Spatial Interpolation. Inverse Distance Weighting The Variogram Kriging Much thanks to Bill Harper for his insights in Practical Geostatistics 2000 and personal conversation. Objectives. In this session we will evaluate a dataset and attempt to:
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Inverse Distance Weighting
The Variogram
Kriging
Much thanks to Bill Harper for his insights in Practical Geostatistics 2000 and personal conversation
Iron concentration
Need to interpolate iron content for unsampled areas
General Statistics
47 samples
Mean value: 36.3
S.D.: 3.73
Data SetTherefore, we are 90% confident that a point drawn at random would be:
30 < T < 42.6
This is based on consulting a students t distribution with 47 samples
Data SetUpper left side
General Statistics
7 samples
Mean value: 40
S.D.: 2.82
Getting somewhat better
Subset of Area (northwest area)Therefore, we are 90% confident that a point drawn at random would be:
34.2 < T < 45.7
This is based on consulting a students t distribution with 7 samples
Now, the question is, do some of the points exhibit more influence than others?
Probably, so lets evaluate the point taking nearness into account
Interpolated value is 39.9
So, our calculation is the same as that in ArcGIS – its just math….
IDWStandard error 2.75
Therefore, we are 90% confident that a point drawn at random would be:
34.7 < T < 45.1
This is based on consulting a students t distribution with 7 samples
IDW – standard ErrorCaveat: we are treating IDW like weighted mean, and the standard deviation like a weighted standard deviation. In reality, you shouldn’t develop confidence intervals for data that is autocorrelated
Power = 2, search = 230
Power = 2, search = 600
So which is best???
Power = 2, search = 150
Power = 4, search = 600
1Clark and Harper Practical Geostatistics 2000. Ecosse North America, Llc
The Variogram
1
d1
1
å

+
2
[Z(xi)
Z(xi
h)]
2N(h)
=
i
1
VariogramsVariogram: g(h) = ½ var [ Z(x) – Z(x+h) ]
= ½ E [ {Z(x) – Z(x+h)}2 ]
In practice:
g(h) =
Where:
pairs of observations
separated by a distance h.
the variance of the errors.
Variogram models must be “positive definite” so that the covariance matrix based on it can be inverted (which occurs in the kriging process). Because of this, only certain models can be used.
We can enter some numbers in Mathcad and see how the variogram changes.
Variogram with a lag size of 5m and a lag tolerance of 2.5m.
Variogram with a lag size of 10m and a lag tolerance of 5m.
Practical Geostatistics 2000
ArcGIS Geostatistical Analyst
3 components: structural (constant
mean), random spatially correlated
component and residual error.
Z(x) = m(x) + g(h) + e”
37.6 < 41.14 < 44.68 %Fe
Kriging
IDW
1Clark and Harper Practical Geostatistics 2000. Ecosse North America, Llc
1Clark and Harper Practical Geostatistics 2000. Ecosse North America, Llc