Modeling Surfaces Using Triangulated Irregular Network Raster Interpolation

Modeling Surfaces Using Triangulated Irregular Network Raster Interpolation From Points The C. W. Post College of Long Island University, Arc. GIS 3 -D Analyst, http: //www. esri. com ( 307 -acre campus located 25 miles east of Manhattan , Brookville, New York)

Surface terrain model for city of Austin, TX Arc. GIS 3 -D Analyst Shoal creek Waller creek

Triangulated Irregular Network (TIN) Algorithm for interpolating irregularly-spaced data to represent terrain heights for terrain modeling UT Campus

TIN • Digital representation of the terrain • Preserves details of a shape on the terrain, more accurate representation of urban area • Break lines represent significant terrain features like a lake or cliff that cause a change in slope • Relatively few points are required to represent large, flat, or smoothly continuous areas • Requires a much smaller number of points than a gridded DTM (The digital terrain model) in order to represent the surface terrain with equal accuracy

Steps to Form a Surface From TIN • A triangular mesh is drawn on the control and determined data points • A perimeter around the data points is established. • The points are connected in a manner that smallest triangle formed from any three points is constructed (convex hull) • To connect the interior points, Delaunay triangulation is used (lines from one triangle do not cross lines of another) • A surface is created by integrating all of the triangles over the domain • Additional elevation data such as spot elevations at summits and depressions and break lines are also collected for the TIN model

A Mesh of Triangles in 2 -D Points Face Edge Points Lines Triangle is the only polygon that is always planar in 3 -D Surfaces Each triangle defines a terrain surface, or facet assumed to be of uniform slope and aspect over the triangle

TIN Triangles in 3 -D (x 3, y 3, z 3) (x 1, y 1, z 1) (x 2, y 2, z 2) z y Projection in (x, y) plane x

Delauney Triangulation • • Developed around 1930 to design the triangles efficiently Geometrically related to theissen tesselations Maximize the minimum interior angle of triangles that can be formed No point lies within the circumcircle of a triangle that is contained in mesh Yes More uniform representation of terrain No

Inputs for Creating a TIN Mass Points Soft Breaklines Hard Breaklines • Mass Points define points anywhere on landscape • Hard breaklines define locations of abrupt surface change (e. g. streams, ridges, road kerbs, building footprints, dams) • Soft breaklines are used to ensure that known z values along a linear feature are maintained in the tin.

TIN with Linear Surface Features Classroom Waller Creek UT Football Stadium City of Austin digitized all the buildings to get emergency vehicles quickly

A Portion of the TIN in Large View

Input data for this portion Mass Points not inside building Soft Breaklines along the hills Hard Breaklines along the roads

TIN Vertices and Triangles

ESRI TIN Engine Integrated Terrain Model, ARCGIS 9. 2 • Creates varying levels of conditions and points to produce pyramid style TINs on the fly • Provides an efficient methodology for working with mass data • Results in a single dataset that can rapidly deploy and visualize TIN based surfaces at multiple scale Courtesy, http: //gis. esri. com

TIN Surface Model Waller Creek Street and Bridge

Data Sources to Develop TINs • LIDAR (Light Detection and Ranging; or Laser Imaging Detection and Ranging) • Aerial photogrammetry

LIDAR • An optical remote sensing technology • Masures properties of scattered light to find range and/or other information of a distant target • LIDAR sensor was mounted on-board • During the flight, the LIDAR sensor pulses a narrow, high frequency laser pulse toward the earth through a port opening in the bottom of the aircraft's fuselage • The LIDAR sensor records the time difference between the emission of the laser beam and the return of the reflected laser signal to the aircraft • Range to an object is determined by measuring the time delay between transmission of a pulse and detection of the reflected signal to the aircraft • Points are distributed across the space, push-broom sensor • Amazing degrees of details. Resolution is 1/9 arc second • 1 arc second DEM = 30 m • 1/3 arc second DEM = 10 m

LIDAR Terrain Surface for Powder River, Wyoming Source: Roberto Gutierrez, UT Bureau of Economic Geology

NCALM National Center for Airborne Laser Mapping • Sponsored by the National Science Foundation (NSF) (http: //www. ncalm. org) • Operated jointly by the Department of Civil and Coastal Engineering, College of Engineering, University of Florida (UF) and the Department of Earth and Planetary Science, University of California- Berkeley (UCB) • Invites proposals from graduate students seeking airborne laser swath mapping (ALSM) observations covering limited areas (generally no more than 40 square kilometers) for use in research to earn an M. S. or Ph. D degree. • Proposals must be submitted on-line by November 30, 2006

Aerial photogrammetry (Stereo photographic coverage) • The aerial photos are taken using a stereoscopic camera • Two pictures of a particular area are simultaneously taken, but from slightly different angles, overlapping photographs • The overlapping area of the two resulting photos is called a stereo pair • Using a computer, stereoplotter, the stereo pair can be viewed as a single image with the appearance of depth or relief

Aerial photogrammetry (Stereo photographic coverage) • The aerial photos are taken using a stereoscopic camera • Two pictures of a particular area are simultaneously taken, but from slightly different angles, overlapping photographs • The overlapping area of the two resulting photos is called a stereo pair • Using a computer, stereoplotter, the stereo pair can be viewed as a single image with the appearance of depth or relief • Ground control points are established based on ground surveys or aerial triangulation and are viewed in the stereoplotter in conjunction with the stereo pair • The image coordinates of any (x, y, z) point in stereoscopic image pair can be determined and randomly selected and digitized

3 -D Arc. Scene, Austin, TX Aerial photogrammetry

3 -D Scene with Buildings

Some advantages of TINS • Fewer points are needed to represent the topography---less computer disk space • Points can be concentrated in important areas where the topography is variable and a low density of points can be used in areas where slopes are constant. • Points of known elevation such as surveyed benchmarks can easily be incorporated • Areas of constant elevation such as lakes can easily be incorporated • Lines of slope inflection such as ridgelines and steep canyons streams can be incorporated as breaklines in TINS to force the TIN to reflect these breaks in topography

Geostatistics Why interpolate? Analogy: Spatially distributed objects are spatially correlated; things that are close together tend to have similar characteristics

Interpolation using Rasters • Interpolation in Spatial Analyst – Inverse distance weighting (IDW) – Spline – TOPOGRID, Topo to Raster (creation of hydrologically correct digital elevation models) – Kriging (utilize the statistical properties of the measured points & quantify the spatial autocorrelation among measured points ) • Interpolation in Geostatistical Analyst

Arc. GIS Spatial Analyst to create a surface using IDW interpolation IDW weights assigned arbitrarily

Example for a linear IDW interpolation

Using the Arc. GIS Spatial Analyst to create a surface using IDW interpolation • Each input point has a local influence that diminishes with distance • It weights the points closer to the processing cell greater than those farther away • With a fixed radius, the radius of the circle to find input points is the same for each interpolated cell • By specifying a minimum count, within the fixed radius, at least a minimum number of input points is used in the calculation of each interpolated cell • A higher power puts more emphasis on the nearest points, creating a surface that has more detail but is less smooth • A lower power gives more influence to surrounding points that are farther away, creating a smoother surface. Search is more globally

Topo to Raster interpolation

Arc. GIS Spatial Analyst to create a surface using Topo to Raster interpolation • Designed for the creation of hydrologically correct digital elevation models • Interpolates a hydrologically correct surface from point, line, and polygon • Based on the ANUDEM program developed by Michael Hutchinson (1988, 1989) • The Arc. GIS 9. x implementation of Topo. Grid from Arc. Info Workstation 7. x • The only Arc. GIS interpolator designed to work intelligently with contour inputs • Iterative finite difference interpolation technique • It is optimized to have the computational efficiency of local interpolation methods, such as (IDW) without losing the surface continuity of global interpolation methods, such as Kriging and Spline

Using the Arc. GIS Spatial Analyst to create a surface using Spline interpolation • Best for generating gently varying surfaces such as elevation, water table heights, or pollution concentrations • Fits a minimum-curvature surface through the input points • Fits a mathematical function to a specified number of nearest input points while passing through the sample points • The REGULARIZED option usually produces smoother surfaces than those created with the TENSION • For the REGULARIZED, higher values used for the Weight parameter produce smoother surfaces • For the TENSION, higher values for the Weight parameter result in somewhat coarser surfaces but with surfaces that closely conform to the control points • The greater the value of Number of Points, the smoother the surface of the output raster

Example for a 2 -D Spline interpolation

Interpoloation using Kriging • Things that are close to one another are more alike than those farther away : spatial autocorrelation • As the locations get farther away, the measured values will have little relationship with the value of the prediction location Kriging weights based on semivariogram

Semi. Variagram Captures spatial dependence between samples by plotting semivariance against seperation distance • Sill The height that the semivariogram reaches when it levels off. • Range: The distance at which the semivariogram levels off to the sill • Nugget effect: a discontinuity at the origin (the measurement error and microscale variation )

Semi. Variagram h = separation distance between i an j

Case study White Hat Niobrara Missouri tributaries Elkhorn North Platte Loup Lower Platte South Platte Middle Platte Big Blue Little Blue Republican Legend High : 5422 meters Low : 836 meters River basin boundary State boundary Automatic weather station Nemaha