spatial interpolation

Spatial Interpolation

GLY 560: GIS and Remote Sensing for Earth Scientists

Class Home Page: http://www.geology.buffalo.edu/courses/gly560/

04/22/23 GLY560: GIS and RS

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)


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


Classification of Interpolators

•Area / Point

•Global / Local

•Exact / Approximate

•Deterministic / Stochastic

•Gradual / Abrupt


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


Area Based Interpolation

Centroid:1. find centroid of area

2. assign total value of data in area to centroid

3. treat as point interpolation.

Overlay:1. overlay of target and source zones

2. determine the proportion of each source zone that is assigned to each target zone

3. apportion the total value of the attribute for each source zone to target zones


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


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


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


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


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.


Example Interpolators

•Theissen Polygons

• Inverse Distance Weighted

•Splines

•Radial Basis Functions

•Global Polynomial

•Kriging


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


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


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.


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


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


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


Variogram

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

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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


Variogram Models


Examples of Kriging

Universal Exponential Circular


Summary of Interpolators(from ESRI Geostatistical Analyst)


Theissen Polygon


Inverse Distance Weighting


Kriging


Conclusions

• Interpolation method depends upon• Character of data

• Your assumptions of data behavior

• When possible, best way to compare methods is to1. try several methods

2. make sure you understand theory

3. refine best method

spatial interpolation

Documents

set of data

data pointse

weight data points

data of different typesunits

data local interpolatorcan

total set of points

splinestochastic interpolators

proximity of points