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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 Learning Opportunities (Waning Class Moments) 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) 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 Traditional GIS (Revolution) 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) GIS and Map-ematical Perspectives (Spatial Statistics) GIS as “Technical Tool” (Where is What) vs. “Analytical Tool” (Why, So What and What if) Grid Layer Map Stack 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 Unique spatial operations (Berry) Statistical Perspective: 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) Spatial Statistics (Linking Data Space with Geographic Space) Roving Window (weighted average) Geo-registered Sample Data Geographic Distribution Spatial Statistics Discrete Sample Map Non-Spatial Statistics Continuous Map Surface Surface Modeling techniques are used to derive a continuous map surface Standard Normal Curve from discrete point data– fits a Surface to the data (maps the variation). Average = 22.6 In Geographic Space, the typical value forms a horizontal plane implying the average is everywhere to form a horizontal plane StDev = 26.2 Histogram …lots of NE locations exceed Mean + 1Stdev X + 1StDev = 22.6 + 26.2 = In Data Space, a standard normal curve can be fitted to the data to identify the “typical value” (average) 0 10 20 30 40 50 Numeric Distribution (Berry) 60 70 80 Unusually high values X= 22.6 +StDev Average 48.8 Spatial Statistics (clustering, correlation) Map Clustering: Elevation vs. Slope Scatterplot Data Pairs Cluster 2 Cluster 1 Cluster 2 Cluster 3 Plots here in… Data Space Elevation Geographic Space (Feet) + Slope + Slope (Percent) Slope draped on Elevation Cluster 1 Elev Three Clusters X axis = Elevation (SNV Normalized) Y axis = Slope (SNV Normalized) Advanced Classification (Clustering) Map Correlation: Roving Window Data Space Spatially Aggregated Correlation Scalar Value – one value represents the overall nonspatial relationship between the two map surfaces …1 large data table Entire Map Elevation (Feet) with 25rows x 25 columns = 625 map values for map wide summary r= …where x = Elevation value and y = Slope value and n = number of value pairs Slope …625 small data tables (Percent) within 5 cell reach = 81map values for localized summary Localized Correlation Predictive Statistics (Correlation) (Berry) Two Clusters Map Variable – continuous quantitative surface represents the localized spatial relationship between the two map surfaces Geographic Space Spatial Statistics (T-test) Cell-by-cell paired values are subtracted Spatially Evaluating the “T-Test” Yield Monitor GPS Precisionis evaluated by first calculating a The T-statistic equation Traditional Agriculture Research Agriculture Numeric map of the Difference (Step 1) and then calculating maps of Distribution the Mean (Step 2) and Standard Deviation (Step 3) of the Difference within a “roving window.” Sample The T-statistic Plots is calculated using the derived Mean and StDev maps using the standard equation (step 4) — point spatially Discrete datalocalized assumed solution. Geo-registered Grid Map Layers Spatial Distribution 5-cell radius “roving window” …containing 73 paired values that are summarized and assigned to center cell to be spatially independent— “randomly or uniformily” distributed in geogaphic space Map Comparison (Statistical (Statistical Tests) Tests) Calculate the “Localized” T-statistic (using a 5-cell roving window) for each grid cell location Step 4. T_test …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. Evaluate the T-statistic Equation (Berry) T_statistic An Analytic Framework for GIS Modeling Surface Modelling operations involve creating continuous spatial distributions from point sampled data. (GIS Modeling Framework paper) (Berry) 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) Keystone concept is… “Spatial Autocorrelation” Inverse Distance Weighted (IDW) Surface Modeling techniques are used to derive a continuous map surface from discrete point data– fits a Surface to the data. Geographic Distribution Discrete Point Map spatial interpolation assigned distanceweighted average of sample points Continuous Map Surface • the conditions of other variables at that location, termed Next week Spatial Correlation (inter-variable dependence) (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) 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, 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) Point Sampling Design Concerns (pattern) Sampling Pattern-- appropriate arrangement of samples considering both spatial interpolation and statistical inference (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)