the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 21 Oct 2026
A general probabilistic framework for prediction of extreme flow events at river confluences
Abstract. River confluences are critical nodes within river systems and are often exposed to increased flood risk, as the co-occurrence of peak streamflows and thus the superposition of flood waves from two or more tributaries may amplify their magnitudes, leading to compound extreme flow events. However, a systematic and holistic framework to analyse and predict extreme streamflows downstream of river confluences remains largely absent from the literature. This study investigates the interaction between upstream and downstream streamflows during extreme events at river confluences and provides a computationally efficient probabilistic framework for estimating extreme downstream flows based on combinations of upstream conditions from individual tributaries. In this work, a confluence is defined as the junction of two upstream river branches that merge into a downstream channel. We introduce a novel probabilistic-based methodology to describe and predict concurrent streamflows, including uncertainty quantification across the three branches when an extreme event occurs in at least one of them. Building on this, a copula-based probabilistic model is developed with a twofold objective: (i) to predict downstream flow given different combinations of upstream conditions, and (ii) to determine combinations of upstream flows associated with downstream flows exceeding specified safety thresholds. The proposed framework is demonstrated and tested using three contrasting river confluences: two confluences in the Rhine river system in Germany i.e., Rhine–Nahe at Bingen and Rhine–Main at Mainz, and the Mississippi-Missouri confluence at St. Louis in the United States. The results demonstrate that confluences sharing the same main river can behave fundamentally differently under extreme conditions. In the Rhine-Nahe system the influence of Nahe to the downstream streamflow is limited due to its smaller size compared to the Rhine. On the other hand, for the Rhine-Main confluence which is characterized by an intermediate difference between the magnitudes of the two contributing river branches, the results show that the superimposition of the flood waves remains a major control of downstream extreme floods. In contrast, for the Mississippi–Missouri confluence, where upstream streamflows of the two contributing river branches are of comparable magnitude, the simultaneous occurrence of moderate peaks is a primary driver of downstream extreme streamflows. In other words, under such conditions, the superimposition of individual flood waves is not the only dominant control on downstream extreme floods, but has the potential to lead to extreme flow events that may have catastrophic consequences. The probabilistic model developed here demonstrates high predictive skill in reproducing downstream streamflows from upstream observations with Nash–Sutcliffe Efficiency, NSE > 0.97 in all cases, despite the contrasting mechanisms underlying the generation of flood waves at the three study confluences, indicating its generalizability and robustness for estimating and predicting upstream flow combinations that may lead to downstream streamflows beyond safety thresholds.
Competing interests: At least one of the (co-)authors is a member of the editorial board of Hydrology and Earth System Sciences.
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
(9956 KB) - Metadata XML
-
Supplement
(5247 KB) - BibTeX
- EndNote
Status: open (until 02 Dec 2026)
- CC1: 'Comment on egusphere-2026-5737', Sivarajah Mylevaganam, 07 Nov 2026 reply
-
CC2: 'Comment on egusphere-2026-5737', Sivarajah Mylevaganam, 07 Nov 2026
reply
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2026/egusphere-2026-5737/egusphere-2026-5737-CC2-supplement.pdf
-
CC3: 'Comment on egusphere-2026-5737', Sivarajah Mylevaganam, 07 Nov 2026
reply
-
The conventional practice of broad academic citation introduces a compounding ambiguity into scientific literature. To ensure transparency regarding the exact data or concepts being referenced, it would be significantly more precise if citations included the specific page or line numbers of the source manuscript. In the absence of such granular attribution, the current peer-review and citation framework risk perpetuating vague references—a systemic issue that has persisted for decades without adequate correction.While substantial research is dedicated to refining methodologies and analytical frameworks to advance the understanding of hydrological systems, minimal critical attention has been directed toward rectifying this unstandardized citation practice. For instance, in the current draft of the manuscript, the authors cite both Leonard et al. (2014) and Zscheischler and Seneviratne (2017) to substantiate the assertion made in lines 30–32. The necessity of a dual citation for this single statement remains unclear. It is impossible to discern whether the referenced concepts are split across the two papers or if both independently corroborate the identical claim. If both sources refer to the exact same finding, the inclusion of multiple manuscripts raises significant questions regarding the purpose of the citation, suggesting a superfluous expansion of the references rather than purposeful academic attribution. A specific line-level attribution from the authors is necessary to resolve this ambiguity and ensure clarity.Ultimately, this lack of clarity makes it difficult to evaluate the precise foundation of the authors' argument. Resolving how these citations are handled is left to the discretion of the editor to determine the appropriate standard.
Citation: https://doi.org/10.5194/egusphere-2026-5737-CC3 -
-
CC4: 'Comment on egusphere-2026-5737', Sivarajah Mylevaganam, 07 Nov 2026
reply
-
In Section 3.2, regarding the sampling of annual maxima at individual gauges, the authors adopt the methodology implemented by Papalexiou and Koutsoyiannis (2013). According to this methodology, two criteria are established to determine the Annual Maximum Flow (AMQ) (see Lines 138–145). The second criterion states that a AMQ is rejected if the fraction of missing values within that year is greater than or equal to 1/3.
However, this data-filtering methodology can be fundamentally flawed when evaluating annual aggregates or statistical representatives, as a simple illustrative example demonstrates.
Instead of a full 365-day year, let us consider a simplified case with a true dataset of six sequential values: [100, 100, 100, 100, 10, 10]. The true sum of this dataset is 420.
Case 1 (Rejection of a highly representative year): Assume the last two entries are missing, leaving the observed data as [100, 100, 100, 100, missing, missing]. Because exactly 1/3 (=2/6) of the data is missing, the adopted rule rejects this entire year’s AMQ. Yet, the available data captures the vast majority of the true underlying signal (contributing 400 out of the true 420 total).
Case 2 (Acceptance of an unrepresentative year): Assume instead that the second entry is missing, yielding [100, missing, 100, 100, 10, 10]. Because only 1/6 of the data is missing, it falls well below the 1/3 threshold, and the methodology accepts this year. However, the observed sum drops significantly to 320 (farther from the true value of 420 than Case 1).
This demonstrates that a rigid fraction-of-missing-data rule fails to account for the magnitude and distribution of the missing values. A year with fewer missing values can yield a much less accurate statistical representation than a year rejected for exceeding the 1/3 threshold. Therefore, the adopted screening methodology is unreliable for this type of hydrological analysis.
Citation: https://doi.org/10.5194/egusphere-2026-5737-CC4 -
-
CC5: 'Comment on egusphere-2026-5737', Sivarajah Mylevaganam, 07 Nov 2026
reply
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2026/egusphere-2026-5737/egusphere-2026-5737-CC5-supplement.pdf
-
CC6: 'Comment on egusphere-2026-5737', Sivarajah Mylevaganam, 07 Nov 2026
reply
In Section 3.3 of the manuscript, the authors outline the methodology employed to identify extreme events at river confluences. However, the proposed framework is sub-optimal and lacks sufficient justification from a hydrological perspective. The discrepancies between the authors' methodology and physically meaningful hydrological processes are detailed below.
“As stated in lines 147–148, the authors define a confluence extreme event as one characterized by an AMQ occurring at one of the gauges (either upstream or downstream), concurrent with the corresponding streamflow conditions in the remaining gauges. Based on this assumption, the authors establish three distinct scenarios wherein one gauge is fixed at its AMQ while the other two reflect their concurrent flows."
In catchment hydrology, however, a downstream extreme event does not strictly depend on an individual gauge reaching its AMQ. Critical confluence extremes frequently manifest under sub-maximal upstream flows due to joint probability behavior, hydrograph translation, and wave attenuation during routing. For example, while an AMQ of 1000 m³/s at Up1 paired with a minor flow of 100 m³/s at Up2 might appear to dominate the system, a joint event featuring 800 m³/s at Up1 and 700 m³/s at Up2 may generate a substantially more severe extreme downstream due to wave superposition. Consequently, the current framework propagates a fundamental methodological flaw that misrepresents compounding flood risks
To address this limitation, the authors are in a better position to map the synchronized streamflows across all three gauges simultaneously. Given that Up1, Up2, and the downstream gauge are temporally aligned, plotting Up1 and Up2 on a horizontal plane against the downstream discharge on the vertical axis yields a continuous 3D domain containing the full empirical data pool. Utilizing this multivariate space, a probability approach could be deployed to objectively isolate true extreme events at the confluence.
Citation: https://doi.org/10.5194/egusphere-2026-5737-CC6 -
CC7: 'Comment on egusphere-2026-5737', Sivarajah Mylevaganam, 08 Nov 2026
reply
The credibility of the co-authors' contributions to this manuscript is highly questionable. Although nearly all co-authors claim credit for reviewing and editing, the current version of the manuscript contains numerous fundamental flaws. Had the co-authors thoroughly reviewed and edited the text, it is highly unlikely that the manuscript would remain in its current state. While many such issues exist throughout the text, two specific instances highlight this lack of oversight:
Premature Variable Usage: The definitions for Up1 and Up2 are introduced on page 5 (lines 126–128). However, these variables are already utilized earlier in the manuscript within Table 1 on page 4.
Inconsistent Definitions: The abbreviation "AMQ" is propagated throughout the manuscript (e.g., line 126, line 56, and line 135) without clear consistency. For example, line 126 introduces confusion as to whether it stands for "average annual maximum streamflow" or "annual maximum streamflow" (the latter of which seems intended).
Furthermore, if the primary author is a graduate student, the actual contribution of the co-authors becomes highly questionable. Claiming authorship credit on a student's independent work is unethical and devalues the integrity of the degree awarded to the student. Because supervisors serve as evaluating members, they should not automatically claim co-authorship. The current status of the manuscript gives the impression that the student is merely being abetted in their research, rather than executing a rigorously supervised, independent study.
Citation: https://doi.org/10.5194/egusphere-2026-5737-CC7 -
CC8: 'Comment on egusphere-2026-5737', Sivarajah Mylevaganam, 08 Nov 2026
reply
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2026/egusphere-2026-5737/egusphere-2026-5737-CC8-supplement.pdf
-
CC9: 'Comment on egusphere-2026-5737', Sivarajah Mylevaganam, 08 Nov 2026
reply
The authors rely on the research works by Bender et al. (2016) and Svensson et al. (2005) to define the maximum travel time from the upstream boundary (Up1) to the downstream gauge based on the Incremental Catchment Area (ICA). Specifically, the methodology [lines 180-190] dictates that for any ICA less than or equal to 5,000 km², the maximum travel time is uniformly fixed at 1 day. While this threshold-based rule may have been operationally adequate for the specific datasets in the cited studies—where the lower bound of catchments evaluated may have closely approached the 5,000 km² limit—applying it universally introduces severe scale distortions that run counter to standard hydrologic theory.
Setting a maximum travel time of 1 day creates a non-physical generalization when applied across a broad spectrum of catchment sizes. For instance, evaluating an ICA of 0.1 km² using the exact same travel time parameter as an ICA of 4,900 km² overlooks fundamental routing principles. In reality, travel time is a function of stream network length, channel geometry, roughness, slope, and the catchment shape factor rather than a simple drainage area threshold.
In this study, the catchments of interest are situated at the lower bounds of this area range (e.g., the Rhine-Nahe confluence, where the ICA is roughly 1,000 km²). At this scale, treating the entire system under a maximum travel time of 1 day oversimplifies the true wave propagation velocity and hydrodynamic behavior along the stream network.
A critical limitation of this approach is that it bypasses the physical reality of the confluence point itself. Before reaching the downstream gauge, the upstream flow must pass through a confluence zone. The reach from this confluence point to the downstream gauge functions as an open, undisturbed stream network. By grouping the entire routing path from Up1 directly to the downstream gauge into a maximum travel time of 1 day, the methodology effectively bypasses the most critical zone—the confluence.
Citation: https://doi.org/10.5194/egusphere-2026-5737-CC9
Viewed
| HTML | XML | Total | Supplement | BibTeX | EndNote | |
|---|---|---|---|---|---|---|
| 271 | 94 | 30 | 395 | 62 | 29 | 23 |
- HTML: 271
- PDF: 94
- XML: 30
- Total: 395
- Supplement: 62
- BibTeX: 29
- EndNote: 23
Viewed (geographical distribution)
| Country | # | Views | % |
|---|
| Total: | 0 |
| HTML: | 0 |
| PDF: | 0 |
| XML: | 0 |
- 1
The concept of dependency has often been introduced using the interplay between dependent and independent variables. This philosophical framing has been well utilized for many decades across various scientific fields, and hydrology has similarly adopted it within its domain to reflect the causal and statistical dependencies among hydrological variables. Although a variable may be categorized as independent in one context, it can become dependent when the analysis is shifted or refined into another domain or problem. Therefore, what is perceived as independent can transform into a dependency depending on the contextual boundary conditions. Ultimately, true independence is governed by nature, which defines the physical boundaries of this concept and shapes the episodes of empirical data on this planet.
In this manuscript, the authors project this concept of dependency and interdependency into the framework of river confluences in hydrology. The authors claim that predicting extreme streamflows downstream of river confluences remains largely unaddressed in the literature, and they have consequently undertaken a collaborative, multinational effort to investigate this apparent research gap. Based on the current version of the manuscript, the following points are brought forward for discussion:
In the current version of the manuscript, the definition of "confluence" found in the introduction [line 30] is incorrect. Furthermore, because the current phrasing suggests a misunderstanding of the concept, the authors should include an authoritative citation to support the definition. Currently, on line 30, the authors state that confluences are "junctions in river networks where two or more tributaries merge to form a downstream river." This definition is inaccurate. A confluence also occurs along a main river whenever a tributary connects to the main stem. For example, in the Mekong River Basin, an international transboundary river basin, the main river features confluences at junctions introduced by tributaries, such as the Tonle Sap. The definition needs to be revised to accurately reflect river network hydrology, and a relevant reference should be cited.
Lines 30–31: The authors state that streamflow downstream of a confluence is a non-linear combination of upstream contributions and categorize this process as a compound event. From the perspective of river hydrology and established nomenclature, this definition is conceptually incomplete and potentially misleading.
While downstream streamflow inherently incorporates upstream inputs, natural river hydrodynamics dictate that flow at a confluence is governed by a mutual interaction of both upstream and downstream hydraulic conditions. A classic illustration of this mechanism is the Tonle Sap in the Mekong River Transboundary Basin, where pronounced seasonal flow reversibility causes the tributary to alternate between acting as an upstream tributary and a downstream recipient. Similar backwater effects, backwater-induced flow attenuation, and seasonal reversals are well-documented across numerous global river networks, including major systems in the United States.
Consequently, the authors must cross-check the cited literature to support this statement [line 30-31] to ensure the context has not been misrepresented. If the statement accurately reflects the cited source, the source itself may be conceptually flawed or insufficient to support such a generalized claim. Relying uncritically on peer-reviewed literature can propagate systemic errors, as even highly cited papers in high-impact journals occasionally contain overlooked methodological or conceptual oversights. The authors should revise this definition to align with standard hydrodynamic principles or provide robust, contextually appropriate literature to justify their framework.