Hypsometry, which is the distribution of area in different elevation bands, encodes useful information about the tectonic and climatic forcing of a given landscape. We develop a unified mathematical theory of the elevation distribution of the surface of steady-state rivers and glaciers, when erosion and rock uplift exactly balance each other out. For a steady climate, a constant and uniform rock uplift, a homogeneous bedrock, and a stream-power erosion rule, the theory reproduces the known logarithmic longitudinal profile of steady-state bedrock rivers. For steady-state mountain glaciers, it predicts a Gaussian surface-elevation distribution, which matches 1-d simulation results. Our theory implies that changes in relief due to given changes in climate or tectonic forcing are stronger in purely fluvial landscapes, compared to purely glacial ones. This new approach may prove useful for extracting quantitative signatures of tectonic and climatic forcing from the surface-elevation distribution of rivers and glaciers in various landscapes.
