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This study presents a new flood routing method integrating the modified Muskingum (NLM7_Aqlat) method with hybrid natural optimization algorithms (hybrid of Humboldt squid optimization algorithm [HSOA] and gradient‐based optimizer [GBO] and hybrid of Pine cone optimization algorithm [PCOA] and GBO). In the NLM7_Aqlat, the lateral flow is applied to a seven‐parameter nonlinear Muskingum model (NLM7), and hybrid natural‐based optimization algorithms optimize the parameters. In Karahan flood routing, the standard value of the mean sum of squared deviations (SSQmean) for integrating the NLM7_Aqlat model and PCOA_GBO was calculated to be 96.06% less than the other 10 algorithms (such as GA and GBO). In Wilson flood routing, the PCOA_GBO algorithm in the NLM7 model calculated the SSQmean criterion value 99% lower than other optimization algorithms. The HSOA_GBO algorithm in the NLM7_Aqlat model provided the best flood routing for Weisman‐Lewis, enhancing hydrograph accuracy. In Karun flood routing, the PCOA algorithm estimated the SSQmean in the NLM7 model to be 89% lower than other algorithms. The new flood routing method showed competitive results versus NLM7. Hybrid optimization algorithms outperformed standalone ones, prompting authors to recommend this methodology for enhancing early flood warning systems.

The Muskingum model has been widely utilized for flood routing by water resources engineers for decades. Since the relation between channel storage and weighted summation of inflow and outflow seems to be nonlinear, a constant exponent parameter is used to account for this nonlinearity. On the other hand, the nonlinear Muskingum models with constant parameters cannot address variation of Muskingum parameters during flood period. In this paper, fourteen new Muskingum models with variable parameters are proposed. In these models, the routing period is divided into two or three sub-periods and the proposed versions of the Muskingum models can possess parameters with different values in these sub-periods. This capability enhances the Muskingum flood routing approach to better capture the reach characteristics and subsequently improve the routing results. The flood routing results for the selected data set demonstrate that three variable-parameter model reduces the SSQ value more than 89% comparing with the best three constant-parameter Muskingum model in the literature. Additionally, it was concluded that considering x-parameter as a variable parameter during a flood period affects the parameter estimation more than imposing the K- and m-parameters to be variable.

Continuous water and sediment flow monitoring across river cross sections is essential for the management of flood- and sediment-related problems in watersheds. The sediment rating curve (SRC) estimates missing or uncertain sediment flow via its corresponding water discharge. Generally, a power form of relationship correlates the two quantities. The log-transformed water discharge and sediment discharge data were used to depict the SRCs developed in the present study. SRC parameter estimation via least squares regression using at-site dataset pairs can be found in the literature. However, the availability of reliable datasets at the site limits model applicability. This method does not describe the SRC on the basis of the continuity aspects of river system flow characteristics. Therefore, the current study proposes integrated SRC estimation models (Model 2 and Model 3) using modified Muskingum equations abiding by the spatial and temporal continuity of the entire river system state. These models are derived from streamflow storage balance criteria and ensure flow continuity norms. Moreover, Model 3 considers an inverse power form of the relationship depicting the water flow characteristics that govern the sediment transport phenomena through the river system. Standalone models for SRC parameter estimation (Model 1) were also developed for comparison among all three models via the root mean square error (RMSE), NRMSE (normalized root mean square error) and coefficient of determination (R2). The Mahanadi River system within Chhattisgarh state, India comprises five sections at tributaries, and the main channel was considered for the study. The improved NRMSE by Model 2 (7.53%) and Model 3 (7.14%) at Rajim and Model 3 (3.44%) at Bamnidhi in comparison to Model 1 at Rajim (9.19%) and Bamnidhi (4.80%) encouraged the application of integrated models for SRC estimation in river systems. Moreover, Model 3 outperformed Model 2 in some cases where the sediment transport process may be governed by water flow characteristics. [Display omitted] •Sediment rating curve estimate for entire river network replacing standalone model.•Muskingum model applications ensuring flow continuity, is recommended to adopt.•Water flow characteristics parameters influence sediment water relationship in river.•Both integrated models outperformed standalone model at upstream bounding section.

River flood routing is an important issue in current water resources management. As a popular hydrological flood routing method, Muskingum model has always been the dominant method of flood routing. This paper reviews the development of Muskingum model and the research status of its parameter estimation. The characteristics and relationships of different types of Muskingum models are compared, and it is found that the combination of mathematical techniques and evolutionary algorithms has shown good results in parameter estimation in recent years. In addition, this paper also gives a brief overview of six accuracy evaluation criteria and nine research case data sets commonly used in the literature. It also introduces some challenges of the Muskingum model and new trends in future research, which should interest researchers and engineers.

Flood is one of the natural hazards that its prediction and control is of great importance. One of the most important models in the field of flood routing is the Muskingum model. In this study, Muskingum four-parameter model is used for flood routing. The existence of unknown parameters causes the Kidney algorithm to be used as a new evolutionary algorithm based on reabsorption and filter operators for flood routing. The operators make the Kidney algorithm accelerate the convergence process and improve the quality of responses. Three floods were selected based on Kidney algorithm. The results indicated that the amount of sum squared deviation reduced by 84, 90, 35 and 86% for the Wilson flood algorithm compared to the Honey bee mating optimization, pattern search, particle swarm optimization (PSO) and harmony search (HS) methods for flood routing based on observational and simulated discharge. Also, the results indicated that the Kidney algorithm is more accurate based on the Muskingum four-parameter model than the Muskingum three-parameter model and the Muskingum two-parameter model. The sum absolute deviation value for the HS, genetic algorithm and PSO methods is 94, 88 and 82% higher than Kidney algorithm for Karahan flood. In addition, the predicted peak discharge for the Karahan flood and the predicted time for peak discharge were more accurate than other evolutionary algorithms. Also, the results of the Kidney algorithm for the Viessman and Lewis floods indicated that the Kidney algorithm well reduces the error indicators. Therefore, the Kidney algorithm as a suitable algorithm based on Muskingum four-parameter model had higher accuracy.

Flood is one of the most destructive natural disasters that damages people’s lives dramatically. Thus, it is crucial for researchers and politicians to research flood routing. The non-linear Muskingum model has been significantly considered by engineers and researchers in flood routing. In this study, in order to increase the accuracy of outflow prediction, the new non-linear Muskingum model, with four variable parameters, is proposed for the first time. In the proposed model, the inflows are divided into three sub-regions, and each of the four hydrologic parameters has a various value in each sub-region. How to select the sub-regions, as well as the values of the hydrologic parameters, is determined by combining both the Particle Swarm Optimization and Genetic Algorithm. The proposed model is studied in four case studies. Compared to the non-linear Muskingum model with three parameters, the amount of sum squared deviation (SSQ) decreased 52 and 6.9% for the first and second case studies, respectively. Compared to the best variable parameter model, the SSQ for the third and fourth case studies reduced 76 and 62%, respectively. The results showed that the SSQ was considerably decreased significantly in all of the four case studies, and the proposed model has superiority over other non-linear Muskingum models, which have been used by other researchers so far.

The BFGS method is a common and effective method for solving unconstrained optimization problems in quasi-Newton algorithm. However, many scholars have proven that the algorithm may fail in some cases for nonconvex problems under Wolfe conditions. In this paper, an adaptive projection BFGS algorithm is proposed naturally which can solve nonconvex problems, and the following properties are shown in this algorithm: ➀ The generation of the step size α j satisfies the popular Wolfe conditions; ➁ a specific condition is proposed which has sufficient descent property, and if the current point satisfies this condition, the ordinary BFGS iteration process proceeds as usual; ➂ otherwise, the next iteration point x j + 1 is generated by the proposed adaptive projection method. This algorithm is globally convergent for nonconvex problems under the weak-Wolfe-Powell (WWP) conditions and has a superlinear convergence rate, which can be regarded as an extension of projection BFGS method proposed by Yuan et al. (J. Comput. Appl. Math. 327:274-294, 2018 ). Furthermore, the final numerical results and the application of the algorithm to the Muskingum model demonstrate the feasibility and competitiveness of the algorithm.

The Muskingum model is one of the most widely used hydrological methods in flood routing, and calibrating its parameters is an ongoing research challenge. We optimized Muskingum model parameters to accurately simulate hourly output hydrographs of three flood-prone rivers in the Karun watershed, Iran. We evaluated model performance using the correlation coefficient (CC), the ratio of the root-mean-square error to the standard deviation of measured data (PSR), Nash–Sutcliffe efficiency (NSE), and index of agreement (d). The results show that the gray wolf optimization (GWO) algorithm, with CC = 0.99455, PSR = 0.155, NSE = 0.9757, and d = 0.9945, performed better in simulating the flood in the first study area. The Kalman filter (KF) improved these measures by + 0.00516, − 0.1246, + 0.02328, and + 0.00527, respectively. Our findings for the second flood show that the gravitational search algorithm (GSA), with CC = 0.9941, PSR = 0.1669, NSE = 0.9721, and d = 0.9921, performed better than all other algorithms. The Kalman filter enhanced each of the measures by + 0.00178, − 0.0175, + 0.0055 and + 0.0021, respectively. The gravitational search algorithm also performed best in the third flood, with CC = 0.9786, PSR = 0.2604, NSE = 0.9321, and d = 0.9848, and with improvements in accuracy using the Kalman filter of + 0.01081, − 0.0971, + 0.394, and + 0.0078, respectively. We recommend the use of GWO-KF for flood routing studies with flood events of high volumes and hydrograph base times, and use of GSA-KF for studies with flood events of high volumes and hydrograph base times.

Machine learning (soft) methods have a wide range of applications in many disciplines, including hydrology. The first application of these methods in hydrology started in the 1990s and have since been extensively employed. Flood hydrograph prediction is important in hydrology and is generally done using linear or nonlinear Muskingum (NLM) methods or the numerical solutions of St. Venant (SV) flow equations or their simplified forms. However, soft computing methods are also utilized. This study discusses the application of the artificial neural network (ANN), the genetic algorithm (GA), the ant colony optimization (ACO), and the particle swarm optimization (PSO) methods for flood hydrograph predictions. Flow field data recorded on an equipped reach of Tiber River, central Italy, are used for training the ANN and to find the optimal values of the parameters of the rating curve method (RCM) by the GA, ACO, and PSO methods. Real hydrographs are satisfactorily predicted by the methods with an error in peak discharge and time to peak not exceeding, on average, 4% and 1%, respectively. In addition, the parameters of the Nonlinear Muskingum Model (NMM) are optimized by the same methods for flood routing in an artificial channel. Flood hydrographs generated by the NMM are compared against those obtained by the numerical solutions of the St. Venant equations. Results reveal that the machine learning models (ANN, GA, ACO, and PSO) are powerful tools and can be gainfully employed for flood hydrograph prediction. They use less and easily measurable data and have no significant parameter estimation problem.

This research introduces a novel nonlinear Muskingum model for river flood routing, aiming to enhance accuracy in modeling. It integrates lateral inflows using the Whale Optimization Algorithm (WOA) and employs a distributed Muskingum model, dividing river reaches into smaller intervals for precise calculations. The primary goal is to minimize the Sum of Square Errors (SSE) between the observed and modeled outflows. Our methodology is applied to six distinct flood hydrographs, revealing its versatility and efficacy. For Lawler's and Dinavar's flood data, the single-reach Muskingum model outperforms multi-reach versions, demonstrating its effectiveness in handling lateral inflows. For Lawler's data, the single-reach model (NR = 1) yields optimal parameters of K = 0.392, x = 0.027, m = 1.511, and β = 0.010, delivering superior results. Conversely, when fitting flood data from Wilson, Wye, Linsley, and Viessman and Lewis, the multi-reach Muskingum model exhibits better overall performance. Remarkably, the model excels with the Viessman and Lewis flood data, especially with two reaches (NR = 2), achieving a 21.6% SSE improvement while employing the same parameter set. This research represents a significant advancement in flood modeling, offering heightened accuracy and adaptability in river flood routing.