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 (Density Analysis, Interpolation and Map Generalization); Interpolation techniques (IDW and Krig)
Spatial Autocorrelation
Assessing interpolation results
Joseph K. Berry, Keck Visiting Scholar

2. Learning Opportunities (Waning Class Moments)
DU GIS Modeling -- Surface Modeling/Analysis
The last of the “Learning Opportunities” that remain are…
Exercise #8 on Surface Modeling (or paper) for 50 points
Exercise #9 on Spatial Data Mining (or paper) for 50 points
Exam #2 on Surface Modeling, Spatial Data Mining and Future Directions
material for 150 points
Optional Exercises for up to 50 extra credit points (can only improve your grade)
2nd Exam Study Questions …posted Monday 3/11 by 12:00noon . Class initiative to “group study” to collectively address the 30+ study questions
2nd Exam …you will download and take the 2-hour exam online (honor system) sometime between 10:00 am, Friday, March 15 and 5:00 pm, Tuesday, March 19
Special, special offer provided you fully participate in the study question “group study” you can choose not to take the second exam—
Fine print: I will simply allocate the points for the exam according to the current percentage of all of your graded materials which means not taking the exam has no effect on your grade.
If you choose to take the exam and get a grade below your current percentage of all graded materials, the exam grade will be ignored …therefore taking the exam can only improve your grade.
(Berry)
Joseph K. Berry, Keck Visiting Scholar

3. 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

4. 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

5. 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

6. GIS and Map-ematical Perspectives (Spatial Statistics)
DU Mini-Workshops on GIS Modeling -- Surface Modeling/Analysis
GIS and Map-ematical Perspectives (Spatial Statistics)
GIS as “Technical Tool” (Where is What) vs. “Analytical Tool” (Why, So What and What if)
Spatial Statistics
Map Stack
Grid Layer
Spatial Statistics seeks to map the spatial variation in a data set instead of focusing on
a single typical response (central tendency) ignoring its spatial distribution/pattern,
and thereby provides a statistical framework for map analysis and modeling of the
Numerical Spatial Relationships within and among grid map layers
Map Analysis Toolbox
Basic Descriptive Statistics (Min, Max, Median, Mean, StDev, etc.)
Basic Classification (Reclassify, Contouring, Normalization)
Map Comparison (Joint Coincidence, Statistical Tests)
Unique Map Statistics (Roving Window and Regional Summaries)
Surface Modeling (Density Analysis, Spatial Interpolation)
Advanced Classification (Map Similarity, Maximum Likelihood, Clustering)
Predictive Statistics (Map Correlation/Regression, Data Mining Engines)
Statistical Perspective:
Unique spatial
operations
(Berry)
Joseph K. Berry, Keck Visiting Scholar

7. Spatial Statistics (Linking Data Space with Geographic Space)
DU Mini-Workshops on GIS Modeling -- Surface Modeling/Analysis
Spatial Statistics (Linking Data Space with Geographic Space)
Continuous Map Surface
Geographic Distribution
Surface Modeling techniques are used to derive a continuous map surface from discrete point data– fits a Surface to the data (maps the variation).
Roving Window (weighted average)
Geo-registered Sample Data
Discrete Sample Map
Spatial
Statistics
Standard Normal Curve
Average = 22.6
Numeric Distribution
StDev =
26.2
Non-Spatial Statistics
In Data Space, a
standard normal curve can
be fitted to the data to identify
the “typical value” (average)
Histogram 70 60 50 40 30 20 10 80
In Geographic Space, the typical value forms a horizontal plane implying
the average is everywhere to
form a horizontal plane
X= 22.6
…lots of NE locations exceed Mean + 1Stdev
X + 1StDev =
= 48.8
Unusually high values
+StDev
Average
(Berry)
Joseph K. Berry, Keck Visiting Scholar

8. Spatial Statistics (clustering, correlation)
DU Mini-Workshops on GIS Modeling -- Surface Modeling/Analysis
Slope
(Percent)
Map Clustering:
Elevation
(Feet)
X axis = Elevation (SNV Normalized)
Y axis = Slope (SNV Normalized)
Elevation vs. Slope Scatterplot
Data Space
Slope draped
on Elevation
Slope
Elev
Data Pairs
Cluster 1
Cluster 2
Cluster 3
Two Clusters
Three Clusters
Geographic Space
Cluster 1
Cluster 2 +
Plots here in…
Data
Space +
Geographic
Space
Advanced Classification (Clustering)
Advanced Classification (Clustering)
Entire Map
Spatially Aggregated Correlation
Scalar Value – one value represents the overall non-spatial relationship between the two map surfaces
…where x = Elevation value and y = Slope value
and n = number of value pairs
r =
…1 large data table
with 25rows x 25 columns =
625 map values for map wide summary
Map Correlation:
Slope
(Percent)
Elevation
(Feet)
Roving Window
Localized Correlation
Map Variable – continuous quantitative surface represents the localized spatial relationship between the two map surfaces
…625 small data tables
within 5 cell reach =
81map values for localized summary
Predictive Statistics (Correlation)
Predictive Statistics (Correlation)
(Berry)
Joseph K. Berry, Keck Visiting Scholar

9. Spatial Statistics (T-test)
DU Mini-Workshops on GIS Modeling -- Surface Modeling/Analysis
Precision
Agriculture
GPS
Yield
Monitor
Cell-by-cell paired values
are subtracted
Geo-registered Grid Map Layers
The T-statistic equation is evaluated by first calculating a map of the Difference (Step 1) and then calculating maps of the Mean (Step 2) and Standard Deviation (Step 3) of the Difference within a “roving window.” The T-statistic is calculated using the derived Mean and StDev maps using the standard equation (step 4) — spatially localized solution.
Spatially Evaluating the “T-Test”
Spatial
Distribution
Numeric
Discrete point data assumed
to be spatially independent—
“randomly or uniformily” distributed in geogaphic space
Sample Plots
Traditional Agriculture Research
Step 4. Calculate the “Localized” T-statistic
(using a 5-cell roving window) for each grid cell location
Evaluate the
T-statistic Equation
5-cell radius “roving window”
…containing 73 paired values that are summarized and assigned to center cell
Col 33
Row 53
Map Comparison (Statistical Tests)
Map Comparison (Statistical Tests)
T_test
T_statistic
…the result is map of the T-statistic indicating how different the two map variables are throughout geographic space and
a T-test map indicating where they are significantly different.
(Berry)
Joseph K. Berry, Keck Visiting Scholar

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

11. Workshop on GIS Modeling (Part 3)
Spatial Dependency
Spatial Variable Dependence — what occurs at a location in geographic space is related to:
the conditions of that variable at nearby locations, termed Spatial Autocorrelation (intra-variable dependence)
Next week
Discrete Point Map
Keystone concept is…
“Spatial Autocorrelation”
Continuous Map Surface
Geographic Distribution
Surface Modeling techniques are used to derive a continuous map surface from discrete point data– fits a Surface to the data.
Inverse Distance Weighted (IDW)
spatial interpolation assigned distance-weighted average of sample points
the conditions of other variables at that location, termed Spatial Correlation (inter-variable dependence)
(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. 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

30. 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)

31. 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.

32. 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.

33. Point Sampling Design Concerns (stratification, size, grid)
DU GIS Modeling -- Surface Modeling/Analysis
Point Sampling Design Concerns (stratification, size, grid)
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

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

35. 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