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

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) 4/16/2017 GLY560: GIS and RS

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 4/16/2017 GLY560: GIS and RS

Classification of Interpolators Area / Point Global / Local Exact / Approximate Deterministic / Stochastic Gradual / Abrupt 4/16/2017 GLY560: GIS and RS

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 4/16/2017 GLY560: GIS and RS

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 4/16/2017 GLY560: GIS and RS

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 4/16/2017 GLY560: GIS and RS

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 4/16/2017 GLY560: GIS and RS

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 4/16/2017 GLY560: GIS and RS

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 4/16/2017 GLY560: GIS and RS

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. 4/16/2017 GLY560: GIS and RS

Example Interpolators Theissen Polygons Inverse Distance Weighted Splines Radial Basis Functions Global Polynomial Kriging 4/16/2017 GLY560: GIS and RS

Theissen Polygons Also called “proximal” method Attempts to weight data points by area Commonly used for precipitation data 4/16/2017 GLY560: GIS and RS

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 4/16/2017 GLY560: GIS and RS

4/16/2017 GLY560: GIS and RS

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 4/16/2017 GLY560: GIS and RS

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. 4/16/2017 GLY560: GIS and RS

4/16/2017 GLY560: GIS and RS

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 4/16/2017 GLY560: GIS and RS

4/16/2017 GLY560: GIS and RS

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 4/16/2017 GLY560: GIS and RS

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 4/16/2017 GLY560: GIS and RS

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 4/16/2017 GLY560: GIS and RS

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 4/16/2017 GLY560: GIS and RS

Variogram Models 4/16/2017 GLY560: GIS and RS

Examples of Kriging Universal Exponential Circular 4/16/2017 GLY560: GIS and RS

Summary of Interpolators (from ESRI Geostatistical Analyst) 4/16/2017 GLY560: GIS and RS

Summary of Interpolators (from ESRI Geostatistical Analyst) 4/16/2017 GLY560: GIS and RS

Theissen Polygon 4/16/2017 GLY560: GIS and RS

Inverse Distance Weighting 4/16/2017 GLY560: GIS and RS

Kriging 4/16/2017 GLY560: GIS and RS

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 4/16/2017 GLY560: GIS and RS