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Spatial Analysis. Longley et al., Ch 14,15. Transformations . Buffering (Point, Line, Area) Point-in-polygon Polygon Overlay Spatial Interpolation Theissen polygons Inverse-distance weighting Kriging Density estimation. Basic Approach. Map. New map. Transformation. Point-in-polygon.

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Spatial Analysis

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Spatial analysis l.jpg

Spatial Analysis

Longley et al., Ch 14,15

Transformations l.jpg


  • Buffering (Point, Line, Area)

  • Point-in-polygon

  • Polygon Overlay

  • Spatial Interpolation

    • Theissen polygons

    • Inverse-distance weighting

    • Kriging

    • Density estimation

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Basic Approach


New map


Point in polygon l.jpg


Select point known to be outside

Select point to be tested

Create line segment

Intersect with all boundary


Count intersections



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Create a buffer: Raster

Create a buffer vector l.jpg

Create a Buffer: vector

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Combining maps


  • As long as maps have same extent, resolution, etc, overlay is direct (pixel-to-pixel)

  • Otherwise, needs interpolation

  • Use map algebra (Tomlin)

  • Tomlin’s operators

    • Focal, Local, Zonal

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Combining Maps


  • A problem

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Max EgenhoferTopological Overlay Relations

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Creating new zones

Town buffer

River buffer

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Other spatial analysis methods

  • Centrographic analysis (mean center)

  • Dispersion measures (stand. Dist)

  • Point clustering measures (NNS)

  • Moran’s I: Spatial autocorrelation (Clustering of neighboring values)

  • Fragmentation and fractional dimension

  • Spatial optimization

    • Point

    • Route

  • Spatial interpolation

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Moran’s I

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Spatial autocorrelation

  • Correlation of a field with itself




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Spatial optimization

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Spatial interpolation

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Linear interpolation



Half way from A to B,

Value is (A + B) / 2


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Nonlinear Interpolation

  • When things aren't or shouldn’t be so simple

  • Values computed by piecewise “moving window”

  • Basic types:1. Trend surface analysis / Polynomial

    2. Minimum Curvature Spline

    3. Inverse Distance Weighted 4. Kriging

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1. Trend Surface/Polynomial

  • point-based

  • Fits a polynomial to input points

  • When calculating function that will describe surface, uses least-square regression fit

  • approximate interpolator

    • Resulting surface doesn’t pass through all data points

    • global trend in data, varying slowly overlain by local but rapid fluctuations

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1. Trend Surface cont.

  • flat but TILTED plane to fit data

    • surface is approximated by linear equation (polynomial degree 1)

    • z = a + bx + cy

  • tilted but WARPED plane to fit data

    • surface is approximated by quadratic equation (polynomial degree 2)

    • z = a + bx + cy + dx2 + exy + fy2

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Trend Surfaces

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2. Minimum Curvature Splines

  • Fits a minimum-curvature surface through input points

  • Like bending a sheet of rubber to pass through points

    • While minimizing curvature of that sheet

  • repeatedly applies a smoothing equation (piecewise polynomial) to the surface

    • Resulting surface passes through all points

  • best for gently varying surfaces, not for rugged ones (can overshoot data values)

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3. Distance Weighted Methods

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3. Inverse Distance Weighted

  • Each input point has local influence that diminishes with distance

  • estimates are averages of values at n known points within window R

where w is some function of distance (e.g., w = 1/dk)

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  • IDW is popular, easy, but problematic

  • Interpolated values limited by the range of the data

  • No interpolated value will be outside the observed range of z values

  • How many points should be included in the averaging?

  • What about irregularly distributed points?

  • What about the map edges?

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IDW Example

  • ozone concentrations at CA measurement stations

    1. estimate a complete field, make a map

    2. estimate ozone concentrations at specific locations (e.g., Los Angeles)

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Ozone in S. Cal: Text Example

measuring stations and concentrations

(point shapefile)

CA cities (point shapefile)

CA outline (polygon shapefile)

DEM (raster)

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IDW Wizard in Geostatistical Analystdefine data source

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Power of distance

4 sectors

Further define interpolation method

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Cross validation

  • removing one of the n observation points and using the remaining n-1 points to predict its value.

  • Error = observed - predicted

Result l.jpg


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4. Kriging

  • Assumes distance or direction betw. sample points shows a spatial correlation that help describe the surface

  • Fits function to

    • Specified number of points OR

    • All points within a window of specified radius

  • Based on an analysis of the data, then an application of the results of this analysis to interpolation

  • Most appropriate when you already know about spatially correlated distance or directional bias in data

  • Involves several steps

    • Exploratory statistical analysis of data

    • Variogram modeling

    • Creating the surface based on variogram

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  • Breaks up topography into 3 elements:Drift (general trend), small deviations from the drift and random noise.

To be stepped over

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Explore with Trend analysis

  • You may wish to remove a trend from the dataset before using kriging. The Trend Analysis tool can help identify global trends in the input dataset.

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Kriging Results

  • Once the variogram has been developed, it is used to estimate distance weights for interpolation

  • Computationally very intensive w/ lots of data points

  • Estimation of the variogram complex

    • No one method is absolute best

    • Results never absolute, assumptions about distance, directional bias

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Kriging Example

Surface has no constant mean

Maybe no underlying trend

surface has a constant mean,

no underlying trend

allows for a trend

binary data

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Analysis of Variogram

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Fitting a Model, Directional Effects

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How Many Neighbors?

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Cross Validation

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Kriging Result

  • similar pattern to IDW

  • less detail in remote areas

  • smooth

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IDW vs. Kriging

  • Kriging appears to give a more “smooth” look to the data

  • Kriging avoids the “bulls eye” effect

  • Kriging gives us a standard error



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