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GeostatisticsGeostatistics
Mike Goodchild
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Spatial interpolationSpatial interpolation
A field– variable is interval/ratio– z = f(x,y)– sampled at a set of points
How to estimate/guess the value of the field at other points?
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Characteristics of interpolated surfacesCharacteristics of interpolated surfaces
Representation– raster, isolines, TIN
Form– rugged or smooth– exact or approximate– continuity
• 0-order• 1-order• 2-order
Uncertainty– variance estimators?
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Linear interpolationLinear interpolation
Along a line– geocoding with address ranges
x2,y2
address2
x1,y1
address1
x,y
address
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In a triangleIn a triangle
20
30
40
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In a rectangleIn a rectangle
Bilinear interpolation
20
30
30
40
(24)
(34)
(29)
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Characteristics of linear interpolationCharacteristics of linear interpolation
Exact 0-order continuity Contours are straight
– but not parallel in bilinear case
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IDWIDW
Advantages– quick, universal, theory-free
Disadvantages– theory-free– directional effects
• non-spatial
– characteristics of a weighted average• when all weights are non-negative
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0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
1 4 7
10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97
100
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Characteristics of IDW surfacesCharacteristics of IDW surfaces
Pass through each data point (exact)– if negative power distance function– 1/0b =
0-, 1-, 2-order continuous– except at data points
Underestimate peaks– volcanoes – unless peak is observation point
Extrapolate to the global mean Noisy extrapolations
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KrigingKriging
Geostatistics as theoretical framework Estimation of parameters from data Use of estimated model to control
interpolation Many versions
– not a simple black box– highlights– demonstration
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The variogramThe variogram
Relationship between variance and distance Formalization of Tobler's First Law Estimated from data
– how well can a given data set estimate variogram?– distribution of sample points is critical
• at peaks and pits• samples the range of possible distances• uniform spacing not desirable• but often out of the user's control
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EstimationEstimation
Data points zi(xi) Interpolate at x
– stochastic process– multiple realizations
• variance obtained from variogram
A set of weights i unique to x– chosen such that the estimate is
• unbiased• minimum variance
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Kriging prediction
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Results of KrigingResults of Kriging
A mean surface A variance surface
– minimum at observation points Mean surface is smoother than any
realization– is not a possible realization
• a mean map is not a possible map
– compare a univariate process– average rainfall versus rainfall from a single storm– conditional simulation
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Kriging standard error
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Kriging variantsKriging variants
Co-Kriging– interpolation process guided by another
variable (field)– hard and soft data– observations of interpolated data are hard– guiding variable is soft
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55
70
83
68
z = f (elevation)
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Co-KrigingCo-Kriging
Linear relationship f Point observations are hard
– accurate, sparse Elevation observations are soft
– inaccurate (errors in measurement or prediction)
– dense
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Co-Kriging prediction
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Co-Kriging standard error
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Indicator KrigingIndicator Kriging
Binary field– c {0,1}
Obtained by thresholding an interval/ratio field– c=1 if z>t else c=0– estimate variogram from observations of c– z is hidden
The multivariate case– sequential assignment
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Indicator KrigingIndicator Kriging
Assign Class 1, notClass 1 Among notClass 1, assign Class 2,
notClass 2 Continue to Class n-1
– notClass n-1 = Class n
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Universal KrigingUniversal Kriging
Simple Kriging is all second order– trend results from random walk
Stochastic process plus trend– trend is first order– remove trend before analysis– restore trend after analysis
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Advantages and disadvantagesAdvantages and disadvantages
Theoretically based Not a black box Statistical
– variance estimates Sensitivity to sample design