Slope Measures

Gradient

Most GIS are able to calculate gradient automatically from some form of digital elevation model (DEM). In the simplest case, gradient can be calculated along a linear profile as follows:

where dz represents the difference in elevation between two neighboring points on the profile and dx represents their horizontal distance of separation. In practice, gradient is often calculated by taking the average change in elevation on either side of the point of interest, in other words, the average of the upslope and downslope gradients.

Most GIS will calculate gradient from a twodimensional surface rather than along a onedimensional profile. In the two-dimensional case, the gradient at any point tends to be calculated in the direction of steepest slope and can be found as follows:

where dz/dx and dz/dy represent the rate of change in elevation in directions parallel to the x- and y-axes of the coordinate system. In practice, this can be calculated simply by comparing the height of a point with some of its eastern and western neighbors and with some of its northern and southern neighbors. Different GIS may select and weight those neighbors in slightly different ways, so the final estimation of steepest gradient may vary a little between systems.

Gradient can be expressed in a variety of ways. If calculated by a computer as described above, gradient is expressed as a number between zero (horizontal) and infinity (vertical). This is sometimes expressed as a ratio (e.g., 1 in 5, or 20%), although most GIS will express gradient in degrees from horizontal.

Choosing the neighbors with which to compare elevation is not always as obvious as it first appears. In particular, gradient is likely to vary depending on the spatial extent over which it is calculated. By selecting very close neighbors, measures of gradient will record many minor undulations in a surface and may produce quite a “noisy” map of gradients. Selecting neighbors at a greater distance of separation for any point will produce a smoother map representing average gradients within a region. The distance of separation of neighbors is partly influenced by the resolution of the DEM used, but it should [Page 390]be appropriate for the application for which gradient is being calculated.

Aspect

Related to the gradient of a point on a surface is the compass direction of steepest slope, usually referred to as aspect and expressed in degrees clockwise from north. It is equally simple to calculate as gradient and, as with that measure, will vary slightly between GIS depending on which neighbors are selected and how they are weighted. Aspect is an important measure for applications that model processes that have some directional relationship with landscape. For example, avalanche prediction and vegetation productivity are both dependent on incident solar radiation that is partly a function of aspect. When mapping aspect, many GIS will use color schemes that approximate shaded relief by using darker shades for southeast-facing slopes and lighter shades for northwest-facing slopes.

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