1. DU GIS Modeling -- Surface Modeling/Analysis
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
Joseph K. Berry, Keck Visiting Scholar

2. Visualizing Terrain Surface Data (Exercise 8 – Part 1)
DU GIS Modeling -- Surface Modeling/Analysis
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
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)
(Berry)
Joseph K. Berry, Keck Visiting Scholar

3. Visualizing Map Surfaces (Exercise 8 – Part 1)
DU GIS Modeling -- Surface Modeling/Analysis
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
(Berry)
Joseph K. Berry, Keck Visiting Scholar

4. DU GIS Modeling -- Surface Modeling/Analysis
Map Analysis Evolution (Revolution)
DU GIS Modeling -- Surface Modeling/Analysis
Traditional GIS
Points, Lines, Polygons
Discrete Objects
Mapping and Geo-query
Forest Inventory Map
Spatial Analysis
Cells, Surfaces
Continuous Geographic Space
Contextual Spatial Relationships
Store
Travel-Time
(Surface)
Traditional Statistics
Mean, StDev (Normal Curve)
Central Tendency
Typical Response (scalar)
Minimum= 5.4 ppm
Maximum= ppm
Mean= 22.4 ppm
StDev= 15.5
Spatial Statistics
Map of Variance (gradient)
Spatial Distribution
Numerical Spatial Relationships
(Surface)
(Berry)
Joseph K. Berry, Keck Visiting Scholar

5. Concepts in GIS -- Topic #1
An Analytic Framework for GIS Modeling
Surface Modelling operations involve creating continuous spatial distributions from point sampled data.
See
(Berry)
Joseph K. Berry, BA_SIS, Inc. All Rights Reserved.

6. DU GIS Modeling -- Surface Modeling/Analysis
Comparing Apples and Oranges (Elevation and Slope)
and
Apples
(Rosales)
Oranges
(Sapindales)
‘you can't compare apples and oranges’
and
Elevation
(Feet)
Slope
(Percent)
…particularly when it comes
to map analysis —
they’re totally different things, right?
(Berry)
Joseph K. Berry, Keck Visiting Scholar

7. DU GIS Modeling -- Surface Modeling/Analysis
Comparing Apples and Oranges (SNV Normalization)
G#1
G#2
G#3
SNV Map 1
Normalized Distribution
R#1
R#2
R#3
SNV Map 2
SNV “Mixed Fruit” Scale
Standard Normal Variable (SNV)
Normalize (SNV)
SNV = ((mapValue - Mean) / Stdev) * 100
Slope
(raw data)
Elevation
Apples
(Rosales)
Oranges
(Sapindales)
G#1= -100, R#1= -100
(-100) – (-100) = |0|
G#1= -100, R#2= 0
(-100) – (0) = |-100|
G#1= -100, R#3= 200
(-100) – (200) = |-300|
100
200
300
400
Increasing differences in distribution matches
SNV Map 1
SNV Map 2
Compare the SNV maps by…
…subtracting SNV maps and taking the absolute value to generate a map of the relative difference between the two maps at every map location
(Berry)
Joseph K. Berry, Keck Visiting Scholar

8. DU GIS Modeling -- Surface Modeling/Analysis
Comparing Apples and Oranges (SNV Difference)
Elevation (SNV normalized)
…absolute difference between the SNV normalized Elevation and Slope maps
Slope (SNV normalized)
Median = 52
Grey = exactly the same position on the SNV distribution
Blue tones = .5 StDev or less difference
…indicates that the two maps are fairly similar–
50% of the map area is .5 StDev or less along the two SNV distributions
(Berry)
Joseph K. Berry, Keck Visiting Scholar

9. DU GIS Modeling -- Surface Modeling/Analysis
Linking Numeric & Geographic Distributions
…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
…simply different ways to organize and analyze “mapped data”
(See Beyond Mapping III, “Topic 7” for more information)
(Berry)
Joseph K. Berry, Keck Visiting Scholar

10. Surface Modeling (Point Density Analysis)
Workshop on GIS Modeling (Part3)
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)
Joseph K. Berry, BA_SIS, Inc. All Rights Reserved.

11. Identifying Unusually High Density
Workshop on GIS Modeling (Part3)
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)
Joseph K. Berry, BA_SIS, Inc. All Rights Reserved.

12. Non-Spatial Statistics (Central Tendency; typical response)
DU GIS Modeling -- Surface Modeling/Analysis
Non-Spatial Statistics (Central Tendency; typical response)
…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)
(Berry)
Joseph K. Berry, Keck Visiting Scholar

13. Assumptions in Non-Spatial Statistics
DU GIS Modeling -- Surface Modeling/Analysis
Assumptions in Non-Spatial Statistics
…uniformly distributed in geographic space (horizontal plane at average; +/- constant)
(Berry)
Joseph K. Berry, Keck Visiting Scholar

14. Geographic Distribution (surface modeling)
DU GIS Modeling -- Surface Modeling/Analysis
Geographic Distribution (surface modeling)
…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
(Berry)
Joseph K. Berry, Keck Visiting Scholar

15. Adjusting for Spatial Reality (masking for discontinuities)
DU GIS Modeling -- Surface Modeling/Analysis
Adjusting for Spatial Reality (masking for discontinuities)
…accounting for known geographic discontinuities or other spatial relationships
(Berry)
Joseph K. Berry, Keck Visiting Scholar

16. Generating a Map of Percent Change (map-ematics)
DU GIS Modeling -- Surface Modeling/Analysis
Generating a Map of Percent Change (map-ematics)
…maps are organized sets of numbers supporting a robust range of Map Analysis operations that can be used to relate spatial variables (map layers)
(Berry)
Joseph K. Berry, Keck Visiting Scholar

17. Spatial Relationships (coincidence , proximity, etc.)
DU GIS Modeling -- Surface Modeling/Analysis
Spatial Relationships (coincidence , proximity, etc.)
…spatial relationships can be utilized to extend understanding
(Berry)
Joseph K. Berry, Keck Visiting Scholar

18. Standard Normal Variable Map
DU GIS Modeling -- Surface Modeling/Analysis
Standard Normal Variable Map
…relates every location to the typical response (Average and StDev) to determine how typical it is
(Berry)
Joseph K. Berry, Keck Visiting Scholar

19. DU GIS Modeling -- Surface Modeling/Analysis
The Average is Hardly Anywhere
Arithmetic Average knows nothing of Geographic Space
Field Collected Data
#15
87 = P2 sample value
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)
(Berry)
Joseph K. Berry, Keck Visiting Scholar

20. DU GIS Modeling -- Surface Modeling/Analysis
Surface Modeling (Map generalization)
Arithmetic Average balances “half” of the data on either side of a Line—
Yavg
Xavg
Line
Plane
Curved
Plane
Line
Spatial Average balances “half” of the data above and below a Horizontal Plane—
Map Generalization – fits standard functional forms to the data, such as a Nth order polynomial for curved surfaces
with several peaks and valleys
(similar to curve fitting techniques in traditional statistics)
(Berry)
Joseph K. Berry, Keck Visiting Scholar

21. DU GIS Modeling -- Surface Modeling/Analysis
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…
…Spatial Interpolation techniques utilize summary of data in a roving window
(Localized Variation)
…repeated smoothing slowly “erodes” the data surface to a flat plane
= AVERAGE
Digital slide show SStat2.ppt
(Berry)
Joseph K. Berry, Keck Visiting Scholar

22. Workshop on GIS Modeling (Part3)
Surface Modeling Methods (Surfer)
Map Generalization — Mathematical Equation/Surface Fitting
Map Generalization — Geometric facets
Spatial Interpolation — “roving window” localized average
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)
Joseph K. Berry, BA_SIS, Inc. All Rights Reserved.

23. DU GIS Modeling -- Surface Modeling/Analysis
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 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)
Exacting Solution— exacting solutions result in the sample value being retained (Krig); non-exacting estimate sample locations (IDW)
Map Generalization (Equation) — these techniques seek the general trend in the data by Fitting a Polynomial Equation to the entire set of sample data (1st 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
(Berry)
Joseph K. Berry, Keck Visiting Scholar

24. DU GIS Modeling -- Surface Modeling/Analysis
Spatial Interpolation (Mapping spatial variability)
…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
(Berry)
Joseph K. Berry, Keck Visiting Scholar

25. DU GIS Modeling -- Surface Modeling/Analysis
Spatial Interpolation (Comparing Average and IDW results)
Comparison of the interpolated surface to the whole field average shows large differences in localized estimates
(Berry)
Joseph K. Berry, Keck Visiting Scholar

26. DU GIS Modeling -- Surface Modeling/Analysis
Spatial Interpolation (Comparing IDW and Krig results)
Comparison of the IDW and Krig interpolated surfaces shows small differences in in localized estimates
(Berry)
Joseph K. Berry, Keck Visiting Scholar

27. Creating and Comparing Map Surfaces
DU GIS Modeling -- Surface Modeling/Analysis
Creating and Comparing Map Surfaces
Use SURFER to Create and Compare map surfaces (Exercise 8, Part 2)
IDW
Krig
Create
IDW - Krig
Compare
(Berry)
Joseph K. Berry, Keck Visiting Scholar

28. DU GIS Modeling -- Surface Modeling/Analysis
Inverse Distance Weighted Approach
(Berry)
Joseph K. Berry, Keck Visiting Scholar

29. DU GIS Modeling -- Surface Modeling/Analysis
Inverse Distance Weighted Calculations
Geometry determines weights (function of distance)
…Weight
…weight * value = weighted Product
…SUMwProducts / SumWeights = wAverage
…but first things first, how do you calculate a weighted average?
(Berry)
Joseph K. Berry, Keck Visiting Scholar

30. Spatial Autocorrelation (Kriging)
DU GIS Modeling -- Surface Modeling/Analysis
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
Data relationships determine weights (function of distance and data patterns)
…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/DPower Geometric Equation
(Berry)
Joseph K. Berry, Keck Visiting Scholar

31. Spatial Interpolation Techniques
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
(Berry)

32. Spatial Interpolation (Evaluating performance)
Workshop on GIS Modeling (Part3)
Spatial Interpolation (Evaluating performance)
Assessing Interpolation Results – Residual Analysis
…the best map is the
one that has the “best
guesses”
(See Beyond Mapping III, Topic 2
for more information)
(Berry)
AVG= 23
Joseph K. Berry, BA_SIS, Inc. All Rights Reserved.

33. Spatial Interpolation (Characterizing error)
Workshop on GIS Modeling (Part3)
Spatial Interpolation (Characterizing error)
A Map of Error (Residual Map)
…shows you where your estimates are likely good/bad
(Berry)
Joseph K. Berry, BA_SIS, Inc. All Rights Reserved.

34. Point Sampling Design Concerns
DU GIS Modeling -- Surface Modeling/Analysis
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)
Joseph K. Berry, Keck Visiting Scholar

35. Point Sampling Design Concerns
DU GIS Modeling -- Surface Modeling/Analysis
Point Sampling Design Concerns
Sampling Pattern-- appropriate arrangement of samples considering both spatial interpolation and statistical inference
(Berry)
Joseph K. Berry, Keck Visiting Scholar

36. Assessing Spatial Autocorrelation (SAC indices)
DU GIS Modeling -- Surface Modeling/Analysis
Assessing Spatial Autocorrelation (SAC indices)
Geary’s C
Moran’s I
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 wij (xi – xj)2] / [(2 SUM wij) SUM (xi – m)2]
Moran’s I = [n SUM wij (xi – m) (xj – m)] / [(SUM wij) SUM (xi – m)2]
where, n = number of cells in the grid
m = the mean of the values in the grid
xi = value of cell in group i and xj = value of cell in group j
wij = switch set to 1 if the cells are adjacent; 0 if not (diagonal)
Expect…
0 < C < Strong Positive SAC I > 0
C > Strong Negative SAC I < 0
C = 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
See Beyond Mapping III, Topic 8, Investigating Spatial Dependency for more information
(Berry)
Joseph K. Berry, Keck Visiting Scholar

37. Spatial Autocorrelation/Variogram Revisited
DU GIS Modeling -- Surface Modeling/Analysis
Spatial Autocorrelation/Variogram Revisited
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?
(Berry)
Joseph K. Berry, Keck Visiting Scholar

38. DU GIS Modeling -- Surface Modeling/Analysis
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
A (50,50), (100,200)
B (50,50), (500,100)
C (100,200), (500,100)
…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
Dist = ( a2 + b2 ) .5
= ( (500-50)2 + (100-50)2 ) .5 =
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 90 45
315
270
135
180
225
Polar Variogram
100
200
300
400
500 A B C
(Berry)
Joseph K. Berry, Keck Visiting Scholar

39. Optional Opportunities
DU GIS Modeling -- Surface Modeling/Analysis
Optional Opportunities
Surfer Tutorials – experience with basic Surfer capabilities
Interpolation Techniques – additional experience with griding tools
Different Data
Different Techniques
Sampling Patterns – understanding alternative sampling pattern considerations
(Berry)
Joseph K. Berry, Keck Visiting Scholar