1. Chapter 16 - Spatial Interpolation
Triangulation
Inverse-distance
Kriging (optimal interpolation)

2. What is “Interpolation”?
Predicting the value of attributes at “unsampled” sites from measurements made at point locations within the same area or region
Predicting the value outside the area - “extrapolation”
Creating continuous surfaces from point data - the main procedures

3. Types of Spatial Interpolation
Global or Local
Global-use every known points to estimate unknown value.
Local – use a sample of known points to estimate unknown value.
Exact or inexact interpolation
Exact – predict a value at the point location that is the same as its known value.
Inexact (approximate) – predicts a value at the point location that differs from its known value.
Deterministic or stochastic interpolation
Deterministic – provides no assessment of errors with predicted values
Stochastic interpolation – offers assessment of prediction errros with estimated variances.

4. Classification of Spatial Interpolation Methods
Global
Local
Stochastic
Deterministic
Stochastic
Deterministic
Regression (inexact)
Kriging (exact)
Trend surface (inexact)
Thiessen (exact)
Density estimation(inexact)
Inverse distance weighted (exact)
Splines (exact)

5. Global Interpolation
Global use all available data to provide predictions for the whole area of interest, while local interpolations operate within a small zone around the point being interpolated to ensure that estimates are made only with data from locations in the immediate neighborhood.
Two types of global: Trend surface and regression methods

6. Trend Surface Analysis
Approximate points with known values with a polynomial equation.
See Box 16.1
Local polynomial interpolation – uses a sample of known points, such as convert TIN to DEM

7. Local, deterministic methods
Define an area around the point
Find data point within
neighborhood
Choose model
Evaluate point value

8. Thiessen Polygon (nearest neighbor)
Any point within a polygon is closer to the polygon’s known point than any other known points.
One observation per cell, if the data lie on a regular square grid, then Thiessen polygons are all equal, if irregular then irregular lattice of polygons are formed
Delauney triangulation - lines joining the data points (same as TIN - triangular irregular network)

9. Thiessen polygons
Delauney Triangulation

10. Example data set
soil data from Mass near the village of Stein in the south of the Netherlands
all point data refer to a support of 10x10 m, the are within which bulked samples were collected using a stratified random sampling scheme
Heavy metal concentration measured

11. Exercise: create Thiessen polygon for zinc concentration
Create a new project
Copy “g:\classes\4650_5650\data\3-22\Soil_poll.dbf” and import it to the project.
After importing the table into the project, you need to create an event theme based on this table
Go to Tools > Add XY Data and make sure the “Easting” is shown in “X” and “Northing” is in “Y”. (Don’t worry the “Unknown coordinate”
Click on OK then the point theme will appear on your project.

12. This is what you might see on screen

13. Create a polygon theme
The next thing you need to do is provide the Thiessen polygon a boundary so that the computing of irregular polygons can be reasonable
Use ArcCatalog to create a new shapefile and name it as “Polygon.shp”
Add this layer to your current project.
Use “Editor” to create a polygon.

14. Creating Polygon Theme

15. Notes: 1)Remember to stop Edits, otherwise your polygon theme will be under editing mode all the time 2)Remember to remove the “selected” points from the “Soil_poll_data.txt”. If you are done so, your Thiessen polygons will be based on the selected points only.

16. Extent and Cell Size
Go to “Spatial Analyst > Options” and click on tab and use “Polygon” as the “Analysis Mask”.
If the Analysis Mask is not set, the output layer will have rectangular shape.

17. Thiessen Polygon from Spatial Analyst
Select Spatial Analyst > Distance > Allocation.
In “Assign to”, select “soil_poll Event” and
Change the default cell size to “0.1”
click OK to create cell in temporary folder.

19. Join Tables
Join soil_poll Events to Alloc3 grid file by “ObjectID” in Alloc3 and “OID” in soil_poll Events.
Click “Advanced” button. Two options are available for joining tables.
Open Attribute of “Alloc3” (name may vary) and view the joined fields.

20. Zinc Concentration
Symbolize the grid with two-color ramp based on Zn concentration

21. Density Estimation
Simple method – divide total point value by the cell size
Kernel estimation – associate each known point with a kernel function, a probability density function.

22. Exercise Compute density of the sampling points from previous dataset.
If you use the xy-event points for calculation, you might receive error message.
Convert this layer to shape file before using Density function from Spatial Analyst.
Select your cell size (such as 1,) and search radius as 5.

23. Density function output

24. Inverse Distance Weighted
the value of an attribute z at some unsampled point is a distance-weighted average of data point occurring within a neighborhood, which compute:
=estimated value at an unsampled point
n= number of control points used to estimate a grid point
k=power to which distance is raised
d=distances from each control points to an unsampled point

25. Computing IDW 6 Z1=40 Z2=60 4 Z4=40 Z3=50 2 Use k = 1 2 4 6 X
Do you get 49.5 for the red square?

26. Exercise - generate a Inversion distance weighting surface and contour
Spatial Analyst > Interpolate to Raster > Inverse Distance Weighted
Make sure you have set the Output cell size to 0.1

27. Contouring
create a contour based on the surface from IDW

28. IDW and Contouring

29. Problem - solution
Unsampled point may have a higher data value than all other controlled points but not attainable due to the nature of weighted average: an average of values cannot be lesser or greater than any input values - solution:
Fit a trend surface to a set of control points surrounding an unsampled point
Insert X and Y coordinates for the unsampled point into the trend surface equation to estimate a value at that point

30. Splines
draughtsmen used flexible rulers to trace the curves by eye. The flexible rulers were called “splines” - mathematical equivalents - localized
piece-wise polynomial function p(x) is

31. Spline - math functions
piece-wise polynomial function p(x) is
p(x)=pi(x) xi<x<xi+1
pj(xi)=pj(xi) j=0,1,,,,
i=1,2,,,,,,k-1
i+1 x1
xk+1 x0 xk
break points

32. Spline
r is used to denote the constraints on the spline (the functions pi(x) are polynomials of degree m or less
r = 0 - no constraints on function

33. Exercise: create surface from spline
have point data theme activated
select “Surface > Interpolate Grid
Define the output area and other parameters
Select “Spline” in Method field, “Zn” for Z Value Field and “regularized” as type

34. Kriging
Comes from Daniel Krige, who developed the method for geological mining applications
Rather than considering distances to control points independently of one another, kriging considers the spatial autocorrelation in the data

35. semivariance 20
Z1 Z2 Z3 Z4 Z5 10
Zi = values of the attribute at control points
h=multiple of the distance between control points
n=number of sample points

36. Semivariance h=1, h=2 h=3 h=4 21.88 91.67 156.25 312.50 100 25 175 8
(Z1-Z1+h)2
100 25
175 8
225
100
425 6
400
225
625 4
625 2
(Z2-Z2+h)2
(Z3-Z3+h)2
(Z4-Z4+h)2
sum
2(n-h)

37. Modifications (in real world)
Tolerance - direction and distance needed to be considered
10m 1m
20o 5m A

38. semivariance
the semivariance increases as h increases : distance increases -> semivariance increases
nearby points to be more similar than distant geographical data

39. data no longer similar to nearby values
sill
range h

40. kriging computations we use 3 points to estimate a grid point
again, we use weighted average
=w1Z1 + w2Z2+w3Z3
= estimated value at a grid point
Z1,Z2 and Z3 = data values at the control points
w1,w2, and w3 = weighs associated with each control point

41. In kriging the weighs (wi) are chosen to minimize the difference between the estimated value at a grid point and the true (or actual) value at that grid point.
The solution is achieved by solving for the wi in the following simultaneous equations
w1(h11) + w2(h12) + w3(h13) = (h1g)
w1(h12) + w2(h22) + w3(h23) = (h2g)
w1(h13) + w2(h32) + w3(h33) = (h3g)

42. w1(h11) + w2(h12) + w3(h13) = (h1g)
Where (hij)=semivariance associated with distance bet/w control points i and j.
(hig) =the semivariance associated with the distance bet/w ith control point and a grid point.
Difference to IDW which only consider distance bet/w the grid point and control points, kriging take into account the variance between control points too.

43. Example distance 1 2 3 g Z1(1,4)=50 1 2 3 g 3.16 2.24 Z3(3,3)=25 2.24
3.16
2.24
Z3(3,3)=25
2.24
1.00
1.41
Z(2,2)=?
Z2(2,1)=40
w10.00+w231.6+w322.4=22.4
w131.6+w20.00+w322.4=10.0
w122.4+w222.4+w30.00=14.1 
=10h h

44. =0.15(50)+0.55(40) (25)
= 37

45. Homework 6 – due next Thursday midnight.
Task 1: Chapter 16 tasks
Task 2:
Calculate volume of contaminated Pb soil in Thiessen polygon exercise based on range of every 50 ppm, assuming soil density of 1.65 g/cm3 and only the top 1-foot soil is considered.
Use IDW to compute the volume of the contaminated Pb
Use Kriging (if it’s working) to compute concentration of Pb
Compare these three methods and see the differences (use same output cell size for all three methods)
In Doc file, describe your selection of cell size, search radius and results from different choice of cell sizes (if you have time to create layers with different cell sizes.