1. Interpolation & Contour Maps

2. Interpolation
We want to draw inferences about data based on the observed data
Visiting every location in a study area to measure the height, magnitude, or amount of a phenomenon is usually difficult or expensive.
Instead, measure the phenomenon at strategically dispersed sample locations, and predicted values can be assigned to all other locations.

3. Spatial Data Perspectives
Deterministic perspective: if we understand the system, we can predict the outcome with certainty
Stochastic perspective: uncertainty remains in the best-designed model or experiment

4. Interpolation Methods
Deterministic perspective: assign values to locations based on the surrounding measured values and on specified mathematical formulas that determine the smoothness of the resulting surface
Stochastic perspective: assign values based on statistical models that include autocorrelation (the statistical relationship among the measured points)

5. Types of Interpolation Methods
Deterministic perspective:
Inverse Distance Weighted: points weighted by distance
Spline: passes exactly through points with constraining equations
Polynomial/trend analysis (nearest/natural neighbor)
Stochastic perspective:
Kriging: weighted by fitted semivariogram (next slides)

6. Natural Neighbor
The colored circles, which represent the interpolating weights, are generated using the ratio of the shaded area to that of the cell area of the surrounding points. The shaded area is due to the insertion of the point to be interpolated into the Voronoi tessellation (Theissen polygons).

7. Inverse Distance Weighted
point i
known value zi
location xi
weight wi distance di
unknown value (to be interpolated)
location x
The estimate is a weighted average

8. Trend Surface/Polynomial
Flat but TILTED plane to fit data
(1st order polynomial)
Z = a + bx + cy
Tilted but WARPED plane to fit data
( 2nd order polynomial )
Z = a + bx + cy + dx2 + exy + fy2

9. Trend Surfaces
Simplifies the surface representation to allow visualization of general trends.
Higher order polynomials can be used
Robust regression methods can be used

10. Kriging Spatial prediction of variable Z at location x
Z(x) = m(x) + γ(h) + ε
Three components:
structural (constant mean),
random spatially correlated component
residual error

11. Variogram
Nugget
Range
Sill
Separation Distance
Semi-Variogram
function
Plot of the correlation of data (g) as a function of the distance between points (h)
12/10/2018

12. Types of Kriging
• Simple Kriging: assumes mean is constant and known • Ordinary Kriging: assumes mean is constant but unknown (MapWindow) • Universal Kriging: assumes mean is varying and unknown - Modeled by a constant, linear, second or third order equation

13. Advantages of Kriging It handles (embraces) spatial autocorrelation
Less sensitive to preferential sampling in specific areas
Allows uncertainty to be estimated (Kriging error)

14. Example: Illinois Precipitation
Formatted into a CSV with Latitude, Longitude, and Precipitation

15. Example: Illinois Precipitation
Imported to MapWindow

16. Example: Illinois Precipitation
Imported to MapWindow

17. Example: Illinois Precipitation
Imported to MapWindow

18. Example: Illinois Precipitation
Assign Projection (comes in Geographic Coordinates), use a Geographic Coordinate System (WGS 84)

19. Example: Illinois Precipitation
Assign Projection (comes in Geographic Coordinates), use a Geographic Coordinate System (WGS 84)

20. Example: Illinois Precipitation
Kriging requires a projected shapefile. Reproject to WGS 84 UTM 16 N

21. Example: Illinois Precipitation
Kriging requires a projected shapefile. Reproject to WGS 84 UTM 16N

22. Example: Illinois Precipitation
Let’s Krige! Add Layers PRECIP_STATIONS_16N.shp and IL_BOUND_16N.shp

23. Example: Illinois Precipitation

24. Example: Illinois Precipitation
Enable Kriging Plugin, Interpolation Methods Menu

25. Example: Illinois Precipitation
Select PRECIP_STATIONS_16N layer

26. Example: Illinois Precipitation
Select PRECIP field

27. Example: Illinois Precipitation
Semivariogram:) – Select distance classes (spatial lags) – Calculate semivariance across lags – Plot semivariance on Y axis, distance on X axis – Fit theoretical model (linear, gaussian)

28. Example: Illinois Precipitation
Apply model (Gaussian, Range = 1,000,000, Nugget = 0, Sill = 85)

29. Example: Illinois Precipitation
Generate surface, save as .asc

30. Example: Illinois Precipitation
Generate contour precipitation map for the kriged surface, raster operation

31. Example: Illinois Precipitation
Choose a location, and an interval for spacing contours

32. Example: Illinois Precipitation

33. Cease the training of impossible hedges around this life
For as fast as you sow them, serendipity’s thickets will appear,
And outgrow them.
Lorna Goodison