the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 22 Sep 2025
Discrete differential geometry of fluvial landscapes
Abstract. Geomorphology as a discipline is often defined by the use of topographic geometry to understand surface processes on Earth and other planets. In practice this requires drawing quantitative connections between metrics of surface geometry and rates of exhumation, while also understanding the spatial partitioning of different erosion processes and the feedbacks between them. Many landscape evolution studies leverage curvature calculated as the scalar output of the Laplacian operator, which does not leverage all the information contained in the surface curvature tensor and which admits systematic error (up to ~300 % percent) when applied directly to map-view topographic projections. In this study we use a formal surface theory approach to compute intrinsic and extrinsic curvature metrics, and associated shape-class distributions, of approximate steady-state fluvial topography of the Oregon Coast Range, USA. This workflow, including careful spectral filtering to isolate wavelengths of interest, provides a nuanced view of landscape structure, while simultaneously eliminating systematic errors arising from map-view approaches to topographic analysis. We leverage two invariants of the curvature tensor at a point – the Mean and Gaussian curvatures – to identify novel systematic structure of topographic geometry in channel and ridge networks that captures the full compliment of documented process regime transitions. Finally, we show remarkable symmetries in the distribution of Mean curvature and associated shape classes, specifically an equipartition of the landscape between concave-down and concave-up elements. These results suggest that formal surface theory approaches could prove valuable in maximizing the utility of digital elevation data and understanding the processes driving the evolution and organization of fluvial landscapes.
Competing interests: NK is a member of the editorial board of Geomorphica.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.- Preprint
(19946 KB) - Metadata XML
- BibTeX
- EndNote
Status: final response (author comments only)
-
RC1: 'Comment on egusphere-2025-4431', Benjamin Kargere, 29 Oct 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-4431/egusphere-2025-4431-RC1-supplement.pdfCitation: https://doi.org/
10.5194/egusphere-2025-4431-RC1 -
RC2: 'Comment on egusphere-2025-4431', Anonymous Referee #2, 10 Dec 2025
For this review, I am going to focus on the big concerns, not point-by-point minor comments. I do this because there are major concerns from my end. I will be direct here: this is a difficult paper to read because several claims and concepts are introduced somewhat abruptly, and the sections do not connect well conceptually. I welcome the authors' explanations, but my understanding is that the manuscript needs a major rewrite; it should not be published in its current form.
My first and biggest concern is the way the manuscript discusses the "error" in the Laplacian. As written, the text implies that the Laplacian is incorrect because it does not match the true 3D curvature. In my view, this is a conceptual misunderstanding. The Laplacian in Figure 7b and the percent error in Equation 39 are measuring exactly what the Laplacian is supposed to measure. It is not intended to represent the geometric curvature of a 3D surface, so the discrepancy shown is not an "error" in the operator itself. It is merely an expected difference. The manuscript repeatedly frames this difference as if geomorphologists expect ∇²z to equal true surface curvature, but I do not think this is accurate. Principal curvature is a geometric property of the surface and is not part of the hillslope diffusion equation; it becomes relevant only when computing intrinsic surface curvature, not when solving diffusion of z(x,y). Yes, geomorphologists sometimes use the shallow slope approximation where Laplacian curvature approximates surface curvature near a hilltop, and the paper rightly shows that this breaks down in steep or complex terrain. However, no quantitative geomorphologist would claim that the Laplacian is exactly equal to 3D mean curvature, and framing the deviation as "error" is misleading. The Laplacian and the surface curvature simply describe different things.
A second major concern is that the manuscript sometimes conflates 2D planform and 3D surface quantities. For example, the comparison between the horizontally-projected catchment area and true surface area is presented in a way that suggests one is correct and the other is not. These are simply different definitions used for different purposes. The tone, as written, comes across stronger than needed. A concrete example is the discussion of the specific drainage area as defined in Bonetti 2018 and related work: this is explicitly a horizontally projected quantity (see figure one in that paper). That does not make it wrong, just different. There is also some confusion between the specific drainage area, which is a pointwise quantity, and the total drainage area, which is defined for a contour width, and the manuscript occasionally cites papers on one while discussing the other. For a paper that aims to clarify differential geometry in fluvial landscapes, this distinction needs to be handled carefully.
Regarding structure: Section 5 is essentially textbook differential geometry, but it appears after Section 4 on spectral filtering and before earlier conceptual sections are fully settled. This makes the manuscript feel meandering and difficult to follow. The historical context in the early part also feels excessive; a condensed version in the discussion might be more effective. Sections 6 and 7, which are the main contribution (application to real topography), come way later.
Another example where the writing overreaches is line approximately 570: "This observation could be interpreted as reflecting a Minimal Surface condition... in which total Mean curvature is minimized." If taken literally, this would imply that steady state landscapes tend toward zero mean curvature everywhere, which is clearly not the case. A plane would be the solution. This statement exemplifies a broader tendency in the manuscript to make speculative claims without engaging fully with the existing literature on landscape organization.
My recommendation: focus the paper on what is genuinely novel, applying well-established differential geometry tools to compute principal curvatures on DEMs and demonstrating what we can learn about landscape segmentation from doing this rigorously. That contribution could be valuable on its own. I suggest removing or significantly toning down the claims about Laplacian "error" and landscape self-organization, as these distract from the real strengths of the paper. As it stands, the main message is difficult to discern.
I hope these comments help in improving the manuscript.
Citation: https://doi.org/10.5194/egusphere-2025-4431-RC2
Model code and software
TopoCurve Matlab Repository Nathaniel Klema https://github.com/ntklema/TopoCurve_Matlab
Viewed
| HTML | XML | Total | BibTeX | EndNote | |
|---|---|---|---|---|---|
| 585 | 64 | 15 | 664 | 25 | 29 |
- HTML: 585
- PDF: 64
- XML: 15
- Total: 664
- BibTeX: 25
- EndNote: 29
Viewed (geographical distribution)
| Country | # | Views | % |
|---|
| Total: | 0 |
| HTML: | 0 |
| PDF: | 0 |
| XML: | 0 |
- 1