the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 22 Jul 2026
A stochastic model of hierarchical sediment transport in watershed
Abstract. Debris flows often occur as intermittent surges in succession; a single event may comprise tens to hundreds of separate surges. While previous studies have focused primarily on isolated surge dynamics, the temporal evolution and statistical properties of the surge sequences as an entirety remain poorly understood. Here, we present a comprehensive analysis of global debris flow datasets, including velocity, discharge, sediment volume, and inter-surge intervals. We quantitatively characterize the surge sequences by the statistic features of the key parameters, revealing 1) a general decay trend in fluctuating discharge, 2) universal distributions for flow velocity (Weibull), discharge/sediment volume (exponentially-modified power-law), and inter-surge intervals (exponential), and 3) self-similar structure in sequence organization. These findings motivate the development of a novel stochastic modeling framework that conceptualizes surge sequences through three key components: 1) The cascading thinning of mass sequences originating from Poisson processes of source failures and inter-processes of bank and bed erosions driven by hydrologic process, 2) a selection coefficient generated through Gaussian process to integrate the effective contributions of mass from multi-sources, 3) model parameter calibration using local monitoring data and experimental validation. Numerical simulations demonstrate the model's ability to accurately replicate observed surge sequences. Notably, the framework spontaneously generates a unified distribution of discharge and sediment volume in the form of a truncated power-law with an exponential cutoff. This provides a novel magnitude–frequency relationship for mass movements and can also be applied to large-scale sediment transport studies.
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CC1: 'Comment on egusphere-2026-3226', Jules Maurice habumugisha, 30 Jul 2026
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AC1: 'Reply on CC1', Jun Zhang, 01 Aug 2026
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The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2026/egusphere-2026-3226/egusphere-2026-3226-AC1-supplement.pdf
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AC2: 'Reply on CC1', Jun Zhang, 01 Aug 2026
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The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2026/egusphere-2026-3226/egusphere-2026-3226-AC2-supplement.pdf
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AC1: 'Reply on CC1', Jun Zhang, 01 Aug 2026
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CC2: 'Comment on egusphere-2026-3226', Pritam Kumar, 31 Jul 2026
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This paper presents a systematic analysis of debris-flow surge sequences based on observational data from Jiangjia Gully (JJG) and 12 global watersheds, and proposes a three-layer stochastic model to simulate surge sequence generation. The study reveals that surge magnitude and inter-surge intervals follow a power-law distribution with exponential cutoff and an exponential distribution, respectively. The model successfully reproduces key statistical characteristics of observed sequences. Overall, this research addresses an important gap in debris-flow research at the "sequence scale," demonstrating significant theoretical innovation and practical application potential. The dataset is substantial and the modeling approach is well conceived.
Recommendation
Minor Revision
Major Comments
- The core assumption that source-area failures follow a Poisson process would benefit from additional supporting evidence
The manuscript assumes that landslide/failure events in source areas follow a Poisson process based on the exponential distribution of inter-failure intervals. This assumption is fundamental to the model framework. While the authors cite Guo et al. (2023) as experimental support, the description of this evidence is relatively brief.
Suggestions:
- In Section 3.4 or the Supplementary Information, expand the description of the experimental results from Guo et al. (2023), clarifying the experimental design, observation duration, and the number of failure events that support the Poisson nature of source failures;
- If the authors have access to direct monitoring data from JJG source areas (e.g., displacement sensors, micro-seismic records), consider adding a time-series plot for a representative period to visually demonstrate the random independence of failure events;
- If direct monitoring data are indeed limited, a brief acknowledgment in Section 5.5 would be helpful, noting that future work with dense seismic arrays or InSAR observations could further validate this assumption.
- The physical meaning of the Gaussian selection coefficient ζ could be clarified further
The selection coefficient ζ∼N(0,1) is introduced to represent the combined effects of intermediate processes (channel erosion, sediment entrainment, etc.). While mathematically elegant, readers may wonder whether this normal distribution is universally applicable across all watersheds and events, and whether its mean and variance should be fixed or site-dependent.
Suggestions:
- Consider adding a brief sensitivity analysis (potentially as Supplementary Material) showing how variations in the variance of ζ affect simulation outcomes. If results are relatively insensitive to ζ parameters, this would strengthen confidence in the simplification;
- In the Discussion, the authors could note that, with more comprehensive data, the distribution parameters of ζ could potentially be linked to watershed characteristics such as drainage area or main-channel slope. This would acknowledge the current simplification while presenting it as a direction for future development.
- Model validation could be strengthened by examining temporal structure
The current validation relies primarily on the KS test (distributional similarity) and visual comparison of the moving average decay trend. The KS test does not assess temporal correlations (e.g., autocorrelation structure), yet the manuscript notes in Section 3.2 that surge sequences exhibit significant autocorrelation and high Hurst exponents. Examining whether simulated sequences also reproduce these temporal structures would strengthen the validation.
Suggestions:
- Consider adding a simple additional validation metric: compute and compare the first several autocorrelation coefficients (e.g., lag-1 to lag-10) or Hurst exponent estimates between observed and simulated sequences;
- This would not require any modification to the model itself, only an additional post-hoc evaluation. If the simulated sequences match the observations on these metrics, it would further demonstrate model effectiveness; if discrepancies exist, candidly discussing them would still provide valuable insight.
- The scope of model applicability could be more clearly defined
The cross-watershed validation using Chalk Cliffs and Gadria Creek is encouraging. However, these sites have relatively short surge sequences (<50 surges) and, like JJG, are primarily "source-failure-dominated" debris flows. Readers may naturally wonder whether the model applies to "runoff-dominated" debris flows (e.g., Cancia in Italy or post-fire sites in the southern US).
Suggestions:
The authors need not conduct extensive new validation for such sites within a minor revision; however, a brief addition to Section 5.5 (Limitations) would be valuable, clearly stating that the model is best suited for cases where source-area failures are the primary sediment source. For runoff-erosion-dominated debris flows, additional hydrological modules may be required. This would honestly delineate the model's applicability and point toward productive future work.
Minor Comments
- Terminology and symbols could be unified
- “Sediment delivery” and “sediment volume” are used interchangeably throughout the manuscript to refer to the same physical quantity S. It is recommended to adopt a single term consistently— “sediment volume” may be more appropriate since S represents a volume —or clearly state at first use that the two terms are used synonymously.
- The symbol ζ is used both as the Pareto shape parameter (near Eq. 8) and as the selection coefficient (Eqs. 10-11). While context generally disambiguates, it is advisable to change the Pareto exponent to ξ and keep ζ for the selection coefficient to avoid potential confusion.
- A few language expressions could be slightly refined
- A few long sentences in the Abstract and Introduction could be broken down for improved readability. For example, lines 11-14 (“While previous studies... remain poorly understood”) would benefit from being split into two sentences.
- The sentence “this approach prioritizes operational feasibility over detailed mechanistic understanding” (end of Section 3.4) is somewhat colloquial. Consider rephrasing to "this approach trades detailed mechanistic representation for operational feasibility" or a similar more formal expression.
- Brief explanatory notes for the tables in Supplementary Information would be helpful
Table S5 lists simulation parameters for global watersheds but does not include a note explaining the physical meaning of each parameter (particularly the α values) and their potential relationship with watershed characteristics. Adding one or two sentences in the table note — for instance, identifying possible controlling factors that differentiate sites with larger α (e.g., CC) from those with smaller α (e.g., GC) — would help readers better interpret the cross-watershed variation in parameters.
Additional Comments
- Practical guidance for hazard warning could be slightly expanded
Section 5.4 discusses the model's application to hazard assessment, but the description remains rather general. It would be helpful for engineering practitioners if the authors could add one or two sentences specifying, in a real-time warning context, what input data the model would require (e.g., rainfall intensity, antecedent soil moisture) and what the approximate prediction time window would be.
- The theoretical significance of the power-law with exponential cutoff deserves stronger emphasis
The distribution given by Eq. 3 — a power law with an exponential cutoff — is an important theoretical contribution of this work. The authors may wish to highlight more explicitly in the Conclusions that this distribution contrasts with the pure power laws commonly observed for landslides and earthquakes, suggesting a "finite-size" characteristic of debris-flow systems (limited by the total available erodible material within the watershed). This may have broader implications for understanding the scale behavior of mountain hazard systems.
Conclusion
This is a well-conceived and valuable study, supported by substantial observational data and a clear modeling framework. The authors address an important gap in the debris-flow literature by focusing on surge sequences as integrated systems rather than isolated events. The statistical findings are robust, the model is innovative, and the cross-watershed validation enhances the generalizability of the conclusions. The comments raised above are primarily aimed at strengthening the physical justification of key assumptions, expanding the validation metrics, and improving presentation clarity. They do not challenge the core methodology or main conclusions. I look forward to seeing the revised manuscript and believe that with these refinements it will be an excellent contribution to NHESS.
Citation: https://doi.org/10.5194/egusphere-2026-3226-CC2 -
AC3: 'Reply on CC2', Jun Zhang, 01 Aug 2026
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The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2026/egusphere-2026-3226/egusphere-2026-3226-AC3-supplement.pdf
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CC3: 'Comment on egusphere-2026-3226', Aaditya Ojha, 11 Sep 2026
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This manuscript presents a novel stochastic framework for modeling debris-flow surge sequences,
addressing a significant gap in the literature. The authors compile an impressive global dataset,
identify universal statistical characteristics across diverse watersheds, and develop a parsimonious
model that successfully reproduces observed sequence statistics. The discovery of a unified
magnitude–frequency distribution represents an important theoretical advance. The methodology is
robust, the validation is thorough, and the potential for operational hazard forecasting is clearly
demonstrated. Overall, this work makes a substantial contribution to probabilistic hazard assessment
and is well-suited for publication in NHESS pending minor revisions.Comments
- The KS test results are inconsistent. The authors claim that D < D0.01 for all events, but in the Supplementary Material, event 000704 has a p-value of 0.004, which would reject the null hypothesis at α = 0.01.
- The description of simulation results in the main text is overly absolute. The authors state that “simulated results show no significant difference from observed data,” but in the Supplementary Material, event 000704 (p = 0.004) and event 990716 (p = 0.075) exhibit fit quality near or below the significance threshold, indicating inconsistency between the main text and the Supplementary Material.
- The parameterization relating source failure to rainfall intensity lacks quantitative validation. The goodness-of-fit (e.g., R2, RMSE, confidence intervals) for λ = exp(kT·IR) is not provided in the manuscript.
- The evidence for self-similarity is insufficient. The authors base their demonstration on a single event (E910709) divided into four subsequences; evidence from one event is inadequate to support such a universal conclusion.
- The application of the model to unmonitored watersheds lacks guidance on parameterization. For watersheds without historical debris-flow records, the manuscript does not provide clear operational guidance on how to estimate key parameters such as λ, m0, and ζ.
- Notation is inconsistent. ζ is used for both the Pareto exponent and the Gaussian selection coefficient; the bold symbols SF(j) and M in Eq. (10) are not explicitly defined in the text.
- Figures 6 and 8 contain too many subpanels, with small fonts and crowded legends, reducing the readability of key results. The statistical parameter comparison subpanels occupy substantial space while conveying limited information.
- Line 97: “systematical” should be “systematic.”
- Line 129: “lead fish sampler” is unclear; a more standard term or an explanation is recommended.
- Line 337: “the same” should be “similar.”
- Reference formatting is inconsistent: some references include DOIs while others do not; journal names are a mix of full names and abbreviations.
- Line 585–586: The critical value c(0.01) = 1.63 is given without specifying the sample size conditions to which it applies.
- The term “thinning” is not clearly defined when first introduced.
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AC4: 'Reply on CC3', Jun Zhang, 13 Sep 2026
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This manuscript presents a novel stochastic framework for modeling debris-flow surge sequences, addressing a significant gap in the literature. The authors compile an impressive global dataset, identify universal statistical characteristics across diverse watersheds, and develop a parsimonious model that successfully reproduces observed sequence statistics. The discovery of a unified magnitude–frequency distribution represents an important theoretical advance. The methodology is robust, the validation is thorough, and the potential for operational hazard forecasting is clearly demonstrated. Overall, this work makes a substantial contribution to probabilistic hazard assessment and is well-suited for publication in NHESS pending minor revisions.
Comments
- The KS test results are inconsistent. The authors claim that D < D0.01 for all events, but in the Supplementary Material, event 000704 has a p-value of 0.004, which would reject the null hypothesis at α = 0.01.
Reply: Following your advice, We have re-checked Table S2 and confirm that for event 000704, D = 0.300 > D0.01 = 0.195 and p = 0.004, so the null hypothesis is rejected at α = 0.01. The original statement “D < D0.01 across all cases” was inaccurate.
We have made the following minimal revisions:
(1) Section 4.1: changed “across all cases” to “for six of the seven analyzed events”, and added a brief clause noting that event 000704 is an exception (D = 0.300, p = 0.004) attributed to its long sequence and pronounced high-magnitude fluctuations not fully captured by the simplified inter-process representation. Also changed “p-values substantially exceed 0.05” to “for the other six events, the p-values exceed 0.01”. (Lines 672-680).
(2) Section 4.2.1: limited “all events” to “all CC and GC events” to avoid conflict with the JJG exception (Lines 749-750).
(3) Abstract and Conclusions: added the qualifier “for the majority of analyzed events” to avoid overstatement (Line 26 and 935).
(4) Table S2: added a one-line note stating D>D0.01 and p = 0.004 for event 000704 (Line 330).
We believe these minimal changes resolve the inconsistency while preserving the overall conclusions. We thank the reviewer again for this careful check.
- The description of simulation results in the main text is overly absolute. The authors state that “simulated results show no significant difference from observed data,” but in the Supplementary Material, event 000704 (p = 0.004) and event 990716 (p = 0.075) exhibit fit quality near or below the significance threshold, indicating inconsistency between the main text and the Supplementary Material.
Reply: We agree that the description of the simulation results in the original manuscript was overly absolute, and that events 000704 (p = 0.004) and 990716 (p = 0.075) exhibit fit quality near or below the significance threshold. We have revised both the main text and the Supplementary Information to present the validation results more accurately and transparently.
(1) Section 4.1 (main text): We have replaced absolute statements such as “no significant difference” and “strong agreement” with more measured wording (“generally agrees well”, “reasonable agreement”). We now explicitly state that “D < D0.01 for six of the seven analyzed events” and identify event 000704 (D = 0.300 > D0.01 = 0.195, p = 0.004) and event 990716 (p = 0.075) as cases with fit quality near or below the significance threshold (Lines 665-667).
(2) Section 4.2.1 (main text): We have clarified that “all events” refers to “all CC and GC events” (Table S3), avoiding conflict with the JJG edge cases (Line 341).
(3) Table S2 (Supporting Information): We have added a note explicitly listing D, D0.01, and p for events 000704 and 990716, and stating that they are near or below the significance threshold (Lines 330-331).
We believe these revisions resolve the inconsistency between the main text and the Supplementary Information, and present the model’s performance honestly without overstating its accuracy.
- The parameterization relating source failure to rainfall intensity lacks quantitative validation. The goodness-of-fit (e.g., R2, RMSE, confidence intervals) for λ = exp(kT·IR) is not provided in the manuscript.
Reply: We acknowledge that the original manuscript presented the parameterization λ=exp(kT·IR) without explicitly documenting its goodness-of-fit, and we apologize for this omission.
The exponential relationship between Poisson intensity and rainfall intensity is not a new assumption introduced in this study, but is derived from our previously published field experiments in Jiangjia Gully (Guo et al., 2021, 2023). In those experiments, seven sets of artificial rainfall tests were conducted on JJG source slopes under rainfall intensities ranging from 12 to 60 mm/h, and the failure rate was found to increase systematically with rainfall intensity, consistent with the exponential form.
To address the reviewer's concern, we have made the following revisions:
(1) Section 3.5 (main text): We have added explicit references to the experimental basis of the λ-IR relationship, stating that it is supported by the seven rainfall experiments in JJG and citing Guo et al. (2021, 2023) (Lines 596-598).
(2) Supplementary Information: We have added a table summarizing the experimental conditions (rainfall intensities, failure counts, mean intervals) from Guo et al. (2021), enabling readers to verify the exponential trend. The original goodness-of-fit statistics are reported in the cited publication (Line 590).
- The evidence for self-similarity is insufficient. The authors base their demonstration on a single event (E910709) divided into four subsequences; evidence from one event is inadequate to support such a universal conclusion.
Reply: We agree that demonstrating self-similarity on the basis of a single event (E910709) is insufficient to support a universal conclusion, and we have revised the manuscript accordingly. We acknowledge that a full multi-event test is beyond the scope of the present study. We have therefore revised the manuscript to present the E910709 analysis as a representative example rather than a universal proof. Specifically:
(1) In Section 3.2, we now state that “we present this analysis as a representative example rather than a universal proof; a systematic multi-event test is identified as a direction for future work” (Lines 322-324).
(2) In the summary of Section 3.2, we have added that “while our demonstration is based on a single representative event, the same statistical features are recovered in the global datasets (Section 4.2), supporting the view that this similarity is a general characteristic of surge sequences” (Lines 369-371).
(3) In the Abstract, we have changed “self-similar structure in sequence organization” to “a self-similar structure in sequence organization identified in a representative long sequence” (Line 19-20).
(4) In Section 5.1 and the Conclusions, we have added the qualifier “observed in long sequences” to avoid overstating universality (Lines 929-930).
(5) In Section 5.5 (Limitations), we have added a sentence noting that “the self-similarity analysis is currently demonstrated on a single long sequence; a systematic test across all available multi-surge events would further consolidate this finding” (Lines 902-904).
(6) In the Supporting Information, we have added a one-line note clarifying that the analysis is based on E910709 as a representative long sequence (Lines 84-86).
- The application of the model to unmonitored watersheds lacks guidance on parameterization. For watersheds without historical debris-flow records, the manuscript does not provide clear operational guidance on how to estimate key parameters such as λ, m0, and ζ.
Reply: We agree that the original manuscript lacked operational guidance for unmonitored watersheds. In Section 3.5, we have added a concise guideline: λ is estimated from the source area (DEM/remote sensing) and local rainfall intensity (regional records), with kT adopted from nearby instrumented catchments; m0 and ξ are obtained from regional landslide inventories or published ranges; and ζ∼N(0,1) requires no site-specific calibration. We also note in Section 5.4 that a first-order simulation for an ungauged catchment requires only the source-area extent, local rainfall intensity, and an order-of-magnitude estimate of m0 and ξ. We believe this provides the clear operational guidance requested by the reviewer while keeping the manuscript concise (Lines 618-625).
We further note in Section 5.4 that a first-order simulation for an ungauged catchment therefore requires only (i) the source-area extent from DEM/remote sensing, (ii) local rainfall intensity, and (iii) an order-of-magnitude estimate of m0 and ξ from regional inventories. The model is thus transferable to ungauged catchments with minimal data requirements (Lines 878-882).
- Notation is inconsistent. ζ is used for both the Pareto exponent and the Gaussian selection coefficient; the bold symbols SF(j) and M in Eq. (10) are not explicitly defined in the text.
Reply: We have carefully checked the manuscript and made the following corrections.
(1) ζ used for both the Pareto exponent and the selection coefficient. We confirm that these are two distinct symbols in the manuscript: the Pareto exponent of the failure-mass distribution is denoted by ξ, whereas the Gaussian selection coefficient is denoted by ζ. The apparent inconsistency likely arose from the visual similarity between ξ and ζ in the PDF, and from the fact that the two symbols were not explicitly contrasted at their first occurrence. To eliminate any ambiguity, we have:
- Checked every occurrence of ξ and ζ throughout the main text and the Supporting Information, and corrected any misused instances;
- Added an explicit clarification at the first occurrence of ξ in Section 3.4: “where ξ is the Pareto exponent of the failure-mass distribution” (Line 442);
- Added a parenthetical clarification in Section 3.5: “the selection coefficient ζ (distinct from the Pareto exponent ξ)” (Line 608);
- Added both symbols to the Appendix A Notation table: ξ — exponent of the failure power-law (Pareto) distribution; ζ — selection coefficient, modeled as a standard normal variate N(0,1) (Line 977).
(2) Undefined symbols in Eq. (10). We have added explicit definitions immediately following Eq. (10):
- SF(j) denotes the mass of the j-th surge in the final sequence in the main channel (Line 553);
- M is the mass vector from the source-failure sequence {S0} (Line 554).
We have also replaced the generic statement “where the symbols are the same as in Eq. 10” after Eq. (11) with explicit definitions of SF(j), S0(τ(j)), and ζ(j).
We believe these corrections fully resolve the notation inconsistencies and improve the clarity of the model formulation. We thank the reviewer for this careful reading.
- Figures 6 and 8 contain too many subpanels, with small fonts and crowded legends, reducing the readability of key results. The statistical parameter comparison subpanels occupy substantial space while conveying limited information.
Reply: We have revised both figures accordingly. The font sizes of the axis labels, tick labels, and legends in Figures 6 and 8 have been enlarged (Line 706 and 760), and the legends have been simplified to reduce visual clutter. These changes make the core results—the sediment fluctuations, the decay trend of ⟨S⟩n, the scaling distributions, and the sediment-volume comparison—substantially easier to read.
- Line 97: “systematical” should be “systematic.”
Reply: revised.
- Line 129: “lead fish sampler” is unclear; a more standard term or an explanation is recommended.
Reply: we have revised it into “sampling pole” (Line 163).
- Line 337: “the same” should be “similar.”
Reply: revised.
- Reference formatting is inconsistent: some references include DOIs while others do not; journal names are a mix of full names and abbreviations.
Reply:
- Line 585–586: The critical value c(0.01) = 1.63 is given without specifying the sample size conditions to which it applies.
Reply: We have carefully checked and revised the reference lists in both the main text (Lines 1003-1249) and the Supporting Information (Lines 377-470), and have standardized them fully in accordance with the NHESS (Copernicus) reference style.
- The term “thinning” is not clearly defined when first introduced.
Reply: We have added an explicit definition at its first occurrence in Section 3.4:
“… this can be reduced to a thinning—a standard operation on a Poisson process in which each event is independently retained with a certain probability, so that the retained events themselves form a new Poisson process with reduced intensity (Kingman, 1993)—from {S0} to {SF}, i.e., …” (Lines 564-566)
We have also added the brief parenthetical clarification “thinning (probabilistic selection)” at the first mention in the Abstract (Line 22), to ensure that the meaning of the term is clear to readers throughout the manuscript.
Data sets
Worldwide Debris-Flow Dataset and Pareto-Poisson Simulation Code Jun Zhang https://doi.org/10.5281/zenodo.18743612
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