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Lecture 07: Terrain Analysis Geography 128 Analytical and Computer Cartography Spring 2007 Department of Geography University of California, Santa Barbara 3D Transformations 3D data often for land surface or bottom of ocean Need three coordinates to determine location (X, Y, Z) Part of analytical cartography concerned with analysis of fields is terrain analysis Include terrain representation and symbolization issues as they relate to data Points, TIN and grids are used to store terrain Interpolation to a Grid Given a set of point elevations (x, y, z) generate a new set of points at the nodes of a regular grid so that the interpolated surface is a reasonable representation of the surface sampled by the points. Imposes a model of the true surface on the sample "Model" is a mathematical model of the neighborhood relationship Influence of a single point = f(1/d) Can be constrained to fit all points Should contain z extremes, and local extreme values Most models are algorithmic local operators Work cell-to-cell. Operative cell = kernel Weighting Methods R Impose z = f (1/d) Computational Intensive, e.g. 200 x 200 cells 1000 points = 40 x 10^6 distance calculations If all points are used and sorted by distance, called "brute force" method Possible to use sorted search and tiling Distance can be weighted and powered by n = friction of distance Can be refined with break lines Zi , j Z d p 1 R p d p 1 n p n p Clarke’s Classic IDW Algorithm Assigns points to cells Averages multiples For all unfilled cells, search outward using an increasingly large square neighborhood until at least n points are found Apply inverse distance weighting Trend Projection Methods Way to overcome high/low constraint Assumes that sampling missed extreme values Locally fits trend, trend surface or bi-cubic spline Least squares solution Useful when data are sparse, texture required Search Patterns Many possible ways to define interpolated "region" Can use # points or distance Problems in – – – Sparse areas Dense areas Edges Bias can be reduced by changing search strategy Kriging Interpolation "Optimal interpolation method" by D.G. Krige Origin in geology (geostatistics, gold mining) Spatial variation = f(drift, random-correlated, random noise) To use Kriging – – – – Model and extract drift Compute variogram Model variogram Compute expected variance at d, and so best estimate of local mean Several alternative methods. Universal Kriging best when local trends are well defined Kriging produces best estimate and estimate of variance at all places on map For more info: http://www.geog.ucsb.edu/~good/176b/n10.html Alternative Methods Many ways to make the point-togrid interpolation Invertibility? Can results be compared and tested analytically Use portion of points and test results with remainder Examine spatial distribution of difference between methods Best results are obtained when field is sampled with knowledge of the terrain structure and the method to be used Surface-Specific Point Sampling Landscape Morphometric Features Terrain "Skeleton" Surface-Specific Point Sampling (cnt.) If the structure of the terrain is known, then intelligent design of sampling and interpolation is best Terrain Skeleton determines most of surface variance Knowledge of skeleton often critical for applications Surface-Specific Point Sampling (cnt.) Source of much terrain data is existing contour maps Problems of contour->TIN or Grid are many, e.g. the wedding cake effect Sampling along contour "fills in" interpolated values Surface Models Alternative to LOCAL operators is to model the whole surface at once Often must be an inexact fit, e.g. when there are many points Sometimes Model is surface is sufficient for analysis Polynomial Series – Least squares fit of polynomial function in 2D. – Simplest form is the linear trend surface, e.g. z = bo + b1x + b2y – Most complex forms have bends and twists Fourier Series – Fit trigonometric series of cosine waves with different wavelengths and amplitudes. – Analytically, can generalize surface by "extracting" harmonics Polynomial Surface Surface Filtering Convolution of filter matrix with map matrix Filter has a response function Filter weights add to one Can enhance properties, or generalize Volumetric Transformations - Slope and Aspect Many possible analytical transformations of 3D data that show interesting map properties Simplest is slope (first derivative, the steepest downhill slope) and aspect (the direction of the steepest downhill slope) Volumetric Transformations - Slope and Aspect (ArcGIS) Volumetric Transformations (cnt.) Terrain partitioning: Often to extract VIPs or a TIN from a grid. Terrain Simulation (many methods e.g. fractals) Intervisibility, e.g. viewshed Terrain Symbolization - Analytical Hill Shading Simulate illumination from an infinite distance light source Light source has azimuth and zenith angle Surface can be reflected light or use log transform Can add shadows for realism, or multiple light sources Terrain Symbolization - Gridded Perspective & Realistic Pespective Create view from a particular camera geometry Can include or excluded perspective Colors should include shading Multiple sequences can generate fly-bys and fly-thrus Next Lecture Map Transformation