Transcript No Slide Title

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
Visualizing Terrain Surface Data (Exercise 8 – Part 1)
Question 1
Access SURFER then enter Map Contour Map New Contour Map \Samples Helens2.grd
Mount St. Helens dataset
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)
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)
Map Analysis Evolution (Revolution)
Traditional GIS
Spatial Analysis
Store
Travel-Time
(Surface)
Forest Inventory
Map
• Points, Lines, Polygons
• Cells, Surfaces
• Discrete Objects
• Continuous Geographic Space
• Mapping and Geo-query
• Contextual Spatial Relationships
Traditional Statistics
Spatial Statistics
Spatial
Distribution
(Surface)
Minimum= 5.4 ppm
Maximum= 103.0 ppm
Mean= 22.4 ppm
StDev= 15.5
• Mean, StDev (Normal Curve)
• Map of Variance (gradient)
• Central Tendency
• Spatial Distribution
• Typical Response (scalar)
• Numerical Spatial Relationships
(Berry)
An Analytic Framework for GIS Modeling
Surface Modelling operations involve
creating continuous spatial distributions
from point sampled data.
See www.innovativegis.com/basis/Download/IJRSpaper/
(Berry)
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)
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)
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)
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)
Assumptions in Non-Spatial Statistics
…uniformly distributed in geographic space (horizontal plane at average; +/- constant)
(Berry)
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)
Adjusting for Spatial Reality (masking for discontinuities)
…accounting for known geographic discontinuities or other spatial relationships
(Berry)
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)
Spatial Relationships (coincidence , proximity, etc.)
…spatial relationships can be utilized to extend understanding
(Berry)
Standard Normal Variable Map
…relates every location to the typical response (Average and StDev)
to determine how typical it is
(Berry)
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)
Surface Modeling (Map generalization)
Arithmetic Average balances
“half” of the data on either
side of a Line—
Line
Xavg
Yavg
Spatial Average balances
“half” of the data above and
below a Horizontal Plane—
Plane
Curved
Plane
Curved
Line
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)
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)
Surface Modeling Methods (Surfer)
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
Map Generalization — Mathematical Equation/Surface Fitting
— fits
an equation to the entire set of sample points
Map Generalization — Geometric facets
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)
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).
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
Thiessen Polygons
(Berry)
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)
Spatial Interpolation (Comparing Average and IDW results)
Comparison of the interpolated surface to the whole field
average shows large differences in localized estimates
(Berry)
Spatial Interpolation (Comparing IDW and Krig results)
Comparison of the IDW and Krig interpolated surfaces
shows small differences in in localized estimates
(Berry)
Creating and Comparing Map Surfaces
Use SURFER to Create and Compare map surfaces
(Exercise 8, Part 2)
Create
IDW
Krig
Compare
IDW - Krig
(Berry)
Inverse Distance Weighted Approach
(Berry)
Inverse Distance Weighted Calculations
Geometry determines weights (function of distance)
…SUMwProducts / SumWeights = wAverage
…Weight
…weight * value = weighted Product
…but first things first, how do you calculate a weighted average?
(Berry)
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
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)
Spatial Interpolation Techniques
Characterizes the spatial distribution by fitting a mathematical equation
to localized portions of the data (roving window)
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.
AVG= 23 everywhere
Inverse Distance Weighted (IDW)
technique weights the samples
such that values farther away
contribute less to the average
…1/Distance Power
(Berry)
Spatial Interpolation (Evaluating performance)
Assessing Interpolation Results – Residual Analysis
(Berry)
…the best map is the
one that has the “best
guesses”
(See Beyond Mapping III,
Topic 2
for more information)
AVG= 23
Spatial Interpolation (Characterizing error)
A Map of Error (Residual Map)
…shows you where your estimates are likely good/bad
(Berry)
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)
Point Sampling Design Concerns
Sampling Pattern-- appropriate arrangement of samples considering both
spatial interpolation and statistical inference
(Berry)
Assessing Spatial Autocorrelation
(SAC indices)
Geary’s C
Geary’s C and Moran’s I — similarity among center cell and adjacent
neighbors vs. center cell and average for the entire data set…
Moran’s I
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 –
where,
m)2]
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<1
C>1
C=1
Strong Positive SAC
Strong Negative SAC
Random distribution
I>0
I<0
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)
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)
Polar Variogram (Similarity, Distance and Angle)
For the three points shown below, the separation
of the three pairs can be summarized as …
Dist = ( a2 + b2 ) .5
Point Pair
Dist
A (50,50), (100,200)
158.11
B (50,50), (500,100)
452.77
C (100,200), (500,100) 412.31
Angle = ARCTAN ( opp / adj )
= ( (500-50)2 + (100-50)2 ) .5
= 452.77
Angle
71.11
6.34
-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
= ARCTAN ( (100-50) / (500-50) )
= 6.34
90
135
45
500
400
A
300
200
Point Pair B
180
100
B
0
#2 (100,150)
C
#3 (500,100)
a
#1 (50,50)
Geographic Space
b
225
Polar Variogram
315
270
(Berry)
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)