Unified Patterns of Topological Structure of Hydrological Characteristics in Global River Networks

Zhao, Chensong; Ke, Qihua; Cao, Yuan; Liu, Shutong; Wei, Randongfang; Xie, Di; Wang, Jinqiang; Huang, Jingwei; Zhang, Qingqing; Gao, Jiahui; Li, Jing; Wang, Guangqian; Zhong, Deyu

doi:10.5194/egusphere-2026-1128

Preprints

https://doi.org/10.5194/egusphere-2026-1128

Preprints

06 Mar 2026

| 06 Mar 2026

Unified Patterns of Topological Structure of Hydrological Characteristics in Global River Networks

Chensong Zhao, Qihua Ke, Yuan Cao, Shutong Liu, Randongfang Wei, Di Xie, Jinqiang Wang, Jingwei Huang, Qingqing Zhang, Jiahui Gao, Jing Li, Guangqian Wang, and Deyu Zhong

Abstract. Unravelling the coupling between river network structure and hydrological fluxes is essential for understanding basin-scale dynamics. While traditional ordering methods describe macro-scale patterns, they often obscure local functional variations. This study quantifies the universal hydrological patterns of global river networks by integrating the classical Horton–Strahler framework with a hierarchical pyramid decomposition technique. Leveraging the global HydroATLAS dataset, we analysed 228 representative basins spanning diverse hydro-climatic regimes. We extracted rigorously defined network attributes and hydrological fluxes to examine the scaling behaviours of fundamental structural components, defined here as basic units. Our results reveal a striking topological invariance in hydrological characteristics across both varying spatial scales and distinct geographic regions. Specifically, the runoff and discharge ratios of these basic units maintain robust statistical consistency regardless of basin size or climatic conditions ranging from humid to arid. This suggests that network topology functions as a dominant physical control, effectively acting as a low-pass filter that dampens high-frequency climatic variability to produce unified global scaling laws. These findings advance the theoretical understanding of fractal river networks. Furthermore, they open new avenues for prospective research, including the integration of these physics-informed topological priors into next-generation Earth system models to improve discharge predictions and water resource modelling in ungauged basins.

Received: 27 Feb 2026 – Discussion started: 06 Mar 2026

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Chensong Zhao, Qihua Ke, Yuan Cao, Shutong Liu, Randongfang Wei, Di Xie, Jinqiang Wang, Jingwei Huang, Qingqing Zhang, Jiahui Gao, Jing Li, Guangqian Wang, and Deyu Zhong

Status: final response (author comments only)

RC1:
'Comment on egusphere-2026-1128', Anonymous Referee #1, 26 May 2026

In the manuscript, the authors investigate the existence of universal hydrological scaling laws in global river networks by analysing 228 basins form HydroATLAS and combining Horton-Strahler ordering with a hierarchical pyramid decomposition framework. The authors show that runoff and discharge rations exhibit scale-invariant statistical behaviour across climates and basin sizes, suggesting that river network topology acts as a dominant control on hydrological dynamics and filters climatic variability.
The manuscript addresses an interesting question using a large global dataset and generally sound analytical methods. Although the presentation is comprehensive and well structured, the interpretation occasionally overstates the universality and physical significance of the reported scaling relationships. My two major comments are:
1) I found the manuscript a bit dense, with 14 figures (4 of which in the discussion, which really break the flow of thoughts). I would consider moving some of the less important figures in the Supporting Info, only citing the result in the main text when applicable.
2) More care needs to be given to the interpretation of the results and consequent discussion. Being a global dataset, HydroATLAS does a great job at capturing global larger-scale patterns. However, the interpretation of "headwaters" that define first order streams requires much attention. In my experience, real first and second order rivers are mostly missing, and this could affect the interpretation of the results. Even more importantly, I think much more care needs to be given to the interpretation of the provided mean annual runoff and discharge for each river. Being derived from a global model, this inevitably misses local heterogeneity and enhances the cross-scale structure, as defined by the structure of the model itself. Often in these cases the provided values should be interpreted as an order or magnitude estimate, more than an actual mean annual runoff. Therefore, the findings of these papers need to be put in this perspective, care taken when projecting these results back to physical processes, and these limitations need to be acknowledged in the discussion.
The manuscript would benefit from a more rigorous assessment of uncertainty and sensitivity to HydroATLAS preprocessing choices.

ADDITIONAL COMMENTS:

The physical interpretation that topology acts as a “low-pass filter” remains largely conceptual and is not directly demonstrated.
Additional validation against observed gauged river data, synthetic benchmark networks, or independent hydrological products would strengthen confidence that the identified patterns are not artefacts of the chosen decomposition scheme or HydroATLAS topology. I know this would be a lot of work, but at least it should be acknowledged in the paper and deferred to a future work.
It will be more informative to change Table 2 into a figure. It will be nice to have a global map showing the basins (maybe using different color for each region). This would give an indication also of the size variability of the basins around the globe. The data in the current table could be shown as a bar plot if you feel it's useful.
The sentence in lines 135-138 needs some restructuring, as the grammar is missing something. Also, point (4) is a consequence, rather than a rule.
Line 159: I think RMS should actually be in mm/year, since it's annual runoff, therefore cumulative runoff should be m3/year.
Line 166: hypothesis should be reported in the final part of the intro
Figure 2: I am not sure what is the added value of this figure as compared with Figure 3 (which I find more intuitive). Consider removing it and use Figure 3 in the explanations.
Line 234: What is specifically the characteristic length? Is this analysis done for each decomposition level (i.e. Strahler order), or are you combining all units with same length together? Also, since length is a continuous variable, it is highly improbable to have two units with exactly the same length. Did you use length classes (e.g., from 0 to 100m, from 100 to 200m etc)?
Caption of Figure 4: "to enable cross-basin comparison", since cross-scale is within the single basin
Is figure 4c and d the same as Figure 4 but in semilog space? If so, consider joining the two figures.
Line 303: what's an iterator and an iteration in this case? Please use consistent terminology and/or an example to clarify.
Caption of Figure 7: please give a specific definition of variability range (e.g. 3 standard deviations).
Figure 9, but in all the paper too: I feel like reporting median values rather than mean ones would be more informative, since most of the indicators show some skewness in the distribution, and it's evident also here.
Figure 9: in many cases it seems that there is a well-defined range and some outliers. Can you identify the outliers in the map and discuss what could cause this deviation?
Appendices: this is the guide text; remove it if no appendices are used.
Code availability: it is highly suggested to share the code in a public repository, too.

Citation: https://doi.org/10.5194/egusphere-2026-1128-RC1
- AC1: 'Reply on RC1', Chensong Zhao, 09 Jul 2026
  
  Reply on RC1
  We thank Referee #1 for the constructive and helpful comments. We appreciate the reviewer’s positive assessment of the scientific question, global dataset, and analytical framework. We agree that the manuscript would benefit from clearer figure organization, more cautious interpretation of HydroATLAS-derived hydrological attributes, and a more explicit treatment of uncertainty and sensitivity.
  We provide below a point-by-point response and describe the revisions that will be incorporated into the revised manuscript and Supporting Information. We also plan to attach a response supplement containing the key additional figures and analyses referred to below.
  Response to major comment 1: Figure organization
  Reviewer comment: I found the manuscript a bit dense, with 14 figures (4 of which in the discussion, which really break the flow of thoughts). I would consider moving some of the less important figures in the Supporting Info, only citing the result in the main text when applicable.
  Response: We thank the reviewer for this helpful suggestion. We agree that the original manuscript contained too many figures in the main text and that this could interrupt the flow of the argument, especially in the Discussion section. In the revised manuscript, we will reorganize the figure set to keep the main text focused on the essential methodological framework and the principal hydrological-topological results.
  Specifically, figures that mainly provide supplementary methodological detail or additional discussion-level support will be moved to the Supporting Information, while figures that directly support the main quantitative conclusions will be retained in the main text. We will also remove or merge redundant figure content where the same information is already presented in a more analytical form elsewhere.
  Planned revision: The original Figure 2 will be moved to the Supporting Information as Figure S1, because the original Figure 3 provides a more intuitive main-text explanation of the pyramid decomposition framework. The original Figure 12 will be moved to the Supporting Information as Figure S2, because it provides supplementary support for the relationship between characteristic length and normalized hydrological variables. In addition, the original Figure 4 will be removed or integrated into the revised hydrological-scaling figure, because its normalized flux patterns are already represented in the semilog scaling analysis. The corresponding main-text discussion, figure numbering, and in-text references will be revised accordingly.
  Response to major comment 2: HydroATLAS low-order streams and model-derived hydrological attributes
  Reviewer comment: More care needs to be given to the interpretation of the results and consequent discussion. Being a global dataset, HydroATLAS does a great job at capturing global larger-scale patterns. However, the interpretation of "headwaters" that define first order streams requires much attention. In my experience, real first and second order rivers are mostly missing, and this could affect the interpretation of the results. Even more importantly, I think much more care needs to be given to the interpretation of the provided mean annual runoff and discharge for each river. Being derived from a global model, this inevitably misses local heterogeneity and enhances the cross-scale structure, as defined by the structure of the model itself. Often in these cases the provided values should be interpreted as an order or magnitude estimate, more than an actual mean annual runoff. Therefore, the findings of these papers need to be put in this perspective, care taken when projecting these results back to physical processes, and these limitations need to be acknowledged in the discussion.
  Response: We thank the reviewer for this important comment. We agree that the low-order streams in HydroATLAS should not be interpreted as a complete representation of all real headwater channels. The first- and second-order streams analysed in this study are HydroATLAS-defined low-order channels, and their representation depends on DEM resolution, drainage-area thresholds, flow-routing algorithms, and hydrographic preprocessing choices. In the revised manuscript, we will explicitly use this interpretation and avoid implying that HydroATLAS first-order streams correspond to all physically existing headwater channels.
  We also agree that the HydroATLAS runoff and discharge variables should be interpreted with caution. The runoff and discharge estimates are derived from WaterGAP-based model outputs and downscaling procedures, and therefore provide a globally consistent model-derived framework rather than precise local observations at individual river reaches. This consistency is useful for broad-scale statistical comparison, but local heterogeneity and model-structure effects remain important limitations. In the revised manuscript, we will clarify this point in the Methods, Discussion, and Conclusions.
  To further address this concern, we have completed an additional independent validation using GRADES-hydroDL V2.0 discharge and MERIT-Basins river topology. This analysis is independent of the HydroATLAS/WaterGAP attributes used in the main analysis. It was conducted for MERIT-Basins Pfafstetter level-1 region 4, which mainly covers Asia, using 464,949 MERIT-Basins reaches and 2015-2024 GRADES-hydroDL discharge data. For orders 3-7, the adjacent-order discharge ratio averaged 4.22, while the cumulative runoff-like ratio estimated from local discharge increments averaged 0.476. The order-2 runoff-like deviation also appeared in this independent dataset, suggesting that the low-order deviation is not unique to HydroATLAS preprocessing. We will present this analysis as a supplementary robustness check, while still making clear that it does not replace broader validation using gauged observations, synthetic benchmark networks, or other independent hydrographic products.
  Planned revision: We will revise the Methods to clarify the source, validation status, and interpretation of the HydroATLAS runoff and discharge attributes. We will revise the Discussion to state that the findings should be interpreted as statistical hydrological-topological patterns within a globally consistent model-derived dataset, rather than as direct proof of universal local hydrological processes. We will also add a Supporting Information section describing the independent GRADES-hydroDL/MERIT-Basins validation, including the data linkage, calculation procedure, diagnostics, and results.
  Response to uncertainty and sensitivity
  Reviewer comment: The manuscript would benefit from a more rigorous assessment of uncertainty and sensitivity to HydroATLAS preprocessing choices.
  Response: We agree with the reviewer. A complete sensitivity analysis to HydroATLAS preprocessing choices would require alternative river-network extraction thresholds, DEM resolutions, independent global hydrographic products, and preferably observed hydrological data. Such a full analysis is beyond the scope of the present revision. However, we will add and clarify several feasible robustness checks that directly address the main sources of uncertainty raised by the reviewer.
  First, the original manuscript already included a comparison of H-S runoff and discharge ratios with and without order-2 ratios, which provides a sensitivity check for the influence of low-order streams. In the revised manuscript, we will make the purpose and interpretation of this existing analysis clearer. Second, because several basic-unit indicators show skewed distributions, Fig. 9 will be revised to report medians and interquartile ranges as robust statistics, with means retained as complementary information. Third, outlying basins will be identified using Tukey's 1.5 x IQR criterion and mapped in the Supporting Information. Fourth, as described above, we have completed an independent GRADES-hydroDL/MERIT-Basins validation of the main H-S order-based scaling patterns.
  Planned revision: The revised manuscript will discuss low-order stream sensitivity, median/IQR statistics, outlying basins, and the new independent validation. The limitations discussion will also explicitly state that a full sensitivity analysis to HydroATLAS preprocessing choices remains an important direction for future work.
  Additional comments
  
  Comment 1: Low-pass filter interpretation
  Reviewer comment: The physical interpretation that topology acts as a "low-pass filter" remains largely conceptual and is not directly demonstrated.
  Response: We agree with this comment. The low-pass-filter interpretation will be retained only as a conceptual interpretation consistent with the observed smoothing of local variability, not as a directly demonstrated physical mechanism. We will avoid implying that the present statistical analysis proves a physical filtering process.
  Planned revision: The low-pass-filter wording will be softened in the Abstract, Discussion, and Conclusions. The revised text will present this idea as an interpretive framework rather than a directly tested mechanism.
  Comment 2: Additional independent validation
  Reviewer comment: Additional validation against observed gauged river data, synthetic benchmark networks, or independent hydrological products would strengthen confidence that the identified patterns are not artefacts of the chosen decomposition scheme or HydroATLAS topology. I know this would be a lot of work, but at least it should be acknowledged in the paper and deferred to a future work.
  Response: We agree that independent validation is valuable. In response, we have completed a supplementary independent validation using GRADES-hydroDL V2.0 discharge and MERIT-Basins topology. This validation is independent of the HydroATLAS/WaterGAP attributes used in the main analysis. GRADES reach discharge was linked to MERIT-Basins river reaches using COMID, and H-S order-based discharge and runoff-like ratios were recalculated.
  For discharge, the adjacent-order discharge ratio averaged 4.22 for orders 3-7. For runoff, because GRADES-hydroDL does not provide the same local runoff attribute as HydroATLAS, we estimated a cumulative runoff-like contribution from local discharge increments, defined as the difference between each reach discharge and the summed discharge of its immediate upstream reaches. The resulting cumulative runoff-like ratio averaged 0.476 for orders 3-7. These values are close to the characteristic values reported in the main HydroATLAS-based analysis.
  Planned revision: A new Supporting Information section will describe this independent validation, including data sources, linkage by COMID, calculation procedure, diagnostics, and results. The Discussion will refer to this analysis as an external robustness check, while still noting that broader validation using gauged observations, synthetic benchmark networks, and additional hydrographic products remains important future work.
  Comment 3: Table 2 as a map
  Reviewer comment: It will be more informative to change Table 2 into a figure. It will be nice to have a global map showing the basins, maybe using different color for each region. This would give an indication also of the size variability of the basins around the globe. The data in the current table could be shown as a bar plot if you feel it is useful.
  Response: We agree. A global map will provide a clearer view of the spatial coverage and basin-size variability than the original table.
  Planned revision: The original Table 2 will be replaced by a figure showing the global distribution of the 228 representative basins. Basins will be colored by HydroBASINS geographic region, and an accompanying bar plot will show the number of representative basins in each region. The corresponding Methods text will be revised to refer to the new figure.
  Comment 4: Lines 135-138
  Reviewer comment: The sentence in lines 135-138 needs some restructuring, as the grammar is missing something. Also, point (4) is a consequence, rather than a rule.
  Response: We thank the reviewer for pointing this out. We agree that the original text incorrectly presented the highest network order as part of the H-S ordering rules, whereas it is more properly a consequence of applying these rules throughout the network.
  Planned revision: The sentence in Section 2.2 will be restructured to list only the three H-S ordering rules. The highest network order, Omega, will then be described as a consequence of applying these rules. The corresponding schematic in Fig. 1a will also be revised by removing the original fourth rule.
  Comment 5: Line 159
  Reviewer comment: I think RMS should actually be in mm/year, since it is annual runoff, therefore cumulative runoff should be m3/year.
  Response: We agree that the unit description should be clarified. The HydroATLAS runoff attribute used in the manuscript is annual runoff depth, and cumulative runoff is obtained by combining runoff depth with catchment area.
  Planned revision: The unit description of RMC will be corrected to mm yr-1, and cumulative runoff will be clarified as being expressed in m3 yr-1. Related text and equations will be checked for consistency.
  Comment 6: Line 166
  Reviewer comment: Hypothesis should be reported in the final part of the introduction.
  Response: We agree. The hypothesis should be introduced before the Methods section rather than first appearing within the methodological derivation.
  Planned revision: The final part of the Introduction will be revised to state that cumulative runoff and discharge are hypothesized to exhibit recurring scaling patterns across H-S stream orders, and that the hydrological characteristics of basic units are hypothesized to follow stable topological relationships within the river-network pyramid framework. The corresponding sentence in Section 2.2 will be revised to begin with "To test this hypothesis..."
  Comment 7: Figure 2 versus Figure 3
  Reviewer comment: Figure 2: I am not sure what is the added value of this figure as compared with Figure 3, which I find more intuitive. Consider removing it and use Figure 3 in the explanations.
  Response: We agree that the original Figure 3 provides a more intuitive main-text explanation of the pyramid decomposition framework.
  Planned revision: The original Figure 2 will be moved to the Supporting Information as Figure S1. The main-text explanation of the pyramid decomposition will be based on the more intuitive main-text figure, and the figure numbering and in-text references will be revised accordingly.
  Comment 8: Characteristic length definition
  Reviewer comment: Line 234: What is specifically the characteristic length? Is this analysis done for each decomposition level, or are you combining all units with same length together? Also, since length is a continuous variable, it is highly improbable to have two units with exactly the same length. Did you use length classes, e.g., from 0 to 100 m, from 100 to 200 m etc.?
  Response: We thank the reviewer for identifying this ambiguity. The characteristic length in this study is not a continuous geometric length. It is an integer topological characteristic length, denoted by lambda, and is defined by the number of inner links that comprise a basic unit. Therefore, no metric length classes such as 0-100 m or 100-200 m are used.
  Basic units extracted from different pyramid decomposition levels are assigned to the same characteristic-length group when they share the same integer lambda. The hydrological statistics are then averaged within each lambda group.
  Planned revision: Sections 2.3 and 2.4 will be revised to explicitly define lambda as the integer topological characteristic length of a basic unit. The caption of the relevant figure will also clarify that normalized length is obtained from the integer topological characteristic length after basin-wise normalization.
  Comment 9: Figure 4 caption
  Reviewer comment: Caption of Figure 4: "to enable cross-basin comparison", since cross-scale is within the single basin.
  Response: We agree with this correction. The normalization was used to support cross-basin comparison rather than cross-scale comparison within a single basin.
  Planned revision: Because the original Figure 4 will be removed or integrated during figure reorganization, the ambiguous phrase "cross-scale comparison" will be removed. The related text will refer to cross-basin comparison where appropriate.
  Comment 10: Figure 4c and 4d
  Reviewer comment: Is Figure 4c and d the same as Figure 4 but in semilog space? If so, consider joining the two figures.
  Response: We agree that the original presentation was partially redundant. The normalized runoff and discharge patterns and the semilog fitting results are closely related and should be presented more compactly.
  Planned revision: The original Figure 4 will be removed from the main text and the discussion of normalized runoff and discharge will be integrated into the revised hydrological-scaling figure corresponding to the original Figure 5. The corresponding text, caption, and figure numbering will be revised.
  Comment 11: Line 303
  Reviewer comment: What is an iterator and an iteration in this case? Please use consistent terminology and/or an example to clarify.
  Response: We agree that the original wording was unclear. The intended meaning was self-similar river-network growth, not a computational iteration.
  Planned revision: The sentence will be revised to avoid the terms "iterator" and "iteration". The revised wording will refer to "stream numbers and stream lengths across successive orders" in self-similar river-network growth.
  Comment 12: Figure 7 caption
  Reviewer comment: Caption of Figure 7: please give a specific definition of variability range, e.g. 3 standard deviations.
  Response: We agree that the variability range should be explicitly defined.
  Planned revision: The caption of Fig. 7 will be revised to state that the solid curves represent global mean trends, while the shaded regions indicate the 5th-95th percentile ranges across individual river networks.
  Comment 13: Mean versus median
  Reviewer comment: Figure 9, but in all the paper too: I feel like reporting median values rather than mean ones would be more informative, since most of the indicators show some skewness in the distribution, and it is evident also here.
  Response: We agree that medians provide a more robust summary for skewed distributions. We will revise Fig. 9 to report medians and interquartile ranges as the primary robust statistics, while retaining arithmetic means as complementary information.
  The updated analysis shows that runoff-related parameters have very similar mean and median values, whereas discharge-related parameters show stronger mean-median separation, consistent with their greater skewness. This supports the main conclusion while more transparently representing distributional asymmetry.
  Planned revision: Fig. 9 will be revised so that the global median is shown as the main horizontal reference line and the median and interquartile range are reported in each panel. Arithmetic means will be shown as secondary dashed reference lines. The figure caption and corresponding Results text will be revised accordingly.
  Comment 14: Outliers in Figure 9
  Reviewer comment: Figure 9: in many cases it seems that there is a well-defined range and some outliers. Can you identify the outliers in the map and discuss what could cause this deviation?
  Response: We agree. We will add an outlier analysis based on the basin-level expected values of the Fig. 9 parameters. Outliers will be identified using Tukey's 1.5 x IQR criterion applied separately to each basic-unit parameter. The identified outliers will then be mapped and classified according to whether they involve discharge-related parameters or only runoff-related parameters.
  The analysis indicates that most outliers are discharge-related, consistent with the stronger mean-median separation observed for discharge parameters. We interpret these deviations as potentially reflecting the greater sensitivity of discharge ratios to downstream routing, contributing-area aggregation, basin size, network complexity, regional hydrological heterogeneity, and uncertainties in the WaterGAP-derived discharge estimates. Arid, high-latitude, glacier-influenced, and topologically complex basins may be particularly prone to such deviations.
  Planned revision: A supplementary outlier map and explanatory text will be added to the Supporting Information. A brief statement will also be added to the main text to direct readers to the supplementary outlier analysis.
  Comment 15: Appendices
  Reviewer comment: Appendices: this is the guide text; remove it if no appendices are used.
  Response: We agree. The remaining template guide text should not appear in the manuscript.
  Planned revision: The appendix guide text will be removed from the main manuscript. Supplementary schematic and supporting analyses will be provided in a separate Supporting Information file.
  Comment 16: Code availability
  Reviewer comment: Code availability: it is highly suggested to share the code in a public repository, too.
  Response: We agree and have prepared a public code release for the analysis workflow.
  Planned revision: The Code availability statement will be revised to state that the analysis and figure-generation code is available in a public GitHub repository at https://github.com/P1AYERChris/river-topology-hydrology and has been archived on Zenodo with DOI https://doi.org/10.5281/zenodo.20424684. The repository documents the required source datasets and input tables, while the HydroATLAS, RiverATLAS, and HydroBASINS source data are not redistributed because of their size.
  Closing note
  The response supplement attached to this author comment will include the key supporting figures for the planned revisions, including the global basin map, the Fig. 9 outlier map, and the GRADES-hydroDL/MERIT-Basins independent validation figures.
  
  Citation: https://doi.org/10.5194/egusphere-2026-1128-AC1
RC2:
'Comment on egusphere-2026-1128', Anonymous Referee #2, 09 Jul 2026

My comments are summarized in the attached file. Please refer to it for details.

Citation: https://doi.org/10.5194/egusphere-2026-1128-RC2
- AC2: 'Reply on RC2', Chensong Zhao, 14 Jul 2026
  
  We thank Referee #2 for the positive and constructive assessment of the manuscript. We agree that the suggested changes will improve readability, notation clarity, and conceptual consistency. We provide below a point-by-point response and describe the revisions that will be incorporated into the revised manuscript and Supporting Information.
  Comment 1: Global map of selected basins
  Reviewer comment: Although Table 2 suggests that the 228 study basins were selected across multiple continents, their actual spatial distribution is not immediately clear. I recommend adding a global map showing the location and extent of all selected basins. Such a figure would help readers more readily assess the geographical coverage of the analysis and identify any potential regional clustering or gaps in basin representation.
  Response: We agree with this suggestion. This point is also consistent with a related comment from Referee #1. A global map will make the spatial coverage of the 228 representative basins much clearer than the original table.
  Planned revision: The original Table 2 will be replaced by a new figure showing the global distribution of the 228 representative basins. Basins will be colored by HydroBASINS geographic region, and an accompanying bar plot will summarize the number of representative basins in each region. The corresponding Methods text will be revised to refer to the new figure instead of Table 2. The response supplement will include this planned figure.
  Comment 2: Summary table of symbols
  Reviewer comment: The manuscript uses a relatively large number of notations, with subscripts, superscripts, overbars, and related modifiers distinguishing different concepts based on the same symbols. Although these notations are generally defined in the text, the overall notation system can still be difficult to follow, particularly across different sections of the manuscript. I therefore recommend adding a summary table listing the main symbols and their definitions. This would improve readability and make it easier for readers to distinguish among closely related quantities.
  Response: We agree. The manuscript introduces several related hydrological and topological quantities across the H-S ordering framework and the pyramid-based basic-unit framework. A symbol table will improve readability and help readers distinguish among closely related quantities.
  Planned revision: We will add a summary table of the main symbols and definitions to the Supporting Information. The table will include the main H-S ordering symbols, hydrological variables, runoff and discharge ratios, fitted ratios, pyramid basic-unit variables, characteristic length, and geometric/topological ratios such as R_B, R_C, and R_A. The Methods section will include a short cross-reference to this table. The response supplement will also include the planned symbol table so that the reviewer can see how the notation will be clarified.
  Comment 3: Definition of characteristic length λ
  Reviewer comment: Line 235: The characteristic length of a basic unit is conceptually defined in the preceding section, but the symbol λ does not appear to be explicitly introduced as its notation before being used here. I suggest explicitly stating that λ denotes the characteristic length of a basic unit.
  Response: We agree that the notation should be introduced more explicitly. The characteristic length used in this study is not a continuous metric length. It is an integer topological characteristic length of a basic unit.
  Planned revision: We will revise the relevant Methods text to explicitly state that λ denotes the integer topological characteristic length of a basic unit. We will also clarify that basic units extracted from different pyramid decomposition levels are assigned to the same characteristic-length group when they share the same integer λ. This clarification will also be included in the planned symbol table in the Supporting Information.
  Comment 4: Definition of R_C and R_A
  Reviewer comment: Lines 398-403: It is not clear whether R_C and R_A have been explicitly defined earlier in the manuscript. Although Horton-type ratios are familiar concepts in river-network analysis, these quantities should still be clearly defined within the manuscript to ensure that the notation is fully self-contained. I suggest providing explicit definitions and, where appropriate, relevant references. Additional details could also be included in an Appendix if a more extensive explanation is needed.
  Response: We agree. Even though these ratios are related to standard Horton-type scaling concepts, the manuscript should define them explicitly and remain self-contained.
  Planned revision: In the main text near Eqs. (8) and (9), we will add a concise definition stating that R_C(ω) denotes the confluence-area ratio between consecutive H-S orders, and R_A(ω) denotes the watershed-area ratio between consecutive H-S orders. The planned Supporting Information symbol table will provide the full definitions of R_B, R_C, R_A, and the associated hydrological ratios, together with relevant references where appropriate.
  Comment 5: Assumptions in Eq. (8)
  Reviewer comment: Equation (8): The derivation appears to assume that mean precipitation and runoff coefficients are approximately invariant across consecutive stream orders. This may be a strong assumption for the relatively large basins considered in this study, where precipitation can vary substantially in space. I therefore have two related questions. (1) The runoff and discharge attributes provided by HydroATLAS are, to my understanding, derived from hydrological modelling that accounts for spatially heterogeneous climatic forcing. If so, the interpretation of Eq. (8) may be limited unless the assumption of order-wise similarity in precipitation and runoff coefficients is demonstrated. (2) Can the authors show that α_ω and P_ω do not vary systematically across stream orders despite spatially heterogeneous precipitation? Such evidence would strengthen the interpretation of Eq. (8).
  Response: We thank the reviewer for this important and technically precise comment. We agree that Eq. (8) should not be interpreted as a strict physical derivation requiring precipitation and runoff coefficients to be exactly invariant across stream orders. The intended role of Eq. (8) is to provide a first-order topological approximation linking runoff ratios to branch and contributing-area ratios, under the simplifying assumption that order-wise differences in mean precipitation and runoff coefficients do not dominate the scaling relationship.
  We also agree that HydroATLAS runoff and discharge attributes are derived from hydrological modelling and therefore already include spatially heterogeneous climatic forcing. For this reason, the empirical relationship shown in the manuscript should be interpreted as a statistical relationship observed in a model-derived global dataset, not as proof that P_ω and α_ω are constant across stream orders. Variability in precipitation, runoff coefficients, local storage, evaporation, and routing processes can contribute to scatter around the first-order relationship.
  To address the reviewer’s question more directly, we have completed a supplementary diagnostic based on the available HydroATLAS hydro-climatic attributes. We summarized the order-wise behaviour of mean precipitation P_ω and runoff coefficient α_ω, where α_ω was estimated from runoff depth divided by precipitation. The diagnostic aggregates all RiverATLAS reaches belonging to each representative basin and H-S order, using catchment area as a weighting factor, so that the calculation represents the full basin-order group rather than a single outlet or representative reach.
  The diagnostic included 5,054,231 RiverATLAS reaches from the 228 representative basins. For orders 1-7, which contain nearly complete basin-order samples, the basin-normalized median precipitation P_ω/P̄ ranged from 0.94 to 1.03, while the basin-normalized median runoff coefficient α_ω/ᾱ ranged from 0.98 to 1.01. Higher orders showed larger uncertainty because they were represented by fewer basins. These results indicate that precipitation and runoff coefficients are not strictly invariant, but they do not show a strong systematic order-wise shift over the main range of H-S orders used in the scaling analysis. We will present this as a diagnostic supporting the use of Eq. (8) as a first-order approximation, not as proof of exact invariance.
  The main-text comparison between R_r and R_C/R_B, together with the planned median/IQR and outlier analyses, will then be interpreted as showing whether the topological approximation captures a broad statistical tendency despite climatic and hydrological heterogeneity. In addition, the independent GRADES-hydroDL/MERIT-Basins validation will be used as an external robustness check for the order-based scaling pattern, while noting that it does not directly demonstrate invariance of P_ω or α_ω.
  Planned revision: We will add a clarification near Eq. (8) stating that the equation is a first-order topological approximation and that order-wise precipitation and runoff-coefficient heterogeneity may contribute to deviations from the theoretical relationship. We will add the supplementary diagnostic of P_ω and α_ω across H-S orders to the Supporting Information, including the area-weighted aggregation procedure and the normalized order-wise results. We will also soften wording that implies a strict physical law and will explicitly state that the HydroATLAS-derived runoff and discharge variables should be interpreted as model-derived estimates suitable for broad-scale statistical comparison. The response supplement will include the planned symbol table and the new Eq. (8) diagnostic figure.
  Comment 6: Appendix section number
  Reviewer comment: Line 538: This is an Appendix section, but an unnecessary chapter number appears to have been included. Please remove the chapter number.
  Response: We agree. The appendix or guide-text formatting should not appear in the manuscript.
  Planned revision: The unnecessary appendix chapter number and template guide text will be removed. Supplementary schematic material, symbol definitions, outlier analysis, and independent validation will be placed in a separate Supporting Information file.
  Comment 7: Conceptual consistency between optimality, deterministic self-organization, and stochastic network theory
  Reviewer comment: In the Introduction, the authors state that “these universal hydrological patterns emerge as a direct consequence of the optimal channel network (OCN) hypothesis...” and that river network evolution is governed by “deterministic self-organization principles rather than stochastic randomness”. I therefore interpreted the manuscript as placing greater emphasis on optimality and deterministic self-organization than on chance or stochastic processes. However, later sections appear to introduce a somewhat different perspective. Section 3.2 refers to self-similarity emerging during “random iterations” (Mantilla et al., 2010), while Section 4.4 states that stable Horton-ratio statistics have a theoretical basis in “stochastic network theory” (Wang and Waymire, 1991). These statements suggest that similar self-similar patterns may also emerge from stochastic processes. I fully acknowledge that the relative roles of chance, self-organization, and optimality remain open to scientific interpretation, and the manuscript should reflect the authors’ own view. Nevertheless, the current presentation may cause some confusion because the Introduction appears to favor a relatively strong optimality-based interpretation, whereas later sections recognize stochastic explanations for similar scaling behavior. I therefore recommend either improving the consistency of the conceptual discussion throughout the manuscript or slightly softening the wording in the Introduction to acknowledge that stochastic mechanisms may provide an alternative (or possibly complementary) explanation.
  Response: We thank the reviewer for this careful reading. We agree that the original wording in the Introduction may have overemphasized optimality and deterministic self-organization, while later sections acknowledged stochastic explanations for similar scaling behaviour. Our intention is not to claim that the observed hydrological-topological patterns uniquely prove the OCN hypothesis or exclude stochastic network mechanisms. Rather, we interpret the observed patterns as being consistent with broader theoretical ideas of river-network self-similarity, which may arise from complementary mechanisms including optimality-based organization, deterministic self-organization, and stochastic network growth.
  This point is also consistent with the comment from Referee #1 that the manuscript should avoid overstating universality and physical significance. In the revised manuscript, we will therefore soften the strongest Introduction wording and make the conceptual discussion more internally consistent.
  Planned revision: We will revise the Introduction to avoid presenting OCN or deterministic self-organization as the sole explanation for the observed scaling patterns. Expressions such as “direct consequence” and wording that contrasts deterministic self-organization too strongly with stochastic randomness will be softened. The revised text will acknowledge that stochastic network theory may provide an alternative or complementary explanation for Horton-type scaling and related self-similar patterns. We will also revise the terminology in Section 3.2 by replacing unclear wording such as “random iterations” with more precise language such as “stochastic network growth” or “self-similar network growth”.
  Closing note
  The response supplement attached to this author comment will include materials directly relevant to the planned revisions for Referee #2, including the global basin map, the planned symbol table, and a short clarification of the Eq. (8) assumption. Where the planned revisions overlap with those described in the response to Referee #1, the revised manuscript will implement them consistently across the Methods, Discussion, Conclusions, and Supporting Information.
  
  Citation: https://doi.org/10.5194/egusphere-2026-1128-AC2

Chensong Zhao, Qihua Ke, Yuan Cao, Shutong Liu, Randongfang Wei, Di Xie, Jinqiang Wang, Jingwei Huang, Qingqing Zhang, Jiahui Gao, Jing Li, Guangqian Wang, and Deyu Zhong

Data sets

Global hydro-environmental sub-basin and river reach characteristics at high spatial resolution S. Linke et al. https://doi.org/10.1038/s41597-019-0300-6 ‍

Chensong Zhao, Qihua Ke, Yuan Cao, Shutong Liu, Randongfang Wei, Di Xie, Jinqiang Wang, Jingwei Huang, Qingqing Zhang, Jiahui Gao, Jing Li, Guangqian Wang, and Deyu Zhong

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Short summary

To understand how river network structures affect water movement, we analysed 228 diverse global basins. We found that river layout acts as a natural filter, creating highly consistent flow patterns regardless of local climate or basin size. These universal rules offer valuable physical insights that can improve computer models for predicting water flow and managing water resources, especially in unmonitored regions.


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