Modelling the topographic surface

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Modelling the topographic surface
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Digital Terrain Modelling process (Weibel and
Heller, 1991)

• 1. Generation sampling of original terrain data
  and model construction
• 2. DTM manipulation modification and refinement
  of DTMs, derivation of intermediate models

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Process (cont.)

• 3. interpretation DTM analysis, information
  extraction from DTMs
• 4. visualization graphical rendering of DTMs and
  derived information
• 5. application development of appropriate
  application models for specific disciplines

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Data structures for Digital Terrain Modelling

• Grids
• elevation matrix
• simple data handling of matrices
• modelling algorithms simple
• may not reflect terrain complexity

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TINs (Triangulated Irregular Networks)

• triangular elements
• vertices at sample points
• reflect density of data points
• topology recorded computed explicitly
• more complex and difficult to handle

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The TIN model
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The TIN landscape
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TIN model description

• irregular points (specially selected)
• points connected by lines form triangles
• plane within each triangle
• each triangle fits neighbour
• continuous surface of triangular facets

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TINs in Vector GIS

• 3 vertices with height attributes
• 3 edges with slope/direction attributes
• polygons with slope, aspect and area attributes

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Triangulating a TIN

• points selected (vertices of triangle network)
• fat 60 degree triangles preferable
• all locations within triangle as close to
  vertices as possible

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Delaunay Triangulation

• Definition
• Three points form a Delaunay triangle if, and
  only if, the circle which passes through them
  contains no other point

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Delaunay Triangles
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Creating Delaunay triangles

• assign all locations to nearest vertex
• boundaries form Thiessen polygons
• two vertices connected if they share common edge
• produced preferred fat triangles

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Thiessen polygons and Delaunay triangles
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Random to grid based terrain modelling methods
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Interpolation

• deriving value of a point with no known value
• eg. contouring
• computer based (procedures formalised)
• additional factors may be added
• minimise amount of data

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Rationale

• Tobler's Law of Geography
• points close together in space are more likely
  to have similar values than points far apart

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Extrapolation

• Interpolation - between data points
• Extrapolation - outside data points
• Assumes trend of surface continues beyond limit
  of data values
• Cannot extrapolate far beyond boundaries
• Confidence level correlates with spacing of points

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Interpolation methods

• 1) Pointwise
• interpolation of height at grid node from
  neighbouring measured points
• 2) Global
• single complex 3D surface through whole data set
• grid nodes interpolated from surface
• 3) Patchwise
• 3D surfaces or patches established

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1) Pointwise methods

• Nearest neighbour search
• Area search - define radius distance
• Number of neighbours (no control over
  distribution)
• Quadrant search
• Octant search

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Radius distance
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Nearest neighbours
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Quadrant search
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Octant search
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Comments

• Sectored searches are rarely used - complicates
  programming
• Limit extrapolation
• Encourage practice of additional data points
  outside area to be mapped
• Straight weighted distance preferred

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Weighted distance

• points closer to interpolation point should have
  more influence (Tobler)
• assumes high degree of autocorrelation
• if this cannot be assumed - isarithmic mapping is
  inappropriate

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Variations

• 1. Inverse of distance
• 2. Inverse of square of distance (decreases
  weight of points further away)
• Formula Swizi/Swi
• Possible values where D is distance

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Example

• Interpolating a height point using W 1/D
• POINT DISTANCE Z VALUE W WZ
• 1 300 105 1/300 0.3499
• 2 200 70 1/200 0.35
• 3 100 55 1/100 0.55
• Swi 0.0183 Swizi 1.2499
• Substituting in formula 1.2499 0.0183
• Z 68.1764 using 1/D
• Z 62.85 using 1/D2
• Z 57.96 using 1/D3

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Comments

• fast
• simple to understand and control
• tendency to produce bulls eye contours around
  data points

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Inverse distance 1/D
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Kriging

• weighted distances applied as a constant
• kriging enables variable weighting to be applied
• weighting dependent on variation of height with
  distance
• variance between points changes over space
• expressed in a variogram

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Deriving the variogram

• divide range of distance into discrete intervals
  (e.g. 10 intervals between 0 and max. distance)
• for every pair of points compute distance and
  difference in height
• assign each pair to one of thedistance ranges
• compute average variance in each distance range
• plot this value at the midpoint distance of each
  range

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Computing the estimates

• variogram used to estimate distance weights
• estimates are statistically unbiased
• computationally intensive
• visually appealing models
• expresses trends in data
• preserves high points as ridges rather than
  bulls eyes

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Kriging
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2. Global polynomial surface fit

• fit entire area by polynomial expansion
• derivative of time series analysis
• ideally surface should pass through all original
  data points exactly
• least squares fit used to gain best fit of
  polynomial to data points

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Polynomial equations

• a) Linear equation (first order)
• z a bx cy
• describe a tilted plane surface
• b) Quadratic equation (second order)
• z a bx cy dx2 exy fy2
• describes a simple hill or valley
• c) Cubic equation (third order)
• z a bx cy dx2 exy fy2 gx3 hx2y
  ixy2 jy3

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Disadvantages

• 1. simplicity of polynomial surfaces
• 2. accelerate upward or downward in areas of no
  control (edges of DTM)
• 3. high order polynomials require high
  computation time on large data sets
• 4. unpredictable oscillations may produce poor
  interpolation values at grid nodes

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Application

• Statistical surfaces
• Geological applications

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Patchwise methods

• Local polynomial surface fit
• local trend surfaces (patchwise method)
• equal size patches
• separate functions calculated for each patch
  (typically polynomial)

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Advantages

• low order terms can be used
• derived points easily calculated

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Disadvantages

• needs more organisation of data and processing
• subdivision needs to be carried out with care
• poor distribution of data points near patch
  corners - affects computed parameters derived
  node values

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Projection of surface dips

• Method
• Dip of surface at control point found by least
  squares fit of plane to surrounding control
  points
• Dips are found for all control points
• Dips are projected from control points to grid
  nodes and average calculated

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Objections to grid based methods

• McCullagh (1990)
• Lack of flexibility responding to local
  variations in data density
• Non-honouring of data points due to
  insufficiently fine grid chosen to reduce
  computing time
• Difficulty in representing cliff and breakline
  information adequately on a continuous surface