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
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Influence of a single point = f(1/d)

Can be constrained to fit all points
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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