Introduction to GIS ModelingIntroduction to GIS Modeling Week 8 — Surface ModelingWeek 8 — Surface Modeling

GEOG 3110 –University of DenverGEOG 3110 –University of Denver

Presented by Presented by Joseph K. BerryJoseph K. BerryW. M. Keck Scholar, Department of Geography, University W. M. Keck Scholar, Department of Geography, University

of Denverof Denver

Digital Elevation Model (DEM)Digital Elevation Model (DEM)Basic surface modeling Basic surface modeling (Density Analysis, Interpolation and Map Generalization)(Density Analysis, Interpolation and Map Generalization); ;

Interpolation techniques Interpolation techniques (IDW and Krig)(IDW and Krig) Spatial AutocorrelationSpatial Autocorrelation

Assessing interpolation resultsAssessing interpolation results

Learning Opportunities Learning Opportunities (Waning Class Moments)(Waning Class Moments)

(Berry)

The last of the The last of the ““Learning OpportunitiesLearning Opportunities” ” that remain are…that remain are…

• Exercise #8 on Exercise #8 on Surface ModelingSurface Modeling (or paper) for 50 points (or paper) for 50 points• Exercise #9 on Exercise #9 on Spatial Data MiningSpatial Data Mining (or paper) for 50 points (or paper) for 50 points• Exam #2Exam #2 on Surface Modeling, Spatial Data Mining and Future Directions on Surface Modeling, Spatial Data Mining and Future Directions material for 150 pointsmaterial for 150 points• Optional ExercisesOptional Exercises for up to 50 extra credit points for up to 50 extra credit points (can only improve your grade)(can only improve your grade)

Special, special offer Special, special offer provided you fully participate in the study question “group study” provided you fully participate in the study question “group study” you can choose not to take the second examyou can choose not to take the second exam— —

Fine printFine print: I will simply allocate the points for the exam according to the current percentage of all of your graded : I will simply allocate the points for the exam according to the current percentage of all of your graded materials which means materials which means not taking the exam has no effect on your gradenot 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 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 …grade will be ignored …therefore taking the exam can only improve your gradetherefore taking the exam can only improve your grade. .

22ndnd Exam Study Questions Exam Study Questions ……posted Monday 3/11 by 12:00noonposted Monday 3/11 by 12:00noon . Class initiative to . Class initiative to “group study” to collectively address the 30+ study questions“group study” to collectively address the 30+ study questions

22ndnd Exam Exam ……you will download and take the 2-hour exam online (honor system) you will download and take the 2-hour exam online (honor system) sometime between sometime between 10:00 am, Friday, March 15 and and 5:00 pm, Tuesday, March 19

Visualizing Terrain Surface Data Visualizing Terrain Surface Data (Exercise 8 – Part 1)(Exercise 8 – Part 1)

Mount St. Helens datasetMount St. Helens dataset

Question 1Question 1Access SURFER Access SURFER then enter Mapthen enter Map Contour Map Contour Map New Contour Map New Contour Map \Samples \Samples Helens2.grd Helens2.grd

(Berry)

There are numerous websites that allow you to There are numerous websites that allow you to

download a DEM and usedownload a DEM and use SURFERSURFER to visualizeto visualize

……a generally useful procedure that you can a generally useful procedure that you can use for lots of reports use for lots of reports (Optional Exercise)(Optional Exercise)

Visualizing Map Surfaces Visualizing Map Surfaces (Exercise 8 – Part 1)(Exercise 8 – Part 1)

Questions 2 and 3Questions 2 and 3

Use SURFER Use SURFER to Create a 2D Contour map and a 3-D Wireframe mapto Create a 2D Contour map and a 3-D Wireframe map

(Berry)(Berry)

Map Analysis EvolutionMap Analysis Evolution (Revolution)(Revolution)

(Berry)(Berry)

Traditional GISTraditional GIS

• Points, Lines, PolygonsPoints, Lines, Polygons

• Discrete ObjectsDiscrete Objects

• Mapping and Geo-queryMapping and Geo-query

Forest Inventory Forest Inventory MapMap

Spatial AnalysisSpatial Analysis

• Cells, Surfaces Cells, Surfaces

• Continuous Geographic SpaceContinuous Geographic Space

• ContextualContextual Spatial Relationships Spatial Relationships

StoreStoreTravel-TimeTravel-Time

(Surface)(Surface)

Traditional StatisticsTraditional Statistics

• Mean, StDev (Normal Curve)Mean, StDev (Normal Curve)

• Central TendencyCentral Tendency

• Typical Response (scalar) Typical Response (scalar)

Minimum= 5.4 ppmMinimum= 5.4 ppmMaximum= 103.0 ppmMaximum= 103.0 ppm

Mean= 22.4 ppmMean= 22.4 ppmStDevStDev= 15.5= 15.5

Spatial StatisticsSpatial Statistics

• Map of Variance Map of Variance (gradient)(gradient)

• Spatial DistributionSpatial Distribution

• NumericalNumerical Spatial Relationships Spatial Relationships

Spatial Spatial DistributionDistribution(Surface)(Surface)

GIS and Map-ematical Perspectives GIS and Map-ematical Perspectives (Spatial Statistics)(Spatial Statistics)

(Berry)

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

GIS as “Technical Tool” (Where is What) vs. “Analytical Tool” (Why, So What and What if)

Map StackGrid Layer

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

Spatial Statistics Spatial Statistics (Linking Data Space with Geographic Space)

(Berry)

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).

Geo-registered Sample Data

Discrete Sample Map

SpatialStatistics

Histogram

706050403020100 80

In Geographic Space, the typical value forms a horizontal plane implying

the average is everywhere toform a horizontal plane

X= 22.6

…lots of NE locations exceed Mean + 1Stdev

X + 1StDev= 22.6 + 26.2

= 48.8

Unusually high

values+StDev

Average

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)

Roving Window (weighted average)

Spatial Statistics Spatial Statistics (clustering, correlation)(clustering, correlation)

X axis = Elevation (SNV Normalized)Y axis = Slope (SNV Normalized)

Elevation vs. Slope Scatterplot

Data Space

Slope draped on Elevation

Slo

pe

Elev

Entire Map

Spatially Aggregated CorrelationScalar Value – one value represents the overall non-spatial relationship between the two map surfaces

…where x = Elevation value and y = Slope valueand n = number of value pairs

r =

…1 large data table with 25rows x 25 columns =

625 map values for map wide summary

Cluster 1

Cluster 2 Cluster 1

Cluster 2Cluster 3

Two Clusters

Three Clusters

Geographic Space

(Berry)

Slope(Percent)

Map Clustering:

Elevation(Feet)

Data Pairs

+

Plots here in…

Data Space

+Geographic

Space

Advanced Classification (Clustering)

Predictive Statistics (Correlation)

Advanced Classification (Clustering)

Predictive Statistics (Correlation)

Map Correlation:

Slope(Percent)

Elevation(Feet)

Roving Window

Localized CorrelationMap 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

Spatial Statistics Spatial Statistics (T-test)(T-test)

Discrete point data assumed to be spatially independent—

“randomly or uniformily” distributed in geogaphic space

Sample PlotsSample Plots

Traditional Agriculture ResearchPrecisionAgriculture

GPSYield Monitor

T_statistic

T_test

SpatialDistribution

NumericDistribution

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”Cell-by-cell paired values are subtracted

Geo-registered Grid Map Layers

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)

Step 4. Calculate the “Localized” T-statistic (using a 5-cell roving window) for each grid cell location

Evaluate the T-statistic Equation

…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.

Map Comparison (Statistical Tests)

(Berry)

An Analytic Framework for GIS ModelingAn Analytic Framework for GIS Modeling

(Berry)(Berry)

Surface ModellingSurface Modelling operations involve operations involve creating continuous spatial distributions creating continuous spatial distributions from point sampled data.from point sampled data.

(GIS Modeling Framework paper)(GIS Modeling Framework paper)

Spatial DependencySpatial Dependency

Spatial Variable DependenceSpatial Variable Dependence — what occurs at a location — what occurs at a location in geographic space is related to:in geographic space is related to:

• the conditions of that variable at nearby locations, termed the conditions of that variable at nearby locations, termed Spatial Spatial AutoAutocorrelation correlation ((intra-variableintra-variable dependence) dependence)

(Berry)(Berry)

Keystone concept is…

“Spatial Autocorrelation” 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 the conditions of other variables at that location, termed Spatial CorrelationSpatial Correlation ( (inter-variableinter-variable dependence) dependence) Next weekNext week

Non-Spatial Statistics Non-Spatial Statistics (Central Tendency; typical response)(Central Tendency; typical response)

(Berry)(Berry)

……seeks to reduce a set of data to seeks to reduce a set of data to a single value that is typical a single value that is typical of the entire data set of the entire data set (Average) and generally assess how typical the typical is (StDev)(Average) and generally assess how typical the typical is (StDev)

Assumptions in Non-Spatial StatisticsAssumptions in Non-Spatial Statistics

(Berry)(Berry)

……uniformly distributeduniformly distributed in geographic space (horizontal plane at average; +/- constant) in geographic space (horizontal plane at average; +/- constant)

Geographic Distribution Geographic Distribution (surface modeling)(surface modeling)

(Berry)(Berry)

……analogous to fitting a curveanalogous to fitting a curve (Standard Normal Curve) in numeric space except (Standard Normal Curve) in numeric space except fitting a map surface fitting a map surface in geographic space to explain variation in the data in geographic space to explain variation in the data

Adjusting for Spatial RealityAdjusting for Spatial Reality (masking for discontinuities)(masking for discontinuities)

(Berry)(Berry)

……accounting for known geographic discontinuities or other spatial relationshipsaccounting for known geographic discontinuities or other spatial relationships

Generating a Map of Percent ChangeGenerating a Map of Percent Change (map-ematics)(map-ematics)

(Berry)(Berry)

……maps are organized sets of numbers supporting a robust range of maps are organized sets of numbers supporting a robust range of Map Analysis Map Analysis operations that can be used to relate spatial variables (map layers)operations that can be used to relate spatial variables (map layers)

Spatial RelationshipsSpatial Relationships (coincidence , proximity, etc.)(coincidence , proximity, etc.)

(Berry)(Berry)

……spatial relationships can be utilized to extend understandingspatial relationships can be utilized to extend understanding

Standard Normal Variable MapStandard Normal Variable Map

(Berry)(Berry)

……relates every location to the typical response (Average and StDev) relates every location to the typical response (Average and StDev) to determine how typical it isto determine how typical it is

The Average is Hardly AnywhereThe 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

Surface ModelingSurface Modeling (Map generalization)(Map generalization)

(Berry)

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)

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—

Yavg

XavgLine

Plane

CurvedPlane

CurvedLine

Surface ModelingSurface Modeling (Iterative Smoothing)(Iterative Smoothing)

The “The “iterative smoothingiterative smoothing” process is similar to slapping a big chunk ” process is similar to slapping a big chunk of modeler’s clay over the “data spikes,” then taking a knifeof modeler’s clay over the “data spikes,” then taking a knife

and cutting away the excess to leave a and cutting away the excess to leave a continuous surfacecontinuous surface that encapsulates that encapsulates the the peakspeaks and and valleysvalleys implied in the original field samples… implied in the original field samples…

(Berry)

……repeated smoothing repeated smoothing slowly “erodes” the data slowly “erodes” the data

surface to a flat planesurface to a flat plane

= = AVERAGEAVERAGE

……Spatial Spatial InterpolationInterpolation

techniques utilize techniques utilize summary of data summary of data

in a roving in a roving windowwindow

((Localized VariationLocalized Variation))

Digital slide show SStat2.pptDigital slide show SStat2.ppt

Surface Modeling Methods Surface Modeling Methods (Surfer)(Surfer)

Inverse DistanceInverse Distance to a Power to a Power — weighted average of samples in the summary — weighted average of samples in the summary window such that the influence of a sample point declines with “simple” distancewindow such that the influence of a sample point declines with “simple” distance

Modified Shepard’s Method Modified Shepard’s Method — uses an inverse distance “least squares” method — uses an inverse distance “least squares” method that reduces the “bull’s-eye” effect around sample pointsthat reduces the “bull’s-eye” effect around sample points

Radial Basis Function Radial Basis Function — uses non-linear functions of “simple” distance to — uses non-linear functions of “simple” distance to determine summary weightsdetermine summary weights

KrigingKriging — summary of samples based on distance and angular trends in the data— summary of samples based on distance and angular trends in the data

Natural Neighbor Natural Neighbor —weighted average of neighboring samples where the weights —weighted average of neighboring samples where the weights are proportional to the “borrowed area” from the surrounding points (based on are proportional to the “borrowed area” from the surrounding points (based on differences in Thiessen polygon sets)differences in Thiessen polygon sets)

Minimum Curvature Minimum Curvature — — analogous to fitting a thin, elastic plate through each analogous to fitting a thin, elastic plate through each sample point using a minimum amount of bendingsample point using a minimum amount of bending

Polynomial Regression Polynomial Regression — — fits an equation to the entire set of sample points fits an equation to the entire set of sample points

Nearest NeighborNearest Neighbor— assigns the value of the nearest sample point— assigns the value of the nearest sample point

TriangulationTriangulation— identifies the “optimal” set of triangles connecting — identifies the “optimal” set of triangles connecting all of the sample points all of the sample points Thiessen PolygonsThiessen Polygons

(Berry)(Berry)

Map Generalization — Mathematical Equation/Surface FittingMap Generalization — Mathematical Equation/Surface Fitting

Map Generalization — Geometric facetsMap Generalization — Geometric facets

Spatial Interpolation — “roving window” localized averageSpatial Interpolation — “roving window” localized average

Surface Modeling Approaches Surface Modeling Approaches (using point samples)(using point samples)

Spatial InterpolationSpatial Interpolation— — these techniques use a roving window to identify these techniques use a roving window to identify Nearby SamplesNearby Samples and then and then Summarize the SamplesSummarize the Samples based on some function of their relative nearness to the location based on some function of their relative nearness to the location being interpolated.being interpolated.

Window ReachWindow Reach— — how far away to reach to collect sample points for processinghow far away to reach to collect sample points for processing

Window ShapeWindow Shape— — shape of the window can be symmetrical (circle) or asymmetrical (ellipse)shape of the window can be symmetrical (circle) or asymmetrical (ellipse)

Summary TechniqueSummary Technique— — a weighted average based on proximity using a fixed geometric a weighted average based on proximity using a fixed geometric relationship (inverse distance squared) or a more complex statistical relationship (spatial relationship (inverse distance squared) or a more complex statistical relationship (spatial autocorrelation)autocorrelation)

Exacting SolutionExacting Solution— — exacting solutions result in the sample value being retained (Krig); exacting solutions result in the sample value being retained (Krig); non-exacting estimate sample locations (IDW)non-exacting estimate sample locations (IDW)

(Berry)(Berry)

Map GeneralizationMap Generalization (Equation) (Equation) — — these techniques seek the general trend in the these techniques seek the general trend in the data by data by Fitting a Polynomial EquationFitting a Polynomial Equation to the entire set of sample data (1 to the entire set of sample data (1stst degree polynomial is degree polynomial is a plane).a plane).

ThiessenThiessen Polygons Polygons

Map GeneralizationMap Generalization (Geometric Facets) (Geometric Facets) — — Triangulated Irregular Triangulated Irregular Network (TIN) is a form of the tessellated model based on Network (TIN) is a form of the tessellated model based on TrianglesTriangles. The . The vertices of the triangles form irregularly spaced nodes and unlike the DEM, vertices of the triangles form irregularly spaced nodes and unlike the DEM, the TIN allows dense information in complex areas, and sparse information in the TIN allows dense information in complex areas, and sparse information in simpler or more homogeneous areas. simpler or more homogeneous areas. http://www.jarno.demon.nl/gavh.htm http://www.jarno.demon.nl/gavh.htm

Spatial Interpolation Spatial Interpolation (Mapping spatial variability)(Mapping spatial variability)

(Berry)(Berry)

……the geo-registered soil the geo-registered soil samples form a pattern of samples form a pattern of ““spikesspikes” throughout the ” throughout the field. Spatial field. Spatial Interpolation is similar to Interpolation is similar to throwing a blanket over throwing a blanket over the spikes that conforms the spikes that conforms to the pattern.to the pattern.

……all interpolation algorithms assume all interpolation algorithms assume 1) “1) “nearby things are more alike than nearby things are more alike than distant thingsdistant things” (spatial autocorrelation), ” (spatial autocorrelation), 2) appropriate 2) appropriate sampling intensitysampling intensity, and , and 3) suitable 3) suitable sampling patternsampling pattern

… …maps the spatial variation in point sampled datamaps the spatial variation in point sampled data

Spatial Interpolation Spatial Interpolation (Comparing (Comparing Average and IDWAverage and IDW results) results)

Comparison of the interpolated surface to the whole field Comparison of the interpolated surface to the whole field average shows average shows large differenceslarge differences in localized estimates in localized estimates

(Berry)(Berry)

Spatial Interpolation Spatial Interpolation (Comparing (Comparing IDW and KrigIDW and Krig results) results)

(Berry)(Berry)

Comparison of the IDW and Krig interpolated surfaces Comparison of the IDW and Krig interpolated surfaces shows shows small differencessmall differences in in localized estimates in in localized estimates

Creating and Comparing Map SurfacesCreating and Comparing Map Surfaces

Use SURFER Use SURFER to to CreateCreate and and CompareCompare map surfaces (Exercise 8, Part 2) map surfaces (Exercise 8, Part 2)

(Berry)(Berry)

IDW

Krig

CreateCreate

IDW - Krig

CompareCompare

Inverse Distance Weighted ApproachInverse Distance Weighted Approach

(Berry)

Spatial Autocorrelation Spatial Autocorrelation (Kriging)(Kriging)

Tobler’s First Law of GeographyTobler’s First Law of Geography— — nearby things are more alike than distant thingsnearby things are more alike than distant things

VariogramVariogram— — plot of sample data similarity as a function of distance between samplesplot of sample data similarity as a function of distance between samples

(Berry)

……Kriging uses regional variable theory based on an underlying variogram to develop Kriging uses regional variable theory based on an underlying variogram to develop custom weightscustom weights based on trends in the sample data (proximity and direction) based on trends in the sample data (proximity and direction)

……uses uses Variogram EquationVariogram Equation instead of a fixed 1/D instead of a fixed 1/DPowerPower Geometric Equation Geometric Equation

Data relationships determine weights (function of distance and data patterns)Data relationships determine weights (function of distance and data patterns)

Spatial Interpolation TechniquesSpatial Interpolation Techniques

((BerryBerry))

Characterizes the spatial distribution by fitting a mathematical equation Characterizes the spatial distribution by fitting a mathematical equation to localized portions of the data (roving window)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

Spatial Interpolation Spatial Interpolation (Evaluating performance)(Evaluating performance)

(Berry)(Berry)Assessing Interpolation Results – Residual AnalysisAssessing Interpolation Results – Residual Analysis

……the best map is the the best map is the one that has the “one that has the “bestbestguessesguesses””

(See Beyond Mapping III, Topic 2 (See Beyond Mapping III, Topic 2 for more information)for more information) AVG= 23

Spatial Interpolation Spatial Interpolation (Characterizing error)(Characterizing error)

A Map of Error (Residual Map)A Map of Error (Residual Map)……shows you shows you wherewhere your estimates are likely good/bad your estimates are likely good/bad

(Berry)

Point Sampling Design Concerns Point Sampling Design Concerns (stratification, size, grid)(stratification, size, grid)

StratificationStratification---- appropriate appropriate groupingsgroupings for sampling for samplingSample SizeSample Size---- appropriate sampling appropriate sampling intensityintensity for each stratified for each stratified groupgroupSampling GridSampling Grid---- appropriate appropriate reference reference grid for locating individual grid for locating individual

point samples (nested best)point samples (nested best)

(Berry)

Point Sampling Design Concerns Point Sampling Design Concerns (pattern)(pattern)

Sampling PatternSampling Pattern-- -- appropriate arrangementappropriate arrangement of samples considering of samples considering both spatial interpolation and statistical inferenceboth spatial interpolation and statistical inference

(Berry)

Optional OpportunitiesOptional Opportunities

(Berry)(Berry)

Surfer TutorialsSurfer Tutorials – experience with basic Surfer capabilities– experience with basic Surfer capabilities

Sampling PatternsSampling Patterns – understanding alternative sampling pattern considerations– understanding alternative sampling pattern considerations

Interpolation TechniquesInterpolation Techniques – additional experience with griding tools– additional experience with griding tools

Different Data

Different Techniques