Lecture Notes for Clarke, K. C. Analytical and Computer Cartography
Lecture 14: Terrain Analysis I
3D Transformations
- Not yet considered in a transformational context
- 3D data often for land surface or bottom of ocean
- Need three coordinates to determine location
- 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
- How do we do transformations?
- In all cases we must fill in data where none exists in space = interpolation
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 tru surface on the sample
- "Model" is a mathematical model of the neighborhood relationship
- Influence of a single point = f(1/d)
- Can be contrained to fit all points
- Should contain z extremes, and local extrema
- Most models are algorithmic local operators
- Work cell-to-cell. Operative cell = kernel
Weighting Methods
- Impose z = f (1/d)
- Computationall rather 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 (Hodgson, ERDAS)
- Distance can be weighted and powered by n = friction of distance
- Can be refined with break lines
- Use cos (angle) to prevent shadowing
Inverse Distance Weighting
inverse_d
- Assigns points to cells
- Averages multiples
- for all unfilled cells
- search outward using an increasingly large square neighborhood
- until at least npts are found
- apply inverse distance weighting
Has been parallelized (and found highly efficient) by Armstrong
Trend Projection Methods
- Way to overcome high/low contstraint
- Assumes that sampling missed extrema
- Locally fits trend, trend surface or bicubic spline
- Least squares solution
- Useful when data are sparse, texture required
Search Patterns
- Many possible ways to define interpolated "region" R
- Can use # points or distance
- Problems in
- Sparse areas
- Dense areas
- Edges
- Bias can be reduced by changing search strategy
Kriging
- "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
- Examing 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
Terrain "Skeleton"
Wedding Cake effect
Specific problem when grids are made from contours (e.g. 3 arc second DEMs)
Keith Clarke Last Change 5/19/97 Copyright Prentice Hall, 1995