Lecture 07 Terrain Analysis

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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
Z
p 1
p   dp
n

x 200 cells 1000 points = 40 x
10^6 distance calculations
Zi, j       R
   If all points are used and sorted
by distance, called "brute force"
d
p 1
p
n

method
   Possible to use sorted search
and tiling

   Distance can be weighted
of distance
   Can be refined with break
lines
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
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
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

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