Geomorphometry, or geomorphometrics (Ancient Greek: γῆ, romanized: gê, lit. 'earth' + Ancient Greek: μορφή, romanized: morphḗ, lit. 'form, shape' + Ancient Greek: μέτρον, romanized: métron, lit. 'measure'), is the science and practice of measuring the characteristics of terrain, the shape of the surface of the Earth, and the effects of this surface form on human and natural geography. It gathers various mathematical, statistical and image processing techniques that can be used to quantify morphological, hydrological, ecological and other aspects of a land surface. Common synonyms for geomorphometry are geomorphological analysis (after geomorphology), terrain morphometry, terrain analysis, and land surface analysis. Geomorphometrics is the discipline based on the computational measures of the geometry, topography and shape of the Earth's horizons, and their temporal change. This is a major component of geographic information systems (GIS) and other software tools for spatial analysis. In simple terms, geomorphometry aims at extracting (land) surface parameters (morphometric, hydrological, climatic, etc.) and objects (watersheds, stream networks, landforms, etc.) using input digital land surface model (also known as digital elevation model, DEM) and parameterization software. Extracted surface parameters and objects can then be used, for example, to improve mapping and modeling of soils, vegetation, land use, geomorphological and geological features and similar. With the rapid increase of sources of DEMs today (and mainly due to the Shuttle Radar Topography Mission and LIDAR-based projects), extraction of land surface parameters is becoming more and more attractive to numerous fields ranging from precision agriculture, soil-landscape modeling, climatic and hydrological applications to urban planning, education, and space research. The topography of almost all Earth has been sampled or scanned today so that DEMs are available at resolutions of 100 m or better at a global scale. Today, land surface parameters are successfully used for both stochastic and process-based modeling, the only remaining issue being the level of detail and vertical accuracy of the DEM.
History Although geomorphometry started with the ideas of Brisson (1808) and Gauss (1827), the field did not evolve much until the development of GIS and DEM datasets in the 1970s. Geomorphology (which focuses on the processes that modify the land surface) has a long history as a concept and area of study, with geomorphometry being one of the oldest related disciplines. Geomatics is a more recently evolved sub-discipline, and even more recent is the concept of geomorphometrics. This has only recently been developed since the availability of more flexible and capable geographic information system (GIS) software, as well as higher resolution Digital Elevation Model (DEM). It is a response to the development of this GIS technology to gather and process DEM data (e.g. remote sensing, the Landsat program and photogrammetry). Recent applications proceed with the integration of geomorphometry with digital image analysis variables obtained by aerial and satellite remote sensing. As the triangulated irregular network (TIN) arose as an alternative model for representing the terrain surface, corresponding algorithms were developed for deriving measurements from it.
Surface gradient (derivatives)
Various basic measurements can be derived from the terrain surface, generally applying the techniques of vector calculus. That said, the algorithms typically used in GIS and other software use approximate calculations that produce similar results in much less time with discrete datasets than the pure continuous function methods. Many strategies and algorithms have been developed, each having advantages and disadvantages.
Surface normal and gradient
The surface normal at any point on the terrain surface is a vector ray that is perpendicular to the surface. The surface gradient ( ∇ f {\displaystyle \nabla f} ) is the vector ray that is tangent to the surface, in the direction of steepest downhill slope.
Slope
Slope or grade measures 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 p = r i s e r u n {\displaystyle p={\frac {rise}{run}}} , commonly expressed as a percentage, such that p = tan α. The latter is frequently used in engineering applications like 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. 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:
The partial derivatives are then approximated as weighted averages of the differences between the opposing sides:
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