γ(h) = 4.0 untuk h > 400m

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CALCULATE:

- WEIGHTS FOR EACH SAMPLES and Langrange multiplier

-KRIGING VARIANCE

- ESTIMATED GRADE AT POINT Xo

4.0

h400

sillγ(h)

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SOLUTION:

STEP-1 : CALCULATE COVARIANCE: SAMPLES TO SAMPLES

σ (h) = SILL – γ(h)

For Grid-1

σ 11 = σ 22 = σ 33 = SILL = 4.0

σ 12 = σ 21 = σ 13 = σ 31 = σ  (h) = σ 

(200m) = 4.0 -0.01 (200)= 2.0

σ 23 = σ 32 = σ  (20m) = 4.0 – 0.01 (20)

= 3.8

STEP-2 : CALCULATE COVARIANCE: POINT (Xo) TO

SAMPLES

σ 01 = σ 02 = σ 03 = σ  (100m)

= 4.0 -0.01 (100) = 3.0

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STEP-3 : COMPLETE MATRIX ELEMENTS OF

KRIGING AND CALCULATE WEIGTHS

1 2 2 -1

2 4 3.8 -1

2 3.8 4 -1

1 1 1 0

W1

W2

W3

µ

●=

3

3

3

1

A BX● =

Grid-1

W1 = 0.487

µ = -0.026

W2 = W3 = 0.256

STEP-4 : CALCULATE KRIGING VARIANCE

σ k 2=  σ o2    µ−∑i= 1

3

Wiσ  oi

= 4 - 0.026 – { 0.487(3) + 0.256 (3) + 0.256 (3)}

= 0.974

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FOR Grid-2

1 2 .27 2.27 -1

2.27 4 2.27 -1

2.27 2.27 4 -1

1 1 1 0

W1

W2

W3

µ

● =

3

3

3

1

WEIGHTS : W1 = W2 = W3 = 0.333

µ = - 0.153

σ k2= 0.847

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