Hypsometric Curve Integral Calculator

The hypsometric integral condenses an entire area-elevation distribution into one dimensionless number between 0 and 1. Physically it approximates the fraction of rock volume still remaining beneath the summit envelope and above the outlet, so a basin returning HI of 0.55 has lost roughly 45 percent of the material that once filled the box defined by its total relief and its planimetric area. The elevation-relief form used here, HI = (Hmean - Hmin) / (Hmax - Hmin), works because mean basin elevation is the area-weighted average of the hypsometric curve, which makes the ratio algebraically equivalent to integrating the normalized curve. Practitioners read values above 0.6 as youthful inequilibrium landscapes, 0.35 to 0.6 as mature, and below 0.35 as old peneplained surfaces. Two caveats dominate practical use. The integral is non-unique: a strongly convex upper limb paired with a concave lower limb can return the same value as a straight line, so the full curve should be plotted alongside the number rather than replaced by it. The integral is also scale dependent and generally declines as basin area grows, which is why comparisons are only defensible between catchments of similar size, lithology, and DEM resolution. Glaciated basins add a further complication, producing concave steps at cirque elevations that depress HI for reasons having nothing to do with fluvial maturity.

Area-altitude analysis grew out of nineteenth-century efforts to describe the Earth statistically. The global hypsographic curve, showing the bimodal split between continental platforms near 0 to 1,000 m and abyssal plains 4,000 to 5,000 m below sea level, was in circulation by the 1880s, and Albrecht Penck treated area-altitude distributions in his 1894 work Morphologie der Erdoberflaeche. Applying the idea to individual drainage basins required normalizing away basin size, which is what Arthur Strahler achieved at Columbia University in his 1952 Geological Society of America Bulletin paper on hypsometric analysis of erosional topography. Strahler plotted relative area against relative height so that any basin could be compared with any other, and he linked curve shapes to the inequilibrium, equilibrium, and monadnock stages inherited from William Morris Davis and the cycle of erosion. The single-number shortcut arrived in 1971, when Richard Pike and Stephen Wilson showed in the same journal that the elevation-relief ratio computed from mean, minimum, and maximum elevation is identical to the integral under Strahler's curve, so the index could be obtained without digitizing the curve at all. That result is what let the integral become a routine layer in tectonic geomorphology once digital elevation models made mean elevation trivial to extract.