Mapping Representations
In cartography, elevation is conventionally represented on maps through symbolic and visual techniques that convey terrain relief in two dimensions, enabling users to interpret topography without three-dimensional models. These methods, refined over centuries, balance detail, readability, and scale constraints to depict variations in height across landscapes.[63]
The historical evolution of elevation mapping began in the 18th century with hachured maps, where short lines oriented downslope indicated terrain steepness. Pioneered by cartographers like Johann Georg Lehmann in 1799, hachures used denser, darker lines for steeper slopes to simulate shading and direction of descent, as seen in early Swiss and French topographic surveys such as the Dufour Map. This qualitative approach dominated until the late 19th century, when more quantitative methods emerged. By 1891, Albrecht Penck's proposal for the International Map of the World (IMW) at the Fifth International Geographical Congress established global standards for uniform elevation depiction, including contour lines and layer tints on a 1:1,000,000 scale, promoting consistency across national maps.[64][65]
Contour lines, connecting points of equal elevation, became the cornerstone of modern topographic mapping in the late 18th century. Credited to mathematician Charles Hutton in 1774 during his survey of Schiehallion mountain in Scotland, contours were initially developed to calculate gravitational mass but quickly adopted for general terrain representation, as evidenced in early Ordnance Survey maps. These lines never cross and their spacing indicates slope steepness—closely packed for steep terrain and widely spaced for gentle slopes. Typical intervals range from 10 to 50 meters on medium-scale maps, such as those produced by the U.S. Geological Survey (USGS), where 10-foot (approximately 3-meter) intervals suit flat areas and 80-foot (24-meter) or larger suit mountainous regions; the exact interval is noted in the map's margin. Hachures are sometimes added to contours to emphasize steepness, with lines perpendicular to the contour and varying in length and density.[66][5][67]
Hypsometric tints employ color gradients to represent elevation bands, providing a layered view of relief without lines. Originating in the early 19th century, following the invention of lithography in 1796, with early colored applications in the 1820s, this method uses sequential colors—often green for lowlands, yellow for mid-elevations, and brown for highlands—to simulate natural vegetation and rock transitions, as refined in late-19th-century European maps. By the mid-20th century, tints evolved to blend continuously for smoother gradients, enhancing visual hierarchy on small-scale world maps like those in the IMW series.[64][68]
Spot heights mark precise elevations at key features, such as summits, benchmarks, or control points, supplementing contours for reference. On USGS quadrangle maps, these are denoted by numerals (e.g., "5280") or symbols like "BM" for bench marks, accurate to within one-third of the contour interval, aiding in navigation and surveying.[5]
To counteract the flattening effect of map projections, vertical exaggeration is applied in derived representations like cross-sectional profiles from topographic maps. This technique scales vertical dimensions disproportionately to horizontal ones, often at ratios like 5:1, making subtle relief visible; for instance, on a 1:50,000 horizontal scale map, a vertical scale of 1:10,000 yields 5:1 exaggeration, calculated as the ratio of horizontal to vertical real-world units. Such exaggeration is noted explicitly to avoid misinterpretation of true proportions.[69]
Digital Elevation Models
Digital elevation models (DEMs) are raster datasets that represent the Earth's surface topography as a grid of elevation values, enabling computational geospatial analysis in geographic information systems (GIS). These models provide a continuous surface approximation, where each cell in the grid stores an elevation measurement relative to a reference datum, facilitating automated processing for various analytical tasks. Unlike vector-based representations, DEMs support efficient derivation of terrain attributes through numerical methods, making them essential for quantitative terrain modeling.[70]
Prominent global DEMs include the Shuttle Radar Topography Mission (SRTM) and the ASTER Global Digital Elevation Model (GDEM). The SRTM, conducted in February 2000 aboard the Space Shuttle Endeavour, produced a near-global dataset covering approximately 80% of Earth's land surfaces between 60°N and 56°S latitudes at a 1 arc-second resolution, equivalent to about 30 meters at the equator. This radar-based mission generated the first widely available high-resolution global elevation data, with tiles distributed in 1° × 1° extents. The ASTER GDEM, derived from optical stereo imagery collected by the Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) instrument on NASA's Terra satellite since 1999, offers global coverage of land areas from 83°N to 83°S at a similar 30-meter (1 arc-second) horizontal resolution. It utilizes automated stereophotogrammetric techniques on over 1.4 million stereo-pair scenes to create a consistent elevation surface. More recent prominent global DEMs include the Copernicus DEM GLO-30, released in 2021 by the European Space Agency, providing 30-meter resolution coverage of global land areas with vertical accuracy better than 4 meters in many regions.[71][72][73][74]
DEMs are structured as regular grids where elevation values are sampled at fixed intervals, typically in geographic coordinates with postings of 1 arc-second (approximately 30 meters near the equator). Each grid cell contains a single elevation value, often in meters relative to mean sea level, forming a two-dimensional array that can span large areas when tiled. Common file formats include GeoTIFF, which embeds geospatial metadata such as projection and georeferencing directly into the raster file, ensuring compatibility with GIS software for analysis and visualization. This gridded format allows for straightforward interpolation and processing, with data often void-filled or masked for areas lacking observations, such as oceans or polar regions.[75][76]
From DEMs, secondary terrain derivatives like slope, aspect, and curvature are computed using finite difference methods, which approximate spatial gradients by differencing elevation values between adjacent cells. Slope, representing the steepness of the terrain, is calculated as the maximum rate of change in elevation, often using the formula for the gradient angle θ\thetaθ: