# interpolation kriging

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Kriging

Using Geostatistical Analyst, ESRI

Chang, Kang-tsung, 2006, Introduction toGeographic Information Systems,

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So Far

We dealt with deterministic models that

can not provide estimates foraccuracy/certainty in predictions.

Kriging is a stochastic model that provides

estimates for accuracy/certainty in

predictions.

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Kriging

Spatial variation consists of 3 components

Random spatially correlated component

A drift or structure/trend

Random error term

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How to measure the spatially

correlated component?

Uses Semivariogram to measure spatially

correlated component, also called spatial

autocorrelation

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Creating a Semivariogram?

Where:

= semivariance between point Xi and Xj

h = distance separating Xi and Xj

Z = attribute value (height, ore quality, etc)

First what is semi-variance between two point separated by distance h?

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Semivariance between one point

and all other points

If I calculate all possible

semivariance for point A

(red dot), I get 11 points

A

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Repeat for Semivariance

calculations for every possible pair

Much more semivariance

computations if we repeat

for all possible pairs

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If all pairs are considered, we get a

Semivariogram cloud

Semivariogram allows us to investigate spatial dependence

If there is spatial dependence points that are closer together will have smallsemi-variance and vice versa

Because all pairs are plotted on the semivariogram, the semivariogram

becomes difficult to manage/interpret

Distance

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Binning is the solution

A process that is used to average semivariance data by distance and

direction

First group pairs of sample points into lag classes. If lag size is 2000

meters, lag classes are: (

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Continue

Third: For each group of pairs with similar lag and direction (in thesame grid cell), we compute the average semivariance between

sample points separated by lag h

Where: = average semi-variance between sample points separated by lag h

N = is the number of pairs of sample points sorted by direction in the bin

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Semivariogram after binning

On a semivariogram, if spatial dependence exist, semivariance

is expected to increase with distance

Same lag, different directions

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Anisotropy

Anisotropy is a term describing the existence of directional differences in spatialdependence

From semivariogram data, directional dependence can be extracted if it exists

low

Plume

Pollutant concentration high

Wind direction

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Continue

Fourth: To use the semivariogram as an interpolator, the semivariogramdata must be fitted with a mathematical function/model

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Semivariogram

Measures the variability of data with

respect to spatial distribution Looks at variance between pairs of data

points over a range of separation scales Forms the basis of every geostatistical

study

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Semiveriogram conditions:

Stationarity

The entire dataset can be described with one statistical model

Under the condition of stationarity, you can see the distance ofcorrelation in your data

h

(h)

Correlated at any distance

Correlated at a max distance

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Semiveriogram types

Many functions to choose from; Geostatistical analyst provides 11 models

Examples include: Gaussian, Linear, Spherical, Circular, and Exponential

Popular models are the spherical and exponential

Spatial dependencelevels after a certain

distance

Spatial dependenceDecreases exponentially

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Nugget (C0

):

Semivariance

at distance 0: Represents microscale

variations or measurement errors

Range (a) : Distance at which semivariance

levels off: Represents the spatially correlated portion

Sill (C0

+ C1

):

Semivariance

at which leveling takes place

N.B.: Every data set will have unique model features (model parameters)

C0

C1

C0 &C1

(h)a

Semivariogram: Model Parameters

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Fifth: Extract model parameters (C0, C1, and a) for the dataset under

consideration from the semivariogram. This will allow us to calculate amodeled semivariance between any 2 points knowing the distance hseparating the two points.

Extract model parameters

C0

C1

C0 &C1

(h)a

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Continue

Sixth: Use the model parameters to extract semivariance values for anypoint by solving a set of simultaneous equations

Let us consider the following Problem:

Knowing three points: 1, 2, 3, we want to use krigging to estimate point 0

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How?

Where:

Problem: Knowing three points: 1, 2, 3, we want to estimate

semivariance values and solve equations to determine value at

point 0

(hij) is the semivariance between known point I & J

(hi0) is the semivariance between known point I & point to be estimated

Lagrange multiplier to ensure minimum estimation of errorMore like a fudge factor

(hij) is the semivariance between known point I & J

(hi0) is the semivariance between known point I & point to be estimated

Lagrange multiplier to ensure minimum estimation of errorMore like a fudge factor

(hij) is the semivariance between known point I & J

(hi0) is the semivariance between known point I & point to be estimated

Lagrange multiplier to ensure minimum estimation of errorMore like a fudge factor

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Thus

Derive semivariance between known pairs of known

points (1,2,3) and each of the known point and unknown

point knowing model parameters

Derive weights solving by solving the simultaneous

equations

Plug in weights in the equation below to calculate thevalue of the unknown point (Z0)

Z0

= Z1

W1

+ Z2

W2

+ Z3

W3

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What if we had > 3 points

(1) Apply Matrix algebra to solve simultaneous Equations

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= Estimated value

= Known value at point x

= Weight associated with point x

S = number of sample points used in estimation

Where:

(2) Estimator equation becomes

What if we had > 3 points

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How is this different from Inverse

Distance MethodKrigging IDW

Calculation of weightsinvolve

(1) Variance between point to beestimated and Known points

(2) Variance between Known points

(1) Variance between pointto be estimated and

Known points

Measuring reliability A measure for the reliability of the

estimate

None

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Kriging Types in ArcGIS

Ordinary kriging

Simple kriging Universal kriging

Indicator kriging Probability kriging

Disjunctive kriging Cokriging

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From before

Spatial variation consists of 3 components

Random spatially correlated component

A trend component

Random non correlated component(error term)

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Components of Kriging

The value of z depends on: (1) trend component, (2) random autocorrelatedcomponent, and (3) random non-correlated component (for simplicity notrepresented in figures)

Z(s) = (s) + (s)Where:

Z = Value at point s

= Trend component value at point s (first order or second order polynomial)

= Random, autocorrolated component

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Ordinary Kriging

Assumes there is no trend

Assumes m(s) is unknown and constant

Focuses on the spatially correlated component

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Simple Kriging

Assumes (s) , the mean of data set is known and is constant

Assumes there is no trend component

In the majority of cases this is unrealistic assumption

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Indicator Kriging

(s) is constant, and unknown

Values are binary (1 or 0)

Example, a point is forest or non forest

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Universal Kriging

Assumes z values change because of a drift (tre