Dispersion Coefficient In Natural Streams Research Articles

Application of genetic programming in presenting novel equations for longitudinal dispersion coefficient in natural streams considering rivers geometry - Implementation in assimilation capacity simulation

Precise estimation of the longitudinal dispersion coefficient (LDC) is crucial for the accurate simulation of water quality management tools such as assimilation capacity. Previous research analyzed the LDC of natural streams in two general categories: ignoring or considering the river sinuosity (σ). Genetic programming (GP) is used in this study to investigate both mentioned categories by applying two experimental datasets from 56 to 24 different rivers worldwide. The first proposed LDC equation of this research (without σ) improves the amounts of statistical measures R2 (Determination Coefficient), OI (Overall Index), NSE (Nash-Sutcliffe Efficiency), WI (Willmott's Index of Agreement), RMSE (Root Mean Square Error), and MAE (Mean Absolute Error) by 3.75%, 4.71%, 7.81%, 0.85%, 13.72%, and 0.68%, respectively, compared to the best values of these indicators in the previous investigations. Regarding the second category, relative and absolute sensitivity analyses are conducted, which reveal that σ is the most influential parameter in the accurate prediction of the LDC among all hydraulics and geometric parameters of the river. This part of the investigation presents four unique LDC equations that closely match the experimental results. Significant improvement of the most accurate presented LDC for statistical indices R2, OI, NSE, WI, RMSE, MAE, and accuracy percentage are obtained equal to 3.27%, 2.41%, 3.16%, 0.81%, 35.1%, 24.47%, 3.8%, respectively, in comparison with the best previous relations. Also, a new indicator for measuring the efficiency of mathematical equations called Mean Normalized Statistical Index (MNSI) is introduced and applied in different parts of this research. Finally, the assimilation capacity of the Kashafrud River is determined based on the analytical method of pollution propagation for three types of water demands utilizing the accurately presented LDC in 1993–2020. The average amount of river assimilation capacity using accurate LDC is simulated at 91.93 tons/day, much lower than the currently reported pollution entrance, which equals 540 tons/day.

Comparison of the efficacy of particle swarm optimization and stochastic gradient descent algorithms on multi-layer perceptron model to estimate longitudinal dispersion coefficients in natural streams

Accurate estimation of the longitudinal dispersion coefficient (LDC) is essential for modeling the pollution status in rivers. This research investigates the capabilities of machine-learning methods such as multi-layer perceptron (MLP), multi-layer perceptron trained with particle swarm optimization (MLP-PSO), multi-layer perceptron trained with Stochastic gradient descent deep learning (MLP-SGD) and different regressions including linear and non-linear regressions (LR and NLR) methods for determining the LDC of pollution in natural rivers and evaluates the accuracy of these methods in comparison with real measured data. Furthermore, the correlation coefficient (CC), root mean squared error (RMSE) and Willmott’s Index (WI) were implemented to evaluate the accuracies of the mentioned methods. Comparison of the results showed the superiority of the MLP-SGD model with CC of 0.923, RMSE of 281.4 and WI of 0.954, which indicates the undeniable accuracy and quality of the deep-learning model that can be used as a powerful model for LDC simulation. Also due to the acceptable performance of the PSO algorithm in the hybridization of the MLP model, the use of PSO algorithms is recommended to train machine-learning techniques for LDC estimation.

The prediction of longitudinal dispersion coefficient in natural streams using LS-SVM and ANFIS optimized by Harris hawk optimization algorithm

Accurate calculation of the longitudinal dispersion coefficient (Kx) of pollution is essential in modeling river pollution status. Various equations are presented to calculate the Kx using experimental, analytical, and mathematical methods. Although machine learning models are more reliable than experimental equations in the presence of uncertainties missing data, they have not been widely used in predicting Kx. In this study, the Kx of the river was predicted using machine learning methods, including least square-support vector machine (LS-SVM), adaptive neuro-fuzzy inference system (ANFIS), and ANFIS optimized by Harris hawk optimization (ANFIS-HHO), and the results were compared with that of the experimental methods. Several scenarios were designed by different combinations of input variables, such as the average depth of the flow (H), average flow velocity (U), and shear velocity (u⁎). The results showed that machine learning models had a more efficient performance to predict Kx than experimental equations. The ANFIS-HHO, with a scenario containing all the input variables, performed better than the other two models, with root mean square error, mean absolute percentage error, and coefficient of determination of 17.0, 0.22, and 0.97, respectively. Furthermore, the HHO algorithm slightly increased the prediction performance of the ANFIS. The discrepancy ratio (DR) evaluation criteria showed that experimental equations overestimated the values of Kx, while the machine learning models resulted in higher precision. Also, the results of Taylor's diagram showed the acceptable performance of the ANFIS-HHO model compared to other models. Given the promising results of the present study, it is expected that the proposed approach can be efficiently used for similar environmental modeling problems.

Predicting longitudinal dispersion coefficient in natural streams using minimax probability machine regression and multivariate adaptive regression spline

This article employs minimax probability machine regression (MPMR) and multivariate adaptive regression spline (MARS) for prediction of longitudinal dispersion coefficient in natural streams. The variables of hydraulic features such as channel width (B), flow depth (H), flow velocity (U), shear velocity (u*) and geometric features such as channel sinuosity (σ) and channel shape parameter (β) were taken as the input. The dispersion coefficient Kx was the decision parameter for the proposed machine learning models. MARS does not assume any functional relationship between inputs and output. The MARS model is a non-parametric regression model that splits the data and fits each interval into a basis function. MPMR is a probabilistic model which maximises the minimum probability of predicted output. MPMR also provides output within some bound of the true regression function. The proposed study gives an equation for prediction of longitudinal dispersion coefficient based on the developed MARS. The developed MARS has been compared with proposed MPMR. Finally, the performances of the models have been measured by different performance metrics.

Predicting longitudinal dispersion coefficient in natural streams using minimax probability machine regression and multivariate adaptive regression spline

This article employs minimax probability machine regression (MPMR) and multivariate adaptive regression spline (MARS) for prediction of longitudinal dispersion coefficient in natural streams. The variables of hydraulic features such as channel width (B), flow depth (H), flow velocity (U), shear velocity (u*) and geometric features such as channel sinuosity (σ) and channel shape parameter (β) were taken as the input. The dispersion coefficient Kx was the decision parameter for the proposed machine learning models. MARS does not assume any functional relationship between inputs and output. The MARS model is a non-parametric regression model that splits the data and fits each interval into a basis function. MPMR is a probabilistic model which maximises the minimum probability of predicted output. MPMR also provides output within some bound of the true regression function. The proposed study gives an equation for prediction of longitudinal dispersion coefficient based on the developed MARS. The developed MARS has been compared with proposed MPMR. Finally, the performances of the models have been measured by different performance metrics.

Estimation of natural streams longitudinal dispersion coefficient using hybrid evolutionary machine learning model

Among several indicators for river engineering sustainability, the longitudinal dispersion coefficient () is the main parameter that defines the transport of pollutants in natural streams. Accurate estimation of has been challenging for hydrologists due to the high stochasticity and non-linearity of this hydraulic-environmental parameter. This study presents a new hybrid machine learning (ML) model integrating a Gaussian Process Regression (GPR) and an evolutionary feature selection (FS) approach (i.e. Covariance Matrix Adaptation Evolution Strategy (CMAES)) to estimate in natural streams. The dataset consists of geometric and hydraulic river system parameters from 29 streams in the United States. The modeling results showed that the proposed model outperformed other models in the literature, producing more stable and accurate estimations. The FS approach evidenced the significance of the cross-sectional average flow velocity (U), channel width (B), and channel sinuosity σ to estimate the dispersion coefficient. In quantitative terms, the integrated GPR model with feature selection approach attained the minimum root mean square error () and maximum coefficient of determination (). The proposed hybrid evolutionary ML model arises as robust, flexible and reliable alternative computer aid technology for predicting the longitudinal dispersion coefficient in natural streams.

Derivation of Optimized Equations for Estimation of Dispersion Coefficient in Natural Streams Using Hybridized ANN With PSO and CSO Algorithms

In this paper, a new hybrid model is developed to improve the accuracy in the prediction of the longitudinal dispersion coefficient (K x ) and the derivation of novel optimized explicit equations for natural streams. For this purpose, an artificial neural network (ANN) is hybridized with particle swarm optimization (PSO) and cat swarm optimization (CSO) algorithms. The CSO and PSO are used to find the optimum values of biases and weights in ANN structure and formulate the results as novel explicit predictive equations than the classical black-box methods. The hydraulic parameters of the natural stream and some geometric parameters were utilized for the model developments. Eight different input combinations are used as the input vectors to ANN, ANN-PSO, and ANN-CSO models, whereas the dispersion coefficient (K x ) is the target model output. The developed models are trained and tested by a comprehensive reference data sets measured on streams in the United States, that were used previously by Tayfur and Singh (2005) in ANN models. The main aims, novelty, and contributions of the present study are 1) improving the accuracy of classical ANN-based K x predictions by hybridizing with CSO and PSO. 2) Performing sensitive analysis of ANN, ANN-CSO, and ANN-PSO based on input combinations 3) derivation of novel explicit optimized ANN-CSO, ANN-PSO, equations for predicting K x rather than the classical ANN black-box methods. The results depicted that the highest accuracy and superiority were attained by the ANN-PSO model, with input variables of B, H, U, U * , followed by ANN-CSO and ANN. By using the optimized trained black box ANN models, two novel explicit predictive equations are derived, and their results are compared with the empirical equations. Comparative assessments confirmed significant improvements in the hybrid equations' results than the classical ANN and previously published equations. The developed novel equations can be used to estimate the Kx in one-dimensional pollutant transfer models that are essential for the pollution studies in environmental river engineering practices.

Estimating longitudinal dispersion coefficient in natural streams using empirical models and machine learning algorithms

The longitudinal dispersion coefficient (LDC) plays an important role in modeling the transport of pollutants and sediment in natural rivers. As a result of transportation processes, the concentration of pollutants changes along the river. Various studies have been conducted to provide simple equations for estimating LDC. In this study, machine learning methods, namely support vector regression, Gaussian process regression, M5 model tree (M5P) and random forest, and multiple linear regression were examined in predicting the LDC in natural streams. Data sets from 60 rivers around the world with different hydraulic and geometric features were gathered to develop models for LDC estimation. Statistical criteria, including correlation coefficient (CC), root mean squared error (RMSE) and mean absolute error (MAE), were used to scrutinize the models. The LDC values estimated by these models were compared with the corresponding results of common empirical models. The Taylor chart was used to evaluate the models and the results showed that among the machine learning models, M5P had superior performance, with CC of 0.823, RMSE of 454.9 and MAE of 380.9. The model of Sahay and Dutta, with CC of 0.795, RMSE of 460.7 and MAE of 306.1, gave more precise results than the other empirical models. The main advantage of M5P models is their ability to provide practical formulae. In conclusion, the results proved that the developed M5P model with simple formulations was superior to other machine learning models and empirical models; therefore, it can be used as a proper tool for estimating the LDC in rivers.

Empirical models for longitudinal dispersion coefficient in natural streams

Empirical models for longitudinal dispersion coefficient in natural streams

Prediction of longitudinal dispersion coefficient in natural streams by prediction map

Longitudinal dispersion coefficient can be determined by experimental procedures in natural streams. Many theoretical and empirical equations that are based on hydraulic and geometric characteristics have been developed from the field experiments of longitudinal dispersion coefficient. Regression analysis, which carries some restrictive assumptions such as linearity, normality and homoscedasticity, was used to derive some of these equations. Generally speaking, results obtained from regression analyses are not that accurate as these assumptions are often not satisfied completely. In this study, a method called Prediction Map (PM) is developed based on geostatistics to predict longitudinal dispersion coefficient from measured discharge values, shear velocities, and other conventional parameters of the hydraulic variables and normalized velocity with the objective of overcoming the drawbacks indicated above. As part of this method, a new procedure called Iterative Error Training Procedure (IETP) was developed to minimize prediction error. The prediction error level was reduced after implementing the IETP. PM was compared with various regression models by taking analyzed errors (average relative error percentage and root mean square error), coefficient of efficiency, coefficient of determination and Scatter Index as performance evaluation criteria. The results of the study indicate that the PM approach can perform very well in predicting longitudinal dispersion coefficient by applying IETP. The presented approach yielded lower average relative error percentage, root mean square error and Scatter Indices, and higher coefficient of efficiency and coefficient of determination values compared to the regression models. One of the important advantages of the PM method is that valuable interpretations and a prediction map can be extracted from the resulting contour maps, and as a result, more accurate predictions can be obtained compared to regression analysis.

Gene expression models for prediction of longitudinal dispersion coefficient in streams

Summary Longitudinal dispersion is the key hydrologic process that governs transport of pollutants in natural streams. It is critical for spill action centers to be able to predict the pollutant travel time and break-through curves accurately following accidental spills in urban streams. This study presents a novel gene expression model for longitudinal dispersion developed using 150 published data sets of geometric and hydraulic parameters in natural streams in the United States, Canada, Europe, and New Zealand. The training and testing of the model were accomplished using randomly-selected 67% (100 data sets) and 33% (50 data sets) of the data sets, respectively. Gene expression programming (GEP) is used to develop empirical relations between the longitudinal dispersion coefficient and various control variables, including the Froude number which reflects the effect of reach slope, aspect ratio, and the bed material roughness on the dispersion coefficient. Two GEP models have been developed, and the prediction uncertainties of the developed GEP models are quantified and compared with those of existing models, showing improved prediction accuracy in favor of GEP models. Finally, a parametric analysis is performed for further verification of the developed GEP models. The main reason for the higher accuracy of the GEP models compared to the existing regression models is that exponents of the key variables (aspect ratio and bed material roughness) are not constants but a function of the Froude number. The proposed relations are both simple and accurate and can be effectively used to predict the longitudinal dispersion coefficients in natural streams.

An Empirical Approach for Determining Longitudinal Dispersion Coefficients in Rivers

Determination of the longitudinal dispersion coefficient (LDC) of a river is needed in studies regarding cleaning the water and protecting its quality when nuclear, chemical or biological contaminants are discharged into the river. This study presents the development of an empirical equation for predicting the longitudinal dispersion coefficient in natural streams. Factors affecting the uniformity of the flow directly affect the LDC. Therefore, the hydraulic radius, defined as the ratio of the wetted area to the wetted perimeter, was considered as an important factor in determining the LDC. The presented equation relates the dispersion coefficient to hydraulic and geometric parameters of the flow, and was derived using dimensional and least squares analysis. The comparison of the predictions using 128 field data sets measured in 41 rivers in the USA has indicated that the proposed equation is reliable in predicting longitudinal dispersion coefficients in natural streams.

Differential Evolution for Prediction of Longitudinal Dispersion Coefficients in Natural Streams

Differential evolution (DE) is a population-based evolutionary algorithm widely used for solving multidimensional global optimization problems over continuous spaces, and has been successfully used to solve several kinds of problems. In this paper, a novel expression for the prediction of longitudinal dispersion coefficient in natural streams is proposed to minimize the sum-square error using differential evolution algorithm. The new expression considers the hydraulic and geometric characteristics of rivers. Datasets consisting 65 sets of observations from 29 rivers in the unite states are used to test the proposed algorithm, and results demonstrate the performance and applicability of the proposed differential evolution. Compared with the previous methods, the new expression using differential evolution is superior to other expressions. Moreover, 56.92 % of the prediction using the new expression lie with the 0.5 < Kpre/Kmeas < 1.5 that is better than other expressions.

A framework development for predicting the longitudinal dispersion coefficient in natural streams using an artificial neural network

AbstractThe main objective of the present investigation is to predict longitudinal dispersion coefficient (Kx) in natural streams using artificial neural network (ANN) technique based on most famous training functions such as Trainlm, Trainrp, Trainscg, Trainoss, and so on. To achieve the goal, hydraulic and geometric data (shear velocity, channel width, local flow depth, and mean longitudinal velocity) that are easily obtained in natural streams are used. First, we have tried to review the most well‐known of published work in the field due to find out deficiencies of them. Second, new approach of ANN model based on the famous training functions is applied for predicting Kx in natural streams and then the best architectures for each training functions is selected by trial and error. Finally, Levenberg‐Marquardt training function (Trainlm) is selected as the best choice for training the network parameters. Determination coefficient (R2) and mean absolute error for ANN (Trainrp) model were equal to 0.94 and 33 in the training and 0.95 and 30 in the testing steps, respectively. It is hoped that the presented methodology in the research, can be useful in river water quality management studies. © 2010 American Institute of Chemical Engineers Environ Prog, 2011.

Genetic Programming for Predicting Longitudinal Dispersion Coefficients in Streams

This paper presents a genetic programming (GP) approach to predict the longitudinal dispersion coefficients in natural streams. Published data were compiled from the literature for the dispersion coefficient for a wide range of flow conditions, and they were used for the development and testing of the proposed method. The proposed GP approach produced excellent results (R2 = 0.98 and RMSE = 0.085) compared to the existing predictors (Rajeev and Dutta, Hydrol Res 40(6):544–552, 2009, R2 = 0.345 and RMSE = 1778.6) for dispersion coefficient.

Support vector machine approach for longitudinal dispersion coefficients in natural streams

This paper presents the support vector machine approach to predict the longitudinal dispersion coefficients in natural rivers. Collected published data from the literature for the dispersion coefficient for wide range of flow conditions are used for the development and testing of the proposed method. The proposed SVM approach produce satisfactory results with coefficient of determination = 0.9025 and root mean square error = 0.0078 compared to existing predictors for dispersion coefficient.

An expert system for predicting longitudinal dispersion coefficient in natural streams by using ANFIS

Longitudinal dispersion coefficient in rivers and natural streams usually is estimated by simple inaccurate empirical relations, because of the complexity of the phenomena. So, in this study using adaptive neuro-fuzzy inference system (ANFIS), which have the ability of perception and realization of phenomenon without need for mathematical governing equations, a new flexible tool is developed to predict the longitudinal dispersion coefficient. The process of training and testing of this new model is done using a set of available published filed data. Several statistical and graphical criterions are used to check the accuracy of the model. The dispersion coefficient values predicted by the ANFIS model satisfactorily compared with the measured data. The predicted values were also compared with those predicted using available empirical equations that have been suggested in the literature and it was found that the ANFIS model with R2=0.99 and RMSE=15.18 in training stage and R2=0.91 and RMSE=187.8 in testing stage is superior in predicting the dispersion coefficient than the best accurate empirical equation with R2=0.48 and RMSE=295.7. The presented methodology in this paper is a new approach in estimating dispersion coefficient in streams and can be combined with mathematical models of pollutant transfer or real-time updating of these models.

Predicting longitudinal dispersion coefficient in natural streams by artificial intelligence methods

AbstractIn this study, three artificial neural network methods, i.e. feed forward back propagation, the radial basis function neural network, and the generalized regression neural network are employed to compute the longitudinal dispersion coefficient in order to evaluate its behaviour in predicting dispersion characteristics in natural streams. These methods, which use hydraulic and geometrical data to predict dispersion coefficients, can easily be applied to natural streams and are proven to be superior in explaining their dispersion characteristics more precisely than existing equations. This method of predicting the longitudinal dispersion coefficient in river flows was tested on 65 data sets, obtained by researchers from 30 rivers in the USA. Results using the models are compared with results obtained in many other studies, and are shown to be more accurate than the other methods considered. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Comments on “Development of an empirical equation for the transverse dispersion coefficient in natural streams” by Tae Myoung Jeon, Kyong Oh Baek and Il Won Seo

In a recent paper published in this journal, Jeon et al. ((2007), Environ Fluid Mech 7(4): 317–329) have presented a new empirical equation for the transverse dispersion coefficient in natural streams that was developed based on the hydraulic and geometric parameters using a regression technique. A total of 32 data sets collected in 32 streams was used. Among them, 16 data sets were used for deriving the new equation, and the other 16 were used for verifying the equation. Starting from dimensional analysis the authors found that transverse dispersion coefficient depends on three parameters, such as sinuosity, aspect ratio and a friction term. The robust least square method was applied to estimate regression coefficients resulting in an equation which allows better prediction of transverse dispersion coefficient than previous literature equations. The discussers would like to highlight some points raised in the paper.

Development of an empirical equation for the transverse dispersion coefficient in natural streams

In this study, a new empirical equation for the transverse dispersion coefficient has been developed based on the hydraulic and geometric parameters in natural streams using a regression technique. First, a total of 32 data sets in 16 streams were collected. Among those sets, 16 sets were used for deriving the new equation, and the other 16 sets were used for verifying the equation. Then, through dimensional analysis, it was found that the normalized transverse dispersion coefficient is associated with several parameters such as sinuosity, aspect ratio, and a friction term. The robust least square method was applied to estimate regression coefficients. The newly proposed equation was proven to be superior in explaining the dispersion characteristics of natural streams more precisely compared to the existing equations.