Statistical SurfacesOverviewSurfacesSurface RepresentationSlide 5Surface RepresentationDEMSlide 8Slide 9ContoursSlide 11Slide 12Sampling the Statistical SurfaceIsometric mapIsoplethic MapIsometric maps & samplingSlide 17The DEMDEM Raster SurfacesWhat is Interpolation?InterpolationInterpolation characteristicsSlide 23Linear InterpolationNonlinear Interpolation MethodsNon-linear interpolationSlide 27Slide 28KrigingKrigingSlide 31SemivariogramInterpolation: IssuesSlide 34Slide 35GIS for avalanche hazard mappingComparing Interpolation ResultsStatistical SurfacesGEOG370Instructor: Christine ErlienOverviewSurfaces–Representing them: Points, rasters, contours, TINs–Sampling surfaces to produce GIS data–DEMsInterpolation–What is it–Types of interpolators–Methods of interpolationSurfacesX, and Y coordinate pairs describing an areaZ-value representing magnitude of feature being considered –Can be physical (e.g., precipitation, elevation, temperature OR socioeconomic (e.g., land value, crime rate, income)Fundamental Analysis: spatial interpolationUnderlying Assumptions:1. The surface of Z values is continuous, allowing estimation of the data value at any location if sufficient information about the surface is provided2. The Z value is spatially dependentSurface RepresentationPoints may be irregularly-spacedCommonly used by land surveyors (Terrain constraints, economic considerations prevent land survey at regularly spaced points.Trade-off: data always need to be processed before they can be used for analysis or presentationhttp://www.innovativegis.com/basis/present/GIS02_geobusiness/GIS02_geoBusiness.htmSurface RepresentationSurface may be represented by raster-based data structure with a set of regularly spaced data points that form a gridWidely used in GIS to represent surface because(a) All orthogonal units are perfectly aligned, and thus they can be referenced systematically by row & column counts. (b) The system is hierarchical (each group of four units forms a larger unit)http://www.sgs.org.sa/GIS/surface.htmDEMThis animation shows flood inundation data from Hurricane Floyd overlaid onto a DEM. http://www.csc.noaa.gov/ncflood/an_gifs/tarAnimFlood.htmlSurface RepresentationThe surface may be represented by contours –A contour is a line of equal z valueContour intervals: Contour lines drawn at an interval (change in z value between the contour lines) that you specify. Where lines are closer together, change in values is more rapid. Elevation and barometric pressure are commonly mapped using contours.From Demers (2005) Introduction to Geographic InformationThe areas where the contours are closer together indicate the steeper locations. In this case, correspond with the areas of higher elevation (in white on the input elevation raster).Contourshttp://www.gis.unbc.ca/courses/geog205/labs/lab14/index.phpInput elevation grid Output contour mapSurface RepresentationThe surface may be organized as a triangulated irregular networks (TINs) –The surface is converted into a set of triangular facetsAdvantage: The TIN structure allows the density of points to vary according to terrain.From Geographic Information Systems & Science, Longley et al. (2005)Vector Model to Represent Surfaces: TINSampling the Statistical SurfaceWhy sample?–Continuous surfaces have an infinite # of possible z valuesSampling a surface to produce maps:–Isarithmic: General term for lines connecting points of equal statistical value (e.g., map employing contour lines)•Isometric: Based on control points that have actual observed values–Most commonly applied to natural surface data (e.g., elevation, temperature, barometric pressure)•Isoplethic: Drawn on the basis of areal averages assigned to arbitrary control points (e.g., centroid)–Not directly observable apply to discrete surface data (e.g., population density, incidence rates)Isometric maphttp://www.d.umn.edu/geog/cartfolder/HTML%20Pages/Isarithmic1.htmMean annual sulfate deposition across Pennsylvania and neighboring states before (1983-1994) and after (1995-2004) implementation of Title IV of the Clean Air Act Amendments of 1990.http://www.dep.state.pa.us/DEP/DEPUTATE/airwaste/aq/acidrain/report_caaa/sulfate_dep.pdfIsoplethic MapIsometric maps & samplingRegular lattice: Regular sampling grid; all objects same distance away from one another–Advantage: Simple–Disadvantage: Not appropriate for sampling rapidly changing surface valuesIrregular lattice: Samples not spaced evenly; sampling determined by prior knowledge of surface values–Where surface is smooth (little change/unit distance)  fewer samples–Rough surface (rapidly changing values)  more samplesFrom Demers (2005) Introduction to Geographic InformationThe DEMDEM: Group of methods for storing z-value info–Based on mathematical or image models–Surface represented in raster or vector (TIN)USGS DEM files: Elevations sampled at regularly spaced intervals. –Raster surface–Formats: 7.5-minute, 15-minute, 2-arc-second (also known as 30-minute), and 1-degree –See http://edc.usgs.gov/products/elevation/dem.htmlDEM Raster Surfaces1014 15 10915 16 111117 16 111013 14 912 1313 13Small cell size  better resolution  more detailCell values indicate elevation Worldwide coverage: coarser resolutions (GTOPO30: 1 km, SRTM 90m)30 m DEMs available for most of U.S. (except Alalska)Graphics by Jun Liang, UNC-Chapel Hill, Department of GeographyWhat is Interpolation?The process of estimating unknown values that fall between known valuesInterpolationWhy do it?–we have “sparse” data samples that do not completely cover a study area, and we want complete coverage•Usually done to represent phenomena that are continuous in realityExamples: –turn spot/point samples of elevation heights into a DEM–turn temperature samples from weather stations into a temperature surfaceInterpolation characteristicsExact vs. inexact interpolators–Exact: at sample points, the interpolation returns the same value as the original sample–Inexact: do not return same value at sample points (usually optimize some other characteristic, e.g. minimize squared deviations of interpolated values from sampled values)Global vs. local interpolators–Global: utilize all sample points to create interpolated surface–Local: utilize subset of sample points in neighborhood of each location being estimated to create