The principle of 3D area calculation is expounded first, followed by the application of two methods to calculate true surface area, namely the Jenness extension and TIN.
3.8.1.1 The principle of three-dimensional area calculation
Terrain area is almost always presented as planimetric area, shown as the ‘horizontal plane’ in Figure 3.34. Two-dimensional area is usually calculated simply as length multiplied by breadth. Most software (including GIS) calculates the planimetric area of map features which is basically the
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bird’s-eye view of an area derived from coordinate points referenced to horizontal space anchored to latitude and longitude. For example, a square kilometre planimetrically calculated in the Swartberg range with its crags and ravines represents the same area of land as a square kilometre in the flat Karoo plains. Of course, the real surface area of the former is larger, since it reflects tilted planes. Calculating the total planimetric area of grid based systems is simply done by multiplying the number of regular-sized grid cells by the planimetric area of an individual grid cell on a horizontal plane, that is
(cell length x cell breadth) x number of grid cells.
However, planimetric area represents only the horizontal dimensions of features. When calculating the true surface area, in this study the area that spekboom covers on steep slopes (compare Figure 3.33), the larger vertical aspect or tilted plane of the terrain has to be accounted for as surface area increases with inclination in a grid cell. Figure 3.33 graphically demonstrates this dilemma in aerial representation and calculation, while Figure 3.34 demonstrates the graphical principles involved.
Source: Lopez & Berry (2003: 1) Figure 3.34 Increasing surface area with inclination in a grid cell
By using a digital elevation model (DEM) of the relevant area, the slope of each grid cell is calculated and then used to adjust planimetric area to surface area. Lopez & Berry (2003) and GeoWorld (2002) provide the formula for surface area calculation as:
Surface area = Planimetric area ÷ Cosine (Slope angle)
where, surface area is the area of the tilted plane (in the shape of a parallelogram) on the terrain surface corresponding to a rectangle on the planimetric reference grid;
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planimetric area is the area of the rectangle on the planimetric reference grid; and
slope angle is the inclination of the tilted plane with respect to the horizontal reference grid. A DEM is a regularly spaced grid of elevation points that can be represented as a raster (a grid of squares, also known as a height map). Within DEMs, each cell value reflects the elevation above some reference point – usually sea level – in metres at the central point in that cell. Figure 3.35 is an expression of topography based on shading a 20m resolution DEM for the larger region within which the BLK PNR study terrain is located. The Figure shows the rugged and uneven topography of the area.
Figure 3.35 Topography showing elevation for BLK PNR created from a 20m DEM
A triangular irregular network (TIN) and a variety of slope-aspect, surface area and land- curvature values can be created from the DEM to calculate true surface area. For this study, the surface area was calculated using an extension provided by Jenness (2010) that creates a surface area layer and a TIN model, both in ArcGIS. The two methods are similar in their calculations (they both calculate three-dimensional area and planemetric area). However, they differ slightly in that one is vector-based and the other raster-based. The differences are discussed below. Comparisons of the two methods should inform the choices other users of the methods will need to make.
3.8.1.2 Three-dimensional area calculation via the Jenness extension
The Jenness (2010) extension makes it possible to generate surface area and surface ratio grids from an existing elevation grid or DEM. The cell values for these grids reflect the surface area and (surface area) ÷ (planimetric area) ratio for the land area contained within that cell’s boundaries.
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The ‘surface area tool’ creates a raster with cell values reflecting the surface area within each cell to calculate total true surface area. Jenness (2010) explains more fully how the tool works. Encapsulated, the Jenness tool uses the DEM to generate eight 3D triangles around each cell centre by connecting each cell centre point with the centre points of the eight surrounding cells, then calculates and sums the area of the portions of each triangle that fall within the cell boundary to determine the true surface area of the grid. For example, in Figure 3.36a, which is a 3x3 sample elevation grid, the tool calculates the surface area for the central cell with elevation value 165 based on the elevation values of that cell plus the eight surrounding cells. In Figure 3.36 the central cell and its surrounding cells are pictured in 3D space as a set of adjacent columns, each rising as high as its specified elevation value. Further in Figure 3.36b, the central cell with height value 165 to the centres of the surrounding cells, and the lengths between adjacent surrounding cells, yield the edge lengths for the triangles I–VIII in Figure 3.37a. These triangles form a continuous surface over the nine cells shown in Figure 3.37a. The surface area within the target cell should, however, only reflect the areas of triangles I–VIII in Figure 3.37a.
The centre cell (165) plus 8 adjacent cells are used
The distances between adjacent surrounding cells are used to calculate true surface area Source: Jenness (2010: 831) Source: Jenness (2010: 831) Figure 3.36 Sample grid system to calculate true
surface area
Figure 3.37 Cell distance principle used to calculate tilted surface area
Therefore, the extension trims the triangles to the cell boundaries in Figure 3.37b by dividing all the triangle side lengths by two.
3.8.1.3 Three-dimensional area calculation via TIN data structures
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The second method, the triangulated irregular network (TIN), is a surface model by which elevation, slope and aspect are captured in vector-based GIS data-base format as illustrated in Figure 3.38. The TIN data model is a conception of point-measured elevation connected by lines
Source: Schneider & Robbins 2009: 22 Figure 3.38 Topographic information in a TIN
and planes. By connecting all these elevation points (vertices) with their neighbours, a network of contiguous, non-overlapping Delaunay triangles is formed. The number of points selected by the command is a function of the specified z-tolerance and the smoothness of the input raster. A TIN surface may differ from the cell centre heights of the input raster. A low number results in a
TIN that preserves more of the detail of the raster surface (ArcGIS 2009). A larger number results in a more generalized representation of the surface. Figure 3.39 displays the effect of different tolerances. In the figure on the left the background raster has been converted to the foreground TIN
Source: NMT 2008: 21 Figure 3.39 Display of different Z tolerance
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using a z-tolerance of 50 units. The output TIN has 169 nodes and 269 triangles. (Only nodes and edges are symbolized.) The figure on the right was the same raster but is converted using a z- tolerance of 25 units. Unlike DEMs, these TINs are continuous vector surfaces and can therefore be precisely measured and clipped (Jenness 2004). By using ‘interpolate polygon to multipatch’ (command in GIS), one can create surface-conforming aerial features by extracting those portions of a surface that fall within the extent of input polygons as multipatches (ArcGIS 2009). Area is then displayed in planimetric and true surface area in the attribute table. The principle of 3D area having been examined is now applied to the study area.