2. Introduction to GIS Modeling Week 8 — Surface Modeling GEOG 3110 –University of Denver Presented by Joseph K. Berry W. M. Keck Scholar, Department of Geography, University of Denver Digital Elevation Model (DEM); Basic surface modeling concepts (Density Analysis, Interpolation and Map Generalization) ; Interpolation techniques (IDW and Krig) ; Spatial Autocorrelation; Assessing interpolation results

3. Visualizing Terrain Surface Data (Exercise 8 – Part 1) Mount St. Helens dataset Question 1 Access SURFER then enter Map  Contour Map  New Contour Map  \Samples  Helens2.grd (Berry) There are numerous websites that allow you to download a DEM and use SURFER to visualize …a generally useful procedure that you can use for lots of reports (Optional Exercise)

4. Visualizing Map Surfaces (Exercise 8 – Part 1) Questions 2 and 3 Use SURFER to Create a 2D Contour map and a 3-D Wireframe map SURFER (Berry)

5. Map Analysis Evolution (Revolution) (Berry) Traditional GIS Points, Lines, Polygons Points, Lines, Polygons Discrete Objects Discrete Objects Mapping and Geo-query Mapping and Geo-query Forest Inventory Map Spatial Analysis Cells, Surfaces Cells, Surfaces Continuous Geographic Space Continuous Geographic Space Contextual Spatial Relationships Contextual Spatial Relationships StoreTravel-Time(Surface) Traditional Statistics Mean, StDev (Normal Curve) Mean, StDev (Normal Curve) Central Tendency Central Tendency Typical Response (scalar) Typical Response (scalar) Minimum= 5.4 ppm Maximum= 103.0 ppm Mean= 22.4 ppm StDev= 15.5 Spatial Statistics Map of Variance (gradient) Map of Variance (gradient) Spatial Distribution Spatial Distribution Numerical Spatial Relationships Numerical Spatial Relationships Spatial Distribution (Surface)

6. An Analytic Framework for GIS Modeling (Berry) Surface Modelling operations involve creating continuous spatial distributions from point sampled data. See www.innovativegis.com/basis/Download/IJRSpaper/ www.innovativegis.com/basis/Download/IJRSpaper/

7. Surface Modeling (Point Density Analysis) Point Density analysis identifies the number of customers with a specified distance of each grid location MapCalc “ScanTotal” – ESRI GRID/Spatial Analyst “ZonalTotal” Roving Window (count) (Berry)

8. Identifying Unusually High Density Pockets of unusually high customer density are identified as more than one standard deviation above the mean MapCalc “Renumber” – ESRI GRID/Spatial Analyst “Reclassify” (Berry)

9. …a Histogram depicts the numeric distribution (Mean/Central Tendency/) …a Map depicts the geographic distribution (Variance/Variability) … Data Values link the two views— Click anywhere on the Map and the Histogram interval is highlighted Click on the Histogram interval and the Map locations are highlighted Linking Numeric & Geographic Distributions (Berry) (See Beyond Mapping III, “Topic 7” for more information) Beyond Mapping IIIBeyond Mapping III …simply different ways to organize and analyze mapped data

10. Non-Spatial Statistics (Central Tendency; typical response) (Berry) …seeks to reduce a set of data to a single value that is typical of the entire data set (Average) and generally assess how typical the typical is (StDev)

11. Assumptions in Non-Spatial Statistics (Berry) …uniformly distributed in geographic space (horizontal plane at average; +/- constant)

12. Geographic Distribution (surface modeling) (Berry) …analogous to fitting a curve (Standard Normal Curve) in numeric space except fitting a map surface in geographic space to explain variation in the data

13. Adjusting for Spatial Reality (masking for discontinuities) (Berry) …accounting for known geographic discontinuities or other spatial relationships

14. Generating a Map of Percent Change (map-ematics) (Berry) …maps are organized sets of numbers supporting a robust range of Map Analysis operations that can be used to relate spatial variables (map layers)

15. Spatial Relationships (coincidence, proximity, etc.) (Berry) …spatial relationships can be utilized to extend understanding

16. Standard Normal Variable Map (Berry) …relates every location to the typical response (Average and StDev) to determine how typical it is

17. The Average is Hardly Anywhere (Berry) Arithmetic Average – plot of the data average is a horizontal plane in 3-dimensional geographic space with some of the data points balanced above (green) and some below (red) the “typical” value (uniform estimate of the spatial distribution) Field Collected Data #15 87 = P2 sample value Arithmetic Average knows nothing of Geographic Space

18. Surface Modeling (Map generalization) (Berry) Map Generalization – fits standard functional forms to the data, such as a N th order polynomial for curved surfaces with several peaks and valleys (similar to curve fitting techniques in traditional statistics) Spatial Average balances “half” of the data above and below a Horizontal Plane— Arithmetic Average balances “half” of the data on either side of a Line— Y avg X avg Line Plane Curved Plane Curved Line

19. Surface Modeling (Iterative Smoothing) The “iterative smoothing” process is similar to slapping a big chunk of modeler’s clay over the “data spikes,” then taking a knife and cutting away the excess to leave a continuous surface that encapsulates the peaks and valleys implied in the original field samples… and cutting away the excess to leave a continuous surface that encapsulates the peaks and valleys implied in the original field samples… (Berry) …repeated smoothing slowly “erodes” the data surface to a flat plane = AVERAGE …Spatial Interpolation techniques utilize summary of data in a roving window (Localized Variation) Digital slide show SStat2.ppt SStat2.ppt

20. Surface Modeling Methods (Surfer) Inverse Distance to a Power — weighted average of samples in the summary window such that the influence of a sample point declines with “simple” distance Modified Shepard’s Method — uses an inverse distance “least squares” method that reduces the “bull’s-eye” effect around sample points Radial Basis Function — uses non-linear functions of “simple” distance to determine summary weights Kriging — summary of samples based on distance and angular trends in the data Natural Neighbor —weighted average of neighboring samples where the weights are proportional to the “borrowed area” from the surrounding points (based on differences in Thiessen polygon sets) Minimum Curvature — analogous to fitting a thin, elastic plate through each sample point using a minimum amount of bending Polynomial Regression — fits an equation to the entire set of sample points Nearest Neighbor — assigns the value of the nearest sample point Triangulation — identifies the “optimal” set of triangles connecting all of the sample points Thiessen Polygons (Berry) Map Generalization — Mathematical Equation/Surface Fitting Map Generalization — Geometric facets Spatial Interpolation — “roving window” localized average

21. Surface Modeling Approaches (using point samples) Spatial Interpolation — these techniques use a roving window to identify Nearby Samples and then Summarize the Samples based on some function of their relative nearness to the location being interpolated. Window Reach — how far away to reach to collect sample points for processing Window Reach — how far away to reach to collect sample points for processing Window Shape — shape of the window can be symmetrical (circle) or asymmetrical (ellipse) Window Shape — shape of the window can be symmetrical (circle) or asymmetrical (ellipse) Summary Technique — a weighted average based on proximity using a fixed geometric relationship (inverse distance squared) or a more complex statistical relationship (spatial autocorrelation) Summary Technique — a weighted average based on proximity using a fixed geometric relationship (inverse distance squared) or a more complex statistical relationship (spatial autocorrelation) Exacting Solution — exacting solutions result in the sample value being retained (Krig); non- exacting estimate sample locations (IDW) Exacting Solution — exacting solutions result in the sample value being retained (Krig); non- exacting estimate sample locations (IDW) (Berry) Map Generalization (Equation) — these techniques seek the general trend in the data by Fitting a Polynomial Equation to the entire set of sample data (1 st degree polynomial is a plane). Thiessen Polygons Map Generalization (Geometric Facets) — Triangulated Irregular Network (TIN) is a form of the tessellated model based on Triangles. The vertices of the triangles form irregularly spaced nodes and unlike the DEM, the TIN allows dense information in complex areas, and sparse information in simpler or more homogeneous areas. http://www.jarno.demon.nl/gavh.htm http://www.jarno.demon.nl/gavh.htm

22. Spatial Interpolation (Mapping spatial variability) (Berry) …the geo-registered soil samples form a pattern of “spikes” throughout the field. Spatial Interpolation is similar to throwing a blanket over the spikes that conforms to the pattern. …all interpolation algorithms assume 1) “nearby things are more alike than distant things” (spatial autocorrelation), 2) appropriate sampling intensity, and 3) suitable sampling pattern …maps the spatial variation in point sampled data …maps the spatial variation in point sampled data

23. Spatial Interpolation (Comparing Average and IDW results) Comparison of the interpolated surface to the whole field average shows large differences in localized estimates (Berry)

24. Spatial Interpolation (Comparing IDW and Krig results) (Berry) Comparison of the IDW and Krig interpolated surfaces shows small differences in in localized estimates

25. Creating and Comparing Map Surfaces Use SURFER to Create and Compare map surfaces (Exercise 8, Part 2) SURFER (Berry) IDW KrigCreate IDW - KrigCompare

26. Inverse Distance Weighted Approach (Berry)

27. Inverse Distance Weighted Calculations (Berry) Geometry determines weights (function of distance) …but first things first, how do you calculate a weighted average? …Weight …weight * value = weighted Product …SUMwProducts / SumWeights = wAverage

28. Spatial Autocorrelation (Kriging) Tobler’s First Law of Geography — nearby things are more alike than distant things Variogram — plot of sample data similarity as a function of distance between samples (Berry) …Kriging uses regional variable theory based on an underlying variogram to develop custom weights based on trends in the sample data (proximity and direction) …uses Variogram Equation instead of a fixed 1/D Power Geometric Equation Data relationships determine weights (function of distance and data patterns)

29. Spatial Interpolation Techniques (Berry) Characterizes the spatial distribution by fitting a mathematical equation to localized portions of the data (roving window) AVG= 23 everywhere Spatial Interpolation techniques use “roving windows” to summarize sample values within a specified reach of each map location. Window shape/size and summary technique result in different interpolation surfaces for a given set of field data …no single techniques is best for all data. Inverse Distance Weighted (IDW) technique weights the samples such that values farther away contribute less to the average …1/Distance Power

30. Spatial Interpolation (Evaluating performance) (Berry) Assessing Interpolation Results – Residual Analysis …the best map is the one that has the “best guesses” (See Beyond Mapping III, Topic 2 Beyond Mapping IIIBeyond Mapping III for more information) AVG= 23

31. Spatial Interpolation (Characterizing error) A Map of Error (Residual Map) …shows you where your estimates are likely good/bad (Berry)

32. Point Sampling Design Concerns Stratification-- appropriate groupings for sampling Sample Size-- appropriate sampling intensity for each stratified group Sampling Grid-- appropriate reference grid for locating individual point samples (nested best) (Berry)

33. Point Sampling Design Concerns Sampling Pattern-- appropriate arrangement of samples considering both spatial interpolation and statistical inference (Berry)

34. Assessing Spatial Autocorrelation (SAC indices) Geary’s C and Moran’s I — similarity among center cell and adjacent neighbors vs. center cell and average for the entire data set… Geary’s C = [(n –1) SUM w ij (x i – x j ) 2 ] / [(2 SUM w ij ) SUM (x i – m) 2 ] Moran’s I = [n SUM w ij (x i – m) (x j – m)] / [(SUM w ij ) SUM (x i – m) 2 ] where, n = number of cells in the grid m = the mean of the values in the grid m = the mean of the values in the grid x i = value of cell in group i and x j = value of cell in group j x i = value of cell in group i and x j = value of cell in group j w ij = switch set to 1 if the cells are adjacent; 0 if not (diagonal) w ij = switch set to 1 if the cells are adjacent; 0 if not (diagonal) Expect… 0 0 0 0 C > 1 Strong Negative SAC I 1 Strong Negative SAC I < 0 C = 1 Random distribution I = 0 C = 1 Random distribution I = 0 …for assessment of SAC in the entire data set A map of Localized SAC Similarity can be constructed by comparing “Adjacent Neighbors” (doughnut hole) value with the “Extended Neighbors” value (doughnut)— Deviation for continuous data (gradient) Proportion Similar for categorical data (discrete) …you determine the size/configuration of the doughnut …you determine the size/configuration of the doughnut (Berry) See Beyond Mapping III, Topic 8, Investigating Spatial Dependency for more information Beyond Mapping IIIBeyond Mapping III Geary’s C Moran’s I

35. Spatial Autocorrelation/Variogram Revisited (Berry) The result is an equation that relates overall sample value Similarity and the Distance between sample locations “Nearby things are more alike than distant things” …but what about directional trends in the data?

36. Polar Variogram (Similarity, Distance and Angle) For the three points shown below, the separation of the three pairs can be summarized as … Point Pair Dist Angle Point Pair Dist Angle A (50,50), (100,200) 158.11 71.11 B (50,50), (500,100) 452.77 6.34 C (100,200), (500,100) 412.31-14.04 …the position in the polar grid summarizes the distance and angle between pairs of points. The “average” difference between the values is computed for each polar sector– Polar Variogram 90 0 45 315 270 135 180 225 Polar Variogram 100 200 300 400 500 A B C (Berry) Dist = ( a 2 + b 2 ).5 = ( (500-50) 2 + (100-50) 2 ).5 = 452.77 Angle = ARCTAN ( opp / adj ) = ARCTAN ( (100-50) / (500-50) ) = 6.34 Geographic Space #1 (50,50) #2 (100,150) a b #3 (500,100) Point Pair B

37. Optional Opportunities (Berry) Surfer Tutorials – experience with basic Surfer capabilities Sampling Patterns – understanding alternative sampling pattern considerations Interpolation Techniques – additional experience with griding tools Different Data Different Techniques