An update on Data-Fusion-based Dam Breach Empirical Equations Based on a Worldwide Historical Dam Failure Database

An update on Data-Fusion-based Dam Breach Empirical Equations Based on a Worldwide Historical Dam Failure Database | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article An update on Data-Fusion-based Dam Breach Empirical Equations Based on a Worldwide Historical Dam Failure Database Monte Azmi This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7289241/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 5 You are reading this latest preprint version Abstract This research concentrated on estimating breaching embankment dams with focus on i) the extraction of failure cases from the historical dam failure database, adoptable for dam breach parameters estimates, ii) an update on data-fusion-based (DFM) dam breach equations published by Azmi and Thomson ( 2024 ) using the adopted failure cases, and finally iii) a comprehensive comparative assessment of all models to estimate breach peak discharge, breach formation time, and average breach width. A multi-stage procedure was followed to select the empirical equations that showed higher similarities to observed data for all three dam breach parameters. Statistical evaluations included t- and F-tests, probability distribution analyses (K-S test, Q-Q plots, boxplots, scatter plots), along with the performance criteria (median of percentage error (MPE), root mean square errors (RMSE), coefficient of variation and Nash Sutcliffe efficiency (NSE)), were used for comparative assessments. The final selected methods are: i) Breach peak outflow: DFM 2024 and updated DFM, ii) Final breach average width: XZ9 (Xu and Zhang 2009 ) and updated DFM, and iii) Failure time: F8 (Froehlich 2008 ) and updated DFM. It is critical to note that the proposed equations are not recommended for concrete dams and or embankment dams with extensive safety elements (e.g., wave walls, additional rock mesh protections). dam break dam breach parameters data fusion model dam failure database Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Introduction Despite notable progress in recent decades, dambreak modeling remains a vital area of research for hydraulic engineers due to its significant implications (Aureli et al 2024). Dam failures can lead to catastrophic flooding, causing severe human and economic losses (Zech and Soares-Frazo 2007). The risk is heightened by aging infrastructure, inadequate spillway capacity, and increasing urban development in flood-prone areas (Zech and Soares-Frazo 2007; Zhang et al. 2016 ). On a global scale, up to 2013, the USA and China led with 87,000 and 98,000 dams respectively, the majority being earth-rock structures (USACE 2013 ; NWR and NBS 2013). While reservoirs are essential for water supply and flood control, historical failures have demonstrated their potential for widespread damage. For instance, China reported 3,541 dam break incidents (95% earth-rock dams) by 2018, and the USA documented 158 overtopping failures between 2000 and 2023 (DSMC 2018 ; ASDSO 2023 ). These statistics underscore the ongoing need for robust modeling and risk assessment in dam safety management. Estimating key dam breach parameters (failure time, average width of breach, and breach peak discharge) remains a complex task due to limited reliable data and the intricate interplay of factors involved in breach events. This often results in high uncertainty ranging from ± 0.5 to ± 1 order of magnitude (Wahl 2004 ). Various methods have been developed to address this, including physical and laboratory models, Computational Fluid Dynamics (CFD), empirical equations, and advanced machine learning techniques (Aureli et al. 2024). Among these, empirical equations are widely used, relying on statistical analysis of reported breach parameters such as failure time, breach width, and peak outflow (Froehlich 1995 , 2008 , 2016b ; Xu and Zhang 2009 ; Zhong et al. 2020 ; Azmi and Thomson 2024 ). Existing empirical models face several limitations: they are primarily based on small dams (under 15m), often overlook failure modes, dam types, and material erodibility, and suffer from limited calibration data and high prediction errors (Sattar 2014 ). These challenges highlight the need for improved data quality, comprehensive modeling approaches, and better integration of dam-specific characteristics to enhance the accuracy and reliability of breach parameter estimation. The most referable historical dam breach datasets were initially gathered and presented in 1995 and were gradually completed in 2016 by Froehlich ( 1995 , 2008 , 2016a , b ); these datasets were used for statistical analysis and driving empirical equations for embankment dam breach parameters. Adopted datasets often include outliers or errors, necessitating careful data validation (Azmi and Thomson 2024 ). In 2020, assembled from 196 references, a worldwide database (Bernard-Garcia and Mahdi 2020 ) was compiled and documented with a total of 3,861 cases of historical dam failures around the world and represents the largest compilation of dam failures recorded to date (17-02-2020). This database includes recorded historical dam failures regardless of the type of dams (e.g. man-made dam), the type of structure (e.g. embankment dam, concrete dam), the type of failure (e.g. overtopping failure, landslide) and the properties of the dams (e.g. dam height, dam storage). Through this process, a total of 45 variables have been used to record various information about the failure (e.g. breach dimensions). It is important to note that this database also includes all databases previously presented and documented by Froehlich ( 1995 , 2008 , 2016a , b ). In the current research, out of all recorded data, 104 dam breach cases were selected, which are related to man-made dams, with earthfill and rockfill types. The main aims of this research are i) the extraction of failure cases from the worldwide historical dam failure database, adoptable for estimating earthfill/rockfill dam breach parameters (breach peak discharge, breach formation time, and average breach width), ii) an update on data-fusion-based (DFM) dam breach equations published by Azmi and Thomson ( 2024 ), and finally iii) a comparative assessment of selected commonly used empirical equations. Database This research draws upon the comprehensive global dam breach dataset compiled and validated by Bernard-Garcia and Mahdi ( 2020 ) (Link to the database). This dataset surpasses the widely used collections in the field of dam breach analysis that have been documented over the past few decades (Froehlich 1995 , 2016b ; Xu and Zhang 2009 ; Zhong et al. 2020 ). From the full dataset of 3,861 recorded cases, 104 were selected for this study. These cases specifically involve man-made dams constructed from earthfill and rockfill materials. Although the dataset has undergone thorough peer review and procedures to address missing data, some gaps remain. To make the most of the available data, the following assumptions were applied: i) Dam Height (h d ): In cases where dam height was missing, it was assumed to be equal to the breach height (h b ). This is a common practice in real-world dam breach assessments to ensure that the entire reservoir volume is considered discharged, allowing for a conservative estimation of downstream impacts. ii) Dam Erodibility: Where erodibility data were unavailable, a medium erodibility classification was assumed. This approach aligns with the methodology proposed by Azmi and Thomson ( 2024 ), aiming to avoid overly conservative or optimistic assumptions, unless data from a highly similar dam with complete information were available. iii) Breaching Parameters: For parameters such as failure time, average breach width, and peak breach discharge, where only a range was provided, the midpoint of the range was used. This avoids extreme assumptions unless a closely comparable dam with complete data could inform a more accurate estimate. This dataset includes dam names, dam type (homogeneous dam (HD), core wall dam (CD), concrete-faced dam (FD), or zoned-fill dam (ZD)), failure mode (overtopping (O) or piping (P)), breach peak discharge in m 3 /s (Q p ), final breach average width in m (B ave ), reservoir storage in m 3 (S), the volume of water above breach invert in m 3 (V w ), dam height (h d ), the height of water above breach invert in m (h w ), the height of breach in m (h b ). It has been assumed that the final breach top and bottom widths are measured along the dam crest. The height of the breach is the difference between the elevation of the top and bottom of the dam at the breach location, where the final breach average width is the average of the top and bottom widths. Failure time in hr (T f ), also called “breach formation time”, is the time from the onset of formation to the full completion stage (Froehlich 2008 ). Based on the available data for dam breach cases, three sub-datasets were formed (the compiled database is presented in Appendix A). Each group corresponds to a specific breach parameter (dependent variable), for which relevant empirical equations were applied to estimate values and conduct performance assessments. Group 1 comprises 130 cases used for estimating the average breach width (B ave ). Group 2 includes 65 cases for estimating the failure time (T f ). Group 3 consists of 54 cases used to estimate the breach peak discharge (Q p ), where the average embankment width of the dam (Wave) was also available. For comparison, Azmi and Thomson ( 2024 ) used 67 cases for B ave , 36 for T f , and 41 for Q p , respectively. Methodology Individual Empirical Equations The methodology begins with the collection of widely used empirical equations, with particular attention to their reported accuracy, simplicity, and practicality. This includes the addition of recently introduced data-fusion-based empirical equations by Azmi and Thomason (2024), as shown in Tables 1 to 3 . To ensure the reliability of each selected equation, only those published in fully peer-reviewed sources such as ISI-indexed journals and technical reports/user manuals from globally recognised hydraulic software like HEC-RAS (2016) were considered. It is important to note that the equation by Xu and Zhang (2009) for predicting T f encompasses the entire timeframe from initial erosion to the post-erosion phase. In contrast, other equations (e.g., Froehlich 1995, 2008) define T f as the period from the onset of formation to the point of full completion (Froehlich 2008). To maintain consistency and comparability among the individual equations, the Xu and Zhang (2009) equation was excluded from the pool of individual empirical equations for this breaching parameter. Table 1 Selected empirical equations for breach peak discharge Reference Equation Froehlich ( 2016b ) a \(\:{Q}_{P}=0.0175{k}_{m}{k}_{h}(g{{h}_{w}{V}_{w}{h}_{b}}^{2}/{{W}_{ave})}^{0.5}\) Froehlich ( 1995 ) \(\:{Q}_{P}=0.607{{V}_{w}}^{0.295}{{h}_{w}}^{1.24}\) Webby (1996) \(\:{Q}_{P}=0.0443\sqrt{g}{{V}_{w}}^{0.365}{{h}_{w}}^{1.405}\) Hooshyaripor et al. (2014) \(\:{Q}_{P}=0.0212{{V}_{w}}^{0.5429}{{h}_{w}}^{0.8713}\) Azimi et al. (2015) \(\:{Q}_{P}=0.0166{{(gV}_{w})}^{0.5}{h}_{w}\) Xu and Zhang ( 2009 ) b \(\:{Q}_{P}=0.175\sqrt{g}{{V}_{w}}^{5/6}{{(h}_{d}/{h}_{r})}^{0.199}{({{V}_{w}}^{1/3}/{h}_{w})}^{-1.274}{e}^{{B}_{4}}\) Zhong et al. ( 2020 ) \(\:{Q}_{P}=\sqrt{g}{{h}_{w}}^{-0.5}{V}_{w}\:\times\:\:\:\left\{\begin{array}{c}{\left(\frac{{{V}_{w}}^{0.333}}{{h}_{w}}\right)}^{-1.58}{\left(\frac{{h}_{w}}{{h}_{b}}\right)\:}^{-0.76}\times\:{{h}_{d}}^{0.1}{e}^{-4.55},\:\:for\:HD\\\:{\left(\frac{{{V}_{w}}^{0.333}}{{h}_{w}}\right)}^{-1.51}{\left(\frac{{h}_{w}}{{h}_{b}}\right)}^{-1.09}\times\:{{h}_{d}}^{-0.12}{e}^{-3.61},\:for\:CD\end{array}\right.\) Pierce et al. (2010) \(\:{Q}_{P}=0.038{{V}_{w}}^{0.475}{{h}_{w}}^{1.09}\) Abdulrahman (2022) c \(\:{Q}_{P}=55.36\times\:MF\times\:{{V}_{w}}^{1.081}{{h}_{w}}^{0.413}\) USBR (1988) \(\:{Q}_{P}=19.1{{h}_{w}}^{1.85}\) Azmi and Thomson ( 2024 ) d Q p = 1.23×F 16 − 0.84×H 14 + 0.26×XZ 9 a for overtopping failure mode, K m = 1.85; for piping failure, K m = 1. When h b ≤6.1m, K h = 1; when h b >6.1m, K h =(h b /6.1) 1/8 . b B 4 = b 3 + b 4 + b 5 , where b 3 = − 0.503, -0.591, and − 0.649 for dams with core walls (CD), concrete-faced dams (FD), and homogeneous/zoned-fill dams (HD/ZD), respectively; b 4 = − 0.705 and − 1.039 for overtopping and piping, respectively; b 5 = -0.007, -0.375, and − 1.362 for high, medium, and low dam erodibility, respectively. c MF is the model of failure, it is equal to 1 for non-overtopping mode and 1.414 for overtopping mode. d F 16 : Froehlich ( 2016b ), H 14 : Hooshyaripor et al. (2014), XZ 9 : Xu and Zhang ( 2009 ) Table 2 Selected empirical equations for failure time Reference Equation Froehlich ( 2008 ) \(\:{T}_{f}=63.2({V}_{w}/g{{h}_{b}^{2})}^{0.5}\) / 3600 Froehlich ( 1995 ) \(\:{T}_{f}=0.00254{\:h}_{b}^{-0.9}{\:V}_{w}^{0.53}\) Zhong et al. ( 2020 ) \(\:{T}_{f}=\left\{\begin{array}{c}{\left(\frac{{{V}_{w}}^{0.333}}{{h}_{w}}\right)}^{0.56}{\left(\frac{{h}_{w}}{{h}_{b}}\right)\:}^{-0.85}\times\:{{h}_{d}}^{-0.32}{e}^{-0.20}\:\:,\:for\:HD\\\:{\left(\frac{{{V}_{w}}^{0.333}}{{h}_{w}}\right)}^{1.52}{\left(\frac{{h}_{w}}{{h}_{b}}\right)}^{-11.36}\times\:{{h}_{d}}^{-0.43}{e}^{-1.57},\:for\:CD\end{array}\right.\) USBR (1988) b \(\:{T}_{f}=0.011\:{W}_{ave}\) MacDonald and Langridge-Monopolis (1984) \(\:{T}_{f}=0.0179{\left(0.0261\right({V}_{w}{h}_{w}{)}^{0.769})}^{0.364}\) Azmi and Thomson ( 2024 ) b T f = 3.27×F 16 − 2.05×F 95 + 0.25×XZ 9 a This equation was only applied for cases in which both had observed/measured T f and observed/measured W ave . b F 16 : Froehlich ( 2016a ), F 95 : Froehlich ( 1995 ), XZ 9 : Xu and Zhang ( 2009 ) Table 3 Summary of selected empirical dam breach models for final breach average width Reference Equation Froehlich ( 2008 ) a \(\:{B}_{ave}=0.27\:{K}_{O}{{V}_{w}}^{0.32}{{h}_{b}}^{0.04}\) Froehlich ( 1995 ) b \(\:{B}_{ave}=0.1803\:{K}_{n}{{V}_{w}}^{0.32}{{h}_{b}}^{0.19}\) Xu and Zhang ( 2009 ) c \(\:{B}_{ave}=0.787\:{h}_{b}{\left({h}_{d}/{h}_{r}\right)}^{0.133}{{(V}_{w}^{0.333}/{h}_{w})}^{0.652}\times\:{e}^{{B}_{3}}\) Zhong et al. ( 2020 ) \(\:{B}_{ave}={h}_{b}\:\times\:\:\:\left\{\begin{array}{c}{\left(\frac{{{V}_{w}}^{0.333}}{{h}_{w}}\right)}^{0.84}{\left(\frac{{h}_{w}}{{h}_{b}}\right)\:}^{2.30}\times\:{{h}_{d}}^{0.06}{e}^{-0.90},\:\:for\:HD\\\:{\left(\frac{{{V}_{w}}^{0.333}}{{h}_{w}}\right)}^{0.55}{\left(\frac{{h}_{w}}{{h}_{b}}\right)}^{1.97}\times\:{{h}_{d}}^{-0.07}{e}^{-0.09},\:\:for\:CD\end{array}\right.\) Von Thun and Gillette (1990) d \(\:{B}_{ave}={2.5\times\:h}_{w}+{C}_{b}\) USBR (1988) \(\:{B}_{ave}={3\times\:h}_{w}\) Azmi and Thomson ( 2024 ) e B ave = 0.92 \(\:\times\:\) Z 20 − 0.58 \(\:\times\:\) F 95 + 1.06 \(\:\times\:\) XZ 9 a for overtopping failure, K m =1.3; for piping failure, K m =1. b for overtopping failure, K n =1.4; for piping failure, K n =1. c B 3 = b 3 + b 4 +b 5 , where b 3 = -0.041, 0.026, and − 0.226 for dams with core walls, concrete-faced dams (CD), and homogeneous/zoned-fill dams (HD/ZD), respectively; b 4 = 0.149 and − 0.389 for overtopping and piping, respectively; and b 5 = 0.291, -0.14, and − 0.391 for high (HE), medium (ME), and low (LE) dam erodibility, respectively. d C b =6.096 if dam storage (S) < 1.2335×10 6 m 3 ; C b =18.288 if 1.2335×10 6 m 3 ≤ S < 6.1676 ×10 6 m 3 ; C b =42.672 if 6.1676×10 6 m 3 ≤ S < 1.2335×10 7 m 3 ; C b =54.864 if S ≥ 1.2335×10 7 m 3 . e F 95 : Froehlich ( 1995 ), XZ 9 : Xu and Zhang ( 2009 ), Z 20 : Zhong et al. ( 2020 ) Table 1 Selected empirical equations for breach peak discharge [Indicative location] Table 2 Selected empirical equations for failure time [Indicative location] Table 3 Summary of selected empirical dam breach models for final breach average width [Indicative location] Selection of Individual Empirical Equations to Update Data-Fusion Based Models One of the primary objectives of this study was to update the data-fusion-based equations originally introduced by Azmi and Thomson ( 2024 ), using the comprehensive database described earlier. Following a similar approach to that outlined in their 2024 study, the first step involved selecting the most suitable empirical equation for each breaching parameter from the pool presented in Tables 1 to 3 . It is important to note that the original equations by Azmi and Thomson ( 2024 ) were reserved solely for final comparative performance assessments and were therefore excluded from the model updating process. To identify the most appropriate empirical equations for input into the data fusion models, a three-step procedure was employed for each breaching parameter: i) Clustering Analysis: A dendrogram technique was used to group empirical equation estimates with observed values, helping to identify which equations fell within the same cluster as the observed data. ii) Factor Analysis: Based on Harman ( 1967 ), factor analysis was conducted using the Scree test (Cattell 1966 ) and the criterion of eigenvalues greater than one (Kaiser 1960 ). This step aimed to extract independent groups that sufficiently represent the variability within the dataset (Azmi et al. 2016 ; Azmi and Sarmadi 2016 ). iii) Stepwise Linear Regression: A stepwise regression approach, guided by the “probability of F-test” criterion and incorporating both forward selection and backward elimination, was applied. This analysis used observed values as the dependent variable and empirical equations as independent variables to determine the most relevant predictors. Additionally, the process included a collinearity diagnosis to detect and eliminate redundant predictors, regardless of their goodness-of-fit performance (Curtis and Ghosh 2011 ). The final set of selected empirical equations used as inputs for the data fusion models is presented in Table 4 . Table 4 Selected empirical equations for breaching parameters Breaching Parameters Selected Empirical Equation Breach peak outflow F16 (Froehlich 2016); Z20 (Zhong et al. 2020 ); XZ9 (Xu and Zhang 2009 ) Final breach average width F95 (Froehlich 1995 ); F8 (Froehlich 2008 ); XZ9 (Xu and Zhang 2009 ) Failure time F95; F8 (Froehlich 1995 , 2008 ); MCLM (MacDonald and Langridge-Monopolis 1984) Table 4 Selected empirical equations for breaching parameters [Indicative location] Multivariate Linear Regression to Update Data-Fusion Based Models When selecting a fusion operation, the accuracy of the outcomes is a primary criterion. However, in engineering practice, it is equally important to consider solutions that are simple and practical. Therefore, a trade-off between simplicity and accuracy is essential. In this study, a linear multivariate regression was adopted for breach estimations. To maximise the benefits of exploring all possible data combinations, a cross-validation process was implemented involving bootstrapping with 100,000 iterations, where the dataset was randomly split into two subsets: 80% for training and 20% for testing (Araghinejad et al. 2011 ; Sarmadi et al. 2017 ). The combined Levenberg-Marquardt algorithm (Gavin 2019 ) with the relative error threshold of 1e − 2 was employed for the optimum curve fitting (linear regression parameter extractions). Hereafter, this approach is referred to as the data fusion-based model (DFM). Four performance metrics were used to evaluate the validation stage of the DFM outputs for each iteration: Root Mean Square Error (RMSE) and Median of Percentage Error (MPE%) ( \(\:PE\%=100\times\:\frac{{X}_{m}-{X}_{O}}{{X}_{O}}\) ; where \(\:{X}_{O}\) is the observed value and \(\:{X}_{m}\) is an estimated value) assess the magnitude of residuals, while Pearson correlation, coefficient of determination (r 2 ) and Nash–Sutcliffe Efficiency (NSE) measure the strength of the relationship between observed and estimated values. RMSE and NSE are particularly sensitive to extreme (tail) values, whereas MRE% and Pearson correlation emphasise the model’s overall performance across the dataset. The NSE ranges from zero (indicating poor performance) to one (indicating optimal performance); however, it can also be negative, which signifies that the model performs worse than simply using the mean of the observed data as a predictor. The final regression coefficients were determined based on the median values across all iterations for each breaching parameter. To define a specific range based on the median of a given population, the use of standard deviation is not applicable, as it is based on the deviation from the mean. Therefore, here, the median of absolute deviation (MAD) was adopted as the appropriate statistic, in which MAD = median (|X i - X m |) where X i is the value of variable X for the sample of i, and X m is the median of entire population for variable X. The confidence interval of median of a population based on MAD will be X m ± MAD. This concept is the equivalent of a common confidence interval as µ ± σ, where µ is the population mean and σ is the standard deviation. The median-based range, defined above, provides insight into the uncertainties associated with a given parameter derived from multi-iteration simulations. Once the updated DFMs were finalised, the resulting equations were used to estimate the breaching parameters for each group. This enabled a performance comparison, using the same four previously mentioned metrics, against other selected individual empirical equations (predictors), as well as the DFM equations introduced by Azmi and Thomson ( 2024 ). To test the hypothesis that the models differ significantly from the observed statistics in terms of their means and standard deviations, both t-tests and F-tests were conducted. The Kolmogorov-Smirnov (K-S) test, Q-Q plots, log-scaled scatter plots, and boxplots (based on the ratio of estimated to observed values) were employed to evaluate and compare the distribution shapes, presence of outliers, bias, and behaviour in the high and low tails. Results Updated DFMs - Regression Coefficients & Efficiency of Testing Stage The median values and corresponding ranges of the coefficients derived from the updated DFMs are presented in Fig. 1 and Table 5 . Each histogram displays a single, distinct peak, and the narrow ranges indicate low uncertainty and high accuracy in the estimated values. Similarly, the performance metrics from the test phase, based on 100,000 cross-validation simulations, are illustrated in Fig. 2 and Table 6 . All performance criterion histograms exhibit a dominant single peak, further supporting the reliability of the model outputs. Strong correlations and high NSE values were observed for both breach peak outflow and final breach average width. These variables also showed PPE values below − 20%, suggesting a minor to moderate underestimation in the estimates. In contrast, the failure time variable, while showing an MPE of -15% (comparable to the other two breach-related variables), demonstrated only moderate correlation and a notably negative NSE. This indicates that the reliability and accuracy of failure time predictions are less robust. These concerns were also highlighted by Azmi and Thomson ( 2024 ), who attributed the poor performance in failure time estimation to the low quality of observed/measured data in the datasets. They argued that such data limitations hinder the effectiveness of data-driven models, regardless of their complexity or sophistication. Table 5 Updated DFM equations for estimating breaching parameters based on the median of 100,000 simulations Breaching Parameters Y = a×X 1 + b×X 2 + c×X 3 median median ± MAD Selected Empirical Equation Breach peak outflow a = 0.3048 b = 0.4804 c = 0.1674 (-0.22,0.82) (0.17,0.77) (-0.42,0.74) 0.3048×F16 + 0.4804×XZ9 + 0.1674×Z20 Final breach average width a= -0.8220 b = 1.0020 c = 1.1031 (-1.10,-0.54) (0.68,1.32) (0.95,1.26) -0.8220×F95 + 1.0021×F8 + 1.1031×XZ9 Failure time a= -1.0648 b = 1.5875 c = 0.6189 (-2.10,-0.03) (0.58,2.59) (0.39,0.84) -1.0648×F95 + 1.5875×F8 + 0.6189×MCLM Table 6 Performance criteria in the test stage, based on the median of 100,000 simulations. Breaching Parameters Pearson median (median ± MAD) Spearman median (median ± MAD) NSE median (median ± MAD) MPE % median (median ± MAD) Breach peak outflow 0.92 (0.87,0.97) 0.87 (0.81,0.92) 0.45 (0.18,0.81) -16.42 (-53.54, 20.70) Final breach average width 0.85 (0.81,0.87) 0.81 (0.78,0.85) 0.66 (0.60,0.71) -10.54 (-18.52,-2.56) Failure time 0.35 (0.22,0.47) 0.52 (0.42,0.61) -0.16 (-0.41,0.09) -14.96 (-34.20,4.28) Table 5 Updated DFM equations for estimating breaching parameters based on the median of 100,000 simulations [Indicative location] Table 6 Performance criteria in the test stage, based on the median of 100,000 simulations. [Indicative location] Comparative Assessments he updated DFMs were applied to estimate breaching parameters for each dataset, enabling comparative assessments across models. A Kolmogorov–Smirnov (K-S) test was conducted at a 5% significance level to compare estimated values from various models, including selected empirical models, DFM 2024 (Azmi and Thomson, 2024 ), and the updated DFM, with observed values for each breaching parameter. All models demonstrated acceptable statistical similarity to observed data (i.e., the null hypothesis was accepted), except for the DFM 2024 model in estimating failure time. In this case, the null hypothesis was rejected, with a K-S statistic of 0.246 exceeding the critical value of 0.238. To visually support these findings, Q-Q plots (Fig. 3 ) were presented with 95% confidence intervals. Generally, the more data points that fall outside the shaded area, the lower the similarity between estimated and observed values. The most noticeable deviation was observed in the DFM 2024 model for failure time, while other models showed acceptable alignment, particularly the updated DFM, which closely matched observed data for breach discharge and breach width. To further examine the distribution of medians, quartiles, and outliers between model estimates and observed data, boxplots were used (Fig. 4 ). For comparative analysis, proportion values (estimated divided by observed) were calculated. The updated DFM, along with Froehlich’s equations (1995, 2009, 2016), demonstrated competitive performance, with similar box positions, medians, and quantiles across models. To assess the pairwise relationship between estimated and observed values, scatter plots were used (Fig. 5 ), including the coefficient of determination (R²) for each subplot. Overall, models showed strong correlation (R² >0.8) for breach peak discharge, moderate correlation (R² = 0.5–0.8) for breach width, and weak correlation (R²<0.3) for failure time. Despite variations in R² values across parameters, different models showed similar spatial patterns for each breaching parameter. Table 7 presents the outcome of assessments for all three breaching parameters between selected individual empirical equations DFMs by Azmi and Thomson ( 2024 ), and updated DFMs. For each breaching parameter, initially, the two methods demonstrating the highest performance were selected. However, if these top-performing methods exhibited similar performance patterns (e.g., both excelled in NSE but performed poorly in MPE), the second method was set aside. In its place, a third method, one that showed acceptable overall performance and performs well specifically in the criteria where the top method falls short, was chosen. This strategy ensured that the final two selected methods collectively addressed the full spectrum of evaluation criteria, thereby providing the most comprehensive and reliable range for estimating each breaching parameter. Table 7 Performance criteria of selected empirical equations, DFMs by Azmi and Thomson ( 2024 ) and updated DFMs. Italic bold values are selected top two best performances. Breaching Parameters Methods R 2 NSE RMSE MPE % Median MAD Range Breach peak outflow F16 0.82 0.78 6246 -10.1 31.9 ( -42.0 , 21.8) XZ9 0.81 0.81 5844 -13.2 36.1 (-49.3, 22.9) Z20 0.85 0.85 5214 -26.2 34.0 (-60.2, 7.8 ) DFM 2024 0.83 0.79 6175 -9.8 36.1 ( -45.9 , 26.2) Updated DFM 0.88 0.88 4691 -19.9 32.2 (-52.1, 12.3 ) Final breach average width F95 0.60 0.58 36.4 -5.2 23.2 (-28.4, 17.9 ) F8 0.59 0.57 36.6 -1.8 29.8 (-31.6, 28.0) XZ9 0.71 0.67 32.3 -17.0 23.6 (-40.6, 6.6 ) DFM 2024 0.70 0.70 30.8 23.0 35.0 ( -12.0 , 58.0) Updated DFM 0.72 0.72 29.8 10.8 30.6 ( -19.7 , 41.4) Failure time F95 0.27 0.01 1.15 -11.3 36.6 (-48.0, 25.3 ) F8 0.25 0.02 1.14 -7.3 35.3 (-42.6, 28.0 ) MCLM 0.15 -0.02 1.17 3.4 59.4 (-56.0, 62.9) DFM 2024 0.29 -0.26 1.45 47.9 60.9 ( -13.0 , 99.9) Updated DFM 0.24 0.17 1.05 20.8 45.7 ( -24.8 , 66.6) Table 7 Performance criteria of selected empirical equations, DFMs by Azmi and Thomson ( 2024 ) and updated DFMs. Italic bold values are selected top two best performances. [Indicative location] For breach peak outflow, the updated DFM demonstrated superior performance across all evaluation criteria, except for the median MPE, which was − 20% compared to -9.8% for the best-performing model. The Z20 model ranked second, nonetheless, exhibited a performance pattern similar to that of the updated DFM. Consequently, DFM 2024 was identified as a complementary option. While DFM 2024 correlation and error magnitudes were close to those of the top model, DFM 2024 achieved the lowest median MPE and a narrower MPE range, making it the second preferred model after the updated DFM. In terms of final breach average width, the updated DFM again ranked first, followed by DFM 2024. Due to their similar performance patterns, the XZ9 model was identified as a suitable complement. With its lower MAD and reduced upper tail, XZ9 can effectively enhance the performance of the updated DFM when adopted in combination. For the failure time parameter, no model clearly outperformed the others; however, except for the updated DFM, with NSE of 0.17, all others produced negative NSE values, underscoring their unreliability for extreme estimates. As a result, the updated DFM was once again selected as the preferred model. The F8 model followed, with an MPE below 10%, a relatively small RMSE, and an upper range under 30%. Summary & Conclusion A data fusion methodology employing multivariate linear regression was applied using a global historical dataset (Bernard-Garcia and Mahdi, 2020 ), integrated with selected empirical equations as outlined by Azmi and Thomson ( 2024 ). The study conducted a comprehensive comparative analysis between the updated data fusion model (DFM), Azmi and Thomson’s equations, and individual empirical formulas across three key breaching parameters: peak discharge, final average breach width, and failure time. Statistical evaluations included t- and F-tests, probability distribution analyses (K-S test, Q-Q plots, boxplots, scatter plots), and performance metrics (R², NSE, RMSE, MPE). Although the updated DFM demonstrated partial superiority across most criteria, a secondary equation was selected to complement areas where the DFM was less effective. This dual-model approach ensures comprehensive coverage of assessment criteria and enhances the robustness of predictions for future dam breach scenarios. The final selected methods are: i) Breach peak outflow: DFM 2024 and updated DFM, ii) Final breach average width: XZ9 (Xu and Zhang 2009 ) and updated DFM, and iii) Failure time: F8 (Froehlich 2008 ) and updated DFM. It is important to distinguish the definition of failure time (T f ) used in different empirical equations. The equation proposed by Xu and Zhang ( 2009 ) defines T f as the entire duration from the onset of erosion to the post-erosion phase. In contrast, other widely used equations define T f more narrowly from the initiation of breach formation to its completion. To ensure consistency and comparability across models, the Xu and Zhang ( 2009 ) equation was excluded from T f assessments. Failure time is still the most challenging breaching parameter to estimate, due to high uncertainties in observed/measured datasets. Gathering more information with much more accuracy and reliability is required to help practitioners apply the outcomes in real-world projects. To reiterate from Azmi and Thomson ( 2024 ), the accuracy of dam breach modelling is primarily hindered by the quality of historical data. Enhancing international collaboration to compile validated datasets and simulate dam failures in laboratory settings can improve model reliability. For large dams, integrating detailed geotechnical data is essential. Additionally, safety features like wave walls or rock mesh can alter failure mechanisms, requiring tailored breaching equations. While historical datasets aim for diversity, practitioners must understand dam structures thoroughly and adjust models accordingly to ensure dependable and context-sensitive predictions. Declarations Conflict of interests: The author has no conflict of interest to declare. Acknowledgment We appreciate the constructive comments of reviews that led to improving the quality of this paper. References Araghinejad S, Azmi M, Kholghi M (2011) Application of artificial neural network ensembles in probabilistic hydrological forecasting. J Hydrol (Amst), 407 (1–4). https://doi.org/10.1016/j.jhydrol.2011.07.011 ASDSO (2023) Dam failures and incidents. Association of State Dam Safety Officials. Accessed March 8, 2023. https://www.damsafety.org/incidents Aureli F, Maranzoni A, Petaccia G Advances in Dam-Break Modeling for Flood Hazard Mitigation: Theory, Numerical Models, and Applications in Hydraulic Engineering. Water 2024, 16, 1093. https://doi.org/10.3390/w16081093 Azmi M, Rüdiger C, Walker JP (2016) A data fusion-based drought index. Water Resour Res 1–18. https://doi.org/10.1002/2015WR017834 Azmi M, Sarmadi F (2016) Comparative evaluations of multivariate methods in spatial clustering of precipitation using GPCC V7 gridded data set: application to the Northern Territory of Australia. Arab J Geosci 9(2). https://doi.org/10.1007/s12517-015-2269-6 Azmi M, Thomson K (2024) Dam Breach Parameters: From Data-Driven-Based Estimates to 2-Dimensional Modeling. Nat Hazards 120:4423–4461 Bernard-Garcia M, Mahdi T-F (2020) A Worldwide Historical Dam Failure's Database. Scholars Portal Dataverse V1. https://doi.org/10.5683/SP2/E7Z09B Cattell R (1966) The scree test for the number of factors. Multivar Behav Res 1:245–276 Curtis SM, Ghosh SK (2011) A Bayesian Approach to Multicollinearity and the Simultaneous Selection and Clustering of Predictors in Linear Regression. J Stat Theory Pract 5(4):715–735 DSMC (2018) Dam breach register book of the national reservoirs (In Chinese). Nanjing, China Froehlich DC (1995) Embankment dam breach parameters revisited. Water Resources Engineering Froehlich DC (2008) Embankment Dam Breach Parameters and Their Uncertainties. J Hydraul Eng 121708. https://doi.org/10.1061/ASCE0733-94292008134 Froehlich DC (2016a) Empirical model of embankment dam breaching. The International Conference On Fluvial Hydraulics - River Flow 2016, 1821–1826. Taylor & Francis Group, 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487 – 2742: CRC Press Froehlich DC (2016b) Predicting Peak Discharge from Gradually Breached Embankment Dam. J Hydrol Eng 21(11). https://doi.org/10.1061/(asce)he.1943-5584.0001424 Gavin H (2019) The Levenberg-Marquardt algorithm for nonlinear least squares curve-fitting problems. Department of Civil and Environmental Engineering, Duke University, p 19 Harman H (1967) Modern factor analysis. University of Chicago Press HEC RAS (2016) HEC-RAS River Analysis System Hydraulic Reference Manual. Hydrologic Engineering Center, p 547. February Kaiser H (1960) The application of electronic computers to factor analysis. Educ Psychol Meas 20:141–151 NWR NBS (2013) Bulletin of first national census for water. (In Chinese). China Water and Power, Beijing Sattar AMA (2014) Gene expression models for prediction of dam breach parameters. J Hydroinformatics 16(3):550–571 IWA Publishing. https://doi.org/10.2166/hydro.2013.084 Sarmadi F, Huang Y, Siems ST, Manton MJ (2017) Characteristics of Wintertime Daily Precipitation over the Australian Snowy Mountains. J Hydrometeorol 18(10):2849–2867. https://doi.org/10.1175/JHM-D-17-0072.1 USACE (2013) National inventory of Dams Trifold Brochure Wahl TL (2004) Uncertainty of Predictions of Embankment Dam Breach Parameters. J Hydraul Eng 130(5):389–397. https://doi.org/10.1061/ASCE0733-94292004130:5389 Xu Y, Zhang LM (2009) Breaching Parameters for Earth and Rockfill Dams. J Geotech GeoEnviron Eng 135(12):1957–1970. https://doi.org/10.1061/(asce)gt.1943-5606.0000162 Zech Y, Soares-Fraz.o S (2007) A Database from the European IMPACT Research. J Hydraul Res 45(Suppl S1):5–7Dam-Break Flow Experiments and Real-Case Data Zhang L, Peng M, Chang D, Xu Y (2016) Dam Failure Mechanisms and Risk Assessment. Singapore, John Wiley & Sons Zhong Q, Chen S, Fu Z, Shan Y (2020) New Empirical Model for Breaching of Earth-Rock Dams. Nat Hazards Rev 21(2). https://doi.org/10.1061/(asce)nh.1527-6996.0000374 . American Society of Civil Engineers (ASCE) Cite Share Download PDF Status: Under Review Version 1 posted Reviewers agreed at journal 19 Dec, 2025 Reviewers invited by journal 29 Sep, 2025 Editor invited by journal 29 Sep, 2025 Editor assigned by journal 07 Aug, 2025 First submitted to journal 04 Aug, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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1","display":"","copyAsset":false,"role":"figure","size":122192,"visible":true,"origin":"","legend":"\u003cp\u003eHistograms of regression’ coefficients based on 100,000 cross-validation simulations. Top row: breach peak discharge; Middle row: final breach average width; Bottom row: failure time. Solid red line shows the median of the histogram, and the red shaded area shows the range of median ± MAD\u003c/p\u003e","description":"","filename":"Figure1Regressionhistogramscoefficients1.png","url":"https://assets-eu.researchsquare.com/files/rs-7289241/v1/8dfd43540eb06924bc593914.png"},{"id":93340733,"identity":"49c23930-3119-4b30-9312-267094c5f369","added_by":"auto","created_at":"2025-10-12 14:34:47","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":160181,"visible":true,"origin":"","legend":"\u003cp\u003eHistograms of performance criteria for the test stage based on 100,000 cross-validation simulations. Top row: breach peak discharge; Middle row: final breach average width; Bottom row: failure time. Solid red line shows the median of the histogram, and the red shaded area shows the range of median ± MAD\u003c/p\u003e","description":"","filename":"Figure1Regressionhistogramscoefficients2.png","url":"https://assets-eu.researchsquare.com/files/rs-7289241/v1/c3301f705490e75ba989dc23.png"},{"id":93339530,"identity":"08a723e2-8a27-4555-a880-c9cee983bbd0","added_by":"auto","created_at":"2025-10-12 14:26:47","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":353990,"visible":true,"origin":"","legend":"\u003cp\u003eQ-Q plots between empirical equation estimates and observed data. Top row: breach peak discharge; Middle row: final breach average width; Bottom row: failure time. The shaded area shows the 95% confidence interval and the blue dashed line shows the one-to-one (X=Y) line.\u003c/p\u003e","description":"","filename":"Figure1Regressionhistogramscoefficients3.png","url":"https://assets-eu.researchsquare.com/files/rs-7289241/v1/5c2fbb3073d1ae09a4d8431a.png"},{"id":93339529,"identity":"0866c4a4-b683-4a63-b53b-39e63df0b685","added_by":"auto","created_at":"2025-10-12 14:26:47","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":43926,"visible":true,"origin":"","legend":"\u003cp\u003eBoxplots between empirical equation estimates and observed data. Top row: breach peak discharge; Middle row: final breach average width; Bottom row: failure time.\u003c/p\u003e","description":"","filename":"Figure1Regressionhistogramscoefficients4.png","url":"https://assets-eu.researchsquare.com/files/rs-7289241/v1/6b8a33f4fbdcf1272844ca5a.png"},{"id":93339534,"identity":"c3b5a172-069b-4ae5-ac23-b5293821c9d7","added_by":"auto","created_at":"2025-10-12 14:26:47","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":247336,"visible":true,"origin":"","legend":"\u003cp\u003eScatter plot between empirical equation estimates and observed data. Top row: breach peak discharge; Middle row: final breach average width; Bottom row: failure time.\u003c/p\u003e","description":"","filename":"Figure1Regressionhistogramscoefficients5.png","url":"https://assets-eu.researchsquare.com/files/rs-7289241/v1/487fbaad66be159aa1bc7c64.png"},{"id":93342410,"identity":"c4f359d9-e7e4-4a9c-bb84-9cd9a7856ccd","added_by":"auto","created_at":"2025-10-12 14:42:50","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1836803,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7289241/v1/4f073f0e-6c53-4b47-9833-a5ebdd4ce4b3.pdf"}],"financialInterests":"","formattedTitle":"An update on Data-Fusion-based Dam Breach Empirical Equations Based on a Worldwide Historical Dam Failure Database","fulltext":[{"header":"Introduction","content":"\u003cp\u003eDespite notable progress in recent decades, dambreak modeling remains a vital area of research for hydraulic engineers due to its significant implications (Aureli et al 2024). Dam failures can lead to catastrophic flooding, causing severe human and economic losses (Zech and Soares-Frazo 2007). The risk is heightened by aging infrastructure, inadequate spillway capacity, and increasing urban development in flood-prone areas (Zech and Soares-Frazo 2007; Zhang et al. \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2016\u003c/span\u003e). On a global scale, up to 2013, the USA and China led with 87,000 and 98,000 dams respectively, the majority being earth-rock structures (USACE \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; NWR and NBS 2013). While reservoirs are essential for water supply and flood control, historical failures have demonstrated their potential for widespread damage. For instance, China reported 3,541 dam break incidents (95% earth-rock dams) by 2018, and the USA documented 158 overtopping failures between 2000 and 2023 (DSMC \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; ASDSO \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). These statistics underscore the ongoing need for robust modeling and risk assessment in dam safety management.\u003c/p\u003e\u003cp\u003eEstimating key dam breach parameters (failure time, average width of breach, and breach peak discharge) remains a complex task due to limited reliable data and the intricate interplay of factors involved in breach events. This often results in high uncertainty ranging from ± 0.5 to ± 1 order of magnitude (Wahl \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2004\u003c/span\u003e). Various methods have been developed to address this, including physical and laboratory models, Computational Fluid Dynamics (CFD), empirical equations, and advanced machine learning techniques (Aureli et al. 2024). Among these, empirical equations are widely used, relying on statistical analysis of reported breach parameters such as failure time, breach width, and peak outflow (Froehlich \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e1995\u003c/span\u003e, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2008\u003c/span\u003e, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2016b\u003c/span\u003e; Xu and Zhang \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2009\u003c/span\u003e; Zhong et al. \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Azmi and Thomson \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2024\u003c/span\u003e). Existing empirical models face several limitations: they are primarily based on small dams (under 15m), often overlook failure modes, dam types, and material erodibility, and suffer from limited calibration data and high prediction errors (Sattar \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2014\u003c/span\u003e). These challenges highlight the need for improved data quality, comprehensive modeling approaches, and better integration of dam-specific characteristics to enhance the accuracy and reliability of breach parameter estimation.\u003c/p\u003e\u003cp\u003eThe most referable historical dam breach datasets were initially gathered and presented in 1995 and were gradually completed in 2016 by Froehlich (\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e1995\u003c/span\u003e, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2008\u003c/span\u003e, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2016a\u003c/span\u003e, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003eb\u003c/span\u003e); these datasets were used for statistical analysis and driving empirical equations for embankment dam breach parameters. Adopted datasets often include outliers or errors, necessitating careful data validation (Azmi and Thomson \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2024\u003c/span\u003e). In 2020, assembled from 196 references, a worldwide database (Bernard-Garcia and Mahdi \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) was compiled and documented with a total of 3,861 cases of historical dam failures around the world and represents the largest compilation of dam failures recorded to date (17-02-2020). This database includes recorded historical dam failures regardless of the type of dams (e.g. man-made dam), the type of structure (e.g. embankment dam, concrete dam), the type of failure (e.g. overtopping failure, landslide) and the properties of the dams (e.g. dam height, dam storage). Through this process, a total of 45 variables have been used to record various information about the failure (e.g. breach dimensions). It is important to note that this database also includes all databases previously presented and documented by Froehlich (\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e1995\u003c/span\u003e, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2008\u003c/span\u003e, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2016a\u003c/span\u003e, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003eb\u003c/span\u003e). In the current research, out of all recorded data, 104 dam breach cases were selected, which are related to man-made dams, with earthfill and rockfill types.\u003c/p\u003e\u003cp\u003eThe main aims of this research are i) the extraction of failure cases from the worldwide historical dam failure database, adoptable for estimating earthfill/rockfill dam breach parameters (breach peak discharge, breach formation time, and average breach width), ii) an update on data-fusion-based (DFM) dam breach equations published by Azmi and Thomson (\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2024\u003c/span\u003e), and finally iii) a comparative assessment of selected commonly used empirical equations.\u003c/p\u003e\u003cp\u003e\u003cb\u003eDatabase\u003c/b\u003e\u003c/p\u003e\u003cp\u003eThis research draws upon the comprehensive global dam breach dataset compiled and validated by Bernard-Garcia and Mahdi (\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) (Link to the database). This dataset surpasses the widely used collections in the field of dam breach analysis that have been documented over the past few decades (Froehlich \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e1995\u003c/span\u003e, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2016b\u003c/span\u003e; Xu and Zhang \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2009\u003c/span\u003e; Zhong et al. \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2020\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eFrom the full dataset of 3,861 recorded cases, 104 were selected for this study. These cases specifically involve man-made dams constructed from earthfill and rockfill materials. Although the dataset has undergone thorough peer review and procedures to address missing data, some gaps remain. To make the most of the available data, the following assumptions were applied: i) Dam Height (h\u003csub\u003ed\u003c/sub\u003e): In cases where dam height was missing, it was assumed to be equal to the breach height (h\u003csub\u003eb\u003c/sub\u003e). This is a common practice in real-world dam breach assessments to ensure that the entire reservoir volume is considered discharged, allowing for a conservative estimation of downstream impacts. ii) Dam Erodibility: Where erodibility data were unavailable, a medium erodibility classification was assumed. This approach aligns with the methodology proposed by Azmi and Thomson (\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2024\u003c/span\u003e), aiming to avoid overly conservative or optimistic assumptions, unless data from a highly similar dam with complete information were available. iii) Breaching Parameters: For parameters such as failure time, average breach width, and peak breach discharge, where only a range was provided, the midpoint of the range was used. This avoids extreme assumptions unless a closely comparable dam with complete data could inform a more accurate estimate.\u003c/p\u003e\u003cp\u003eThis dataset includes dam names, dam type (homogeneous dam (HD), core wall dam (CD), concrete-faced dam (FD), or zoned-fill dam (ZD)), failure mode (overtopping (O) or piping (P)), breach peak discharge in m\u003csup\u003e3\u003c/sup\u003e/s (Q\u003csub\u003ep\u003c/sub\u003e), final breach average width in m (B\u003csub\u003eave\u003c/sub\u003e), reservoir storage in m\u003csup\u003e3\u003c/sup\u003e (S), the volume of water above breach invert in m\u003csup\u003e3\u003c/sup\u003e (V\u003csub\u003ew\u003c/sub\u003e), dam height (h\u003csub\u003ed\u003c/sub\u003e), the height of water above breach invert in m (h\u003csub\u003ew\u003c/sub\u003e), the height of breach in m (h\u003csub\u003eb\u003c/sub\u003e). It has been assumed that the final breach top and bottom widths are measured along the dam crest. The height of the breach is the difference between the elevation of the top and bottom of the dam at the breach location, where the final breach average width is the average of the top and bottom widths. Failure time in hr (T\u003csub\u003ef\u003c/sub\u003e), also called “breach formation time”, is the time from the onset of formation to the full completion stage (Froehlich \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2008\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eBased on the available data for dam breach cases, three sub-datasets were formed (the compiled database is presented in Appendix A). Each group corresponds to a specific breach parameter (dependent variable), for which relevant empirical equations were applied to estimate values and conduct performance assessments. Group 1 comprises 130 cases used for estimating the average breach width (B\u003csub\u003eave\u003c/sub\u003e). Group 2 includes 65 cases for estimating the failure time (T\u003csub\u003ef\u003c/sub\u003e). Group 3 consists of 54 cases used to estimate the breach peak discharge (Q\u003csub\u003ep\u003c/sub\u003e), where the average embankment width of the dam (Wave) was also available. For comparison, Azmi and Thomson (\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2024\u003c/span\u003e) used 67 cases for B\u003csub\u003eave\u003c/sub\u003e, 36 for T\u003csub\u003ef\u003c/sub\u003e, and 41 for Q\u003csub\u003ep\u003c/sub\u003e, respectively.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e"},{"header":"Methodology","content":"\u003cp\u003e\u003cstrong\u003eIndividual Empirical Equations\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe methodology begins with the collection of widely used empirical equations, with particular attention to their reported accuracy, simplicity, and practicality. This includes the addition of recently introduced data-fusion-based empirical equations by Azmi and Thomason (2024), as shown in Tables \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e to \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e. To ensure the reliability of each selected equation, only those published in fully peer-reviewed sources such as ISI-indexed journals and technical reports/user manuals from globally recognised hydraulic software like HEC-RAS (2016) were considered.\u003c/p\u003e\n\u003cp\u003eIt is important to note that the equation by Xu and Zhang (2009) for predicting T\u003csub\u003ef\u003c/sub\u003e encompasses the entire timeframe from initial erosion to the post-erosion phase. In contrast, other equations (e.g., Froehlich 1995, 2008) define T\u003csub\u003ef\u003c/sub\u003e as the period from the onset of formation to the point of full completion (Froehlich 2008). To maintain consistency and comparability among the individual equations, the Xu and Zhang (2009) equation was excluded from the pool of individual empirical equations for this breaching parameter.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n \u003ctable id=\"Tab1\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eSelected empirical equations for breach peak discharge\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eReference\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eEquation\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eFroehlich (\u003cspan class=\"CitationRef\"\u003e2016b\u003c/span\u003e) \u003csup\u003ea\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Q}_{P}=0.0175{k}_{m}{k}_{h}(g{{h}_{w}{V}_{w}{h}_{b}}^{2}/{{W}_{ave})}^{0.5}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\u003cp\u003eFroehlich (\u003cspan class=\"CitationRef\"\u003e1995\u003c/span\u003e)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Q}_{P}=0.607{{V}_{w}}^{0.295}{{h}_{w}}^{1.24}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eWebby (1996)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Q}_{P}=0.0443\\sqrt{g}{{V}_{w}}^{0.365}{{h}_{w}}^{1.405}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eHooshyaripor et al. (2014)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Q}_{P}=0.0212{{V}_{w}}^{0.5429}{{h}_{w}}^{0.8713}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eAzimi et al. (2015)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Q}_{P}=0.0166{{(gV}_{w})}^{0.5}{h}_{w}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\u003cp\u003eXu and Zhang (\u003cspan class=\"CitationRef\"\u003e2009\u003c/span\u003e) \u003csup\u003eb\u003c/sup\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Q}_{P}=0.175\\sqrt{g}{{V}_{w}}^{5/6}{{(h}_{d}/{h}_{r})}^{0.199}{({{V}_{w}}^{1/3}/{h}_{w})}^{-1.274}{e}^{{B}_{4}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eZhong et al. (\u003cspan class=\"CitationRef\"\u003e2020\u003c/span\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Q}_{P}=\\sqrt{g}{{h}_{w}}^{-0.5}{V}_{w}\\:\\times\\:\\:\\:\\left\\{\\begin{array}{c}{\\left(\\frac{{{V}_{w}}^{0.333}}{{h}_{w}}\\right)}^{-1.58}{\\left(\\frac{{h}_{w}}{{h}_{b}}\\right)\\:}^{-0.76}\\times\\:{{h}_{d}}^{0.1}{e}^{-4.55},\\:\\:for\\:HD\\\\\\:{\\left(\\frac{{{V}_{w}}^{0.333}}{{h}_{w}}\\right)}^{-1.51}{\\left(\\frac{{h}_{w}}{{h}_{b}}\\right)}^{-1.09}\\times\\:{{h}_{d}}^{-0.12}{e}^{-3.61},\\:for\\:CD\\end{array}\\right.\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ePierce et al. (2010)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Q}_{P}=0.038{{V}_{w}}^{0.475}{{h}_{w}}^{1.09}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eAbdulrahman (2022) \u003csup\u003ec\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Q}_{P}=55.36\\times\\:MF\\times\\:{{V}_{w}}^{1.081}{{h}_{w}}^{0.413}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eUSBR (1988)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Q}_{P}=19.1{{h}_{w}}^{1.85}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eAzmi and Thomson (\u003cspan class=\"CitationRef\"\u003e2024\u003c/span\u003e) \u003csup\u003ed\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cem\u003eQ\u003c/em\u003e\u003csub\u003e\u003cem\u003ep\u003c/em\u003e\u003c/sub\u003e = 1.23\u0026times;F\u003csub\u003e16\u003c/sub\u003e\u0026thinsp;\u0026minus;\u0026thinsp;0.84\u0026times;H\u003csub\u003e14\u003c/sub\u003e\u0026thinsp;+\u0026thinsp;0.26\u0026times;XZ\u003csub\u003e9\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003ctfoot\u003e\n \u003ctr\u003e\n \u003ctd colspan=\"2\"\u003e\u003csup\u003ea\u003c/sup\u003e for overtopping failure mode, K\u003csub\u003em\u003c/sub\u003e = 1.85; for piping failure, K\u003csub\u003em\u003c/sub\u003e = 1. When h\u003csub\u003eb\u003c/sub\u003e \u0026le;6.1m, K\u003csub\u003eh\u003c/sub\u003e = 1; when h\u003csub\u003eb\u003c/sub\u003e \u0026gt;6.1m, K\u003csub\u003eh\u003c/sub\u003e =(h\u003csub\u003eb\u003c/sub\u003e/6.1)\u003csup\u003e1/8\u003c/sup\u003e. \u003csup\u003eb\u003c/sup\u003e B\u003csub\u003e4\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;b\u003csub\u003e3\u003c/sub\u003e\u0026thinsp;+\u0026thinsp;b\u003csub\u003e4\u003c/sub\u003e + b\u003csub\u003e5\u003c/sub\u003e, where b\u003csub\u003e3\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;\u0026minus;\u0026thinsp;0.503, -0.591, and \u0026minus;\u0026thinsp;0.649 for dams with core walls (CD), concrete-faced dams (FD), and homogeneous/zoned-fill dams (HD/ZD), respectively; b\u003csub\u003e4\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;\u0026minus;\u0026thinsp;0.705 and \u0026minus;\u0026thinsp;1.039 for overtopping and piping, respectively; b\u003csub\u003e5\u003c/sub\u003e = -0.007, -0.375, and \u0026minus;\u0026thinsp;1.362 for high, medium, and low dam erodibility, respectively. \u003csup\u003ec\u003c/sup\u003e MF is the model of failure, it is equal to 1 for non-overtopping mode and 1.414 for overtopping mode. \u003csup\u003ed\u003c/sup\u003e F\u003csub\u003e16\u003c/sub\u003e: Froehlich (\u003cspan class=\"CitationRef\"\u003e2016b\u003c/span\u003e), H\u003csub\u003e14\u003c/sub\u003e: Hooshyaripor et al. (2014), XZ\u003csub\u003e9\u003c/sub\u003e: Xu and Zhang (\u003cspan class=\"CitationRef\"\u003e2009\u003c/span\u003e)\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tfoot\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"gridtable\"\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003e\u003cbr\u003e\u003c/div\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003e\u003cbr\u003e\u003c/div\u003e\n \u003ctable id=\"Tab2\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eSelected empirical equations for failure time\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eReference\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eEquation\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eFroehlich (\u003cspan class=\"CitationRef\"\u003e2008\u003c/span\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{T}_{f}=63.2({V}_{w}/g{{h}_{b}^{2})}^{0.5}\\)\u003c/span\u003e\u003c/span\u003e / 3600\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\u003cp\u003eFroehlich (\u003cspan class=\"CitationRef\"\u003e1995\u003c/span\u003e)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{T}_{f}=0.00254{\\:h}_{b}^{-0.9}{\\:V}_{w}^{0.53}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\u003cp\u003eZhong et al. (\u003cspan class=\"CitationRef\"\u003e2020\u003c/span\u003e)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{T}_{f}=\\left\\{\\begin{array}{c}{\\left(\\frac{{{V}_{w}}^{0.333}}{{h}_{w}}\\right)}^{0.56}{\\left(\\frac{{h}_{w}}{{h}_{b}}\\right)\\:}^{-0.85}\\times\\:{{h}_{d}}^{-0.32}{e}^{-0.20}\\:\\:,\\:for\\:HD\\\\\\:{\\left(\\frac{{{V}_{w}}^{0.333}}{{h}_{w}}\\right)}^{1.52}{\\left(\\frac{{h}_{w}}{{h}_{b}}\\right)}^{-11.36}\\times\\:{{h}_{d}}^{-0.43}{e}^{-1.57},\\:for\\:CD\\end{array}\\right.\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eUSBR (1988) \u003csup\u003eb\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{T}_{f}=0.011\\:{W}_{ave}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eMacDonald and Langridge-Monopolis (1984)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{T}_{f}=0.0179{\\left(0.0261\\right({V}_{w}{h}_{w}{)}^{0.769})}^{0.364}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eAzmi and Thomson (\u003cspan class=\"CitationRef\"\u003e2024\u003c/span\u003e) \u003csup\u003eb\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003ef\u003c/em\u003e\u003c/sub\u003e = 3.27\u0026times;F\u003csub\u003e16\u003c/sub\u003e\u0026thinsp;\u0026minus;\u0026thinsp;2.05\u0026times;F\u003csub\u003e95\u003c/sub\u003e\u0026thinsp;+\u0026thinsp;0.25\u0026times;XZ\u003csub\u003e9\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003ctfoot\u003e\n \u003ctr\u003e\n \u003ctd colspan=\"2\"\u003e\u003csup\u003ea\u003c/sup\u003e This equation was only applied for cases in which both had observed/measured T\u003csub\u003ef\u003c/sub\u003e and observed/measured W\u003csub\u003eave\u003c/sub\u003e. \u003csup\u003eb\u003c/sup\u003e F\u003csub\u003e16\u003c/sub\u003e: Froehlich (\u003cspan class=\"CitationRef\"\u003e2016a\u003c/span\u003e), F\u003csub\u003e95\u003c/sub\u003e: Froehlich (\u003cspan class=\"CitationRef\"\u003e1995\u003c/span\u003e), XZ\u003csub\u003e9\u003c/sub\u003e: Xu and Zhang (\u003cspan class=\"CitationRef\"\u003e2009\u003c/span\u003e)\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tfoot\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"gridtable\"\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003e\u003cbr\u003e\u003c/div\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003e\u003cbr\u003e\u003c/div\u003e\n \u003ctable id=\"Tab3\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eSummary of selected empirical dam breach models for final breach average width\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eReference\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eEquation\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eFroehlich (\u003cspan class=\"CitationRef\"\u003e2008\u003c/span\u003e) \u003csup\u003ea\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{B}_{ave}=0.27\\:{K}_{O}{{V}_{w}}^{0.32}{{h}_{b}}^{0.04}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eFroehlich (\u003cspan class=\"CitationRef\"\u003e1995\u003c/span\u003e) \u003csup\u003eb\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{B}_{ave}=0.1803\\:{K}_{n}{{V}_{w}}^{0.32}{{h}_{b}}^{0.19}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eXu and Zhang (\u003cspan class=\"CitationRef\"\u003e2009\u003c/span\u003e) \u003csup\u003ec\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{B}_{ave}=0.787\\:{h}_{b}{\\left({h}_{d}/{h}_{r}\\right)}^{0.133}{{(V}_{w}^{0.333}/{h}_{w})}^{0.652}\\times\\:{e}^{{B}_{3}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eZhong et al. (\u003cspan class=\"CitationRef\"\u003e2020\u003c/span\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{B}_{ave}={h}_{b}\\:\\times\\:\\:\\:\\left\\{\\begin{array}{c}{\\left(\\frac{{{V}_{w}}^{0.333}}{{h}_{w}}\\right)}^{0.84}{\\left(\\frac{{h}_{w}}{{h}_{b}}\\right)\\:}^{2.30}\\times\\:{{h}_{d}}^{0.06}{e}^{-0.90},\\:\\:for\\:HD\\\\\\:{\\left(\\frac{{{V}_{w}}^{0.333}}{{h}_{w}}\\right)}^{0.55}{\\left(\\frac{{h}_{w}}{{h}_{b}}\\right)}^{1.97}\\times\\:{{h}_{d}}^{-0.07}{e}^{-0.09},\\:\\:for\\:CD\\end{array}\\right.\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eVon Thun and Gillette (1990) \u003csup\u003ed\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{B}_{ave}={2.5\\times\\:h}_{w}+{C}_{b}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eUSBR (1988)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{B}_{ave}={3\\times\\:h}_{w}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eAzmi and Thomson (\u003cspan class=\"CitationRef\"\u003e2024\u003c/span\u003e) \u003csup\u003ee\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cem\u003eB\u003c/em\u003e\u003csub\u003e\u003cem\u003eave\u003c/em\u003e\u003c/sub\u003e= 0.92\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\times\\:\\)\u003c/span\u003e\u003c/span\u003eZ\u003csub\u003e20\u003c/sub\u003e \u0026minus; 0.58\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\times\\:\\)\u003c/span\u003e\u003c/span\u003eF\u003csub\u003e95\u003c/sub\u003e + 1.06\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\times\\:\\)\u003c/span\u003e\u003c/span\u003eXZ\u003csub\u003e9\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003ctfoot\u003e\n \u003ctr\u003e\n \u003ctd colspan=\"2\"\u003e\u003csup\u003ea\u003c/sup\u003e for overtopping failure, K\u003csub\u003em\u003c/sub\u003e=1.3; for piping failure, K\u003csub\u003em\u003c/sub\u003e=1. \u003csup\u003eb\u003c/sup\u003e for overtopping failure, K\u003csub\u003en\u003c/sub\u003e=1.4; for piping failure, K\u003csub\u003en\u003c/sub\u003e=1. \u003csup\u003ec\u003c/sup\u003e B\u003csub\u003e3\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;b\u003csub\u003e3\u003c/sub\u003e\u0026thinsp;+\u0026thinsp;b\u003csub\u003e4\u003c/sub\u003e+b\u003csub\u003e5\u003c/sub\u003e, where b\u003csub\u003e3\u003c/sub\u003e= -0.041, 0.026, and \u0026minus;\u0026thinsp;0.226 for dams with core walls, concrete-faced dams (CD), and homogeneous/zoned-fill dams (HD/ZD), respectively; b\u003csub\u003e4\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;0.149 and \u0026minus;\u0026thinsp;0.389 for overtopping and piping, respectively; and b\u003csub\u003e5\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;0.291, -0.14, and \u0026minus;\u0026thinsp;0.391 for high (HE), medium (ME), and low (LE) dam erodibility, respectively. \u003csup\u003ed\u003c/sup\u003e C\u003csub\u003eb\u003c/sub\u003e=6.096 if dam storage (S)\u0026thinsp;\u0026lt;\u0026thinsp;1.2335\u0026times;10\u003csup\u003e6\u003c/sup\u003e m\u003csup\u003e3\u003c/sup\u003e; C\u003csub\u003eb\u003c/sub\u003e=18.288 if 1.2335\u0026times;10\u003csup\u003e6\u003c/sup\u003e m\u003csup\u003e3\u003c/sup\u003e\u0026thinsp;\u0026le;\u0026thinsp;S\u0026thinsp;\u0026lt;\u0026thinsp;6.1676 \u0026times;10\u003csup\u003e6\u003c/sup\u003e m\u003csup\u003e3\u003c/sup\u003e; C\u003csub\u003eb\u003c/sub\u003e=42.672 if 6.1676\u0026times;10\u003csup\u003e6\u003c/sup\u003e m\u003csup\u003e3\u003c/sup\u003e\u0026thinsp;\u0026le;\u0026thinsp;S\u0026thinsp;\u0026lt;\u0026thinsp;1.2335\u0026times;10\u003csup\u003e7\u003c/sup\u003e m\u003csup\u003e3\u003c/sup\u003e; C\u003csub\u003eb\u003c/sub\u003e=54.864 if S\u0026thinsp;\u0026ge;\u0026thinsp;1.2335\u0026times;10\u003csup\u003e7\u003c/sup\u003e m\u003csup\u003e3\u003c/sup\u003e. \u003csup\u003ee\u003c/sup\u003e F\u003csub\u003e95\u003c/sub\u003e: Froehlich (\u003cspan class=\"CitationRef\"\u003e1995\u003c/span\u003e), XZ\u003csub\u003e9\u003c/sub\u003e: Xu and Zhang (\u003cspan class=\"CitationRef\"\u003e2009\u003c/span\u003e), Z\u003csub\u003e20\u003c/sub\u003e: Zhong et al. (\u003cspan class=\"CitationRef\"\u003e2020\u003c/span\u003e)\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tfoot\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eTable\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e Selected empirical equations for breach peak discharge\u003c/p\u003e\n\u003cp\u003e[Indicative location]\u003c/p\u003e\n\u003cp\u003eTable\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e Selected empirical equations for failure time\u003c/p\u003e\n\u003cp\u003e[Indicative location]\u003c/p\u003e\n\u003cp\u003eTable\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e Summary of selected empirical dam breach models for final breach average width\u003c/p\u003e\n\u003cp\u003e[Indicative location]\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eSelection of Individual Empirical Equations to Update Data-Fusion Based Models\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eOne of the primary objectives of this study was to update the data-fusion-based equations originally introduced by Azmi and Thomson (\u003cspan class=\"CitationRef\"\u003e2024\u003c/span\u003e), using the comprehensive database described earlier. Following a similar approach to that outlined in their 2024 study, the first step involved selecting the most suitable empirical equation for each breaching parameter from the pool presented in Tables\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e to \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e. It is important to note that the original equations by Azmi and Thomson (\u003cspan class=\"CitationRef\"\u003e2024\u003c/span\u003e) were reserved solely for final comparative performance assessments and were therefore excluded from the model updating process.\u003c/p\u003e\n\u003cp\u003eTo identify the most appropriate empirical equations for input into the data fusion models, a three-step procedure was employed for each breaching parameter: i) Clustering Analysis: A dendrogram technique was used to group empirical equation estimates with observed values, helping to identify which equations fell within the same cluster as the observed data. ii) Factor Analysis: Based on Harman (\u003cspan class=\"CitationRef\"\u003e1967\u003c/span\u003e), factor analysis was conducted using the Scree test (Cattell \u003cspan class=\"CitationRef\"\u003e1966\u003c/span\u003e) and the criterion of eigenvalues greater than one (Kaiser \u003cspan class=\"CitationRef\"\u003e1960\u003c/span\u003e). This step aimed to extract independent groups that sufficiently represent the variability within the dataset (Azmi et al. \u003cspan class=\"CitationRef\"\u003e2016\u003c/span\u003e; Azmi and Sarmadi \u003cspan class=\"CitationRef\"\u003e2016\u003c/span\u003e). iii) Stepwise Linear Regression: A stepwise regression approach, guided by the \u0026ldquo;probability of F-test\u0026rdquo; criterion and incorporating both forward selection and backward elimination, was applied. This analysis used observed values as the dependent variable and empirical equations as independent variables to determine the most relevant predictors. Additionally, the process included a collinearity diagnosis to detect and eliminate redundant predictors, regardless of their goodness-of-fit performance (Curtis and Ghosh \u003cspan class=\"CitationRef\"\u003e2011\u003c/span\u003e). The final set of selected empirical equations used as inputs for the data fusion models is presented in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003e\u003cbr\u003e\u003c/div\u003e\n \u003ctable id=\"Tab4\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eSelected empirical equations for breaching parameters\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eBreaching Parameters\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eSelected Empirical Equation\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eBreach peak outflow\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eF16 (Froehlich 2016); Z20 (Zhong et al. \u003cspan class=\"CitationRef\"\u003e2020\u003c/span\u003e); XZ9 (Xu and Zhang \u003cspan class=\"CitationRef\"\u003e2009\u003c/span\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eFinal breach average width\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eF95 (Froehlich \u003cspan class=\"CitationRef\"\u003e1995\u003c/span\u003e); F8 (Froehlich \u003cspan class=\"CitationRef\"\u003e2008\u003c/span\u003e); XZ9 (Xu and Zhang \u003cspan class=\"CitationRef\"\u003e2009\u003c/span\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eFailure time\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eF95; F8 (Froehlich \u003cspan class=\"CitationRef\"\u003e1995\u003c/span\u003e, \u003cspan class=\"CitationRef\"\u003e2008\u003c/span\u003e); MCLM (MacDonald and Langridge-Monopolis 1984)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eTable\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e Selected empirical equations for breaching parameters\u003c/p\u003e\n\u003cp\u003e[Indicative location]\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMultivariate Linear Regression to Update Data-Fusion Based Models\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWhen selecting a fusion operation, the accuracy of the outcomes is a primary criterion. However, in engineering practice, it is equally important to consider solutions that are simple and practical. Therefore, a trade-off between simplicity and accuracy is essential. In this study, a linear multivariate regression was adopted for breach estimations. To maximise the benefits of exploring all possible data combinations, a cross-validation process was implemented involving bootstrapping with 100,000 iterations, where the dataset was randomly split into two subsets: 80% for training and 20% for testing (Araghinejad et al. \u003cspan class=\"CitationRef\"\u003e2011\u003c/span\u003e; Sarmadi et al. \u003cspan class=\"CitationRef\"\u003e2017\u003c/span\u003e). The combined Levenberg-Marquardt algorithm (Gavin \u003cspan class=\"CitationRef\"\u003e2019\u003c/span\u003e) with the relative error threshold of 1e\u003csup\u003e\u0026minus;\u0026thinsp;2\u003c/sup\u003e was employed for the optimum curve fitting (linear regression parameter extractions). Hereafter, this approach is referred to as the data fusion-based model (DFM).\u003c/p\u003e\n\u003cp\u003eFour performance metrics were used to evaluate the validation stage of the DFM outputs for each iteration: Root Mean Square Error (RMSE) and Median of Percentage Error (MPE%) (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:PE\\%=100\\times\\:\\frac{{X}_{m}-{X}_{O}}{{X}_{O}}\\)\u003c/span\u003e\u003c/span\u003e; where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{X}_{O}\\)\u003c/span\u003e\u003c/span\u003e is the observed value and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{X}_{m}\\)\u003c/span\u003e\u003c/span\u003e is an estimated value) assess the magnitude of residuals, while Pearson correlation, coefficient of determination (r\u003csup\u003e2\u003c/sup\u003e) and Nash\u0026ndash;Sutcliffe Efficiency (NSE) measure the strength of the relationship between observed and estimated values. RMSE and NSE are particularly sensitive to extreme (tail) values, whereas MRE% and Pearson correlation emphasise the model\u0026rsquo;s overall performance across the dataset. The NSE ranges from zero (indicating poor performance) to one (indicating optimal performance); however, it can also be negative, which signifies that the model performs worse than simply using the mean of the observed data as a predictor. The final regression coefficients were determined based on the median values across all iterations for each breaching parameter.\u003c/p\u003e\n\u003cp\u003eTo define a specific range based on the median of a given population, the use of standard deviation is not applicable, as it is based on the deviation from the mean. Therefore, here, the median of absolute deviation (MAD) was adopted as the appropriate statistic, in which MAD\u0026thinsp;=\u0026thinsp;median (|X\u003csub\u003ei\u003c/sub\u003e - X\u003csub\u003em\u003c/sub\u003e|) where X\u003csub\u003ei\u003c/sub\u003e is the value of variable X for the sample of i, and X\u003csub\u003em\u003c/sub\u003e is the median of entire population for variable X. The confidence interval of median of a population based on MAD will be X\u003csub\u003em\u003c/sub\u003e \u0026plusmn; MAD. This concept is the equivalent of a common confidence interval as \u0026micro;\u0026thinsp;\u0026plusmn;\u0026thinsp;\u0026sigma;, where \u0026micro; is the population mean and \u0026sigma; is the standard deviation. The median-based range, defined above, provides insight into the uncertainties associated with a given parameter derived from multi-iteration simulations.\u003c/p\u003e\n\u003cp\u003eOnce the updated DFMs were finalised, the resulting equations were used to estimate the breaching parameters for each group. This enabled a performance comparison, using the same four previously mentioned metrics, against other selected individual empirical equations (predictors), as well as the DFM equations introduced by Azmi and Thomson (\u003cspan class=\"CitationRef\"\u003e2024\u003c/span\u003e). To test the hypothesis that the models differ significantly from the observed statistics in terms of their means and standard deviations, both t-tests and F-tests were conducted. The Kolmogorov-Smirnov (K-S) test, Q-Q plots, log-scaled scatter plots, and boxplots (based on the ratio of estimated to observed values) were employed to evaluate and compare the distribution shapes, presence of outliers, bias, and behaviour in the high and low tails.\u003c/p\u003e"},{"header":"Results","content":"\u003cp\u003e\u003cstrong\u003eUpdated DFMs - Regression Coefficients \u0026amp; Efficiency of Testing Stage\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe median values and corresponding ranges of the coefficients derived from the updated DFMs are presented in Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e and Table \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e. Each histogram displays a single, distinct peak, and the narrow ranges indicate low uncertainty and high accuracy in the estimated values. Similarly, the performance metrics from the test phase, based on 100,000 cross-validation simulations, are illustrated in Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e and Table \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e. All performance criterion histograms exhibit a dominant single peak, further supporting the reliability of the model outputs. Strong correlations and high NSE values were observed for both breach peak outflow and final breach average width. These variables also showed PPE values below − 20%, suggesting a minor to moderate underestimation in the estimates. In contrast, the failure time variable, while showing an MPE of -15% (comparable to the other two breach-related variables), demonstrated only moderate correlation and a notably negative NSE. This indicates that the reliability and accuracy of failure time predictions are less robust. These concerns were also highlighted by Azmi and Thomson (\u003cspan class=\"CitationRef\"\u003e2024\u003c/span\u003e), who attributed the poor performance in failure time estimation to the low quality of observed/measured data in the datasets. They argued that such data limitations hinder the effectiveness of data-driven models, regardless of their complexity or sophistication.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n \u003ctable id=\"Tab5\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eUpdated DFM equations for estimating breaching parameters based on the median of 100,000 simulations\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eBreaching Parameters\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eY = a×X\u003csub\u003e1\u003c/sub\u003e + b×X\u003csub\u003e2\u003c/sub\u003e + c×X\u003csub\u003e3\u003c/sub\u003e\u003c/p\u003e\n \u003cp\u003emedian\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003emedian ± MAD\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eSelected Empirical Equation\u003c/p\u003e\n \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eBreach peak outflow\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003ea = 0.3048\u003c/p\u003e\n \u003cp\u003eb = 0.4804\u003c/p\u003e\n \u003cp\u003ec = 0.1674\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e(-0.22,0.82)\u003c/p\u003e\n \u003cp\u003e(0.17,0.77)\u003c/p\u003e\n \u003cp\u003e(-0.42,0.74)\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.3048×F16 + 0.4804×XZ9 + 0.1674×Z20\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eFinal breach average width\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003ea= -0.8220\u003c/p\u003e\n \u003cp\u003eb = 1.0020\u003c/p\u003e\n \u003cp\u003ec = 1.1031\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e(-1.10,-0.54)\u003c/p\u003e\n \u003cp\u003e(0.68,1.32)\u003c/p\u003e\n \u003cp\u003e(0.95,1.26)\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.8220×F95 + 1.0021×F8 + 1.1031×XZ9\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eFailure time\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003ea= -1.0648\u003c/p\u003e\n \u003cp\u003eb = 1.5875\u003c/p\u003e\n \u003cp\u003ec = 0.6189\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e(-2.10,-0.03)\u003c/p\u003e\n \u003cp\u003e(0.58,2.59)\u003c/p\u003e\n \u003cp\u003e(0.39,0.84)\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e-1.0648×F95 + 1.5875×F8 + 0.6189×MCLM\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"gridtable\"\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003e\u003cbr\u003e\u003c/div\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003e\u003cbr\u003e\u003c/div\u003e\n \u003ctable id=\"Tab6\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 6\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003ePerformance criteria in the test stage, based on the median of 100,000 simulations.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eBreaching Parameters\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003ePearson\u003c/p\u003e\n \u003cp\u003emedian\u003c/p\u003e\n \u003cp\u003e(median ± MAD)\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eSpearman\u003c/p\u003e\n \u003cp\u003emedian\u003c/p\u003e\n \u003cp\u003e(median ± MAD)\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eNSE\u003c/p\u003e\n \u003cp\u003emedian\u003c/p\u003e\n \u003cp\u003e(median ± MAD)\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eMPE %\u003c/p\u003e\n \u003cp\u003emedian\u003c/p\u003e\n \u003cp\u003e(median ± MAD)\u003c/p\u003e\n \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eBreach peak outflow\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.92\u003c/p\u003e\n \u003cp\u003e(0.87,0.97)\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.87\u003c/p\u003e\n \u003cp\u003e(0.81,0.92)\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.45\u003c/p\u003e\n \u003cp\u003e(0.18,0.81)\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e-16.42\u003c/p\u003e\n \u003cp\u003e(-53.54, 20.70)\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eFinal breach average width\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.85\u003c/p\u003e\n \u003cp\u003e(0.81,0.87)\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.81\u003c/p\u003e\n \u003cp\u003e(0.78,0.85)\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.66\u003c/p\u003e\n \u003cp\u003e(0.60,0.71)\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e-10.54\u003c/p\u003e\n \u003cp\u003e(-18.52,-2.56)\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eFailure time\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.35\u003c/p\u003e\n \u003cp\u003e(0.22,0.47)\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.52\u003c/p\u003e\n \u003cp\u003e(0.42,0.61)\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.16\u003c/p\u003e\n \u003cp\u003e(-0.41,0.09)\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e-14.96\u003c/p\u003e\n \u003cp\u003e(-34.20,4.28)\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eTable \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e Updated DFM equations for estimating breaching parameters based on the median of 100,000 simulations\u003c/p\u003e\n\u003cp\u003e[Indicative location]\u003c/p\u003e\n\u003cp\u003eTable \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e Performance criteria in the test stage, based on the median of 100,000 simulations.\u003c/p\u003e\n\u003cp\u003e[Indicative location]\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eComparative Assessments\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003ehe updated DFMs were applied to estimate breaching parameters for each dataset, enabling comparative assessments across models. A Kolmogorov–Smirnov (K-S) test was conducted at a 5% significance level to compare estimated values from various models, including selected empirical models, DFM 2024 (Azmi and Thomson, \u003cspan class=\"CitationRef\"\u003e2024\u003c/span\u003e), and the updated DFM, with observed values for each breaching parameter. All models demonstrated acceptable statistical similarity to observed data (i.e., the null hypothesis was accepted), except for the DFM 2024 model in estimating failure time. In this case, the null hypothesis was rejected, with a K-S statistic of 0.246 exceeding the critical value of 0.238.\u003c/p\u003e\n\u003cp\u003eTo visually support these findings, Q-Q plots (Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e) were presented with 95% confidence intervals. Generally, the more data points that fall outside the shaded area, the lower the similarity between estimated and observed values. The most noticeable deviation was observed in the DFM 2024 model for failure time, while other models showed acceptable alignment, particularly the updated DFM, which closely matched observed data for breach discharge and breach width.\u003c/p\u003e\n\u003cp\u003eTo further examine the distribution of medians, quartiles, and outliers between model estimates and observed data, boxplots were used (Fig. \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e). For comparative analysis, proportion values (estimated divided by observed) were calculated. The updated DFM, along with Froehlich’s equations (1995, 2009, 2016), demonstrated competitive performance, with similar box positions, medians, and quantiles across models.\u003c/p\u003e\n\u003cp\u003eTo assess the pairwise relationship between estimated and observed values, scatter plots were used (Fig. \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e), including the coefficient of determination (R²) for each subplot. Overall, models showed strong correlation (R² \u0026gt;0.8) for breach peak discharge, moderate correlation (R² = 0.5–0.8) for breach width, and weak correlation (R²\u0026lt;0.3) for failure time. Despite variations in R² values across parameters, different models showed similar spatial patterns for each breaching parameter.\u003c/p\u003e\n\u003cp\u003eTable \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e presents the outcome of assessments for all three breaching parameters between selected individual empirical equations DFMs by Azmi and Thomson (\u003cspan class=\"CitationRef\"\u003e2024\u003c/span\u003e), and updated DFMs. For each breaching parameter, initially, the two methods demonstrating the highest performance were selected. However, if these top-performing methods exhibited similar performance patterns (e.g., both excelled in NSE but performed poorly in MPE), the second method was set aside. In its place, a third method, one that showed acceptable overall performance and performs well specifically in the criteria where the top method falls short, was chosen. This strategy ensured that the final two selected methods collectively addressed the full spectrum of evaluation criteria, thereby providing the most comprehensive and reliable range for estimating each breaching parameter.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n \u003ctable id=\"Tab7\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 7\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003ePerformance criteria of selected empirical equations, DFMs by Azmi and Thomson (\u003cspan class=\"CitationRef\"\u003e2024\u003c/span\u003e) and updated DFMs. Italic bold values are selected top two best performances.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003eBreaching Parameters\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003eMethods\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003eR\u003csup\u003e2\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003eNSE\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003eRMSE\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\" colspan=\"3\"\u003e\n \u003cp\u003eMPE %\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eMedian\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eMAD\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eRange\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"5\"\u003e\n \u003cp\u003eBreach peak outflow\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eF16\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.82\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.78\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e6246\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e-10.1\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e31.9\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e(\u003cstrong\u003e-42.0\u003c/strong\u003e, 21.8)\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eXZ9\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.81\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.81\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e5844\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e-13.2\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e36.1\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e(-49.3, 22.9)\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eZ20\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.85\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.85\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e5214\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e-26.2\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e34.0\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e(-60.2, \u003cstrong\u003e7.8\u003c/strong\u003e)\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan type=\"BoldItalicUnderline\" class=\"BoldItalicUnderline\" name=\"Emphasis\"\u003eDFM 2024\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.83\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.79\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e6175\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e-9.8\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e36.1\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e(\u003cstrong\u003e-45.9\u003c/strong\u003e, 26.2)\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan type=\"BoldItalicUnderline\" class=\"BoldItalicUnderline\" name=\"Emphasis\"\u003eUpdated DFM\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.88\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.88\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e4691\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e-19.9\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e32.2\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e(-52.1, \u003cstrong\u003e12.3\u003c/strong\u003e)\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"5\"\u003e\n \u003cp\u003eFinal breach average width\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eF95\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.60\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.58\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e36.4\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e-5.2\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e23.2\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e(-28.4, \u003cstrong\u003e17.9\u003c/strong\u003e)\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eF8\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.59\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.57\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e36.6\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e-1.8\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e29.8\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e(-31.6, 28.0)\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan type=\"BoldItalicUnderline\" class=\"BoldItalicUnderline\" name=\"Emphasis\"\u003eXZ9\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.71\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.67\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e32.3\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e-17.0\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e23.6\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e(-40.6, \u003cstrong\u003e6.6\u003c/strong\u003e)\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eDFM 2024\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.70\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.70\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e30.8\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e23.0\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e35.0\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e(\u003cstrong\u003e-12.0\u003c/strong\u003e, 58.0)\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan type=\"BoldItalicUnderline\" class=\"BoldItalicUnderline\" name=\"Emphasis\"\u003eUpdated DFM\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.72\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.72\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e29.8\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e10.8\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e30.6\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e(\u003cstrong\u003e-19.7\u003c/strong\u003e, 41.4)\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"5\"\u003e\n \u003cp\u003eFailure time\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eF95\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.27\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.01\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e1.15\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e-11.3\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e36.6\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e(-48.0, \u003cstrong\u003e25.3\u003c/strong\u003e)\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan type=\"BoldItalicUnderline\" class=\"BoldItalicUnderline\" name=\"Emphasis\"\u003eF8\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.25\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.02\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e1.14\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e-7.3\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e35.3\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e(-42.6, \u003cstrong\u003e28.0\u003c/strong\u003e)\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eMCLM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.15\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.02\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e1.17\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e3.4\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e59.4\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e(-56.0, 62.9)\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eDFM 2024\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.29\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.26\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e1.45\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e47.9\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e60.9\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e(\u003cstrong\u003e-13.0\u003c/strong\u003e, 99.9)\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan type=\"BoldItalicUnderline\" class=\"BoldItalicUnderline\" name=\"Emphasis\"\u003eUpdated DFM\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0.24\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.17\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e1.05\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e20.8\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e45.7\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e(\u003cstrong\u003e-24.8\u003c/strong\u003e, 66.6)\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eTable \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e Performance criteria of selected empirical equations, DFMs by Azmi and Thomson (\u003cspan class=\"CitationRef\"\u003e2024\u003c/span\u003e) and updated DFMs. Italic bold values are selected top two best performances.\u003c/p\u003e\n\u003cp\u003e[Indicative location]\u003c/p\u003e\n\u003cp\u003eFor breach peak outflow, the updated DFM demonstrated superior performance across all evaluation criteria, except for the median MPE, which was − 20% compared to -9.8% for the best-performing model. The Z20 model ranked second, nonetheless, exhibited a performance pattern similar to that of the updated DFM. Consequently, DFM 2024 was identified as a complementary option. While DFM 2024 correlation and error magnitudes were close to those of the top model, DFM 2024 achieved the lowest median MPE and a narrower MPE range, making it the second preferred model after the updated DFM. In terms of final breach average width, the updated DFM again ranked first, followed by DFM 2024. Due to their similar performance patterns, the XZ9 model was identified as a suitable complement. With its lower MAD and reduced upper tail, XZ9 can effectively enhance the performance of the updated DFM when adopted in combination. For the failure time parameter, no model clearly outperformed the others; however, except for the updated DFM, with NSE of 0.17, all others produced negative NSE values, underscoring their unreliability for extreme estimates. As a result, the updated DFM was once again selected as the preferred model. The F8 model followed, with an MPE below 10%, a relatively small RMSE, and an upper range under 30%.\u003c/p\u003e\n\n\n\n"},{"header":"Summary \u0026 Conclusion","content":"\u003cp\u003eA data fusion methodology employing multivariate linear regression was applied using a global historical dataset (Bernard-Garcia and Mahdi, \u003cspan class=\"CitationRef\"\u003e2020\u003c/span\u003e), integrated with selected empirical equations as outlined by Azmi and Thomson (\u003cspan class=\"CitationRef\"\u003e2024\u003c/span\u003e). The study conducted a comprehensive comparative analysis between the updated data fusion model (DFM), Azmi and Thomson’s equations, and individual empirical formulas across three key breaching parameters: peak discharge, final average breach width, and failure time. Statistical evaluations included t- and F-tests, probability distribution analyses (K-S test, Q-Q plots, boxplots, scatter plots), and performance metrics (R², NSE, RMSE, MPE). Although the updated DFM demonstrated partial superiority across most criteria, a secondary equation was selected to complement areas where the DFM was less effective. This dual-model approach ensures comprehensive coverage of assessment criteria and enhances the robustness of predictions for future dam breach scenarios. The final selected methods are: i) Breach peak outflow: DFM 2024 and updated DFM, ii) Final breach average width: XZ9 (Xu and Zhang \u003cspan class=\"CitationRef\"\u003e2009\u003c/span\u003e) and updated DFM, and iii) Failure time: F8 (Froehlich \u003cspan class=\"CitationRef\"\u003e2008\u003c/span\u003e) and updated DFM.\u003c/p\u003e\u003cp\u003eIt is important to distinguish the definition of failure time (T\u003csub\u003ef\u003c/sub\u003e) used in different empirical equations. The equation proposed by Xu and Zhang (\u003cspan class=\"CitationRef\"\u003e2009\u003c/span\u003e) defines T\u003csub\u003ef\u003c/sub\u003e as the entire duration from the onset of erosion to the post-erosion phase. In contrast, other widely used equations define T\u003csub\u003ef\u003c/sub\u003e more narrowly from the initiation of breach formation to its completion. To ensure consistency and comparability across models, the Xu and Zhang (\u003cspan class=\"CitationRef\"\u003e2009\u003c/span\u003e) equation was excluded from T\u003csub\u003ef\u003c/sub\u003e assessments. Failure time is still the most challenging breaching parameter to estimate, due to high uncertainties in observed/measured datasets. Gathering more information with much more accuracy and reliability is required to help practitioners apply the outcomes in real-world projects.\u003c/p\u003e\u003cp\u003eTo reiterate from Azmi and Thomson (\u003cspan class=\"CitationRef\"\u003e2024\u003c/span\u003e), the accuracy of dam breach modelling is primarily hindered by the quality of historical data. Enhancing international collaboration to compile validated datasets and simulate dam failures in laboratory settings can improve model reliability. For large dams, integrating detailed geotechnical data is essential. Additionally, safety features like wave walls or rock mesh can alter failure mechanisms, requiring tailored breaching equations. While historical datasets aim for diversity, practitioners must understand dam structures thoroughly and adjust models accordingly to ensure dependable and context-sensitive predictions.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eConflict of interests:\u003c/strong\u003e\u003cp\u003eThe author has no conflict of interest to declare.\u003c/p\u003e\u003c/p\u003e\u003ch2\u003eAcknowledgment\u003c/h2\u003e\u003cp\u003eWe appreciate the constructive comments of reviews that led to improving the quality of this paper.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eAraghinejad S, Azmi M, Kholghi M (2011) Application of artificial neural network ensembles in probabilistic hydrological forecasting. 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American Society of Civil Engineers (ASCE)\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"natural-hazards","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"nhaz","sideBox":"Learn more about [Natural Hazards](https://www.springer.com/journal/11069)","snPcode":"11069","submissionUrl":"https://submission.nature.com/new-submission/11069/3","title":"Natural Hazards","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"dam break, dam breach parameters, data fusion model, dam failure database","lastPublishedDoi":"10.21203/rs.3.rs-7289241/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7289241/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis research concentrated on estimating breaching embankment dams with focus on i) the extraction of failure cases from the historical dam failure database, adoptable for dam breach parameters estimates, ii) an update on data-fusion-based (DFM) dam breach equations published by Azmi and Thomson (\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2024\u003c/span\u003e) using the adopted failure cases, and finally iii) a comprehensive comparative assessment of all models to estimate breach peak discharge, breach formation time, and average breach width. A multi-stage procedure was followed to select the empirical equations that showed higher similarities to observed data for all three dam breach parameters. Statistical evaluations included t- and F-tests, probability distribution analyses (K-S test, Q-Q plots, boxplots, scatter plots), along with the performance criteria (median of percentage error (MPE), root mean square errors (RMSE), coefficient of variation and Nash Sutcliffe efficiency (NSE)), were used for comparative assessments. The final selected methods are: i) Breach peak outflow: DFM 2024 and updated DFM, ii) Final breach average width: XZ9 (Xu and Zhang \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2009\u003c/span\u003e) and updated DFM, and iii) Failure time: F8 (Froehlich \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2008\u003c/span\u003e) and updated DFM. It is critical to note that the proposed equations are not recommended for concrete dams and or embankment dams with extensive safety elements (e.g., wave walls, additional rock mesh protections).\u003c/p\u003e","manuscriptTitle":"An update on Data-Fusion-based Dam Breach Empirical Equations Based on a Worldwide Historical Dam Failure Database","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-10-12 14:26:42","doi":"10.21203/rs.3.rs-7289241/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"reviewerAgreed","content":"","date":"2025-12-19T06:45:09+00:00","index":0,"fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-09-29T15:14:59+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"Natural Hazards","date":"2025-09-29T13:12:18+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-08-07T08:39:05+00:00","index":"","fulltext":""},{"type":"submitted","content":"Natural Hazards","date":"2025-08-04T19:27:39+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"natural-hazards","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"nhaz","sideBox":"Learn more about [Natural Hazards](https://www.springer.com/journal/11069)","snPcode":"11069","submissionUrl":"https://submission.nature.com/new-submission/11069/3","title":"Natural Hazards","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"e9e202bd-3b5f-4eea-81c3-46bd06480f82","owner":[],"postedDate":"October 12th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-05-19T12:43:07+00:00","versionOfRecord":[],"versionCreatedAt":"2025-10-12 14:26:42","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7289241","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7289241","identity":"rs-7289241","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}