1. Lecture 6: Point Interpolation
Department of Geography and Urban Studies, Temple University GUS 0265/0465 Applications in GIS/Geographic Data Analysis
Lecture 6: Point Interpolation

2. Interpolating a Surface From Sampled Point Data
Assumes a continuous surface that is sampled
Interpolation
estimating the attribute values of locations that are within the range of available data using known data values
Extrapolation
estimating the attribute values of locations outside the range of available data using known data values

3. Interpolation
Estimating a point here: interpolation
Sample data

4. Extrapolation
Sample data
Estimating a point here: extrapolation

5. Sampling Strategies for Interpolation

6. Linear Interpolation If Sample elevation data A = 8 feet and
B = 4 feet
then
C = (8 + 4) / 2 = 6 feet
Sample elevation data A C B
Elevation profile

7. Non-Linear Interpolation
Sample elevation data
Often results in a more realistic interpolation but estimating missing data values is more complex A C B
Elevation profile

8. Exact Interpolation Sample elevation data
Interpolated surface passes through the original data points
Elevation profile

9. Inexact Interpolation
Sample elevation data
Interpolated surface does not pass through all of the original data points
Elevation profile

10. Global Interpolation
Uses all known sample points to estimate a value at an unsampled location
Sample data

11. Local Interpolation
Uses a neighborhood of sample points to estimate a value at an unsampled location
Sample data
Uses a local neighborhood to estimate value, i.e. closest n number of points, or within a given search radius

12. Trend Surface Global method Inexact Can be linear or non-linear
Multiple regression (e.g. predicting a z elevation value [dependent variable] with x and y location values [independent variables])

13. 1st Order Trend Surface z = b0 + b1x + e
In one dimension: z varies as a linear function of x z
z = b0 + b1x + e x

14. 1st Order Trend Surface z = b0 + b1x + b2y + e
In two dimensions: z varies as a linear function of x and y z y
z = b0 + b1x + b2y + e x

15. 2nd Order Trend Surface z = b0 + b1x + b2x2 + e
In one dimension: z varies as a non-linear function of x
z = b0 + b1x + b2x2 + e z x

16. Higher Order Trend Surfaces in Two Dimensions

17. Splines Local method Can be exact or inexact Non-linear
Derived from the use of flexible rulers for drafting
Fits a piece-wise polynomial function to a neighborhood of sample points
Typically produces a smooth surface

18. Thin Plate Spline
Minimizes curvature while ensuring interpolated surface passes through the sampled points
Can adjust the tension of the spline to address ‘overshoot’ problem where estimates are too high due to the modeled curvature of the surface

19. Splines
In one dimension
In two dimensions z y y x x

20. Inverse Distance Weighted (IDW)
Local method
Exact
Can be linear or non-linear
The weight (influence) of a sampled data value is inversely proportional to its distance from the estimated value

21. Inverse Distance Weighted (IDW)
In English:
Find the neighboring sample points of the target location (i.e. through n nearest neighbors or a search radius)
Find the distance from each sample point to the target location
Weight each sample point according to the inverse of its distance from the target location taken to the r exponent
Average the weighted attribute values of the sample points and assign the resulting value to the target location

22. Inverse Distance Weighted (IDW): Example
100
IDW: Closest 3 neighbors, r = 2 4 3
160 2
200

23. Inverse Distance Weighted (IDW): Example
Weights
A BC
1 / (42) = / (32) = / (22) = .2500
A = 100 4
B = 160 3 2
C = 200

24. Inverse Distance Weighted (IDW): Example
Weights
A BC
1 / (42) = / (32) = / (22) = .2500
The weight = inverse of the distance squared
A = 100 4
B = 160 3 2
C = 200

25. Inverse Distance Weighted (IDW): Example
Weights
Weights * Value
A BC
1 / (42) = / (32) = / (22) = .2500
.0625 * 100 = * 160 = * 200 = 50.00
Total = .4236
A = 100 4
= 74.01
B = 160 3
74.01 / = 175 2
C = 200

26. Kriging

27. Kriging
A geostatistical method of interpolation that incorporates three components of variation:
A spatially correlated component
A ‘drift’ or trend structure
Random error

28. The Nature of Spatial Variation

29. Calculating Semivariance
Kriging uses the semivariogram
Semivariance is a measure of the degree of spatial dependence between samples
The sampled semivariance is the average difference in value between observations a certain distance apart.

30. Semivariogram

31. Semivariogram Models

32. Ordinary Kriging…
estimates the value at an unsampled location by a weighted average of the values of nearby sampled points.
weights the sampled points by the semivariance of the distance between each of the sampled points and the unsampled location, AND the semivariances of the distances among all paired combinations of sampled points.
results in the estimation of kriging variance, typically mapped as the kriging standard deviation, which may be interpreted as the reliability of the estimate at each location.

33. Ordinary Kriging

34. Ordinary Kriging

38. Other Types of Kriging Block Kriging Non-Linear Kriging
Ordinary Kriging with Anisotropy
Co-Kriging
Universal Kriging
Indicator Kriging