CPSC 695. Methods for interpolation and analysis of continuing surfaces in GIS Dr. M. Gavrilova

Transcription

1 CPSC 695 Methods for interpolation and analysis of continuing surfaces in GIS Dr. M. Gavrilova

2 Overview Data sampling for continuous surfaces Interpolation methods Global interpolation Local interpolation Continuous surface analysis

3 Visualization and analysis of continuous data Visualization of continuous data (terrain models, soil samples, etc) requires a variety of methods from: Computer graphics Statistics Mathematics Computational geometry

4 Interpolation Interpolation is a first step for representing a continuous surface when only sample data is available. It is used when: The discretized surface has a different level of resolution, cell size or orientation When a continuous surface is represented by a different data model that required When the data do not cover the domain of interest completely

5 Data sources for interpolation Aerial photos Scanners Satellite images Point samples Digitized maps

6 Sampling Different types of sampling are used to collect data

7 Methods for interpolation Interpolation methods Global interpolation Classification models Trend surfaces on geometric coordinates Regression models on multiple attributes Local interpolation Delaunay triangulation based Linear and inverse distance weighting Thin plate splines

8 Global classification models Used for prediction of values of attributes. ANOVA standard analysis of variance method: z(x 0 )=m+alpha k +epsilon, Where z is the attribute value at location x 0, m is the mean of z values, alpha k is the deviation between m and the mean of z values over a smaller unit k, and epsilon is residual error of measurement (noise). Assumption: no correlation between measurements within units, changes take place on the boundaries that are sharp, within units data is very consistent.

9 Global regression model using trend surfaces Regression model uses continuous function to predict unknown values: Can be linear z(x)=b 0 +b 1 x+e, Quadratic z(x)=b 0 +b 1 x+ b 2 x 2 +e or any higher degree polynomial (used in 3D modeling), e is an error. Trend surfaces that result are also suitable for smoothing the data.

10 Regression models on multiple attributes Regression model can be based on two or more attributes, especially is their relationship is easy to establish: z(x)=b 0 +b 1 P 1 +b 2 P 2 +e, where P 1, P 2 are independent properties. Ex: level of zinc in the soil and distance to river

11 Local interpolation methods Based on proximity Thiessen polygons (Voronoi regions) Interpolate over polygons (Voronoi regions), more precise predictor!

12 Voronoi regions

13 Inverse distance interpolation Inverse distance interpolators combine notion of proximity (VD) and the gradual change of trend surfaces by adding some distance-weighted attributes to the interpolation formula. Results more smooth interpolation.

14 Splines (a) local adjustments result in local changes (b) sharp corners can be improved by using exact splines (c) selecting break points has significant impact on splines

15 Splines Splines are piece-wise function. Interpolation values can be calculated more quickly. Thin plate splines sometimes produce unrealistically smooth surfaces.

16 Inverse distance vs. spline Inverse distance interpolations (ac), spline interpolation (d)

17 DEM

18 DEM

19 DEM DEM generated from splines (left) and Delaunay triangulation (right)

20 Spatial filtering

21 Spatial filtering Low-pass filter the value for the cell is computed as average of other cells High-pass-continuous surface low pass

22 Spatial filtering Window size has effect on filtering

23 Smoothing of maps

24 Other spatial operations Slope, profile convexity and other derivatives of the surface can be extracted Surface topology, networks can be derived Improvements to visual quality of maps can be achieved