GE3502GE5502 Geographic and Land Information Systems

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GE3502/GE5502Geographic and LandInformation
Systems
Lecture 13 Basic DEM production and interpolation
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Digital Elevation Models

• Although the landform is a continually varying
  surface, it can be represented by contours (ie.
  topographic map).
• Any digital representation of the continuous
  variation of relief over space is called a
  digital elevation model (DEM)

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Uses of DEMs

• Storage of elevation data in national databases

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Uses of DEMs

• Cut-and fill problems in landslide analysis,
  road design etc.

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• Analysis of cross-country visibility (viewsheds)

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• Viewsheds (cont)

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• 3-dimensional display of landforms for visual
  purposes (landscape architecture)
• Background for displaying thematic information

eg. Rainfall overlain onto a DEM
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• Computing slope, aspect, slope distances
  (important in calculating runoff and erosion)

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• Substitute z-axis with other variables of choice
  such as climate, population, noise, pollution or
  groundwater variables
• eg. Rainfall surface.

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Other applications

• Planning road routes, power line locations
• Statistical analysis and comparisons of terrain
  types
• Important input to more complex modelling
• Fire risk
• Erosion/soil loss models
• Land suitability analyses
• Many more.

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Methods of representing DEMs

• A. Mathematical methods
• Global - Fourier series/multiquadratic
  polynomials
• Local - Regular and irregular patches
• B. Image methods
• Line models - Horizontal/vertical
  slices/critical lines (ridges, saddles, stream
  courses, shorelines, slope breaks)
• Point models
• Regular rectangular grid of uniform and
  variable density (Altitude matrices)
• Irregular network using triangulation
  (Triangulated Irregular Network - TIN) or
  proximity analysis

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Tessellation

• Vector model
• A tessellation of a plane is the filling of the
  plane with repetitions of figures (or polygons)
  in such a way that no figures overlap and there
  are no gaps. These polygons can be regular or
  irregular
• Given a set of two or more distinct points, all
  locations in that space are associated with the
  closest member(s) of the point set with respect
  to the Euclidean distance.
• Not true interpolation

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Tessellation of the plane
Voronoi polygons
Points

• The result is a tessellation of the plane into a
  set of regions called ordinary Voronoi polygons
• If this tessellation is triangles this is called
  a Delaunay triangulation

Clusters
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Voronoi tessellation
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Triangular Methods

• Vector model
• Triangles are used to represent a surface defined
  by z-values located at irregularly spaced (x,y)
  points.
• With reference to a DEM this is called a TIN
  (triangular irregular network).
• From this we can produce contour lines.

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TIN
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Spatial interpolation

• a process of predicting unknown values by using
  known values at multiple locations around the
  unknown value.
• Spatial interpolation involves finding a function
  f(z) which best represents the entire surface of
  z-values (called data values) associated with
  irregularly located (x, y) points (called data
  sites)
• In addition this function predicts z-values for
  other regularly spaced locations. Such a function
  is referred to as an interpolant
• There are two types of interpolants, exact and
  approximate (data smoothing)

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Contour generation and 3-D surface plotting

• Contour and 3-D surface plotting routines in a
  GIS require that (x,y,z) data points be placed in
  a regularly-spaced grid. Most input data are not
  regularly spaced.
• Such data must be "gridded", i.e. interpolated,
  into regularly spaced (x,y) points having
  z-values estimated at the lattice points of the
  rectangular grid.
• Numerous mathematical techniques available for
  gridding irregularly spaced data
• Linear Interpolation (Thiessen Triangulation)
• Non-linear interpolation
• C1 Spline
• Weighted Moving Average
• Kriging

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Modelling gradients
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Linear Interpolation

• Based on the Thiessen triangulation method. The
  fundamental basis of the procedure is to connect
  adjacent irregularly-located (x,y,z) data points
  into a grid of triangles.
• The three data points connected by each triangle
  define a plane (as in a TIN).
• A vertical (constant x and y) line is extended
  from each point in the output lattice.
• The intersection of the vertical line with the
  plane formed by the vertices of the triangle
  defines the z-value of the surface at each (x,y)
  lattice grid point.

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C1 spline function

• Spline function is the mathematical equivalent of
  a flexible ruler
• Piece-wise functions approximate a curved line or
  surface by a sequence of straight line segments
  fit to a small number of points
• Joins between one part of the curve and another
  are continuous.
• C1 surface interpolation has the property that it
  has no sharp discontinuities, and generally
  yields a smooth, pleasing contour map
• In regions of sparse data excessive "peaks" and
  "valleys" may be generated with some data sets

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Weighted Moving Average Method

• Values of the function for the set of n regular
  grid points are computed based on the value of z
  at some number of the closest adjacent irregular
  data points
• The value of z for each actual data point is
  distance-weighted so that closer real points lend
  more weight
• A search radius is defined as well as the number
  of closest data points to be used to compute each
  z-value at a grid point.
• The final form of the resulting map is strongly
  dependent on the value of r, on the clustering of
  the actual data points, and on the presence of
  outliers

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Weighted moving average
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Kriging

• Natural data are difficult to model using smooth
  functions because normally random fluctuations
  and measurement error combine to cause
  irregularities in sampled data values
• Kriging was developed to model those stochastic
  concepts. It is based on the concept of a
  regionalized variable that has three components
• STRUCTURAL - This may be represented by the mean
  or a constant trend.
• SPATIALLY CORRELATED - Data often exhibit
  positive spatial correlations.
• RANDOM NOISE - Measurement errors, other errors.

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Data points
Structural
Random noise
Correlated
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Kriging
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Kriging

• This method requires extensive user knowledge and
  as such is most subject to misuse. This method
  will generate invalid or useless output data when
  used incorrectly
• Kriging is a weighted average method of gridding
  which determines weights based on the location of
  the data and the degree of spatial continuity
  present in the data as expressed by a
  semi-variogram
• The weights are determined so that the average
  error of estimation is zero and the variance of
  estimation minimized
• There are two types of kriging
• ordinary kriging the expected value of the
  underlying trend of z is assumed to be constant
  over the entire (x,y) grid area
• universal kriging allows the specification of a
  drift polynomial describing the underlying
  trend in the data

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Rainfall minus terrain
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Rainfall with terrain