Lecture 20: Spatial Interpolation III Topics: Point Estimation: 2. Methods: 2.1 Nearest Neighbor (Thiessen Polygon) 2.2 Triangulation (fitting a plane) References: Chapter 11, Isaaks, E. H., and R. M. Srivastava, 1989. Applied Geostatistics, Oxford University Press, New York Chapter 5, Burrough, P.A. and R.A. McDonnell, 1998. Principles of Geographical Information Systems, Oxford University Press, New York, pp. 113-117.

Outlines 3. Methods: 3.1 The Nearest Neighbor Approach (Thiessen Polygon Approach) 3.1.1 The Idea: To assign the value of the nearest sample to the site to be estimated 3.1.2 Implementation: 1) Nearest neighbor search Calculate the distances between the sample points and the site Assign the value of the sample with shortest distance to the site (The Shortest Distance Diagram) 2) Thiessen polgyong approach (polygon of influence) Construct a polygon around each sample so that any location in the polygon is closest to the sample of the polygon than to any other sample points (The Thiessen Polygon Construction Diagrams) (…) Any location in a Thiessen polygon will get the attribute value of the sample point of that Thiessen polygon

3.1.3 The issues: 1) Number of sample points: 1 sensitive to the accuracy of one point 2) Distribution of sample points: N/A 3) Weight allocation: all onto one sample point 4) Uncertainty information: none 5) Discontinuous property value at the boundaries (Discontinuities) The step function effect (The Step Function Diagram) 3.2 Triangulation: 3.2.1 The Idea: To fit a plane to three sample points and use the fitted plane to estimate value of any point bounded by these three sample points

3.2 Triangulation: (continued …) 3.2.2 Implementation: 1) Linear multivariate combination of location and attributes z’ = ax + by + c where x is easting and y is northing (Figure 11.3) what are the values for a, b, and c for a given coupling of three points (x1, y1), (x2, y2), (x3, y3)? We can solve the following simultaneous equations to obtain these values: z1 = ax1 + by1 +c z2 = ax2 + by2 +c z3 = ax3 + by3 +c For each coupling of three points, we need a new equation, that is, we need a new set of a, b, and c.

3.2 Triangulation: (continued …) 3.2.2 Implementation: (continued …) 1) Linear multivariate combination of location and attributes (continued …) Example: For the three sample points in Figure 11.3 (Figure11.3): (63, 140) (64, 129) (71, 140) we have the following three equations: 63a + 140b + c = 696 64a + 129b + c = 227 71a + 140b + c = 606 Solving these, we have: a = -11.25, b = 41.614, c = -4421.159 We have an estimation equation: z’ = -11.25x + 41.614y – 4421.159 For point (65, 137), we have an estimate of 548.7

3.2 Triangulation: (continued …) 3.2.2 Implementation: (continued …) 2) Weighted combination of three sample values (Figure 11.5) For point (65, 137), we have:

3.2.3 The issues: 1) Number of sample points: 3 less sensitive to the accuracy of one point 2) Distribution of sample points: N/A 3) Weight allocation: based on distance 4) Uncertainty information: none 5) First derivative is discontinuous at the triangle edges (Discontinuous First Derivative) 6) How to define the coupling of three points (triangles) (Which three points should be used?) Delaunay Triangulation: (Delaunay Diagram) 7) Interpolation only for points within the triangle

Questions 1. What is the idea behind the nearest neighbor approach? How is information on spatial autocorrelation used in this approach? 2. How is the Thiessen polygon method used in nearest neighbor approach of spatial interpolation? 3. What are the major issues related to the nearest neighbor approach? 4. What is the idea behind the triangulation approach? How is information on spatial autocorrelation used in this approach? 5. What are the major issues related to the triangulation approach?