Point Estimation:

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Lecture 23 Spatial Interpolation VI
Topics
Point Estimation 3. Methods 3.6
Ordinary Kriging
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.

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Outlines
3. Methods 3.6 Ordinary Kriging 3.6.1
The Idea (1) Interpolation should
minimize the uncertainty associated with
the interpolation. (2) The spatial
continuity should be explicitly used to determine
the weights in such a way that the
uncertainty is minimized. 3.6.2
Implementation (1) Define an error
variance model (the objective function)
The model should have the weights as
variables and should be based on
quantitative definition of spatial continuity
expressed as a semi-variogram model.
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(1) Define an error variance model
(continued) The semi-variogram based error
variance model
Where
? is the Lagrange parameter (dummy
variable) The idea is to select a set of
weights which produce a minimal value from
the model.
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(2) Minimize the objective function (the error
variance model) Setting the partial
derivatives of the objective function with
respect to each of the variables to zero will
produce a set of (n1) simultaneous
equations
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(3) Produce the interpolation Using the
linear combination equation
(4) Compute the minimized estimation variance
(uncertainty)
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3.6.3 Examples (1) Sample configuration
around a point of estimation (Figure
12.1) (2) An exponential semivariogram model
with a nugget of zero, rang of 10,
sill of 10 (Figure 12.2).
(3) Use the error variance model to determine
the weights
(4) Solve for the weights 0.173,
0.318, 0.129, 0086, 0.151, 0.057, 0.086
(Figure 12.3)
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3.6.3 Examples (continued ) (5) Estimated
value at point 0 is V0 (0.173)(477)
(0.318)(696) (0.129)(277)
(0.086)(646) (0.151)(606) (0.057)(791)
(0.086)(783)
592.7 ppm (6) Minimized estimation variance is
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3.6.4 Observations and Issues (1) the sets
of weights minimize the modeled variance (2)
sample configuration is considered in determine
the weights (Figure 12.3) (3)
spatial continuity is quantitatively included in
estimation a) the effect of scale
(Figure 12.5 and Figure 12.6) b) the
effect of shape (Figure 12.7 and Figure 12.8)
c) the effect of nugget (Figure 12.9 and
Figure 12.10) d) the effect of range
(Figure 12.11 and Figure 12.12) e) the
effect of anisotropy (Figure 12.13 and Figure
12.14) (Figure 12.16) (4)
uncertainty information is provided (5)
undesired drastic change of attribute values at
sample points (The Attribute value
change around sample point) (6) data hungry
(7) requirement of stationarity assumption
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Questions
1. What is the idea behind the kriging method?
How is information on spatial autocorrelation
used in this approach? 2. How does kriging
resolve the sample clustering problem? 3. How is
weight value related to distance with
kriging? 4. Why do people often call kriging the
optimal interpolation techniques? 5. From the
way how the ordinary kriging method using spatial
autocorrelation, explain the importance of
meeting the stationarity assumption. 5. What are
the major problems with the kriging method?