Geomorphometry

A surface with a sample of normal vectors

The surface normal at any point on the terrain surface is a vector ray that is perpendicular to the surface. The surface gradient (${\displaystyle \nabla f}$) is the vector ray that is tangent to the surface, in the direction of steepest downhill slope.

The geometry of calculating slope

Slope or grade is a measure of how steep the terrain is at any point on the surface, deviating from a horizontal surface. In principle, it is the angle between the gradient vector and the horizontal plane, given either as an angular measure α (common in scientific applications) or as the ratio ${\displaystyle p={\frac {rise}{run}}}$, commonly expressed as a percentage, such that p = tan α. The latter is commonly used in engineering applications, such as road and railway construction.

Deriving slope from a raster digital elevation model requires calculating a discrete approximation of the surface derivative based on the elevation of a cell and those of its surrounding cells, and several methods have been developed.[12] For example, the Horne method, implemented in ArcGIS, uses the elevation of a cell and its eight immediate neighbors, spaced by the cell size or resolution r:[13][14]

The partial derivatives are then approximated as weighted averages of the differences between the opposing sides:

${\displaystyle {\frac {\partial z}{\partial x}}\approx {\frac {e_{NE}+2e_{E}+e_{SE}-e_{NW}-2e_{W}-e_{SW}}{8r}}}$

${\displaystyle {\frac {\partial z}{\partial y}}\approx {\frac {e_{NE}+2e_{N}+e_{NE}-e_{SW}-2e_{S}-e_{SW}}{8r}}}$

The slope (in percent) is then calculated using the Pythagorean theorem:

${\displaystyle \tan \alpha ={\sqrt {({\frac {\partial z}{\partial x}})^{2}+({\frac {\partial z}{\partial y}})^{2}}}}$

The second derivative of the surface (i.e., curvature) can be derived using similarly analogous calculations.

The aspect of the terrain at any point on the surface is the direction the slope is "facing," or the cardinal direction of the steepest downhill slope. In principle, it is the projection of the gradient onto the horizontal slope. In practice using a raster digital elevation model, it is approximated using one of the same partial derivative approximation methods developed for slope.[12] Then the aspect is calculated as:[15]

${\displaystyle \tan \beta ={\frac {\frac {\partial z}{\partial y}}{-{\frac {\partial z}{\partial x}}}}}$

This yields a counter-clockwise bearing, with 0° at east.

Shaded relief map of New Jersey

Another useful product that can be derived from the terrain surface is a shaded relief image, which approximates the degree of illumination of the surface from a light source coming from a given direction. In principle, the degree of illumination is inversely proportional to the angle between the surface normal vector and the illumination vector; the wider the angle between the vectors, the darker that point on the surface is. In practice, it can be calculated from the slope α and aspect β, compared to a corresponding altitude φ and azimuth θ of the light source:[16]

${\displaystyle i=\cos \phi \cos \alpha +\sin \phi \sin \alpha \cos(\theta -\beta )}$

The resultant image is rarely useful for analytical purposes, but is most commonly used as an intuitive visualization of the terrain surface, because it looks like an illuminated three dimensional model of the surface.

Natural terrain features, such as mountains and canyons, can often be recognized as patterns in elevation and its derivative properties. The most basic patterns include locations where the terrain changes abruptly, such as peaks (local elevation maxima), pits (local elevation minima), ridges (linear maxima), channels (linear minima), and passes (the intersections of ridges and channels).

Due to limitations of resolution, axis-orientation, and object-definitions the derived spatial data may yield meaning with subjective observation or parameterisation, or alternatively processed as fuzzy data to handle the varying contributing errors more quantitatively – for example as a 70% overall chance of a point representing the peak of a mountain given the available data, rather than an educated guess to deal with the uncertainty.[17]

In many applications, it is useful to know how much the surface varies in each local area. For example, one may need to distinguish between mountainous areas and high plateaus, both of which are high in elevation, but with different degrees of "ruggedness." The local relief of a cell is a measurement of this variability in the surrounding neighborhood (typically the cells within a given radius), for which several measures have been used, including simple summary statistics such as the total range of values in the neighborhood, an interquantile range, or the standard deviation. More complex formulas have also been developed to capture more subtle variation.[18]

In many situations, terrain can have a profound effect on local environments, especially in semi-arid climates and mountainous areas, including well-known effects such as Altitudinal zonation and the Slope effect. This can make it a significant factor in modeling and mapping microclimates, vegetation distribution, wildlife habitat, and precision agriculture.

Due to the simple fact that water flows downhill, the surface derivatives of the terrain surface can predict surface stream flow. This can be used to construct stream networks, delineate drainage basins, and calculate total flow accumulation.

Mountains and other landforms can block the visibility between locations on opposite sides. Predicting this effect is a valuable tool for applications as varied as military tactics and locating cell sites. Common tools in terrain analysis software include computing the line-of-sight visibility between two points, and generating a viewshed, the region of all points that are visible from a single point.[20]

Map depicting cut and fill areas for a construction site.

Many construction projects require significant modification of the terrain surface, including both the removal and addition of material. By modeling the current and designed surface, engineers can calculate the volume of cuts and fills, and predict potential issues such as slope stability and erosion potential.