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Copy of paper presented at the Paper presented at the 8th International Symposium on Spatial Data Handling, Vancouver, Canada, July 11-15th, 1998.
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| Full paper (Word 97 doc.) | Full paper (Zipped Word 97 doc.) | LandSerf software |
A new method of modelling surface form based the identification of a conic section's semi-axes is presented. It characterises surface behaviour over a local region rather than at a point. It is shown to produce significantly different results for calculations such as slope, aspect and surface flow direction. It is suggested that this method is more appropriate for characterisation of surfaces where continuity between neighbouring points in a DEM is important. An algorithm using the method is given, along with results of its implementation. It is shown to be appropriate for multi-scale surface specific feature identification as it allows users to 'filter' scales of interest. It is also demonstrated to be useful in the first stage of creating a topological weighted surface network.
Keywords: DEM, weighted surface network, scale, morphometry.
Note: See the proceedings of SDH '98 (Chrisman, N. and Poiker, T. (eds.), 1998) for the full paper. The following is a summary with full colour images.
To use digital elevation models effectively, some assumption must be made about the surface implied by the cell values of a DEM. Three such assumptions are illustrated in Figure 1.
Figure 1 Three forms of interpolation used to transform discrete DEM cell values
into continuous surface models. (a) Proximal interpolation; (b) linear interpolation;
(c) cubic spline interpolation.
Examples of all three assumptions can be found in the literature. This paper considers a new way of modelling the continuity of surfaces using quadratic interpolation.
Quadratic interpolation models some local patch of the DEM with a bi-variate quadratic function. This has been used widely to measure surface properties such as slope, aspect and curvature. Quadratic surfaces can be associated with between 0 and 2 principal flow directions or axes.
Figure 2 Quadratic surface z = 0.1x2 + 0y2 + 0.2xy -0.1x -0.1y + 10 with
principal flow directions.
Quadratic functions are examples of conic sections whose axes may be identified. If the axes intersect with a given region of interest on a surface, we can identify a surface specific feature.
Figure 3 Elliptic, hyperbolic and parabolic conic sections with their semi-axes
identified.
Figure 4 Three possible intersection cases between a (circular) region of interest
and conic section's semi-axes. (a) two axes intersect with region - pit, peak or pass;
(b) one axis intersects with region - channel or ridge; (c) no intersection - planar.
Using Analytic Geometry, the principal axes can be identified and compared with an arbitrary region of interest in the DEM.
The following algorithm describes the identification of surface specific features and the possible intersection of feature's principal axes with the region of interest. Note that for brevity, the special cases of theta=90 degrees (primary feature axis has infinite gradient) and theta=0 (secondary feature axis has infinite gradient) have been left out of the algorithm description.
GetFeatureAxes Algorithm Part 1
GetFeatureAxes Algorithm Part 2
The procedure described above can be used to identity surface specific features on DEMs of real terrains. The examples below use Ordnance Survey 50m DEMs, Crown Copyright.
Figure 5 Surface-specific features identified from a DEM (Lake District, England).
Peaks are red; ridges are yellow; passes are green; channels are blue; pits are black.
Window size used for estimating the quadratic function is 15x15 cells, region radius
is a 4 cells.
This method along with many others in the literature appears to identify 'spurious pits'. These are in fact not spurious as they frequently represent drainage outlets that are expressed at a finer scale the DEM resolution. Such pits are likely to exist at all scales and should be considered an important part of the surface description.
Figure 6 The identification of pits within a confined drainage network
Figure 7 A topological network of ridges and channels derived from the Lake District
DEM. Linear features are bounded by pits, peaks and passes.
A new method of analysing continuous local regions of a DEM has been described and demonstrated. By considering surface behaviour over some areal extent around some region of interest, more information can be gained than by considering point behaviour alone.
