Geostatistics

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Geostatistics

• Mike Goodchild

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Spatial 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 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 interpolation

• Along a line
• geocoding with address ranges

x2,y2 address2
x,y address
x1,y1 address1
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In a triangle
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In a rectangle

• Bilinear interpolation

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Characteristics of linear interpolation

• Exact
• 0-order continuity
• Contours are straight
• but not parallel in bilinear case

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IDW

• 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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Characteristics 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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Kriging

• 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 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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Estimation

• 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 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 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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z f (elevation)
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Co-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 Kriging

• Binary field
• c 0,1
• Obtained by thresholding an interval/ratio field
• c1 if zgtt else c0
• estimate variogram from observations of c
• z is hidden
• The multivariate case
• sequential assignment

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Indicator 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 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 disadvantages

• Theoretically based
• Not a black box
• Statistical
• variance estimates
• Sensitivity to sample design