1. Spatial Interpolation
GLY 560: GIS and Remote Sensing for Earth Scientists
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2. 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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GLY560: GIS and RS

3. 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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4. Classification of Interpolators
Area / Point
Global / Local
Exact / Approximate
Deterministic / Stochastic
Gradual / Abrupt
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5. 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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6. 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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GLY560: GIS and RS

7. 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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8. 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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GLY560: GIS and RS

9. 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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GLY560: GIS and RS

10. 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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11. 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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12. Example Interpolators
Theissen Polygons
Inverse Distance Weighted
Splines
Radial Basis Functions
Global Polynomial
Kriging
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13. Theissen Polygons Also called “proximal” method
Attempts to weight data points by area
Commonly used for precipitation data
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14. 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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16. 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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GLY560: GIS and RS

17. Radial Basis Functions (RBF’s)
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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19. 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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21. 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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22. 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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23. 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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24. Deriving the Variogram
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
For every pair of points, compute distance and the squared difference in values
Assign each pair to one of the distance ranges, and accumulate total variance in each range
After every pair has been used (or a sample of pairs in a large dataset) compute the average variance in each distance range
Plot this value at the midpoint distance of each range
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25. Variogram Models
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26. Examples of Kriging Universal Exponential Circular 4/16/2017
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27. Summary of Interpolators (from ESRI Geostatistical Analyst)
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28. Summary of Interpolators (from ESRI Geostatistical Analyst)
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29. Theissen Polygon
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30. Inverse Distance Weighting
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31. Kriging
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32. 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
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GLY560: GIS and RS