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Spatial Interpolation

Spatial Interpolation. GLY 560: GIS and Remote Sensing for Earth Scientists. Class Home Page: http://www.geology.buffalo.edu/courses/gly560/. Introduction.

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Spatial Interpolation

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  1. Spatial Interpolation GLY 560: GIS and Remote Sensing for Earth Scientists Class Home Page: http://www.geology.buffalo.edu/courses/gly560/

  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) 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 GLY560: GIS and RS

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

  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 GLY560: GIS and RS

  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 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 GLY560: GIS and RS

  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 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 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 GLY560: GIS and RS

  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. GLY560: GIS and RS

  12. Example Interpolators • Theissen Polygons • Inverse Distance Weighted • Splines • Radial Basis Functions • Global Polynomial • Kriging GLY560: GIS and RS

  13. Theissen Polygons • Also called “proximal” method • Attempts to weight data points by area • Commonly used for precipitation data GLY560: GIS and RS

  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 GLY560: GIS and RS

  15. GLY560: GIS and RS

  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 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. GLY560: GIS and RS

  18. GLY560: GIS and RS

  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 GLY560: GIS and RS

  20. GLY560: GIS and RS

  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 GLY560: GIS and RS

  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 GLY560: GIS and RS

  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 GLY560: GIS and RS

  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 GLY560: GIS and RS

  25. Variogram Models GLY560: GIS and RS

  26. Exponential Universal Circular Examples of Kriging GLY560: GIS and RS

  27. Summary of Interpolators(from ESRI Geostatistical Analyst) GLY560: GIS and RS

  28. Summary of Interpolators(from ESRI Geostatistical Analyst) GLY560: GIS and RS

  29. Theissen Polygon GLY560: GIS and RS

  30. Inverse Distance Weighting GLY560: GIS and RS

  31. Kriging GLY560: GIS and RS

  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 GLY560: GIS and RS

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