1. Spatial Analysis
Longley et al.

2. Transformations Buffering (Point, Line, Area) Point-in-polygon
Polygon Overlay
Spatial Interpolation
Theissen polygons
Inverse-distance weighting
Kriging
Density estimation

3. Basic Approach
Map
New map
Transformation

4. Point-in-polygon Select point known to be outside
Select point to be tested
Create line segment
Intersect with all boundary
segments
Count intersections
EVEN=OUTSIDE
ODD=INSIDE

5. Create a buffer: Raster

6. Create a Buffer: vector

7. Combining maps
RASTER
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

8. Combining Maps
VECTOR
A problem

9. Max Egenhofer Topological Overlay Relations

10. Creating new zones
Town buffer
River buffer

11. 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

12. Moran’s I

13. Spatial autocorrelation
Correlation of a field with itself
Low
High
Maximum

14. Spatial optimization

15. Spatial interpolation

16. Linear interpolation C B
Half way from A to B,
Value is (A + B) / 2 A

17. 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

18. 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

19. 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

20. Trend Surfaces

21. 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)

22. 3. Distance Weighted Methods

23. 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)

24. IDW 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?

25. 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)

26. Ozone in S. Cal: Text Example
measuring stations and concentrations
(point shapefile)
CA cities (point shapefile)
CA outline (polygon shapefile)
DEM (raster)

27. IDW Wizard in Geostatistical Analyst define data source

28. Further define interpolation method
Power of distance
4 sectors

29. Cross validation
removing one of the n observation points and using the remaining n-1 points to predict its value.
Error = observed - predicted

30. Result

31. 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
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
the basis of this 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
Kriging is based on an analysis of the data, then an application of the
results of this analysis to interpolation

32. Kriging
Breaks up topography into 3 elements: Drift (general trend), small deviations from the drift and random noise.
To be stepped over

33. 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.

34. 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
Deriving the variogram
the input data for Kriging is usually an irregularly spaced sample of points
to compute a variogram we need to determine how varianceincreases with distance
begin by dividing 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 z 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
fit one of a standard set of curve shapes to the points
"model" the variogram
Computing the estimates:
interpolated values are the sum of the weighted values of some number of known points where weights depend on the distance between the interpolated and known points
weights are selected so that the estimates are:
unbiased (if used repeatedly, Kriging would give the correctresult on average)
minimum variance (variation between repeated estimates is minimum)

35. Kriging Example surface has a constant mean, no underlying trend
Surface has no constant mean
Maybe no underlying trend
surface has a constant mean,
no underlying trend
allows for a trend
binary data

36. Analysis of Variogram

37. Fitting a Model, Directional Effects

38. How Many Neighbors?

39. Cross Validation

40. Kriging Result similar pattern to IDW less detail in remote areas
smooth

41. IDW vs. Kriging
Kriging
Kriging appears to give a more “smooth” look to the data
Kriging avoids the “bulls eye” effect
Kriging gives us a standard error
IDW