Fractal properties of isolines at varying altitude reveal different dominant geological processes on Earth

Geometrical properties of landscapes result from the geological processes that have acted through time. The quantitative analysis of natural relief represents an objective form of aiding in the visual interpretation of landscapes, as studies on coastlines, river networks, and global topography, have shown. Still, an open question is whether a clear relationship between the quantitative properties of landscapes and the dominant geomorphologic processes that originate them can be established. In this contribution, we show that the geometry of topographic isolines is an appropriate observable to help disentangle such a relationship. A fractal analysis of terrestrial isolines yields a clear identification of trenches and abyssal plains, differentiates oceanic ridges from continental slopes and platforms, localizes coastlines and river systems, and isolates areas at high elevation (or latitude) subjected to the erosive action of ice. The study of the geometrical properties of the lunar landscape supports the existence of a correspondence between principal geomorphic processes and landforms. Our analysis can be easily applied to other planetary bodies.

💡 Research Summary

In this paper the authors present a systematic fractal‑analysis of topographic isolines (contour lines) on Earth and the Moon, with the aim of linking the geometric properties of these lines to the dominant geological and climatic processes that shape planetary surfaces. High‑resolution digital elevation models were used: the SRTM30‑plus dataset for Earth (≈1 km resolution over continents and ≈4 km over the ocean) and the Clementine altimetry data for the Moon (≈30 km resolution). Isolines were extracted at 100 m elevation intervals by interpolating between neighboring grid points that straddle the target height. Three equivalent definitions of an isoline were tested, all yielding identical results.

The fractal dimension D of each isoline was estimated using the classic box‑counting method. Square boxes of side length r were overlaid on the set of points belonging to an isoline, and the number of occupied boxes N(r) was recorded for a range of scales. A linear fit to log N(r) versus log r within a conservative scale window (rmin≈6 km to rmax≈60 km for Earth; rmin≈15 km to rmax≈150 km for the Moon) provided D. The authors explicitly note that isolines are not mathematically strict fractals; the lower bound is set by data resolution, while the upper bound is limited by the need to stay within geologically homogeneous regions. To avoid edge effects, the planetary surfaces were tiled into cells (4°×4° for Earth, 15°×15° for the Moon), each cell being roughly ten times larger than rmax. Within each cell, D was computed for every isoline that contained at least 500 points on Earth (200 on the Moon). The regression error never exceeded 4 %. Robustness checks included varying the resolution, the minimum point threshold, pruning isolated “islands” (tiny peaks or craters), and changing cell size and rmax; these tests produced only minor quantitative shifts (≈0.1 in D) without altering the qualitative patterns.

Statistical analysis focused on the dependence of the average fractal dimension ⟨D⟩ on three variables: longitude, latitude, and elevation. For Earth, ⟨D⟩ showed virtually no systematic variation with longitude or latitude, but displayed a pronounced, non‑monotonic dependence on elevation. In contrast, the Moon exhibited a nearly constant ⟨D⟩≈1.2–1.3 across all three variables, reflecting the relative uniformity of its surface‑forming processes.

Mapping the spatial distribution of D on Earth revealed distinct geomorphological signatures. Low D values (≈1.0, deep blue) correspond to smooth marine regions such as oceanic trenches, continental slopes, and abyssal plains—areas dominated by steep, relatively unfragmented topography or by sedimentary infill. Mid‑ocean ridges and transform‑fault zones display higher D (≈1.4–1.5), indicating a more intricate contour geometry caused by active seafloor spreading and faulting. Continental coastlines are clearly delineated by a jump from D≈1.0 below sea level to higher values on land; along the coasts, a modest increase of D with latitude is observed, likely linked to colder climates and enhanced fluvial or glacial erosion.

On land, the authors identify two major regimes. A yellow‑green band spanning latitudes 40° S to 40° N and elevations 0–2000 m shows moderate D (≈1.2–1.4), reflecting the combined influence of river networks, weathering, and moderate relief. Three isolated high‑D patches (red, D>1.5) correspond to the Siberian highlands, the Himalaya, and the Andes—regions where high elevation and/or high latitude promote extensive glacial activity and intense tectonic uplift, both of which increase contour complexity.

The lunar analysis, performed with the same methodology, yields a remarkably homogeneous D field. Despite the obvious asymmetry in crater density between the lunar near‑side and far‑side, the average D varies by less than 0.1, confirming that the Moon’s surface is largely governed by impact cratering and ancient volcanic resurfacing, with negligible ongoing endogenic or exogenic modification.

Overall, the study demonstrates that the local fractal dimension of topographic isolines is a powerful, scale‑independent metric that can be computed automatically from global DEMs. It captures the imprint of dominant geological agents—tectonic spreading, subduction, fluvial erosion, glaciation, and impact cratering—without requiring prior geological mapping. The authors argue that this approach can be readily extended to other planetary bodies (e.g., Mars, Venus) where high‑resolution topography exists, providing a quantitative bridge between geomorphology and planetary science and offering a new tool for monitoring landscape evolution in response to climate change or tectonic activity.