Geog. 579: GIS and Spatial Analysis - Lecture 21 Overheads 1 Point Estimation: 3. Methods: 3.6...

1Geog. 579: GIS and Spatial Analysis - Lecture 21 OverheadsGeog. 579: GIS and Spatial Analysis - Lecture 21 Overheads

Point Estimation: 3. Methods: 3.6 Ordinary Kriging

Topics:Topics:

Lecture 23: Spatial Interpolation VI

References:References:

- Chapter 12, Isaaks, E. H., and R. M. Srivastava, 1989. Applied Geostatistics, Oxford University Press, New York- Chapter 5 & 6, Burrough, P.A. and R.A. McDonnell, 1998. Principles of Geographical Information Systems, Oxford University Press, New York, pp. 113-117.


OutlinesOutlines

3. Methods:3. Methods: 3.6 3.6 Ordinary Kriging 3.6.1 The Idea: 3.6.1 The Idea: (1) Interpolation should minimize the uncertainty associated with (1) Interpolation should minimize the uncertainty associated with the interpolation. the interpolation. (2) The spatial continuity should be explicitly used to determine (2) The spatial continuity should be explicitly used to determine the weights in such a way that the uncertainty is minimized.the weights in such a way that the uncertainty is minimized. 3.6.2 Implementation:3.6.2 Implementation:

(1) Define an error variance model (the objective function):(1) Define an error variance model (the objective function):

The model should have the weights as variables and should The model should have the weights as variables and should be based on quantitative definition of spatial continuity be based on quantitative definition of spatial continuity expressed as a semi-variogram model.expressed as a semi-variogram model.


is the Lagrange parameter (dummy variable)

The idea is to select a set of weights which produce a minimal value from the model.

~ ~ (~ ~ ) ( ~ ~ ) ( ) R i jj

n

i

n

ij i ii

n

ii

n

w w w w2 2

11

2 20

1 1

2 2 1

Where: ~ 2

1 C Co

(1) Define an error variance model (continued…):(1) Define an error variance model (continued…):

The semi-variogram based error variance model:The semi-variogram based error variance model:


(2) Minimize the objective function (the error variance model)(2) Minimize the objective function (the error variance model)

Setting the partial derivatives of the objective function with Setting the partial derivatives of the objective function with respect to each of the variables to zero will produce a set ofrespect to each of the variables to zero will produce a set of (n+1) simultaneous equations:(n+1) simultaneous equations:

i = 1, …, nw j ij ij

n~ ~

0

1

wii

n

1

1


(3) Produce the interpolation:(3) Produce the interpolation:

Using the linear combination equation:Using the linear combination equation:

n

iii zwZ

10

ˆ

(4) Compute the minimized estimation variance (uncertainty):(4) Compute the minimized estimation variance (uncertainty):

n

iiiR w

10

2 ~~


3.6.3 Examples:3.6.3 Examples: (1) Sample configuration around a point of estimation(1) Sample configuration around a point of estimation ((Figure 12.1)) (2) An exponential semivariogram model with a nugget (2) An exponential semivariogram model with a nugget of zero, rang of 10, sill of 10 (Figure 12.2).of zero, rang of 10, sill of 10 (Figure 12.2).

~ ~ (~ ~ ) ( ~ ~ ) ( ) R i jj

n

i

n

ij i ii

n

ii

n

w w w w2 2

11

2 20

1 1

2 2 1

~( ) ( exp( )) ( ). h C Ch

ae h

0 10 31

310 1

(3) Use the error variance model to determine the weights(3) Use the error variance model to determine the weights

(4) Solve for the weights:(4) Solve for the weights: 0.173, 0.318, 0.129, 0086, 0.151, 0.057, 0.086 0.173, 0.318, 0.129, 0086, 0.151, 0.057, 0.086 ((Figure 12.3)


3.6.3 Examples: (continued …)3.6.3 Examples: (continued …)

(5) Estimated value at point 0 is:(5) Estimated value at point 0 is: V0 =(0.173)(477) + (0.318)(696) + (0.129)(277) V0 =(0.173)(477) + (0.318)(696) + (0.129)(277) +(0.086)(646) + (0.151)(606) + (0.057)(791)+(0.086)(646) + (0.151)(606) + (0.057)(791) + (0.086)(783) + (0.086)(783) = 592.7 ppm= 592.7 ppm (6) Minimized estimation variance is:(6) Minimized estimation variance is:

2

10

2 96.8~~ ppmwn

iiiR


3.6.4 Observations and Issues:3.6.4 Observations and Issues: (1) the sets of weights minimize the modeled variance(1) the sets of weights minimize the modeled variance (2) sample configuration is considered in determine the weights(2) sample configuration is considered in determine the weights ((Figure 12.3)) (3) spatial continuity is quantitatively included in estimation(3) spatial continuity is quantitatively included in estimation a) the effect of scale (a) the effect of scale (Figure 12.5 and Figure 12.6)) b) the effect of shape (b) the effect of shape (Figure 12.7 and Figure 12.8)) c) the effect of nugget (c) the effect of nugget (Figure 12.9 and Figure 12.10)) d) the effect of range (Figure 12.11 and d) the effect of range (Figure 12.11 and Figure 12.12)) e) the effect of anisotropy (e) the effect of anisotropy (Figure 12.13 and Figure 12.14)) (Figure 12.16)(Figure 12.16) (4) uncertainty information is provided(4) uncertainty information is provided (5) undesired drastic change of attribute values at sample points(5) undesired drastic change of attribute values at sample points (The Attribute value change around sample point)(The Attribute value change around sample point) (6) data hungry(6) data hungry (7) requirement of stationarity assumption(7) requirement of stationarity assumption


1. What is the idea behind the kriging method? How is information 1. What is the idea behind the kriging method? How is information on spatial autocorrelation used in this approach? on spatial autocorrelation used in this approach?

2. How does kriging resolve the sample clustering problem?2. How does kriging resolve the sample clustering problem?

3. How is weight value related to distance with kriging?3. How is weight value related to distance with kriging?

4. Why do people often call kriging the optimal interpolation 4. Why do people often call kriging the optimal interpolation techniques?techniques?

5. From the way how the ordinary kriging method using spatial 5. From the way how the ordinary kriging method using spatial autocorrelation, explain the importance of meeting the autocorrelation, explain the importance of meeting the stationarity assumption.stationarity assumption.

5. What are the major problems with the kriging method?5. What are the major problems with the kriging method?

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Geog. 579: GIS and Spatial Analysis - Lecture 21 Overheads 1 Point Estimation: 3. Methods: 3.6...

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