Surface area processing in GIS

Surface area processing in GIS

Calculating Slope from DEM
Onorati, et al. (1992) evaluated the efficiency of several slope calculation methods. The methods they took into consideration were

The Four Contiguous Right Triangles (FCRT) method,
Maximum Downward Gradient (MDG).
Bicubic Spline First Derivatives (BSFD).

The FCRT method takes into account the local elevation pattern and slope is calculated for a sub-pixel. MDG method (Travis et al, 1975) employs a comparison of the elevation of the central pixel in a 3 by 3 window with the neighbouring eight pixels. A variant of this algorithm can be the comparison of the central pixel in a 3 by 3 window with the four straight side pixels. The BSFD technique is based on the bicubic splines of the first derivative of the DEM (Catmull, 1974). In the aforesaid study it was observed that the BSFD method tends to smooth out sharp slopes thereby having a tendency to present more gradient values ranging between 0 to around 45. On the other hand FCRT operator was complicated to process and also the output was four times larger than the original DEM file, which is highly undesirable when dealing with vast data sets. The MDG operator was found to provide a more truthful approximation of slope gradient. Skidmore (1989) reviewed six methods of estimating slope. He concluded that both first derivative method (Zevenbergen and Thorne, 1967) and second derivative method (Horn, 1981) were superior to the MDG method. This was because of the fact that the MDG estimator was prone high errors due to local errors of elevation (Burrough & McDonnell, 1998).

The slope function in Arc/Info follows the second derivative method. Conceptually, the slope function fits a horizontal plane to the z (elevation) values of a 3 by 3 cell neighbourhood around the processing or centre cell. The direction the horizontal plane faces is the aspect for the processing cell. The slope for the cell is calculated from the 3 by 3 neighbourhood using the average maximum technique (Burrough & McDonnell, 1998). If there is a cell location in the neighbourhood with a no data z value the z value of the centre cell will be assigned to the location. At the edge of the grid, at least three cells (outside the grid's extent) will contain no data as their z values. These cells will be assigned the centre cell's z value. The result is a flattening of the 3 by 3 horizontal that is fit to these edge cells, which thus usually leads to a reduction in the slope. This flattening effect did not affect the current study, as the edge cells were not included in the area of interest. For simplicity and conventional reasons we adopt the Arc/Info methodology for slope.

Despite the wide use of GIS for several terrain analyses, including hydrological and agricultural studies, a basic standardised functionality is still lacking in modern commercial GIS packages, which allows the surface area of a terrain to be calculated from a DEM. Slope gradient being an approximation from the neighbouring pixels, its accuracy in determining the surface area but only remains to be verified. On the other hand, since it is near impossible to determine accurately the actual surface area of a region both because of the fact that traditional methods are not feasible Slope Angle, on a smaller scale and that the surface of a terrain is subject to continuous undulations as compared to the featureless plane. This is the basic reason why the performance of using slope for surface area calculation cannot be intrinsically evaluated. Only a comparison of its performance with respect to other algorithms can be performed up to certain level of satisfaction.

Surface Area
Surface area calculations have been attempted before (Strahler, 1952) and several algorithms were developed for the purpose, all of which are based on slope. Surface area, therefore is a second derivative of elevation data. Elghazali et al. (1986), described the areal parameter, which essentially is a global function that produces a ratio between the surface area and plan area. This parameter was used for terrain characterisation but we found it useful as an measure to estimate surface area.

Surface Area from Slope Gradient The slope area for each pixel is calculated and then the summation of all pixels falling within a parcel constitutes the surface area of the parcel. The slope area for each pixel can be approximated from the resolution of the pixel and the slope value for the pixel. In figure 1, area ABEF is the slope area for the pixel ABCD. The slope area for this pixel is given by the expression (AB*CD)/Cos(è ).

Figure 1: Slope area for a Pixel ABDC

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