Lecture 18: Spatial Interpolation I Topics: Point Estimation: 1. Process and Issues: References: Chapter 8, 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. 98-102.

Outlines 1. Process and Issues 1.1 The need for spatial interpolation: (The Purpose Diagram) 1.2 The process: 1.2.1 Determine the level of details needed The spacing of each estimation (spatial resolution) 1.2.2 Collect sample data to support the level of details needed (the Variable Sample Scheme Figure) 1.2.3 Examine the spatial autocorrelation from the data set: a) Degree of the spatial autocorrelation b) Spatial extent of the spatial autocorrelation (The Semivariogram Approach)

1.2 The process: (continued…) 1.2.4 Determine the method to be used for interpolation Depends on the project requirements, sample data and the characteristics of the methods The basic equation for interpolation: Where: Z0’ is the attribute value to be predicted at the unvisited-site Zi is the attribute value at the i point of the nearby locations wi is the weight assigned to the attribute at point i, wi should sum up to 1 (to be unbiased) n is the total number of nearby locations involved 1.2.5 Evaluate the interpolation

1.3 The basic issues: 1.3.1 Determine number of sample points in each estimation: 1) The requirement of the methods 2) The density of samples and spatial autocorrelation a) Defining the search neighborhood (search radius or search window): Spacing of Sample Points a) regular spacing: (Regular Spacing Figure) b) irregular spacing: (Irregular Spacing Figure) b) The magic number: Twice the search spacing About 4 - 12 sample points for each estimation without going out the range of spatial autocorrelation

1.3 The basic issues: (continued…) 1.3.1 Determine number of sample points: (continued..) c) Consequences: if too few sample points: the interpolation could be too sensitive if too many samples: too much computation time and too much redundancy irrelevance in the sample data sets since sample points too far away may be included 1.3.2 Determine the distribution of sample points: 1) the nature of the phenomena to be interpolated 2) the distribution of samples (1) Two questions: a) are there nearby samples that are redundant? The clustering problem (The clustering Figure)

1.3 The basic issues: (continued…) 1.3.2 Determine the distribution (continued…): b) are there nearby samples relevant? Impact of point from different population Relevance has to be determined by the user’s understanding of the sample data. Determination of relevance is case dependent and is much more important than choosing a interpolation techniques. (2) Search strategy: Quadrant Search (Regular Quadrant Search, Figure 14.2) (Revised Quadrant Search) 1.3.3 Determine the weights The main difference among methods is how each of them allocate weights to sample points. 1.3.4 Uncertainty Assessment

Questions 1. Why do we need spatial interpolation? What is the problem of just using the samples? 2. What are the steps involved in spatial interpolation? What are the key issues in making an interpolation? 3. What are the things to consider when deciding how many sample points to be included for an estimation at a given point? What happens if too few or too many points are used? 4. What the things to consider when deciding the spatial distribution of sample points for an estimation at a given point? Why do people say clustering of sample points is not desired? What is quadrant search?