Aspect and Slope Angle Calculator

Slope and aspect are the two first-order derivatives of a terrain surface, and both fall out of the same pair of partial derivatives estimated from a digital elevation model. The centred-difference scheme used here takes the elevation difference between opposite neighbours and divides by twice the cell spacing, so with 30 m cells and neighbours at 500 m east and 510 m west, dz/dx is -0.1667. Slope is the arctangent of the vector magnitude of the two gradients, and aspect is the compass bearing of that same gradient vector measured clockwise from north. Reporting conventions differ: degrees run 0 to 90, while percent grade is the raw rise over run times 100, so 45 degrees is 100 percent and a 100 percent grade is not vertical. Aspect is undefined where slope is zero, and GIS packages conventionally code such flat cells as -1. The values are strongly resolution dependent. Coarser grids average elevation over larger footprints and systematically suppress slope, so a 90 m SRTM tile can report maximum slopes far below what a 1 m lidar DEM shows for the same hillside, and slope-based landslide thresholds are only comparable between studies at matched resolution. Practitioners generally treat 25 to 45 degrees as the band where shallow failures concentrate. The hillshade value returned assumes a sun altitude of 45 degrees and an azimuth of 315 degrees, the standard northwest illumination that avoids the relief-inversion illusion, and it is a visualisation index from 0 to 255 rather than a radiometric quantity.

Quantifying slope from maps long predated computers: geomorphologists measured contour spacing by hand and built slope-frequency histograms through the first half of the twentieth century, a laborious process that limited studies to small areas. The shift came with gridded elevation data. Ian Evans, working at Durham University in the 1970s and publishing an influential general geomorphometry framework in 1980, set out the statistical approach of fitting a local surface to a moving window and taking its derivatives, establishing slope, aspect, and curvature as a coherent family of land-surface parameters. The specific algorithm behind most GIS software today comes from Berthold Horn's 1981 paper on hill shading and the reflectance map in the Proceedings of the IEEE, which used a weighted third-order finite difference across all eight neighbours of a cell. Esri adopted this method for its slope, aspect, and hillshade tools, which is why the Horn formulation became the de facto standard. Lyle Zevenbergen and Colin Thorne offered an alternative in 1987 by fitting a partial quartic surface through the cell neighbourhood, a method favoured for curvature work and implemented in GRASS and SAGA. Data availability then transformed the field, with the Shuttle Radar Topography Mission flown in February 2000 delivering near-global elevation coverage and national lidar programmes such as the USGS 3D Elevation Program pushing routine analysis to metre resolution.