Lecture 07: Terrain Analysis Geography 128 Analytical and Computer Cartography Spring 2007...
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Lecture 07: Terrain Analysis
Geography 128
Analytical and Computer Cartography
Spring 2007
Department of Geography
University of California, Santa Barbara
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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
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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
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Weighting Methods
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
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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
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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
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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
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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
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Alternative Methods
Many ways to make the point-to-grid 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
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Surface-Specific Point Sampling
Landscape Morphometric Features
Terrain "Skeleton"
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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
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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
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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
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Surface Filtering
Convolution of filter matrix with map matrix
Filter has a response function
Filter weights add to one
Can enhance properties, or generalize
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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)
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Volumetric Transformations- Slope and Aspect (ArcGIS)
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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
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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
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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
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Next Lecture
Map Transformation