spatial interpolation, kriging

7/28/2019 Spatial Interpolation, Kriging

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Spatial Interpolation,

Kriging

Thomas K. Windholz


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Outline

Spatial Interpolation Basics

Simple, Ordinary, and Universal Kriging

Flavors of Kriging


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Spatial interpolation allows us predictsvalues at unsampled locations.

In general, a model fitting samples canbe split into first and second ordercomponents.

First order can be captured by, forexample, a trend surface throughregression.

Second order looks at residuals and their

Covariance (e.g., Krigingnamed after


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Outline



Flavors of Kriging


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Simple, Ordinary, and

Universal Kriging

What is the difference?

Derivation of simple kriging

Equations for prediction & kriging

variance

Analysis steps

Augmentation to incorporate trend


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Differences Among Kriging

Methods Remember equation for a spatial random

variable:

Simple Kriging: No trend (only )

Ordinary Kriging: Constant trend

Universal Kriging: Polynomial trend

)()()( sUsxsYT

)(sU


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Derivation of Simple Kriging

Basic Idea:

n

iii sUssU 1 )()()(

s1

ss2

s3

s4

1

3 4

2


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Basic Idea:

How should differ from ?)(

sU

s1

ss2

s3

s4

)(sU

1

3 4

2


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One descriptor is expected mean square

error:

)(

)(2)()(

)()( 22

2

sUsUEsUEsUEsUsUE

n

i

ii

n

i

n

j

jiji ssCsssCss1

2

1 1

),()(2),()()(

)()(2)()( 2 scssCs TT


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One descriptor is expected mean square

error:

)(

)(2)()(

)()( 22

2

sUsUEsUEsUEsUsUE

)()(2)()( 2 scssCs TT

minimize

n

i

ii

n

i

n

j

jiji ssCsssCss1

2

1 1

),()(2),()()(


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Minimization (through differentiation) results

in:

which we can use back in our starting

equation of:

)()( 1 scCs

n

i

ii sUssU1

)()()(


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Note on the side:

),( ji ssC

),( issc

Covariance matrix among all sample sites

Covariance vector between

prediction locations and all sample sitessi


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Kriging variance

The kriging variance results in

(substitute (s) in the expected mean

square error):

)()()()( 1222 scCscsUsUE Te


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Analysis Steps

Remove trend if it exists

Calculate empirical variogram on

residuals

Fit theoretical variogram

Calculate C and c (actually -1 and )

Predict value & add trend

Estimate error


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Augmentation to incorporate

trend

Ordinary kriging incorporates a constant

trend Universal kriging incorporates a trend of

order x


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Ordinary Kriging

General:

Simple Kriging:

Ordinary & Universal Kriging:

)()()( sUsxsY T

n

i

ii sUssU1

)()()(

n

i

ii sYssY1

)()()(


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Ordinary Kriging

Can handle constant trend (mean)

through an augmented matrix C+ and

augmented vectors +(s) and c+(s).

C )(s =

=

)(sc

011

1),(),(

1),(),(

1

111

nnn

n

ssCssC

ssCssC

)(

)(

)(1

s

s

s

n

1

),(

),( 1

nssC

ssC


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Ordinary Kriging

Can handle constant trend (mean)

through an augmented matrix C+ and

augmented vectors +(s) and c+(s).

C )(s =

=

)(sc

011

1),(),(

1),(),(

1

111

nnn

n

ssCssC

ssCssC

)(

)(

)(1

s

s

s

n

1

),(

),( 1

nssC

ssC

Simple kriging


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Ordinary Kriging

Constant trend will be simultaneouslypredicted.

Can estimate variogram from yvalueswithout removing trend (since it isconstant).

Usually works within a neighborhood andnot with entire dataset.

Since it works in a neighborhood trend onlyhas to be constant in the neighborhood (be

cautious with this statement)


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Universal Kriging

Can handle polynomial trend:

C )(s = )(sc

00)()(

00)()(

)()(),(),(

)()(),(),(

1

111

11

111111

npp

n

npnnnn

pn

sxsx

sxsx

sxsxssCssC

sxsxssCssC

)(

)(

)(

)(

1

1

s

s

s

s

p

n

=

)(

)(

),(

),(

1

1

sx

sx

ssC

ssC

p

n


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Universal Kriging

Polynomial trend will be simultaneouslypredicted.

Cannot estimate variogram from yvalues without removing trend first!!!

Thus, trend has to be removed firstanyway to estimate variogram!

Neighborhood not such an issue as withordinary kriging.


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Outline



Flavors of Kriging


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Flavors of Kriging

Block Kriging

Co-Kriging

Others: Robust, Disjunctive, and

Indicator Kriging


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Block Kriging

Uses a blockA rather than locations s.

For example:

),( issCA

dsssC

sAC Ai

i

),(

),(

),( ssC2

),(),(

A

sdsdssCAAC A


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Co-Kriging

Basic idea is to use a second, highly

correlated, variable in locations where

primary variable is (or cannot) bemeasured.

Primary and secondary variable

Samples with:


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Co-Kriging

Basic idea is to use a second, highly

correlated, variable in locations where

primary variable is (or cannot) bemeasured.

Primary and secondary variable

Secondary variable only

Samples with:


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Co-Kriging

Essential component of co-kriging is the

cross-covariogram or cross-variogram:

XYYX sXhsYEhC )()()(

)()()()()(2 sXsXsYhsYEhYX


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Co-Kriging

and the empirical cross-variogram

can be estimated through:

hss

jijiYX

ji

xxyyhn

h ))(()(

1)(2


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Co-Kriging

Thus, we can model the variable Y(s)

through:

With the solution foragain as an

augmented system +.

mn

j

jxi

n

i

iyi sYssYssY11

)()()()()(


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Spatial Interpolation,

Kriging

Thomas K. Windholz

spatial interpolation, kriging

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