Estimating critical depth and discharge over sloping rough end depth using machine learning

The end-depth structure called free overfall happens when water drops from up to down as a free fall because of a sudden drop or sudden change in the channel bed level caused by a free flow of water. Because of this drop, the brink's pressure distribution is not hydrostatic at the subcritical flow. The flow changed from Gradually Varied Flow (GVF) to Rappelled Varied Flow (RVF) GVF to RVF. The relation between depth over the threshold (brink depth, y_b) and standard depth (y_n), end-depth ratio (EDR), is more critical to predict Q over this structure because it is used as a flow measurement device.

Many investigations dealt with hydraulic characteristics of end-depth structures (Mohammed et al. 2007), which have presented a variation of water depth on vertical and skewed free overfall. The results clarified that a tilted outflow free fall is vertically superior, and the depth at the brink for vertical is greater than 11% than skew. The impact of stream channel slope on vertical and inclined free fall was investigated by Mohammed (2009b). The investigation showed that the bed slope affected discharge and the water depth of free overfall. The discharge coefficient for an inclined model is more significant than the vertical 25%. The authors present a theoretical equation to find EDR, water surface profile, and discharge. Mohammed (2009a) investigated a new model of an end-depth structure. He studied triangular shapes with an opposite flow direction with different angles. He compared the results with a standard vertical model. The results showed that the brink depth was more significant than 6% for triangular-shaped compared with vertical. Mohammed (2012) presented a theoretical study to predict EDR and end-depth discharge (EDD) for end-depth models with different end shapes. The study presents a new empirical equation to calculate EDR and EDD with a percentage error not exceeding 8.5% compared with the experimental data.

There are many studies conducted on channels with different shapes (Dey & Kumar 2002; Irzooki & Hasan 2018; Muhsin & Noori 2021), which investigated the free overfall model in a triangular channel. They focused on the effect of side and bed slopes and bed roughness. The study refers to EDR = 0.695 and 0.755, respectively, and this value increases when the side slope increases. Dey et al. (2004), Raikar et al. (2004), and Zeidan et al. (2021) investigated free overfall in an inverted semicircular channel. These studies developed new relationships to calculate discharge as well as EDR of 0.81. There are many studies in which free overfall with bed roughness is investigated (Öztürk 2005; Guo et al. 2006; Mohammed et al. 2011, 2013, 2018; FIRAT 2015). These investigation results referred to EDR values reaching 0.67, the brink-to-y_c ratio increased when the channel slope decreased, and roughness material and distribution increased.

In the last several decades, several investigators have turned to the use of machine learning (ML) algorithms for the analysis of field and laboratory data and have discovered noticeably superior results than those obtained using traditional statistical approaches (Olyaie et al. 2019; Yousif et al. 2019; Suntaranont et al. 2020; Salmasi et al. 2021; Thakur et al. 2021). Researchers have recently focused on the Gaussian process (GP), random forest (RF), random tree (RT), reduced error pruning (REP) tree, and RF in predicting hydraulic features (Sihag et al. 2019, 2020; Salmasi et al. 2021). The current work compares the outcomes with models based on linear regression (LR). It introduces M5P, RF, RT, REP tree, and GP as alternative methods for determining y_c for the end-depth structure and Q. Gharehbaghi et al. (2023) investigated the influence of a submerged multiple-vane system on the dimensions of the flow separation zone. Several data-driven models are accessible, such as gene expression programming, support vector regression (SVR), Radial Basis Function (RBF), and a robust hybrid SVR with an ant colony optimization algorithm (ACO). Based on statistical metrics, the model grading procedure, scatter plot, and the hybrid SVR (RBF)-ACO model are the most accurate and exact models for predicting the maximum relative length and width. The total grades for these models are 6.75 and 5.8, respectively. The ML method was presented by Latif & Ahmed (2023) to predict the reservoir inflow in the Dokan dam in Iraq and the Warragamba dam in Australia using SVR. The RMSE for the Dokan dam daily inflow is 145.7 and R² is 0.85. The findings indicated that the ML had strong performance in Iraq, but its accuracy in Australia was lacking.

There are three channel slopes, 1/100 and 1/200, and leveling cases (0°). The water level was measured using a point gauge with 0.1 mm accuracy for each experiment of the aforementioned instances, estimated standard depth, brink depth, and typical depth above the classic weir. There are 215 runs, 135 for rough, and 80 for smooth models.

The standard measurement weir is installed at the channel end until the water head is constant; then, the water level above the weir and measured water volume to time (head-discharge) measurement and used Equation (1) for actual discharge calculated then raise the weir and continue all sizes to ensure not influence on free overfall model, then repeat these procedures for other discharges.

M5P-tree is a regression problem-solving genetic algorithm initially proposed by Quinlan Basser (1992). This tree approach establishes LR attributes on the station node by categorizing or splitting distinct data sections into numerous spaces. It suits a multiple LR model on each sublocation. The M5P-tree technique is based on continuous class issues rather than discontinuous slices, and it can handle functions with many dimensions. It displays the data for each built-in linear model component, making it possible to evaluate the nonlinear relationship between the datasets. Fault evaluation is conducted with the knowledge of the M5P-tree model tree separation criteria per node. The preset value difference of the class entering the node determines the number of errors. Any node solution is obtained using the feature that optimizes the predicted error reduction. Depending on fault estimates per node, details on the M5P-tree model tree splitting criteria are presented. The standard deviation (SD) of the target class at the node is used to compute the M5P error – overfitting results from this division, creating a large tree-like structure. The huge tree is trimmed in the second step, and the chopped subtrees are replaced with LR functions.

The RF theory was established by Breiman (2001). A regression and categorization machine technique builds a collection of tree alternatives at random and predicts the class, which may be the categorization style and separate trees' regressions. To develop a tree, the RF has an assembly of input values at each node (Singh et al. 2017). A decision tree is vital in a packing-based RF classifier, a sampling strategy in which the identical specimens are utilized several times and then re-inserted into the database. It is a variation of the bootstrap grouping tree approach known as bagging if only sample bootstrapping is utilized for categorized or regressed and no sampling prediction is made.

An irregular forest-trained model is often used to build models because of its ease of use and excellent performance, even with small datasets. RFs were extensively employed in transportation research. Thakur et al. (2021) employed the RF model to forecast the bond strength of Fiber-reinforced plastic (FRP) bars and obtained good results.

The RT method is a regulated teaching technique that creates several separate learners. The collected data are irregularly formed after trying to bag, termed a group teaching algorithm, and every node of an irregular tree is separated. The optimum among the subgroup of predictors is chosen randomly at that node (Aldous 1991). The random trees blend two current ML techniques: binary class trees and RF concepts. Each tree-like bagging is evaluated and regenerated with the training data.

All of the items in the subgroup were irregularly evaluated in the second phase at every node. Following this, the most suitable separation for the subgroup was calculated. For complicated and nonlinear relationships, RT-categorized data and analytics methods can be utilized (Shi et al. 2020). For the investigation, classification and regression approaches were applied. In the clustering algorithm, the classifier receives an input to categorize each tree in a forest group and provides results in a dataset that gets the majority of votes.

The REP tree method is a rapid logical classifier trees methodology that leverages the notion of computer-selected random features to reduce variance inaccuracy (Quinlan 1987; Devasena 2014). The REP tree employs the logistic regression technique and creates several trees via multiple computation processes, from which the most straightforward tree was selected (Devasena 2014). Observing training datasets and reducing the tree's internal structure complexity enables the REP tree to provide a flexible and uncomplicated modeling approach whenever the result is significant (Mohamed et al. 2012). During this approach, the pruning algorithm considers the backward overfitting complexity and uses the post-pruning algorithm to encourage the lowest version of the best precision tree logic (Quinlan 1987; Chen et al. 2019). It only chooses numbers for mathematical characteristics once (Kalmegh 2015).

A methodology for virtual ML called the vector method (GP regression) allows computer systems to adapt and enhance their capabilities. A method that acts directly above the feature space is GP regression, which is based on the idea that nearby observations must exchange data (Kuß 2006). Other GPs also include the expansion of the core of the Gaussian distribution. The mean and covariance show the Gaussian probability density matrix and the kernel-based correlation vector. GP regression models can predict the unknown input data based on the probability theorem. In addition, they may give expected accuracy, which raises the statistical significance of the predictive model's findings. A GP involves unlimited random variables; hence, procedures are grounded on multivariate Gaussian models. Since the invention of this approach a few years ago, it has been widely used in many study fields, including chemistry, medicine, construction, etc. (Singh et al. 2017).

Calculating the goodness-of-the-fit indices: In the present investigation, numerous necessary statistical measures such as the CC, normalized error (NE), RMSE, normalized root mean square errors (NRMSE), MAE, legates and McCabe's index (LMI), and Willmott's index (WI) were calculated to quantify the fit among the experimental data and the predicted data using applied models. Formulas of these indicators are listed in Equations (12)–(17).

In this study, the parameters were adapted to predict the y_c (m) and Q flowing through the depth model's terminus (Q (m³/s), free overfall. The results and discussion of LR, M5P, RF, RT, REP tree, and GP-based models' performance are covered in this portion.

Table 3 shows the achievement evaluation parameters for every one of the models utilized in the training and testing phases. The RT study estimates the y_c better than other models when examining performance assessment indicators during training. This model is more effective than the different applied models. Table 3 recommends that the LR model also performs superior to the GP_Poly model for predicting y_c. Typically, throughout the training stage, the models can be categorized from finest to lowest: RT, RF, Pearson VII kernel function-based GP model, REP tree, M5P, GP_RBF, LR, and GP_Poly.

The GP RBF model outperforms other applicable models in the testing process with the lowest RMSE of 0.0021, NRMSE of 0.0361, MAE of 0.0015 and the highest CC of 0.9912, LMI of 0.8839, WI of 0.9956, NE of 0.9823. This model is more effective than the previous one that has been used. Table 3 shows that the LR model outperforms the M5P, RF, RT, REP tree, and GP Poly models for y_c prediction, with CC values of 0.9837, RMSE values of 0.0028, WI values of 0.9927, LMI values of 0.8264, NE values of 0.9712, NRMSE values of 0.0482, and MAE values of 0.0021 for testing stages. The models can be ordered from good to bad throughout the testing stage: GP_RBF, GP_PUK, LR, RF, M5P, RT, REP tree, and GP_Poly. By using various soft computing algorithms, Appendix A displays the consistency plot of natural versus anticipated y_c values for training and testing phases. These figures indicate that the GP_RBF model is the best-applied model to predict the actual depth. All predicted values using the GP_RBF model lie much closer to the line of perfect agreement (y = x), with R² values as 0.989 and 0.982 for the training and testing stages, respectively. Results of single-factor ANOVA suggest that there is an insignificant difference among various groups. Table 4 indicates that all predictive models are suitable for predicting y_c using this dataset.

The performance assessment parameters of each of the models used to predict actual Q (m³/s) in the training and testing stages are listed in Table 5. The measurement of performance evaluation indices in the training stages shows that the RT model predicts the actual Q (m³/s) superior to other models. The RT model has had the minor RMSE = 0, NRMSE = 0, MAE = 0.0, and the most incredible CC = 1, LMI = 1, WI = 1, and Nash-Sutcliffe efficiency (NSE) = 1 in the training phase, according to the performance indicators. This model is more accurate than the other applied model's model. Table 5 suggests that the LR model is also performing better than the polynomial kernel function-based (GP_Poly)-based model for the prediction of actual Q with CC values as 0.9812, RMSE values as 0.0010, WI values as 0.9902, LMI values as 0.8112, NE values as 0.9620, NRMSE values as 0.0806, and MAE values as 0.0008 for the training stage. In general, throughout the training phase, the models can be categorized from the highest to the lowest in this manner: RT, RF, Pearson VII kernel function-based GP_PUK, REP tree, radial basis kernel function-based (GP_RBF), M5P, LR, and polynomial kernel function-based GP_Poly.

During the testing stage, the GP RBF model outperformed other applicable models in predicting accurate Q, with the lowest RMSE = 0.0007, NRMSE = 0.0516, MAE = 0.0004, and the highest CC = 0.9916, LMI = 0.9026, WI = 0.9956, NE = 0.9830. This model is much more effective than the others that have been used. Table 5 proposes that the LR model is also outperforming superior M5P, RF, RT, REP tree, and GP_Poly-based models for the prediction of actual Q with CC values as 0.9796, RMSE values as 0.0011, WI values as 0.9891, LMI values as 0.8189, NE values as 0.9584, NRMSE values as 0.0806, and MAE values as 0.0008 for testing stages. The models can be categorized from the best to lowest throughout the testing phase: GP_RBF, GP_PUK, LR, RF, M5P, RT, GP_Poly, and REP tree. Appendix B shows the agreement plot of actual versus predicted values of real Q with various soft computing techniques for training and testing stages. These figures also indicate that the GP_RBF model is the best among all applied models for the prediction of actual Q. All the predicted values using the GP_RBF model lie very close to a line of perfect agreement (y = x) with R² values as 0.989 and 0.983 for training and testing stages, respectively. Results of single-factor ANOVA suggest that there is an insignificant difference among various groups. Table 6 indicates that all predictive models are suitable for predicting actual Q using this dataset.

Because the GP RBF-based model outperforms all regression and soft programming models for predicting (y_c) and discharge crossing over the end-depth model (Q), a sensitivity analysis was conducted with the GP RBF model to determine the most sensitive feature among input variables for predicting (y_c) and discharge crossing over the terminal depth model (Q). For (y_c) and discharge crossing above the terminal depth model, input configuration models are generated by removing one input parameter in each case, as shown in Tables 7 and 8 (Q). The root-mean-square error, MAE, and CC are considered while evaluating each model's performance. The friction coefficient is the dataset's most significant input variable for evaluating y_c. Table 7 also indicates that the friction coefficient has a substantial influence in predicting the discharge passing over the end-depth model (Q) compared to other input parameters in this study. The friction coefficient and normal depth are the most significant input variables compared to other parameters for predicting (y_c) and discharge crossing above the end-depth model (Q).

This study examines how soft computing and regression approaches can forecast (y_c) and discharge over the terminal depth model (Q). For estimating (y_c) and flow crossing through the end-depth model, LR, M5P, RF, RT, REP tree, and GP-based models are utilized (Q). The behavior of constructed models is assessed using seven distinct goodness-of-fit criteria. According to achievement examination results, it is observed that the radial kernel function-based GP model (GP_RBF) is the most suitable model for the prediction of y_c and Q crossing above the end-depth model (Q) compared to other applied models with lowest RMSE = 0.0021, 0.007, NRMSE = 0.0361, 0.0516, MAE = 0.0015, 0.004 and the highest CC = 0.9912, 0.9916, LMI = 0.8839, 0.9026, WI = 0.9956, 0.9956 and NE = 0.9823, 09830 for y_c and Q crossing above the end-depth model (Q), respectively, with testing stage. Another primary outcome of this investigation is that LR-based equations' performance is better than all applied models except GP_PUK and GP_RBF for y_c and Q.

Sensitivity analysis results indicate that the friction coefficient is the most significant input variable compared to other parameters for predicting y_c and Q crossing above the end-depth model (Q) using this dataset.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.