| 16 Apr 2026
From regular to random: a unifying framework for step-pool spacing
Abstract. Steep streams frequently display a distinctive step-pool structure where water crests over a near-vertical drop and plunges into a deeper depression in a repeated pattern. Because they naturally moderate the flow of water and sediment in hazardous mountain catchments, step-pools are often installed in stream management and restoration projects. However, emulating step-pool sequences is hindered by debate on whether natural step-pools are themselves regularly or randomly spaced. Here we show that the spacing of step-pool sequences spans a continuum between regularity and randomness driven by multiple formation mechanisms. Analyzing a compilation of natural, experimental, and numerically simulated step-pools, we found that natural variability and inherent limits on minimum spacing prevent fully regular or random sequences. While certain mechanisms result in comparatively regular or random spacing, no single mechanism dominates step-pool development. Our results resolve longstanding tension between a plethora of proposed formation mechanisms that yield contrasting predictions. Furthermore, the emergent limits on spacing variability provide testable predictions about the adjustment of sequence spacing following river disturbance that may eventually be used to define concrete targets for stream restoration and hazard management.
I really enjoyed this paper, and appreciated how its approach to data analysis allows one to better understand the implications of step-pool spacing with regard to possible governing processes. As pointed out by the authors, the approach has important implications for restoration designs.
I have made a few, mostly minor, editorial comments that I believe will improve the authors’ ability to communicate their ideas, but otherwise the manuscript is pretty close to being ready for publication.
Jim Pizzuto
Professor Emeritus
University of Delaware
USA
Some detailed comments, keyed to the text:
1. Line 11. Is it possible to describe these inherent limits on minimum spacing, so the reader can understand this without going through the entire article? This would help facilitate acceptance of the authors' ideas, though it might be difficult to achieve.
2. Line 44. Is it possible to more clearly state the goal of this paper? What is a "framework"? Is it a mathematical approach to evaluating step-pool spacing that is independent of formation mechanism? Is it a hypothesis? A type of formal data analysis (i.e., a process for analyzing observations of step-pool spacing)? Something else? It would be helpful to the reader to better understand what the authors are contributing here.
3. Line 81. In the spirit of reproducibility, it might be helpful to include (likely in the supplement) a series of computational steps followed in creating the reference lines. I have some rather dim idea regarding how this is done, but if I wanted to recreate Figure 2, I would be pretty confused regarding the details. Ok, now I see this in the Appendix, so ideally the appendix can be referenced here so the reader knows where to look.
4. Line 86. I think it would be helpful to add a figure to illustrate the concept of "space between reference lines". I am not sure what this concept refers to. Is this the sequence of points within a certain scaled distance of each line, or something like this? Rather than relying on abstract terms, a graphical illustration would be much more effective in communicating the author's concept here. Or better yet (now that I have read ahead), please see comments on Figure 2.
5. Figure 2. At the risk of complicating this figure, it might be helpful to label CV_N = 0 and 1 on the figure. I guess the solid blue line is CV_N = 0 and the solid red line is CV_N = 1, right?
6. Line 113. “Closer” would be a better description than "closest". This change is appropriate because only 2 options are being compared, and also because the data do not generally plot "close" to either end member, but are actually "closest" to the gray line separating the two end members in Figure 3.
7. Line 122. Can this data point be highlighted in Figure 3, so the reader doesn't have to hunt for it by scanning the long and complex legend to the right?