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

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

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-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