Mountains, surfaces and roughness
Mountain profiles often appear related across changes of scale, but terrain is usually modeled as self-affine: horizontal distance and vertical relief scale by different factors.
A roughness exponent
For a self-affine height field, typical elevation differences grow as a power of horizontal separation. The Hurst exponent summarizes this relationship over a scale range and relates to spectral slope and surface dimension under model assumptions.
In a self-affine height field, typical elevation difference grows approximately as Δxᴴ with horizontal separation. Small H gives rougher and large H smoother surfaces. The exponent can be estimated from profiles, variograms or spectra and is meaningful only across a finite range. Cell size and large-scale trend must be handled or regional slope will be confused with local roughness.
Processes leave directional signatures
Tectonics, erosion, glaciation, sediment transport and rock resistance create ridges and valleys with preferred directions and characteristic scales. A single isotropic fractal surface cannot explain all of them.
Rivers incise preferred directions, glaciers widen valleys, layered rock produces ledges and tectonics organizes entire ridges. Such anisotropies contradict an isotropic random field. Direction-dependent measurements can reveal them. A landscape is not simply noise with the right exponent; its forms retain the memory of different processes and timescales.
Synthetic terrain needs more than noise
Fractional noise creates plausible broad-to-fine roughness, but convincing landscapes add drainage, erosion, material layers and scale-aware shading. Fractal synthesis supplies a starting field; geomorphology gives it history.
Plausible synthetic terrain often begins with scale-rich noise but then needs drainage, erosion and material-dependent rules. Domain warping breaks coordinate-grid regularity but cannot guarantee physically plausible rivers. Only when water converges downhill, transports sediment and stabilizes slopes do valleys, ridges and talus develop relationships beyond texture.
Sources and further reading
This article summarizes the following specialist sources in original wording. Accessed and editorially reviewed 12 August 2026.
Continue the idea
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