The Electronic Storefront

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Geog 488/588 GIS I an introduction
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Last week

• Geographic query and analysis
• Interrogation
• Measurement
• Transformation
• Proximity
• Overlay
• Extraction
• Raster Analysis

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This week

• Spatial interpolation
• Density estimation
• Advanced spatial modeling
• Optimization
• Guest speaker Josh Cerra Herrera Inc.

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Spatial Interpolation

• Values of a field have been measured at a number
  of sample points
• There is a need to estimate the complete field
• to estimate values at points where the field was
  not measured
• to create a contour map by drawing isolines
  between the data points
• Methods of spatial interpolation are designed to
  solve this problem

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Spatial Interpolation
ORIGINAL SAMPLE POINTS
Interpolated Values
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Inverse Distance Weighting (IDW)

• The unknown value of a field at a point is
  estimated by taking an average over the known
  values
• weighting each known value by its distance from
  the point, giving greatest weight to the nearest
  points

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point i known value zi location xi weight wi
distance di
unknown value (to be interpolated) location x
The estimate is a weighted average
Weights decline with distance
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Issues with IDW

• The range of interpolated values cannot exceed
  the range of observed values
• it is important to position sample points to
  include the extremes of the field
• this can be very difficult

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A Potentially Undesirable Characteristic of IDW
interpolation
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Kriging

• A technique of spatial interpolation firmly
  grounded in geostatistical theory
• The semi-variogram reflects Toblers Law
• differences within a small neighborhood are
  likely to be small
• differences rise with distance

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• The semi-variogram is a function that relates
  semi-variance (or dissimilarity) of data points
  to the distance that separates them.
• If we can understand the difference between an
  unknown quantity and a known quantity, we we can
  estimate the unknown point

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Stages of Kriging

• Analyze observed data to estimate a
  semi-variogram
• Estimate values at unknown points as weighted
  averages
• obtaining weights based on the semivariogram
• the interpolated surface replicates statistical
  properties of the semivariogram

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IDW vs. Kriging
Kriging

• Kriging appears to give a more natural look to
  the data
• Kriging avoids the bulls eye effect
• Kriging also give us a standard error

IDW
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Density Estimation and Potential

• Spatial interpolation is used to fill the gaps in
  a field
• Density estimation creates a field from discrete
  objects
• the fields value at any point is an estimate of
  the density of discrete objects at that point
• e.g., estimating a map of population density (a
  field) from a map of individual people (discrete
  objects)

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The Kernel Function

• Each discrete object is replaced by a
  mathematical function known as a kernel
• Kernels are summed to obtain a composite surface
  of density
• The smoothness of the resulting field depends on
  the width of the kernel
• narrow kernels produce bumpy surfaces
• wide kernels produce smooth surfaces

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typical kernel function
smooth statistical surface
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Street Intersections
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Street Intersections
2 mile kernel
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Street Intersections
½ mile kernel
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