Spatial Interpolation

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Spatial Interpolation

• GLY 560 GIS and Remote Sensing for
  Earth Scientists

Class Home Page http//www.geology.buffalo.edu/c
ourses/gly560/
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Introduction

• Spatial interpolation is the estimation the value
  of properties at unsampled sites within the area
  covered by existing observations.
• Usual Rationale points close together are more
  likely to have similar values than points far
  apart (Tobler's Law)

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Use of Spatial Interpolation in GIS

• Provide contours for displaying data graphically
• Calculate some property of the surface at a given
  point
• Compare data of different types/units in
  different data layers

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Classification of Interpolators

• Area / Point
• Global / Local
• Exact / Approximate
• Deterministic / Stochastic
• Gradual / Abrupt

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Area Based Interpolation

• Given a set of data mapped on one set of source
  zones, determine the values for a different set
  of target zones
• For example
• given population counts for census tracts,
  estimate populations for electoral districts
• vegetation and soil maps

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Area Based Interpolation

• Centroid
• find centroid of area
• assign total value of data in area to centroid
• treat as point interpolation.
• Overlay
• overlay of target and source zones
• determine the proportion of each source zone that
  is assigned to each target zone
• apportion the total value of the attribute for
  each source zone to target zones

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Point Based Interpolation

• Given points whose locations and values are
  known, determine the values of other points at
  locations
• For example
• weather station readings
• spot heights
• porosity measurements

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Global vs. Local Interpolators

• Global interpolators determine a single function
  which is mapped across the whole region
• e.g. trend surface
• Local interpolators apply an algorithm repeatedly
  to a small portion of the total set of points
• e.g. inverse distance weighted

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Exact vs. Approximate Interpolators

• Exact interpolators honor all data points
• e.g. inverse distance weighted
• Approximate interpolators try to approach all
  data points
• e.g. trend surface

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Deterministic vs. Stochastic

• Deterministic interpolators model a data point at
  a particular position.
• e.g. spline
• Stochastic interpolators try to model probability
  of a data point being at a particular position
• e.g. kriging, fourier analysis

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Gradual/Abrupt Interpolators

• Gradual interpolators assume continuous and
  smooth behavior of data everywhere
• Abrupt interpolators allow for sudden changes in
  data due to boundaries or undefined derivatives.

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Example Interpolators

• Theissen Polygons
• Inverse Distance Weighted
• Splines
• Radial Basis Functions
• Global Polynomial
• Kriging

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Theissen Polygons

• Also called proximal method
• Attempts to weight data points by area
• Commonly used for precipitation data

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Inverse Distance Weighted

• Essentially moving average methods, estimates
  based upon proximity of points known data
• Exact interpolator
• The best results from IDW are obtained when
  sampling is sufficiently dense with regard to the
  local variation you are attempting to simulate.
• If the sampling of input points is sparse or very
  uneven, the results may not sufficiently
  represent the desired surface

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Splines

• The mathematical equivalent of using a flexible
  ruler (called a spline)
• Piecewise polynomials fit through data (local
  interpolator)
• Can be used as an exact or approximate
  interpolator, depending upon the degrees of
  freedom granted (e.g. polynomial order)
• Best for smooth datasets, can cause wild
  fluctuations otherwise

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Radial Basis Functions (RBFs)

• Exact version of spline
• Like bending a sheet of rubber to pass through
  the points, while minimizing the total curvature
  of the surface.
• It fits piecewise polynomial to a specified
  number of nearest input points, while passing
  through the sample points.

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Global Polynomial

• Fit one polynomial through entire dataset.
• Advantages
• Creates very smooth surfaces
• Implies homogenous behavior (model) of dataset
• Disadvantages
• Higher-order polynomials may reach ridiculously
  large or small values outside of data area
• Susceptible to outliers in the data

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Stochastic (Geostatistical) Interpolators

• Geostatistical techniques create surfaces
  incorporating the statistical properties of the
  measured data.
• Produces not only prediction of surfaces, but
  uncertainty estimates of prediction
• Many methods are associated with geostatistics,
  but they are all in the kriging family

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Kriging

• Developed by Georges Matheron, as the "theory of
  regionalized variables", and D.G. Krige as an
  optimal method of interpolation for use in the
  mining industry
• Basis of technique is the rate at which the
  variance between points changes over space
• This is expressed in the variogram which shows
  how the average difference between values at
  points changes with distance between points

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Variogram

• Plot of the correlation of data (g) as a function
  of the distance between points (h)

Range
Semi-Variogram function
Sill
Nugget
Separation Distance
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Deriving the Variogram

1. Divide the range of distance into a set of
   discrete intervals, e.g. 10 intervals between
   distance 0 and the maximum distance in the study
   area
2. For every pair of points, compute distance and
   the squared difference in values
3. Assign each pair to one of the distance ranges,
   and accumulate total variance in each range
4. After every pair has been used (or a sample of
   pairs in a large dataset) compute the average
   variance in each distance range
5. Plot this value at the midpoint distance of each
   range

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Variogram Models
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Examples of Kriging
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Summary of Interpolators(from ESRI
Geostatistical Analyst)
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Summary of Interpolators(from ESRI
Geostatistical Analyst)
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Theissen Polygon
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Inverse Distance Weighting
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Kriging
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Conclusions

• Interpolation method depends upon
• Character of data
• Your assumptions of data behavior
• When possible, best way to compare methods is to
• try several methods
• make sure you understand theory
• refine best method