Ice flood disaster risk assessment of the Inner Mongolia reach in the Yellow River based on unascertained measure theory-variable fuzzy sets.

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Learn more: PMC Disclaimer | PMC Copyright Notice PLoS One . 2026 May 29;21(5):e0350284. doi: 10.1371/journal.pone.0350284 Ice flood disaster risk assessment of the Inner Mongolia reach in the Yellow River based on unascertained measure theory-variable fuzzy sets Yu Deng 1, 2, 3, * , Lu Jiang 1 , Juan Wang 1 Editor: Beata Calka 4 Author information Article notes Copyright and License information 1 School of Water Resources and Transportation, Zhengzhou University, Zhengzhou, Henan, China 2 Yellow River Institute of Hydraulic Research, Zhengzhou, Henan, China 3 Technological Innovation Center for Levee Safety and Disaster Prevention, MWR, Zhengzhou, Henan, China 4 Military University of Technology Faculty of Civil Engineering and Geodesy: Wojskowa Akademia Techniczna im Jaroslawa Dabrowskiego Wydzial Inzynierii Ladowej i Geodezji, POLAND Competing Interests: NO authors have competing interests. ✉ * E-mail: dengyu@vip.sina.cn Roles Yu Deng : Conceptualization, Methodology, Writing – original draft Lu Jiang : Data curation, Formal analysis, Methodology, Writing – original draft Juan Wang : Conceptualization, Writing – review & editing Beata Calka : Editor Received 2025 Aug 17; Accepted 2026 May 12; Collection date 2026. © 2026 Deng et al This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. PMC Copyright notice PMCID: PMC13221040  PMID: 42213727 Abstract Ice flood disasters represent the most severe natural hazards affecting the Yellow River during winter-spring periods, posing significant threats to residents’ safety and impeding regional socioeconomic development in the basin. Despite their critical implications, systematic risk assessment methodologies tailored for Yellow River ice floods remain underdeveloped. Current approaches inadequately address the inherent uncertainty and fuzziness characteristics of ice flood disaster systems, thereby compromising assessment accuracy. To bridge this gap, this study develops an integrated risk assessment model combining Unascertained Measure Theory (UMT) and Variable Fuzzy Set Theory (VFST). The hybrid model was implemented in three typical reaches of the Yellow River’s Inner Mongolia section: Tokto County, Jungar Banner, and Qingshuihe County. Risk evaluation results classified Tokto County and Qingshuihe County as “medium-risk” areas, while Jungar Banner was identified as a “high-risk” zone. Validation analyses confirmed strong consistency between model outputs and historical disaster patterns, demonstrating the framework’s reliability. This research establishes a novel methodological foundation for ice flood risk quantification and provides actionable insights for disaster mitigation strategies in cold-region river systems. The proposed approach offers technical support for enhancing flood management decision-making processes in the Yellow River Basin. Introduction Ice flood disasters predominantly occur during winter and spring seasons, initiated by thermal fluctuations that trigger ice jam formation. These jams obstruct river channels, inducing abrupt upstream water level surges and subsequent flooding—a phenomenon prevalent in high-latitude river systems. In China’s cold climatic zones north of 30°N, approximately 70% of rivers undergo winter-spring freeze-up cycles, with the Yellow River Basin’s Inner Mongolia reach experiencing particularly severe ice flood events. Early research concerning Yellow River ice jams could be traced back to the 1990s. Shen et al. [ 1 ] developed the RICE model, a numerical framework integrating one-dimensional steady-state flow dynamics, heat transfer mechanisms, and ice transport equations. Subsequent enhancements introduced unsteady flow dynamics and ice regime parameters, resulting in the advanced RICEN model for ice regime simulations in China’s Yellow River Basin [ 2 ]. In parallel work, Shen et al. [ 3 ] conducted experimental investigations using low-density particulates to analyze particle morphology effects on ice transport capacity. Their methodology combined Eulerian finite element analysis with smoothed particle hydrodynamics (SPH) to develop 2D numerical models simulating surface ice dynamics and fluvial blockage formation [ 4 ].Chen et al. [ 5 ] proposed a fuzzy-optimized neural network incorporating hydrothermal, hydrodynamic, and geomorphological parameters to predict ice phenology (freeze-up/break-up timing) in the Yellow River’s Inner Mongolia section, demonstrating robust validation results. Lin [ 6 ] implemented fixed-wing unmanned aerial vehicle (UAV) systems for experimental ice monitoring, pioneering UAV applications in Yellow River ice surveillance. Through micromechanical modeling and macro-mechanical property simulations, Deng [ 7 ] established and validated the representative volume unit (RVE) methodology for quantifying Yellow River ice strength. Building on physical testing, Deng [ 8 ] further created a numerical model for three-point bending fracture analysis, elucidating fracture toughness characteristics of river ice. Advancing understanding of Yellow River ice phenomena has revealed two systemic challenges: inherent uncertainty in flood dynamics and complexity in environmental interactions. Recent climate variability and anthropogenic disturbances have exacerbated thermal instability, amplifying ice flood unpredictability. Concurrently, diversified river morphology and expanding hydraulic infrastructure have compounded system uncertainties. These synergistic factors intensify challenges for riverside disaster prevention and mitigation efforts. Risk assessment emerges as a critical tool for ice flood management, enabling scientific quantification of spatial risk levels along river reaches. Strategic implementation of targeted prevention measures and optimized resource allocation, guided by assessment outcomes, significantly enhances flood resilience. Consequently, advancing risk assessment methodologies for Yellow River ice floods holds paramount importance. Wu et al. [ 9 ] created a hybrid risk model combining projection pursuit, fuzzy clustering, and accelerated genetic algorithms, applied to 1991–2010 ice flood data in the Ningxia-Inner Mongolia reach. Luo [ 10 ] employed a dynamic multiindex GM (1, 1) model to forecast ice jam/dam risks during 2013–2015 in the same region. Wang [ 11 ] developed a random forest-based assessment framework for systematic risk evaluation in Inner Mongolia’s flood-prone zones. Hu [ 12 ] addressed composite value inflation limitations in catastrophe theory methods through model modifications, subsequently applying enhanced techniques to Inner Mongolia ice flood risk analysis. Contemporary flood risk assessment research has achieved substantial theoretical advancements across multiple methodological frameworks. Nevertheless, current approaches insufficiently resolve the intrinsic uncertainty and fuzziness inherent to ice flood disaster systems, requiring improved evaluation accuracy. UMT provides an effective mechanism for processing incomplete objective information under data-deficient scenarios. This methodology enhances assessment precision through minimized subjective bias, with demonstrated cross-disciplinary applicability in risk quantification studies [ 13 , 14 ]. Differing fundamentally from classical fuzzy set theory, VFST introduces adaptive coefficients enabling dynamic membership function adjustments. This innovation facilitates enhanced characterization of transient fuzzy system behaviors. The approach has proven particularly advantageous in hydraulic engineering evaluations and water resource risk management applications [ 15 , 16 ]. Drawing upon the aforementioned analysis, this paper combines the UMT with the VFST to conduct in-depth research into the flood risk in the Inner Mongolia section of the Yellow River in Tokto County, Jungar Banner, Qingshuihe County, which are prone to flood danger, and presents a new methodology and perspective for evaluating flood disaster risk in the Inner Mongolia section of the Yellow River by establishing a flood hazard risk assessment model and carrying out a systematic assessment. This study employs the following organizational framework to ensure technical clarity and reader accessibility. Section 2 establishes the conceptual architecture and systematic assessment index system, defining critical parameters for ice flood risk quantification. Section 3 delineates the methodological framework, incorporating theoretical integration of UMT with VFTS principles and operational development of the hybrid evaluation model. Section 4 implements the proposed methodology across three strategically selected regions along the Yellow River’s Inner Mongolia reach (40°15’N–40°45’N, 111°02’E–111°46’E), executing computational analyses to identify critical spatial risk patterns and validate model efficacy. Section 5 synthesizes principal scientific contributions and practical implications derived through systematic investigation, while proposing adaptive flood management strategies for cold-region river systems Procedures The specific steps of this study are as follows: Gather relevant hydrological, meteorological, socio-economic, and engineering data pertaining to the study area. Establish the index system and classification criteria for evaluating ice flood disaster risk. Ascertain the weight of each index via the Analytic Hierarchy Process (AHP). Create a coupled evaluation model grounded in UMT and VFST. Compute the risk levels of each evaluation object through the developed risk assessment model. Ice flood disaster risk assessment constitutes a complex analytical process requiring systematic index framework development as its foundation. The selection of evaluation parameters must follow established criteria including scientific validity, hierarchical organization, systemic coherence, and practical applicability. Drawing upon regional disaster system theory [ 17 ] and incorporating causative mechanisms of ice flood disasters alongside existing assessment frameworks [ 18 ], this study establishes a three-tiered evaluation system ( Fig 1 ). The primary tier (objective layer) focuses on ice flood disaster risk quantification. The secondary tier (criterion layer) comprises four components: disaster-inducing factors, disaster-breeding environment, hazards-bearing, and disaster response capability. Disaster-inducing factors and predisposing conditions collectively characterize hazard magnitude, while hazards-bearing bodies quantify regional vulnerability. Disaster response capability evaluates infrastructural resilience. The tertiary tier (index layer) contains 13 operationalized parameters across these criteria. Each evaluation metric was systematically categorized into distinct classification tiers using longitudinal extremum analysis (encompassing maximum, minimum, and mean values) derived from multi-decadal datasets, with operational thresholds explicitly defined in Table 1 . Fig 1. Risk assessment index system for ice flood disasters. Open in a new tab The figure illustrates the hierarchical framework of the ice flood disaster risk assessment index system developed for the Inner Mongolia reach of the Yellow River, together with its specific indicators at each level. Table 1. Classification standard of risk assessment index of ice flood. Index Risk level Very low risk Low risk Medium risk High risk Very high risk Maximum ice thickness during the freezing period(m) <0.2 0.2-0.4 0.4-0.6 0.6-0.8 >0.8 Flow ice density(%) <30 30-50 50-70 70-90 >90 Average flow the during freezing period(m 3 /s) >900 700-900 500-700 300-500 <300 Water level rise during the freezing period(m) <0.5 0.5-1 1-1.5 1.5-2 >2 Daily average minimum temperature during the ice flow period(°C) >-14 (−14,-16] (−16,-18] (−18,-20] <−20 Longitudinal gradient of river channel(‰) >0.4 0.3-0.4 0.2-0.3 0.1-0.2 <0.1 River channel curvature <1.2 1.2-1.4 1.4-1.6 1.6-1.8 >1.8 River section length(m) <50 000 50 000-100 000 100 000-150 000 150 000-200 000 >200 000 Number of river crossing bridges <1 1-3 3-5 5-7 >7 Population density(person/km 2) <50 50-200 200-400 400-600 >600 Main crop yields(kg/m 2 ) <0.45 0.45-0.5 0.5-0.55 0.55-0.6 >0.6 Protective engineering(Embankment length)(m) <3 × 10 4 3 × 10 4 -6 × 10 4 6 × 10 4 -9 × 10 4 9 × 10 4 -1.2 × 10 5 >1.2 × 10 5 Distance from reservoir or ice separation zone(m) <10 5 10 5 -3 × 10 5 3 × 10 5 -5 × 10 5 5 × 10 5 -7 × 10 5 >7 × 10 5 Open in a new tab Method Analytic hierarchy process In this research, the weights of the indexes are determined using the AHP. Owing to its high degree of comprehensiveness and wide applicability, AHP is widely regarded as a reliable method for assigning weights. The detailed procedures are as follows: Based on the established evaluation index system, pairwise comparisons of the indexes are performed within each level to form the judgment matrix; Solve the judgment matrix in order to calculate the maximum eigenvalue and eigenvector, and carry out consistency testing; Aggregate the weights of each level to acquire the overall weights of the lower-level indexes. The entropy weight method The entropy method is an approach to assigning weights based on the impact of each indicator’s value variation on the overall system. In information theory, entropy measures uncertainty, and the greater the entropy value of an indicator, the smaller its weight. When addressing the problem of multi-indicator weighting, the entropy method helps eliminate biases from subjective assignments, improving the objectivity and accuracy of the evaluation results. The specific steps are as follows: standardize the data according to the indicator type (benefit-oriented or cost-oriented); process the standardized data to calculate the entropy value for each indicator; and determine the weight of each indicator based on its entropy value. Uncertainty measurement theory The ice flood disaster risk assessment framework incorporates 13 evaluation indexes. Let the evaluation object set M ={ M 1 , M 2 , …, M 13 }, where M m represents the m-th evaluation index. Each measurement value of the evaluation index is denoted by x m . The measurement values are classified into five evaluation levels, represented as C 1 , C 2 , …, C 5 , with the condition that C 1 < C 2 < C 3 < C 4 < C 5 . Let u mk = u ( x m ∈ C k ) represent the degree to which the measurement value x m belongs to the evaluation level C k . The uncertainty measure u satisfies conditions (1) – (3) [ 18 ]: 0 ≤ u ( x m ∈ C k ) ≤ 1 (2) u ( x m ∈ U ) = 1 (3) u ( x m ∈ ⋃ l = 1 k C 1 ) = ∑ l = 1 k u ( x m ∈ C 1 ) (4) The measurement matrix for a single index in ice flood disaster risk assessment, denoted as u mk , can be expressed as follows: u m k = ( @ c c c @ u 11 ⋯ u 1 k ⋮ ⋮ ⋮ u m 1 ⋯ u m k ) (5) The comprehensive evaluation vector is then computed through: u = W u m k (6) Where: W is the index weight vector previously determined through entropy weighting. Variable fuzzy set theory Chen proposed the VFST considering the relative nature and dynamic changes of fuzzy concepts, which provides a new method for addressing dynamic fuzzy problems [ 19 ]. The theory is based on two key concepts: the relative membership degree and the relative difference degree: D ― A ~ ( i ) = μ A ~ ( i ) − μ A ~ 𝒞 ( i ) (7) Where: i represents any element in the discourse domain I , i ∈ I ; I is the discourse domain, which is the set of all possible elements under consideration in this evaluation; A ~ and A ~ c represent attractive and repulsive properties respectively; μ A ~ ( i ) and μ A ~ 𝒞 ( i ) represent the relative membership degrees of attractive and repulsive properties respectively, given that μ A ~ ( i ) + μ A ~ 𝒞 ( i ) =1; D A ~ ( i ) is the relative difference between i and A ~ . Therefore, the relative membership degree of i to A ~ can be articulated as: μ A ~ ( i ) = ( 1 + D A ~ ( i ) ) / 2 (8) The comprehensive relative membership degree of evaluation index i to level k can be calculated by the following formula: ν k = F ( ω m , μ m k ) = { 1 + [ ∑ m = 1 13 [ ω m ( 1 − μ m k ) ] p ∑ m = 1 13 ( ω m μ m k ) p ] α p } − 1 (9) UMT-VFST evaluation method Although VFST effectively reflects the dynamic fuzziness of phenomena, the calculation of relative membership degrees necessitates initially ascertaining relative difference degrees, which frequently depend excessively on practical experience. The uncertainty measure in uncertainty measurement theory and the relative membership degree in VFST share analogous meanings to a certain extent: the closer the value is to 1, the higher the degree of consistency between the assessed entity and the specified evaluation tier. By substituting the uncertainty measure vector for the relative membership degree in the VFST, the two methods can be organically integrated, exploiting their complementary strengths to further enhance the scientific rigor and accuracy of the assessment. The linear uncertainty measurement function is selected to construct the single index measurement matrix as follows: { @ c @ u m k = { @ c c c @ − x m / ( a k + 1 − a k ) + a k + 1 / ( a k + 1 − a k ) a k < x m ≤ a k + 1 0 x m ≥ a k + 1 u m ( k + 1 ) = { @ c c c @ 0 x m ≤ a k x m / ( a k + 1 − a k ) − a k / ( a k + 1 − a k ) a k < x m ≤ a k + 1 (10) Where: x m denotes the measured value of evaluation index M m ; a k and a k+1 represent the k -th and k + 1-th critical thresholds on the unascertained measure function. Subsequently, the unascertained measure u mk is used to replace the relative membership degree µ mk , serving as a bridge to couple the unascertained measure theory with the variable fuzzy set theory. μ m k = u m k (11) Subsequently, by utilizing the variable parameter combinations in the VFST, the comprehensive membership degree under each parameter combination is ascertained, dynamically illustrating the risk levels of the evaluation regions. ν k = F ( ω m , μ m k ) = { 1 + [ ∑ m = 1 13 [ ω m ( 1 − u m k ) ] p ∑ m = 1 13 ( ω m u m k ) p ] α p } − 1 (12) Where: ω m is the weight of the m index; α serves as the parameter for the optimization criterion, α is either 1 or 2; p represents the parameter for distance measurement, p is either 1 or 2. There are usually four combinations: ① α = 1, p = 1; ② α = 1, p = 2; ③ α = 2, p = 1; ④ α = 2, p = 2. Considering that there are currently no clear standards or guidelines for the classification of flood disaster risk levels, this paper refers to the classification method used in flood disaster risk assessment [ 20 ]. The ice flood disaster risk is categorized into five levels, and the classification established utilizing level eigenvalues. Initially, the comprehensive membership degree obtained from Equation (10) is normalized as follows: V k = ν k / ∑ k = 1 5 ν k (13) The level eigenvalues are calculated based on Equation (12) as follows: H = ∑ k = 1 5 ( V k k ) (14) The level eigenvalues corresponding to the risk levels are shown in Table 2 . Table 2. The corresponding relationship between the level eigenvalue H and the evaluation level. Evaluation level Level eigenvalue H Very low risk 1-1.5 Low risk 1.5-2.5 Medium risk 2.5-3.5 High risk 3.5-4.5 Very high risk 4.5-5 Open in a new tab Results and discussion Weight calculation Following the established methodological framework, pairwise comparison matrices were constructed for hierarchical indexes. Through structured expert elicitation using Saaty’s 9-point scale, relative importance ratings were systematically obtained. The parametric indices A1-A13 exhibit systematic correspondence with the constituent elements within the evaluation framework schematically illustrated in Fig 1 . Validation analysis demonstrates that all judgment matrices satisfy consistency ratio (CR) thresholds (CR < 0.1), confirming the statistical validity and methodological robustness of the derived weight allocation. By combining the Risk assessment index system for ice flood disasters in Fig 1 and the index data in Table 3 , the entropy weight method is used to calculate the objective weights of each indicator. The comprehensive weight is then obtained by calculating the weighted average of the results from the AHP and entropy weight method, with equal weights (50% each) assigned to both methods.The finalized weight distribution pattern is visually presented in Fig 2 . Table 3. Index value of ice flood disaster risk assessment in each region. Index Tokto County Jungar Banner Qingshuihe county Maximum ice thickness during the freezing period(m) 0.6 0.95 0.85 Flow ice density(%) 40 60 100 Average flow the during freezing period(m 3 /s) 460 455 425 Water level rise during the freezing period(m) 1.4 6.6 6.1 Daily average minimum temperature during the ice flow period(°C) −18 −18.4 −16.8 Longitudinal gradient of river channel(‰) 0.24 0.66 0.38 River channel curvature 1.06 1.92 1.26 River section length(m) 37 500 238 000 65 000 Number of river crossing bridges 4 13 3 Population density(person/km 2) 152 44 32 Main crop yields(kg/m 2 ) 0.511 0.49 0.335 Protective engineering(Embankment length)(m) 78 200 60 646 30 000 Distance from reservoir or ice separation zone(m) 97 000 73 000 45 000 Open in a new tab Fig 2. Weights of indexes at all levels. Open in a new tab The figure presents the magnitude and relative importance of the weights assigned to each indicator, clearly revealing their respective contributions to the overall assessment. Risk grade calculation This investigation focuses on the 2016 ice flood event at the reservoir tail reach of Wanjiazhaias a representative case study. The methodologically triangulated dataset integrates: (a) scholarlyreferences [ 12 ], (b) municipal statistical yearbooks, (c) multi-sensor measurements frommeteorological observatories, and (d) instrumented hydrological monitoring records, assystematically cataloged in Table 3 . (1) Single-index measurement function: First, calculate the measurement function for each index, that is, the single-index measurement function. By taking the maximum ice thickness as an example, the uncertainty measurement function of which is shown in Fig 3 . Based on Equation 5 , yields the single-index measurement for the maximum ice thickness during the freezing period in Jungar Banner as (0,0,0,0,1). Similarly, the single-index measurement matrix for the 13 ice flood disaster risk evaluation indexes in Jungar Banner is presented below: Fig 3. Measure function of maximum ice thickness during freezing period. Open in a new tab The figure depicts the measure function for maximum ice thickness during the freezing period, shown here as a representative example. u 13 * 5 = [ @ c c c c c @ 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0.275 0.725 0 0 0 0.3 0.7 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0.7 0.3 0 0 0 0.478 0.522 0 0 0.81 0.19 0 0 0 ] (15) (2) Relative membership degree: Based on Equation 10 and the single-index measurement matrix previously calculated, the relative membership degree of ice flood disaster risk for Jungar Banner is ascertained. The distribution of the relative membership degrees across different levels is illustrated in Fig 4 . Fig 4. Relative membership degree of all levels of risk of ice flood disaster in Jungar Banner. Open in a new tab Taking Jungar Banner as an example, the figure compares the comprehensive relative membership degrees obtained under the four alternative parameter combinations. (3) Evaluation level determination: Based on Equations 11 and 12 along with the relative membership degree vector of ice flood disaster risk for Jungar Banner previously calculated, yields the level eigenvalue for Jungar Banner’s ice flood disaster risk. In the same manner, the level eigenvalues for Tokto County and Qingshuihe County are computed. The results of level eigenvalues under diverse parameter combinations are presented in Fig 5 . For Tokto County, the ice flood disaster risk level eigenvalue H ― =2.088, indicating a “ medium risk “; For Jungar Banner, the level eigenvalue H ― =3.545, indicating a “high risk”; For Qingshuihe County, the level eigenvalue H ― =2.625, indicating a “medium risk”. Fig 5. Level characteristic values under different parameter combinations in each banner county. Open in a new tab The figure displays the level-eigenvalue profiles calculated for the three study areas under each parameter combination. Evaluation results and analysis According to the evaluation results: (1) Jungar Banner has the highest risk level of ice flood disaster, which is classified as “high risk”. The river in Jungar Banner has a great length, a high degree of curvature, and numerous bridges across it. These characteristics make it easy for ice to cause waterway obstruction, thereby triggering flood disasters. Hence, the risk of ice flood disaster in this area is considerably higher than that in other areas. According to the historical disaster records, the ice jam bursting in Zhungeer Banner caused flooding in nearby villages and inundated residential houses, which corresponds to the assessment results. (2) Tokto County has the lowest risk level of ice flood disaster, which is classified as “medium risk”. Although Tokto County has the highest population density among the three study areas, it features thinner ice, a lower water level rise, less river curvature, and longer levees. These factors effectively reduce the likelihood and risk of ice flood disasters, resulting in a significantly lower risk value for Tokto County compared to other areas. (3) Qingshuihe County, classified as a medium-risk area, has a risk value between the other two. In Qingshuihe County, the ice layer is thicker and the water level rises significantly during the freezing period, resulting in a higher flood risk than in Tokto County. However, due to the short river reach in this area, the river has a small curvature and is close to the downstream Wanjiazhai Reservoir, the flood risk can be promptly alleviated by regulating the reservoir’s flow. Consequently, the risk value of Qingshuihe County lies between those of the other two regions. (4) The results were compared with those from the improved catastrophe evaluation method reported in [ 12 ]. The latter classified Tokto County as a moderate-risk area (0.644), Jungar Banner as a high-risk area (0.743), and Qingshuihe County as a moderate-risk area (0.676). Both methods produced consistent risk classifications and rankings across the three regions, and the results closely correspond to the actual ice flood conditions in each area. This comparison confirms the reliability of the evaluation method proposed in this study. Conclusion (1) Based on the analysis of the disaster-causing factors and disaster formation mechanisms of ice flood hazards in the Yellow River Inner Mongolia section, and with reference to existing relevant studies, a risk assessment index system for ice flood disasters was constructed with the regional disaster system theory. This index system effectively reflects the risk characteristics of ice flood disasters in the Yellow River Inner Mongolia section, providing a theoretical basis for regional risk assessments. (2) Based on the UMT and the VFST, the ice flood risk assessment model for the Yellow River segment in Inner Mongolia was established, and the risk assessment was carried out in 3 typical areas of the Inner Mongolia section. The results showed the following: the ice flood disaster risk level in Tokto County, with a characteristic value of H ― =2.088, was classified as “medium risk”; that in Jungar Banner, with a characteristic value of H ― =3.545, was classified as “high risk”; and that in Qingshuihe County, with a characteristic value of H ― =2.625, was classified as “medium risk”. (3) By utilizing the established risk assessment model, the ice flood disaster risks in the three counties were analyzed and then compared with the results from the improved catastrophe evaluation method. The evaluation results from both methods were consistent regarding the risk rankings, confirming the reliability of the findings. This study offers scientific evidence and effective technical support for risk assessment and flood prevention in the Yellow River Inner Mongolia section, thereby contributing to disaster mitigation efforts. 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Wang C, Wang K, Liu D, Zhang L, Li M, Imran Khan M. Development and application of a comprehensive assessment method of regional flood disaster risk based on a refined random forest model using beluga whale optimization. J Hydrol. 2024;633:130963. doi: 10.1016/j.jhydrol.2024.130963 [ DOI ] [ Google Scholar ] PLoS One. doi: 10.1371/journal.pone.0350284.r001 Decision Letter 0 Sanjay Kumar Author information Copyright and License information Roles Sanjay Kumar : Academic Editor © 2026 Sanjay Kumar This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. 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Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented. --> Reviewer #1: Yes Reviewer #2: Yes Reviewer #3: Yes ********** -->2. Has the statistical analysis been performed appropriately and rigorously? --> Reviewer #1: Yes Reviewer #2: Yes Reviewer #3: Yes ********** -->3. Have the authors made all data underlying the findings in their manuscript fully available? The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.--> Reviewer #1: Yes Reviewer #2: Yes Reviewer #3: Yes ********** -->4. Is the manuscript presented in an intelligible fashion and written in standard English? PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.--> Reviewer #1: Yes Reviewer #2: Yes Reviewer #3: Yes ********** -->5. Review Comments to the Author Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)--> Reviewer #1: Overall, I like the article which I consider an interesting contribution to Disaster Risk Assessment. I do have some minor comments The introduction section has too long paragraphs, please separate them into shorter paragraphs for the good sake of the reader On lines 116-120 authors explain the three-tiered evaluation system, but they use different expressions on those lines and the figure, for instance in line 118-119 they list: disaster-inducing factors, disaster-predisposing conditions, hazard-bearing entities, and disaster mitigation capacity. However, in Figure 2 they list: disaster-inducing factors; disaster-breeding environment, hazard bearing bodies, and disaster response capability. That difference leads to confusion and must be emended Line 144, 161, 225, I do not understand the symbology, please check the edition of the pdf document. Reviewer #2: This paper presents an innovative approach by integrating Uncertainty Measure Theory (UMT) with Variable Fuzzy Set Theory (VFST) to develop a model for assessing ice flood risk in the Inner Mongolia reach of the Yellow River. The study shows strong originality, but some revisions are needed: 1. In UMT-VFST Evaluation Method, the division of risk into five levels lacks a clear reference to specific standards. Please provide the source of the classification (e.g., national standards, industry guidelines, or authoritative literature). 2. The terminology for ice floods is inconsistent (e.g., ice flood vs. ice-jam flood). Consistency in terminology should be ensured throughout. 3. Issues in the tables: The header row uses indexes instead of the singular index, and the overall formatting (column width, alignment, borders) reduces readability. Please unify the use of index and improve formatting to meet academic publishing standards. 4. There is inconsistent use of Chinese and English punctuation throughout the manuscript. For example, in line 231, the Chinese full-width parenthesis “（” is used instead of the half-width “(”. It is recommended to unify the use of half-width punctuation marks, and carefully check all parentheses, commas, periods, etc. throughout the text. Reviewer #3: This paper couples Uncertainty Measure Theory with Variable Fuzzy Set Theory to address the complex characteristics of disaster systems, thereby enabling a more accurate analysis and assessment of ice flood disasters along the Inner Mongolia reach of the Yellow River. The study demonstrates strong originality, but certain aspects still require improvement. 1 In Section 'Weight calculation', the exclusive reliance on the Analytic Hierarchy Process (AHP) for weight determination raises concerns regarding methodological robustness. Given that AHP is inherently subjective due to its dependence on expert judgment, it is recommended to incorporate objective weighting methods (e.g., entropy weight method, CRITIC) to enhance result reliability. 2 In Figure 3, the font sizes are inconsistent—for example, the axis labels, legend, and data annotations use different sizes, which reduces readability. It is recommended to check all figures in the manuscript and ensure uniform font style and size. 3 The use of thousand separators in the data is inconsistent (e.g., in Table 1). Please review and correct this throughout the manuscript. 4 All variables, parameters, and function names should be formatted consistently in either italics or roman type, in accordance with academic conventions. Particular attention should be paid to font usage in the Method section. 5 In the Weight Calculation section, paragraph formatting is inconsistent. Please revise according to the journal’s formatting guidelines. ********** -->6. PLOS authors have the option to publish the peer review history of their article ( what does this mean? ). If published, this will include your full peer review and any attached files. If you choose “no”, your identity will remain anonymous but your review may still be made public. Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy .--> Reviewer #1: Yes: Jose Luis Arumi Reviewer #2: No Reviewer #3: No ********** [NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.] To ensure your figures meet our technical requirements, please review our figure guidelines: https://journals.plos.org/plosone/s/figures You may also use PLOS’s free figure tool, NAAS, to help you prepare publication quality figures: https://journals.plos.org/plosone/s/figures#loc-tools-for-figure-preparation. NAAS will assess whether your figures meet our technical requirements by comparing each figure against our figure specifications. --> PLoS One. 2026 May 29;21(5):e0350284. doi: 10.1371/journal.pone.0350284.r002 Author response to Decision Letter 1 Article notes Copyright and License information Collection date 2026. PMC Copyright notice 10 Dec 2025 Reviewer #1: Overall, I like the article which I consider an interesting contribution to Disaster Risk Assessment.I do have some minor comments 1.The introduction section has too long paragraphs, please separate them into shorter paragraphs for the good sake of the reader Answer: Thank you for your valuable comment. As you pointed out, the introduction in our previous submission contained overly long paragraphs, which may have hindered readability. We have now reorganized and divided the introduction into shorter, more coherent paragraphs to improve clarity and enhance the overall reading experience. 2.On lines 116-120 authors explain the three-tiered evaluation system, but they use different expressions on those lines and the figure, for instance in line 118-119 they list: disasterinducing factors, disaster-predisposing conditions, hazard-bearing entities, and disaster mitigation capacity. However, in Figure 2 they list: disaster-inducing factors; disaster-breeding environment, hazard bearing bodies, and disaster response capability. That difference leads to confusion and must be emended Answer: Thank you for your professional question. We have carefully revised the manuscript to address the inconsistencies in terminology. The expressions disaster-inducing factors, disaster-breeding environment, hazards-bearing body, and disaster response capability are now used consistently throughout the paper. 3.Line 144, 161, 225, I do not understand the symbology, please check the edition of the pdf document. Answer: Thank you for your professional question. In the previous version, we did not provide sufficiently clear explanations for some of the symbols introduced in the manuscript, which may have affected readers’ understanding. We have now added detailed explanations and clarifications to improve the readability of this section. Reviewer #2: This paper presents an innovative approach by integrating Uncertainty Measure Theory (UMT) with Variable Fuzzy Set Theory (VFST) to develop a model for assessing ice flood risk in the Inner Mongolia reach of the Yellow River. The study shows strong originality, but some revisions are needed: 1. In UMT-VFST Evaluation Method, the division of risk into five levels lacks a clear reference to specific standards. Please provide the source of the classification (e.g., national standards,industry guidelines, or authoritative literature). Answer: Thank you for your professional comments! At present, there are no clear norms or unified classification standards for ice flood disaster risk assessment. Therefore, this study refers to the mature classification methods in the field of flood disaster risk assessment, divides the ice flood disaster risk levels into five grades, and has supplemented relevant explanations in the manuscript. 2. The terminology for ice floods is inconsistent (e.g., ice flood vs. ice-jam flood). Consistency in terminology should be ensured throughout. Answer: Thank you for your professional insight. At present, there is no standardized or universally accepted classification system for ice flood disaster risk assessment. Therefore, in this study, we refer to the classification methods commonly used in flood disaster risk assessment and divide ice flood disaster risk into five levels. We have added corresponding explanations in the revised manuscript. 3. Issues in the tables: The header row uses indexes instead of the singular index, and the overall formatting (column width, alignment, borders) reduces readability. Please unify the use of index and improve formatting to meet academic publishing standards. Answer: Thank you for your professional comment. We have carefully reviewed the tables and corrected all non-standard elements accordingly. 4. There is inconsistent use of Chinese and English punctuation throughout the manuscript. For example, in line 231, the Chinese full-width parenthesis “（” is used instead of the halfwidth “(”. It is recommended to unify the use of half-width punctuation marks, and carefully check all parentheses, commas, periods, etc. throughout the text. Answer: Thank you for your professional comment. We have thoroughly reviewed the formatting and layout of the manuscript and revised them in accordance with the journal’s formatting requirements. Reviewer #3: This paper couples Uncertainty Measure Theory with Variable Fuzzy Set Theory to address the complex characteristics of disaster systems, thereby enabling a more accurate analysis and assessment of ice flood disasters along the Inner Mongolia reach of the Yellow River. The study demonstrates strong originality, but certain aspects still require improvement. 1. In Section 'Weight calculation', the exclusive reliance on the Analytic Hierarchy Process (AHP) for weight determination raises concerns regarding methodological robustness. Given that AHP is inherently subjective due to its dependence on expert judgment, it is recommended to incorporate objective weighting methods (e.g., entropy weight method, CRITIC) to enhance result reliability. Answer: Thank you for your professional comment. As you pointed out, relying solely on subjective weighting methods may affect the accuracy of the evaluation results. Therefore, following your suggestion, we incorporated the entropy weight method as an objective weighting approach. By combining both subjective and objective weights, we improved the reliability of the final assessment results. 2. In Figure 3, the font sizes are inconsistent—for example, the axis labels, legend, and data annotations use different sizes, which reduces readability. It is recommended to check all figures in the manuscript and ensure uniform font style and size. Answer: Thank you for your professional comment. We have revised the non-standard formatting issues in the figures and tables to ensure full compliance with the journal’s requirements. 3. The use of thousand separators in the data is inconsistent (e.g., in Table 1). Please review and correct this throughout the manuscript. Answer: Thank you for your professional comment. We have carefully reviewed all data formats in the manuscript and corrected any irregularities. 4. All variables, parameters, and function names should be formatted consistently in either italics or roman type, in accordance with academic conventions. Particular attention should be paid to font usage in the Method section. Answer: Thank you for your professional comment. The formatting of all variables, parameters, and function names in the manuscript has been revised accordingly. 5. In the Weight Calculation section, paragraph formatting is inconsistent. Please revise according to the journal’s formatting guidelines. Answer: Thank you for your professional comment. The formatting of the Weight caught section has been revised accordingly. We would like to express our sincere gratitude to all editors and experts for their careful guidance! We have strictly revised the manuscript in accordance with the review comments. Should there be any aspects that still need improvement, we sincerely request further corrections and look forward to your reply. Attachment Submitted filename: Response to Reviewers.docx pone.0350284.s002.docx (16.8KB, docx) PLoS One. doi: 10.1371/journal.pone.0350284.r003 Decision Letter 1 Beata Calka Author information Copyright and License information Roles Beata Calka : Academic Editor © 2026 Beata Calka This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. PMC Copyright notice 13 May 2026 Ice Flood Disaster Risk Assessment of the Inner Mongolia Reach in the Yellow River Based on Unascertained Measure Theory-Variable Fuzzy Sets PONE-D-25-44421R1 Dear Dr. deng, We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements. Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication. An invoice will be generated when your article is formally accepted. Please note, if your institution has a publishing partnership with PLOS and your article meets the relevant criteria, all or part of your publication costs will be covered. Please make sure your user information is up-to-date by logging into Editorial Manager at Editorial Manager® and clicking the ‘Update My Information' link at the top of the page. For questions related to billing, please contact billing support . If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they’ll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org. Kind regards, Beata Calka, PH.D. Academic Editor PLOS One Additional Editor Comments (optional): Before final acceptance, editorial revisions are required, particularly in Table 1, to ensure full consistency and uniformity in the presentation of data throughout the manuscript. Reviewers' comments: Reviewer's Responses to Questions --> Comments to the Author 1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.--> Reviewer #1: All comments have been addressed Reviewer #3: All comments have been addressed ********** -->2. Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented. --> Reviewer #1: Yes Reviewer #3: Yes ********** -->3. Has the statistical analysis been performed appropriately and rigorously? --> Reviewer #1: Yes Reviewer #3: Yes ********** -->4. Have the authors made all data underlying the findings in their manuscript fully available? The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.--> Reviewer #1: Yes Reviewer #3: Yes ********** -->5. Is the manuscript presented in an intelligible fashion and written in standard English? PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.--> Reviewer #1: Yes Reviewer #3: Yes ********** -->6. Review Comments to the Author Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)--> Reviewer #1: All my previous comments were satisfactory addressed and the article is sound. Therefore, I have no further comments. Reviewer #3: The author has provided reasonable explanations and adequate responses to the five comments raised in the first review. Similarly, the author has made serious, systematic, and comprehensive improvements based on the comments from the other two reviewers. It is suggested that some details of the manuscript be further refined. For instance, in Table 1 the numerical ranges expressed in two forms—basic mathematical notation and statistical/quality management notation. It is suggested that the author use a consistent format for expressing numerical ranges in Table 1. ********** -->7. PLOS authors have the option to publish the peer review history of their article ( what does this mean? ). If published, this will include your full peer review and any attached files. If you choose “no”, your identity will remain anonymous but your review may still be made public. Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy .--> Reviewer #1: No Reviewer #3: No ********** PLoS One. doi: 10.1371/journal.pone.0350284.r004 Acceptance letter Beata Calka Author information Copyright and License information Roles Beata Calka : Academic Editor © 2026 Beata Calka This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. PMC Copyright notice PONE-D-25-44421R1 PLOS One Dear Dr. deng, I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS One. Congratulations! Your manuscript is now being handed over to our production team. At this stage, our production department will prepare your paper for publication. This includes ensuring the following: * All references, tables, and figures are properly cited * All relevant supporting information is included in the manuscript submission, * There are no issues that prevent the paper from being properly typeset You will receive further instructions from the production team, including instructions on how to review your proof when it is ready. Please keep in mind that we are working through a large volume of accepted articles, so please give us a few days to review your paper and let you know the next and final steps. Lastly, if your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org. You will receive an invoice from PLOS for your publication fee after your manuscript has reached the completed accept phase. If you receive an email requesting payment before acceptance or for any other service, this may be a phishing scheme. Learn how to identify phishing emails and protect your accounts at https://explore.plos.org/phishing . If we can help with anything else, please email us at customercare@plos.org. Thank you for submitting your work to PLOS ONE and supporting open access. 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