Most GIS packages report planar area, not surface area. Some packages, such as ArcGIS and Surfer provide the option to compute surface area, although they may not specify in detail how these are calculated. Computation of surface areas will result in a value that is greater than or equal to the equivalent planar projection of the surface. The ratio between the two values provides a crude index of surface roughness.

If a surface is represented in TIN form, the surface area is simply the sum of the areas of the triangular elements that make up the TIN. Let T_j={x_ij,y_ij,z_ij} i=1,2,3 be the coordinates of the corner points of the j^th TIN element. Then the surface area of the planar triangular (or planimetric area) determined when z=constant, can be calculated from a series of simple cross-products:

Note that this expression is the same as the polygon area formula we provided earlier, but for a (n‑1)=3‑vertex polygon:

where i+1=4 is defined as the first point, i=1.

The term inside the brackets is the entire area of a parallelogram, and half this gives us our triangle area. This expression can also be written in a convenient matrix determinant form (ignoring the j’s for now) as:

If the slope of T_j, q_j, is already known or easily computed, the area of T_j can be directly estimated as A_j/cos(q_j). Alternatively the three dimensional equivalent of the matrix expression above can be written in a similar form, as:

If z=0 for all i, the second and third terms in this expression disappear and we are left with the result for the plane. Suppose that we have a plane right angled triangle with sides 1,1 and Ö2, defined by the 3D coordinates: (0,0,0), (1,0,0) and (1,1,0). This plane triangle has area of one half of a square of side length 1, so its area is 0.5 units, as can be confirmed using the first formula above. If we now set one corner, say the point (1,1,0) to have a z-value (height) of 1, its coordinate becomes (1,1,1) and the area increases to 0.7071 units, i.e. by around 40%.

If the surface representation is not in TIN form, but represented as a grid or DEM, surface area can again be computed by TIN-like computations. In this case the 8-position immediate neighbourhood of each cell can be remapped as a set of 8 triangles connecting the centre-points of each cell to the target cell (Figure 4‑4). In each case the triangle area can be computed, as above, and 25% of each area assigned to the central cell (shaded grey in Figure 4‑4). Research has suggested that the resulting total surface area for the cell is close to that produced by TIN models for grids of 250+ cells.

Figure 4‑4 Triangular approximation of surface area

Grid surface areas can also be computed using the slope adjustment method described above assuming that a slope grid has been first been computed. Note that grid resolution is generally coarser than the vertical or z-dimension resolution. For example, a DEM might be defined in 25mx25m squares, with a vertical resolution of 1m. If generating your own slope values, areas etc. from such data the different scales must be included within the calculation. If the units for the vertical resolution do not match the horizontal scale then it may also be necessary to apply a scaling factor to the z-dimension.

Surface areas are computed relative to a reference plane, usually z=0. In some instances it may be necessary to specify the value to be associated with the reference plane, and whether the absolute values, positive values and/or negative values are to be computed. Surfer, for example, supports all three options which it describes as positive planar and surface area, negative planar and surface area and total planar area. For example, consider the surface illustrated in Figure 4‑5. This shows a surface model of a GB Ordnance Survey DEM for tile TQ81NE, which we shall use as a test surface in various sections of this Guide. This is a 5000mx5000m area provided as a 10mx10m grid of elevations, with an elevation range from around 10m‑70m. Using a reference plane of z=30m the surface area above this reference plane is roughly 1.3sq kms and below 30m is roughly 2.1sq kms (3.4sq kms in total compared to 2.5sq kms for the planar area). For details of volume computations, which also use reference surfaces, see Section 6.2.6.