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Applications in GIS(Kriging Interpolation)

Dr. S.M. MalaekAssistant: M. Younesi

Interpolating a Surface From

Sampled Point Data

Interpolating a Surface From Sampled Point Data

Assumes a continuous surface that is sampled

Interpolation

Estimating the attribute values of locations that are within the range of available data using known data values

Extrapolation

Estimating the attribute values of locations outside the range of available data using known data values


Estimating a point here: interpolation

Sample data

Interpolation


Estimating a point here: extrapolation

Sample data

Extrapolation

Sampling Strategies for Interpolation

Regular Sampling Random Sampling



Elevation profile

Sample elevation data

A

B

If

A = 8 feet and

B = 4 feet

then

C = (8 + 4) / 2 = 6 feetC

Linear Interpolation


Non-Linear Interpolation

Elevation profile

Sample elevation data

A

B

C

Often results in a more realistic interpolation but estimating missing data values is more complex


Global InterpolationUses allall known sample points to estimate a value at an unsampled location

Sample data


Local InterpolationUses a neighborhood of sample points to estimate a value at an unsampled location

Sample data

Uses a local neighborhood to estimate value, i.e. closest n number of points, or within a given search radius

Trend Surface

Trend Surface

Global method

Inexact

Can be linear or non-linear

predicting a z elevation value [dependent variable] with x and y location values [independent variables]

Trend Surface

1st Order Trend Surface

In one dimension: z varies as a linear function of x

x

zz = b0 + b1x + e

Trend Surface

1st Order Trend Surface

In two dimensions: z varies as a linear function of x and y

z = b0 + b1x + b2y + e

x

yz

Trend Surface

Inverse Distance Weighted (IDW)

Inverse Distance Weighted

Local method

Exact

Can be linear or non-linear

The weight (influence) of a sampled data value is inversely proportional to its distance from the estimated value

Inverse Distance Weighted(Example)

4

3

2

100

160

IDW: Closest 3 neighbors, r = 2

200

1),(1

),(

1

1

i

with

iin

ip

i

n

ip

i

i

zyxzor

d

dz

yxz


4

3

2

A = 100

B = 160

C = 200

1 / (42) = .0625 1 / (32) = .1111 1 / (22) = .2500

Weights

A BC


1 / (42) = .0625 1 / (32) = .1111 1 / (22) = .2500

.0625 * 100 = 6.25 .1111 * 160 = 17.76 .2500 * 200 = 50.00

Weights Weights * Value

A BC

74.01 / .4236 = 175

Total = .4236

6.25 +17.76 + 50.00 = 74.01 4

3

2

A = 100

B = 160

C = 200

Geostatistics

Geostatistics Geostatistics:The original purpose of geostatistics

centered on estimating changes in ore grade within a mine.

The principles have been applied to a variety of areas in geology and other scientific disciplines.

A unique aspect of geostatistics is the use of regionalized variables which are variables that fall between random variables and completely deterministic variables.

Geostatistics Regionalized variables describe phenomena

with geographical distribution (e.g. elevation of ground surface).

The phenomenon exhibit spatial continuity.

Geostatistics It is notalways possible to sample every location. Therefore, unknown values must be estimated

from data taken at specific locations that can be sampled.

The size, shape, orientation, and spatial

arrangement of the sample locations are termed the support and influence the capability to predict the unknown samples.

Semivariance

Semivariance Regionalized variable theory uses a related

property called the semivariancesemivariance to express the degree of relationship between points on a surface.

The semivariance is simply half the

variance of the differences between all possible points spaced a constant distance apart.

Semivariance is a measure of the degree of spatial dependence between samples (elevation(

Semivariance semivariance :The magnitude of the

semivariance between points depends on the distance between the points. A smaller distance yields a smaller semivariance and a larger distance results in a larger semivariance.

Calculating the Semivariance (Regularly Spaced PointsRegularly Spaced Points(

Consider regularly spaced points distance (d) apart, the semivariance can be estimated for distances that are multiple of (d) (Simple form):

hN

ihii

h

zzN

h1

2)(2

1)(

Semivariance

Zi is the measurement of a regionalized variable taken at location i ,

Zi+h is another measurement taken h intervals away d

Nh is number of separating distance = number of points –Lag (if the points are located in a single profile)

hN

ihii

h

zzN

h1

2)(2

1)(

Calculating the Semivariance

(Irregularly Spaced PointsRegularly Spaced Points( Here we are going to explore directional variograms. Directional variograms is defines the spatial variation among points separated by space lag h. The difference from the omnidirectional variograms is that h is a vector rather than a scalar.

For example, if d={d1,d2}, then each pair of compared samples should be separated in E-W direction and in S-N direction.

Calculating the Semivariance

(Irregularly Spaced PointsRegularly Spaced Points( In practice, it is difficult to find enough sample points which are

separated by exactly the same lag vector [d]. The set of all possible lag vectors is usually partitioned into

classes

Variogram

Variogram

The plot of the semivariances as a function of distance from a point is referred to as a semivariogram or variogram.

Variogram The semivariance at a distance d = 0 should be zero, because there

are no differences between points that are compared to themselves. However, as points are compared to increasingly distant points, the

semivariance increases.

Variogram The range is the greatest distance over which the value at a point

on the surface is related to the value at another point. The range defines the maximum neighborhood over which

control points should be selected to estimate a grid node.

Variogram (Models(

It is a ‘model’ semi-variogram and is usually called the spherical model. a is called the range of influence of a sample.

C is called the sill of the semi-variogram.

ahC

ahah

ah

Ch

where

where3

3

21

23

)(

Variogram (Models(Exponential Model

aheCh 1)(γ

spherical and exponential with the

same range and sill spherical and exponential with the same

sill and the same initial slope

Kriging

Interpolation

Kriging Interpolation

Kriging is named after the South African engineer, D. G. Krige, who first developed the method.

Kriging uses the semivariogram, in calculating estimates of the surface at the grid nodes.

Kriging Interpolation The procedures involved in kriging incorporate measures of error and uncertainty

when determining estimations. In the kriging method, every known data value and every missing data value has

an associated variance. If ‘C’ is constant (i.e. known value exactly), its variance is zero.

Based on the semivariogram used, optimal weights are assigned to known values in order to calculate unknown ones. Since the variogram changes with distance, the weights depend on the known sample distribution.

Ordinary Kriging

Ordinary Kriging Ordinary kriging is the simplest form of kriging.

It uses dimensionless points to estimate other dimensionless points, e.g. elevation contour plots.

In Ordinary kriging, the regionalized variable is assumed to be stationary.

Punctual (Ordinary) Kriging In our case Z, at point p, Ze (p) to be calculated

using a weighted average of the known values or control points:

)()(iie

pzwpz This estimated value will most likely differ from the actual

value at point p, Za(p), and this difference is called the

estimation errorestimation error:

)()( pzpzaep

Punctual (Ordinary) Kriging If no drift exists and the weights used in the estimation sum to

one, then the estimated value is said to be unbiased. The scatter of the estimates about the true value is termed the error or estimation variance,

n

pzpzn

iiiaie

z

1

2

2

)]()([σ

Punctual (Ordinary) Kriging kriging tries to choose the optimal weights that produce the minimum

estimation error . Optimal weights, those that produce unbiased estimates and have a minimum

estimation variance, are obtained by solving a set of simultaneous equations .

)(γ)(γ)(γ)(γ

)(γ)(γ)(γ)(γ

)(γ)(γ)(γ)(γ

3333322311

2233222211

1133122111

p

p

p

hhwhwhw

hhwhwhw

hhwhwhw

1321 www

Punctual (Ordinary) Kriging A fourth variable is introduced called the Lagrange

multiplier

1

)()()()(

)()()()(

)()()()(

321

3333322311

2233222211

1133122111

www

hhwhwhw

hhwhwhw

hhwhwhw

p

p

p

1

)(

)(

)(

0111

1)()()(

1)()()(

1)()()(

3

2

1

3

2

1

333231

232121

131211

p

p

p

h

h

h

w

w

w

hhh

hhh

hhh

Punctual (Ordinary) Kriging Once the individual weights are known, an estimation

can be made by

332211)( zwzwzwpze

And an estimation variance can be calculated by

λ)(γ)(γ)(γσ 313212112 pppz hwhwhw

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