Relations between land surface properties: Altitude, slope and curvature

Abstract

The land surface is studied in terms of point properties, linear (flow-line and break-line) properties and areal properties. These show considerable structure and within-region predictability: the land surface is not (uni-)fractal but varies along multiple statistical dimensions. The simplest point properties are altitude itself and the first and second order derivatives, slope and curvature, of the altitude surface. Two components of each derivative are of proven value: it is best to separate the vertical (slope gradient and gradient change) from the horizontal (aspect and aspect change). The relationships between these five, some of which are not intuitive, are discussed here with illustrations from Germany.

Slope gradient usually varies with altitude, but not monotonically. The strength of the relationship varies, and in dissected plateaus the overall correlation is negative. The effects of mesoclimate (slope climate) mean that geomorphological processes vary with aspect, and there are a number of hypotheses of how this might affect gradient. Such signals are masked by considerable scatter on gradient-aspect plots, and it is important to use appropriate tools for relating linear to circular variables. Surface curvature (convexity taken as positive) is separated into vertical (profile, change of gradient) and horizontal (plan, change of aspect) components. The relationship between these is almost universally positive, but very weak: correlations are around +0.2, whether or not transformations to reduce the effects of extreme values are applied. As might be expected, profile convexity almost universally increases with altitude, although the correlation is rarely stronger than that of gradient with altitude. Plan convexity is least varied (closest to zero) on steep slopes; extreme but real values are found on floodplains.