## Abstract

Most of the studies on labyrinth weir were carried out in the laboratory, and regression models have been developed for discharge coefficient in terms of pertinent independent parameters. It is difficult to obtain an exact analytical solution to the head discharge relationship due to the existence of 3D flow. Consequently, various forms of soft computing techniques are used as an appropriate alternative to achieve greater accuracy in developing a discharge prediction model. In the present study, support vector regression (SVR) has, therefore, been implemented to develop a discharge coefficient prediction model for a triangular labyrinth (TL) weir using a sizeable amount of laboratory data available in the literature. An attempt has also been made to obtain a simple discharge coefficient equation using the same data based on the non-linear regression (NLR) approach for field application. A comparative study has been carried out to assess the accuracy of the discharge coefficient models obtained in the present study and those reported in the literature. Sensitivity analysis has been made to study the influence of individual parameters on the discharge coefficient. The accuracy of different discharge coefficient prediction models was also tested for the data of prototype labyrinth weir and appropriate models were recommended for the field application.

## HIGHLIGHTS

In the present study, SVR has been implemented to develop a Cd prediction model for a TL weir using a sizeable amount of laboratory data available in the literature. An attempt has also been made to obtain a simple Cd equation using the same data based on the NLR approach for field application. A comparative study has been carried out to assess the accuracy of the Cd models obtained in the present study.

### Graphical Abstract

## NOTATIONS

## INTRODUCTION

Gentilini (Hager *et al.* 2015) conducted the first experiment on the sharp-crested oblique weir and triangular labyrinth weir (obtained by connecting an even number of oblique weirs) having vertex angles 60°, 90°, 120°. He changed channel width, crest length, number of oblique weirs, and orientation (normal and inverted) and plotted the graphs between the ratio of discharge coefficients (*C _{d}*/

*C*) of the labyrinth weir to the normal weir and the ratio of head over the weir to channel width (

_{dn}*h*/

*W*), which indicate that the data points corresponding to the same side wall angle lie on the same curve irrespective of their plan shape. Hay & Taylor (1970) extensively investigated the performance of labyrinth weirs of different plan forms like triangular, rectangular, and trapezoidal. They defined the ratio of discharge over labyrinth weir

*Q*to the discharge over normal weir

*Q*operating under similar hydraulic conditions as magnification ratio. They presented the results in the form of magnification ratio (

_{n}*Q*/

*Q*) versus head-to-weir height ratio (

_{n}*h/p*) and found that the discharge magnification decreases with an increase in

*h*/

*p*.

Darvas (1971) introduced an empirical discharge equation along with a trapezoidal labyrinth weir design chart. Hinchliff & Houston (1984) developed design guidelines based on the model study of the Ute dam and Hyrum dam. Lux (1984) proposed a different discharge coefficient equation, including vertical aspect ratio (*w*/*p*) and shaped constant *k*. Based on the experimental investigations, Tullis *et al.* (1995) proposed a procedure for designing a trapezoidal labyrinth weir. They also proposed the discharge coefficient equations as a fourth-order polynomial of *h/p* for different side wall angles. Willmore (2004) also developed equations for *C _{d}* similar to that of Tullis

*et al.*(1995). Tullis

*et al.*(2007) proposed a dimensionless submerged head relationship for the trapezoidal labyrinth weir.

Kumar *et al.* (2011) carried out an experimental study on a single-cycle sharp-crested triangular labyrinth (TL) weir having different vertex angles (*θ* = 30°, 60°, 90°, 120°, 150°, and 180°) and observed that the interference of nappe is more pronounced at the least apex angle. They proposed a fourth-order polynomial fit equation for *C _{d}* as a function

*h/p*for different vertex angles. Gupta

*et al.*(2014) extended the work of Kumar

*et al.*(2011), including three more vertex angles (

*θ*= 75°, 105°, 135°), and proposed another similar equation for

*C*. It is worthwhile to note that using a polynomial of degree n for labyrinth weirs having m configurations (different side wall angles) requires m*(

_{d}*n*+ 1) coefficients leading to a complex form of equation for estimation of discharge coefficient.

Bijankhan & Kouchakzadeh (2017) tested a single-cycle triangular labyrinth weir having vertex angles of (74°, 90°, 120°, and 180°). They developed a criterion to distinguish between free and submerged flow conditions experimentally. They further proposed a mathematical shape for the submerged stage-discharge formula based on the dimensional analysis and self-similarity concept. They argued that the proposed rating curve may be used for free and submerged flow conditions continuously and within the transition zone. Finally, they deduced an expression for the discharge coefficient from the stage-discharge relationship. However, so many coefficients are involved in determining the discharge coefficient.

Some of the well-known predictors are summarized in Table 1. The equations developed for *C _{d}* are applicable only in the range of data covered in a particular study. Therefore, it is necessary to develop a simple and accurate equation to compute

*C*covering a wide range of data available in the literature.

_{d}Author . | Proposed equation . | Remarks . | Weir type . | Crest Type . |
---|---|---|---|---|

Ghodsian (2009) | 0.3 ≤ h/p ≤ 0.7 | Triangular- 2 cycles | SC-QR-HR-FT | |

Kumar et al. (2011) | 0 | Triangular plan form | SC | |

Gupta et al. (2014) | 0< h/p < 0.7 | Triangular labyrinth weir | S.C. | |

Karami et al. (2017) | Triangular plan form | SC | ||

Bonakdari et al. (2020) | y = upstream flow depth | Triangular plan form | SC |

Author . | Proposed equation . | Remarks . | Weir type . | Crest Type . |
---|---|---|---|---|

Ghodsian (2009) | 0.3 ≤ h/p ≤ 0.7 | Triangular- 2 cycles | SC-QR-HR-FT | |

Kumar et al. (2011) | 0 | Triangular plan form | SC | |

Gupta et al. (2014) | 0< h/p < 0.7 | Triangular labyrinth weir | S.C. | |

Karami et al. (2017) | Triangular plan form | SC | ||

Bonakdari et al. (2020) | y = upstream flow depth | Triangular plan form | SC |

Note: S.C. – Sharp crested, Q.R. – Quarter round, H.R. – Half round, F.T. – Flat top.

Recent advancements in soft computing (genetic programming, neural networks, fuzzy genetic programming, decision support systems, neuro-fuzzy systems, support vector machines, etc.) attracted researchers to use them in the complex flow phenomenon over a labyrinth weir and to develop discharge coefficient predictions.

Laboratory data collected by Kumar *et al.* (2011) for one cycle triangular labyrinth weir has been used by many researchers for the development of a discharge coefficient model based on soft computing techniques. Zaji *et al.* (2015) estimated the discharge coefficient employing multiple non-linear particle swarm optimization (MNLPSO), multiple linear particle swarm optimization (MLPSO), and radial basis neural network (RBNN) and found that MLPSO performs well. Zaji *et al.* (2016) applied a firefly optimization algorithm (FFA) to the discharge coefficient prediction model for better accuracy. Parsaie & Haghiabi (2017) reported that the SVM-based model performed better than the RBNN and MLPNN. Karami *et al.* (2017) showed that support vector regression (SVR)-Firefly yielded better results as compared to the classic SVR, principal component analysis (PCA), and response surface methodology (RSM). Karami *et al.* (2018) developed the discharge coefficient model using genetic programming (GP), artificial neural network (ANN), and extreme learning machine (ELM) and showed that the ELM model performed the best. Haghiabi *et al.* (2018) employed multilayer perceptron (MLP) and ANFIS to predict *C _{d,}* and they found that the structure of ANFIS model is more optimal than that of MLP model. Bonakdari

*et al*. (2020) proposed an equation for

*C*using non-linear regression (NLR) and modeled the data with gene expression programming (GEP) to predict

_{d}*C*. The results of GEP were more capable than NLR. Mahmoud

_{d}*et al.*(2021) estimated the flow rate of a sharp-crested triangular labyrinth weir as a function of head-to-weir height ratio

*h/p*and sidewall angle (

*θ*) through several soft computing tools such as the adaptive neuro-fuzzy inference system (ANFIS), MLP, SVR and radial basis function neural network (RBFNN). Emami

*et al.*(2022) used the walnut optimization algorithm employing five parameters (

*L/h*,

*Fr*,

*L/p*,

*θ*,

*h/p*) and found the model containing

*L/p*,

*h/p*, and

*θ*predicts the best value of

*C*. Table 2 summarizes the performance of the soft computing technique for dataset 1. The performance of the SVR (Firefly algorithm) prediction model developed by Karami

_{d}*et al.*(2017) is the best.

Literature . | Method . | RMSE . | R^{2}
. |
---|---|---|---|

Zaji et al. (2015) | MNLPSO | 0.0220 | 0.94 |

Karami et al. (2017) | SVR- FA | 0.0035 | 0.99 |

Karami et al. (2018) | ELM | 0.0060 | 0.97 |

Bonakdari et al. (2020) | GEP | 0.0260 | 0.93 |

Shafiei et al. (2020) | ORELM | 0.0320 | 0.94 |

Mahmoud et al. (2021) | ANFIS | 0.2100 | 0.99 |

Emami et al. (2022) | Walnut-SVR | 0.0040 | 0.99 |

Literature . | Method . | RMSE . | R^{2}
. |
---|---|---|---|

Zaji et al. (2015) | MNLPSO | 0.0220 | 0.94 |

Karami et al. (2017) | SVR- FA | 0.0035 | 0.99 |

Karami et al. (2018) | ELM | 0.0060 | 0.97 |

Bonakdari et al. (2020) | GEP | 0.0260 | 0.93 |

Shafiei et al. (2020) | ORELM | 0.0320 | 0.94 |

Mahmoud et al. (2021) | ANFIS | 0.2100 | 0.99 |

Emami et al. (2022) | Walnut-SVR | 0.0040 | 0.99 |

Note: Bold values belong to the best model.

Support vector machine (SVM) has also successfully employed in various fields of hydraulic engineering such as determination of discharge coefficient of weir (Haghiabi *et al.* 2017; Azamathulla *et al.* 2019), evapotranspiration (Choubin *et al.* 2019), sediment transport (Choubin *et al.* 2018; Kisi *et al.* 2019) and rainfall modelling and river flow prediction (Azamathulla & Wu 2011), scour prediction on grade control structures (Goel & Pal 2009).

Nonetheless, the complex three-dimensional flow pattern over the labyrinth weir makes it challenging to obtain an accurate analytical solution to the head discharge relationship (Crookston & Tullis 2012). As a result, soft computing techniques could be considered an appropriate alternative. The performance of the SVM in the discharge prediction model has been reported as the best among the other commonly used soft computing techniques.

The main objective of the present study is, therefore, to (i) assess the accuracy of SVR in the development of a discharge coefficient prediction model for a triangular labyrinth weir using a sizeable amount of laboratory data available in the literature (ii) to obtain a simple discharge coefficient equation using the same data based on the non-linear regression (NLR) approach for field application (iii) to compare the accuracy of the discharge coefficient models obtained in the present study with those reported in the literature, and (iv) to study the influence of individual parameters on discharge coefficient using sensitivity analysis.

## MATERIAL AND METHODS

The discharge prediction model for the triangular labyrinth weir was developed based on the conventional approach of regression analysis and the most popular soft computing approach using the laboratory data obtained from the literature as described herein**.**

### Dataset and range of data

Laboratory data collected under free flow conditions by Kumar *et al.* (2011), Gupta *et al.* (2014), and Bijankhan & Kouchakzadeh (2017) has been procured from the literature and grouped under four sets, as shown in Table 3. The range of the pertinent parameters is shown in Table 4.

Datasets . | Dataset 1 . | Dataset 2 . | Dataset 3 . | Dataset 4 . |
---|---|---|---|---|

Data source | Kumar et al. (2011) | Kumar et al. (2011) and Gupta et al. (2014) | Kumar et al. (2011) and Bijankhan & Kouchakzadeh (2017) | Kumar et al. (2011), Gupta et al. (2014) and Bijankhan & Kouchakzadeh (2017) |

No. of data | 123 | 123 + 95 = 218 | 123 + 42 = 165 | 123 + 95 + 42 = 260 |

Datasets . | Dataset 1 . | Dataset 2 . | Dataset 3 . | Dataset 4 . |
---|---|---|---|---|

Data source | Kumar et al. (2011) | Kumar et al. (2011) and Gupta et al. (2014) | Kumar et al. (2011) and Bijankhan & Kouchakzadeh (2017) | Kumar et al. (2011), Gupta et al. (2014) and Bijankhan & Kouchakzadeh (2017) |

No. of data | 123 | 123 + 95 = 218 | 123 + 42 = 165 | 123 + 95 + 42 = 260 |

Parameter . | Kumar et al. (2011)
. | Gupta et al. (2014)
. | Bijankhan & Kouchakzadeh (2017) . |
---|---|---|---|

Channel dimension (m) | 12 × 0.28 × 0.41 | 5.36 × 0.26 × 0.45 | 12 × 0.6 × 0.4 |

Weir Type | Triangular | Triangular | Triangular |

Discharge (m^{3}/s) | 0.54–0.80 | 0.38–0.61 | 0.73–0.81 |

Head (m) | 0.008–0.069 | 0.0267–0.0808 | 0.01–0.03 |

Weir height (m) | 0.092–0.108 | 0.10 | 0.10 |

Vertex angle (^{o}) | 30–180 | 60–180 | 90–180 |

No. of runs | 123 | 95 | 42 |

Parameter . | Kumar et al. (2011)
. | Gupta et al. (2014)
. | Bijankhan & Kouchakzadeh (2017) . |
---|---|---|---|

Channel dimension (m) | 12 × 0.28 × 0.41 | 5.36 × 0.26 × 0.45 | 12 × 0.6 × 0.4 |

Weir Type | Triangular | Triangular | Triangular |

Discharge (m^{3}/s) | 0.54–0.80 | 0.38–0.61 | 0.73–0.81 |

Head (m) | 0.008–0.069 | 0.0267–0.0808 | 0.01–0.03 |

Weir height (m) | 0.092–0.108 | 0.10 | 0.10 |

Vertex angle (^{o}) | 30–180 | 60–180 | 90–180 |

No. of runs | 123 | 95 | 42 |

### Regression model

The discharge coefficient basically depends on geometric and flow parameters. Two NLR models of the following form have been adopted in the present study in Table 5.

Models . | Model details . | Remarks . |
---|---|---|

RM1 | a_{i}, b, and _{i}c are coefficients, and _{i}m is the exponents. _{i}η = h/p; θ = vertex angle; β = W/P; W = channel width, P = weir height, h = total head | |

RM2 |

Models . | Model details . | Remarks . |
---|---|---|

RM1 | a_{i}, b, and _{i}c are coefficients, and _{i}m is the exponents. _{i}η = h/p; θ = vertex angle; β = W/P; W = channel width, P = weir height, h = total head | |

RM2 |

The coefficients and exponents of these models may be estimated using the observed set of data through NLR analysis.

### Support vector machine

*ε*for all training data. The non-linear model of the SVR is given bywhere

*f*(

*x*) = output of the model,

*w*= weight vector,

*φ*(

*x*) = nonlinear function in the feature space,

*b*= bias term, and denotes the inner product. Here, a lower value of w denotes the flatness of Equation (1), which is accomplished by reducing the Euclidean norm ‖

*w*‖ (Smola

*et al.*1996). An optimization problem, like the following, can be used to determine the ideal weight vector

*w*and non-linear function

*φ*(

*x*).

introducing slack variables as into Equation (2) to make the constraints feasible, the problem reduces to estimate *w* and *b* that minimise the following function:

*ε*. The values of C and

*ε*are suitably selected by users to find the optimal solution.

## ANALYSIS, RESULTS AND DISCUSSION

Discharge coefficient prediction models for a triangular labyrinth weir using a sizeable amount of laboratory data (260) available in the literature have been developed based on non-linear regression (Minitab Software) and SVR (DTREG Software). *C _{d}* was considered as a dependent variable against the independent variables as

*h/p*,

*θ, N,*and

*W/p*for the SVR model and

*h*/

*p*,

*θ*, and

*N*in the regression model

*.*The results of the analysis are discussed in the following sections.

### Regression models

The performance of these regression models for the entire data has been evaluated and shown in Table 6. This table indicates that performance of RM1 is better than that of RM2.

Models . | Performance parameters . | Dataset 4 . |
---|---|---|

RM1 | MAPE | 3.786058 |

RMSE | 0.034135 | |

RM2 | MAPE | 8.890694 |

RMSE | 0.067149 |

Models . | Performance parameters . | Dataset 4 . |
---|---|---|

RM1 | MAPE | 3.786058 |

RMSE | 0.034135 | |

RM2 | MAPE | 8.890694 |

RMSE | 0.067149 |

S.N. . | Box plot summaries . | Details of parameters of box and whisker plot . |
---|---|---|

1 | Minimum score | Lowest score, excluding outliers |

2 | Lower quartile (Q_{1}) | 25% of scores fall below the lower quartile value |

3 | Median (Q_{2}) | It is the midpoint of the data and divides the box into two parts |

4 | Upper quartile (Q_{3}) | 75% of the scores fall below the upper quartile value |

5 | Maximum score | The highest score, excluding outliers |

S.N. . | Box plot summaries . | Details of parameters of box and whisker plot . |
---|---|---|

1 | Minimum score | Lowest score, excluding outliers |

2 | Lower quartile (Q_{1}) | 25% of scores fall below the lower quartile value |

3 | Median (Q_{2}) | It is the midpoint of the data and divides the box into two parts |

4 | Upper quartile (Q_{3}) | 75% of the scores fall below the upper quartile value |

5 | Maximum score | The highest score, excluding outliers |

(a) Interquartile range (IQR): The box plot shows the middle 50% of scores (i.e., the range between the 25th and 75th percentile). IQR = Q_{3}-Q_{1.}

(b) Upper and lower whiskers: The upper and lower whiskers are the scores outside the middle 50% (i.e., the lower 25% of scores and the upper 25% of scores).

(c) Outliers are those measurements not within 1.5 × IQR of the lower and upper quartiles.

The predicted to observed discharge coefficients ratio (* PODCR)* have been compared in the box-and-whisker diagrams, as shown in Figure 2. It may be clearly observed that the scatter for the RM1 is less than that of RM2, although the mean values for both models are nearly one. This indicates that the prediction of Model 1 is more satisfactory as compared to RM2.

### Support vector regression (SVR) model for prediction of discharge coefficient

Different forms of discharge coefficient predictors have been reported by various investigators (Table 1). Five possible prediction models, as provided in Table 8, were considered herein to test the best model for the present set of data.* _{.}* The performance of each model is assessed through statistical performance indices. Table 9 shows the performance indices of the proposed various SVR prediction models. A perusal of this table indicates that the SVR3 model containing three geometric parameters,

*h/p*,

*θ*, and

*N,*produce the most accurate results. Further, it can be observed that SVR3 gives the highest accuracy with training R

^{2}= 0.997 and testing R

^{2}= 0.994 with the Hyper parameters' gamma = 15.018, C = 1.408, and epsilon = 0.001. It is further observed that the performance of SVR3, SVR4, and SVR5 are also fairly comparable.

Models . | Model detail . |
---|---|

SVR 1 | C_{d}=f (h/p) |

SVR 2 | C_{d}=f (h/p, θ) |

SVR 3 | C_{d}=f (h/p, θ, N) |

SVR 4 | C_{d}=f (h/p, θ, N, W/p) |

SVR 5 | C_{d}=f (h/p, θ, W/p) |

Models . | Model detail . |
---|---|

SVR 1 | C_{d}=f (h/p) |

SVR 2 | C_{d}=f (h/p, θ) |

SVR 3 | C_{d}=f (h/p, θ, N) |

SVR 4 | C_{d}=f (h/p, θ, N, W/p) |

SVR 5 | C_{d}=f (h/p, θ, W/p) |

Models . | Training dataset . | Testing dataset . | ||||
---|---|---|---|---|---|---|

R^{2}
. | RMSE . | MAPE . | R^{2}
. | RMSE . | MAPE . | |

SVR 1 | 0.599 | 0.774 | 10.319 | 0.0.592 | 0.078 | 10.397 |

SVR 2 | 0.744 | 0.062 | 8.503 | 0.734 | 0.063 | 8.667 |

SVR 3 | 0.997 | 0.0069 | 0.649 | 0.994 | 0.0098 | 1.133 |

SVR 4 | 0.997 | 0.0069 | 0.796 | 0.994 | 0.0095 | 1.154 |

SVR 5 | 0.997 | 0.00667 | 0.763 | 0.992 | 0.0106 | 1.220 |

Models . | Training dataset . | Testing dataset . | ||||
---|---|---|---|---|---|---|

R^{2}
. | RMSE . | MAPE . | R^{2}
. | RMSE . | MAPE . | |

SVR 1 | 0.599 | 0.774 | 10.319 | 0.0.592 | 0.078 | 10.397 |

SVR 2 | 0.744 | 0.062 | 8.503 | 0.734 | 0.063 | 8.667 |

SVR 3 | 0.997 | 0.0069 | 0.649 | 0.994 | 0.0098 | 1.133 |

SVR 4 | 0.997 | 0.0069 | 0.796 | 0.994 | 0.0095 | 1.154 |

SVR 5 | 0.997 | 0.00667 | 0.763 | 0.992 | 0.0106 | 1.220 |

Note: Bold values belong to the best model.

### A comparative study of various discharge coefficient predictors

A comparison of existing discharge coefficient predictors for dataset 1 has been made to validate the present approach. Dataset 2, containing the entire data (260), has been used for the development of a discharge coefficient prediction model based on regression and the SVR approach.

#### A comparative study of regression models (RM) and SVR models for dataset 1

*et al.*(2017) and Bonakdari

*et al.*(2020) but inferior to Kumar

*et al.*(2011). However, the equation proposed by Kumar

*et al.*(2011) is complicated as it involves a fourth-order polynomial in

*θ*and is not easy for application. The equation proposed in the present study is simple and easy to use. As per the criteria for the scale of the judgment of predicted accuracy (Lewis scale 1982) in Table 11, all predictors lie in a highly accurate range. Table 12 indicates the performance indices for the SVR model developed in the present study and that is available in the literature for Dataset 1. It may be observed that the present model performs equally well with Karami

*et al.*(2017). A visual comparison of the prediction model has been made through a visual plot, as shown in Figure 4. A similar conclusion may also been drawn from this plot.

Performance parameters . | Present study . | Predictors available in the literature . | |||
---|---|---|---|---|---|

RM1 . | RM2 . | Karami et al. (2017)
. | Bonakdari et al. (2020)
. | Kumar et al. (2011)
. | |

MAPE | 3.92 | 3.945 | 5.36 | 4.66 | 2.11 |

RMSE | 0.041 | 0.038 | 0.046 | 0.044 | 0.011 |

Performance parameters . | Present study . | Predictors available in the literature . | |||
---|---|---|---|---|---|

RM1 . | RM2 . | Karami et al. (2017)
. | Bonakdari et al. (2020)
. | Kumar et al. (2011)
. | |

MAPE | 3.92 | 3.945 | 5.36 | 4.66 | 2.11 |

RMSE | 0.041 | 0.038 | 0.046 | 0.044 | 0.011 |

MAPE . | Judgment of predicted accuracy . |
---|---|

Less than 10% | Highly accurate |

10% to 20% | Good predicted |

21% to 50% | Reasonable |

51% and above | Inaccurate |

MAPE . | Judgment of predicted accuracy . |
---|---|

Less than 10% | Highly accurate |

10% to 20% | Good predicted |

21% to 50% | Reasonable |

51% and above | Inaccurate |

Performance parameters . | Present study . | Karami et al. (2017)
. |
---|---|---|

MAPE | 0.88 | 0.90 |

RMSE | 0.009 | 0.006 |

Performance parameters . | Present study . | Karami et al. (2017)
. |
---|---|---|

MAPE | 0.88 | 0.90 |

RMSE | 0.009 | 0.006 |

Note: Bold value indicates the best model.

#### A comparative study of RM and SVR models for all sets of data in the present study

Model . | Performance parameters . | Dataset 1 . | Dataset 2 . | Dataset 3 . | Dataset 4 . |
---|---|---|---|---|---|

RM1 | MAPE | 3.945 | 4.420815 | 3.19367 | 3.786058 |

RMSE | 0.0378 | 0.037219 | 0.032977 | 0.034135 | |

SVR3 | MAPE | 0.883 | 1.002 | 0.8175 | 0.649 |

RMSE | 0.0089 | 0.0083 | 0.0789 | 0.0069 |

Model . | Performance parameters . | Dataset 1 . | Dataset 2 . | Dataset 3 . | Dataset 4 . |
---|---|---|---|---|---|

RM1 | MAPE | 3.945 | 4.420815 | 3.19367 | 3.786058 |

RMSE | 0.0378 | 0.037219 | 0.032977 | 0.034135 | |

SVR3 | MAPE | 0.883 | 1.002 | 0.8175 | 0.649 |

RMSE | 0.0089 | 0.0083 | 0.0789 | 0.0069 |

Model . | Performance parameters . | Dataset 1 . | Dataset 2 . | Dataset 3 . | Dataset 4 . |
---|---|---|---|---|---|

RM2 | MAPE | 3.945 | 5.360559 | 3.161388 | 8.890694 |

RMSE | 0.0378 | 0.046624 | 0.032904 | 0.067149 | |

SVR5 | MAPE | 0.897 | 1.218 | 0.7337 | 0.763 |

RMSE | 0.00896 | 0.009 | 0.0078 | 0.00677 |

Model . | Performance parameters . | Dataset 1 . | Dataset 2 . | Dataset 3 . | Dataset 4 . |
---|---|---|---|---|---|

RM2 | MAPE | 3.945 | 5.360559 | 3.161388 | 8.890694 |

RMSE | 0.0378 | 0.046624 | 0.032904 | 0.067149 | |

SVR5 | MAPE | 0.897 | 1.218 | 0.7337 | 0.763 |

RMSE | 0.00896 | 0.009 | 0.0078 | 0.00677 |

### Sensitivity analysis

*h/p*have the most,

*θ*moderate, and

*W/p*the least influence on the discharge coefficient. However, the influence of N is also almost in the range of

*W/p. h/p, θ, N*and

*W/p*influence the discharge coefficient in descending order. This can also be observed qualitatively in Figure 7.

Independent variables . | MAPE . | Relative MAPE . |
---|---|---|

All independent variables except W/p | 0.65 | 0.82 |

All independent variables except N | 0.76 | 0.95 |

All independent variables except θ | 4.67 | 5.87 |

All independent variables except h/p | 5.71 | 7.17 |

All independent variables | 0.80 | 1.00 |

Independent variables . | MAPE . | Relative MAPE . |
---|---|---|

All independent variables except W/p | 0.65 | 0.82 |

All independent variables except N | 0.76 | 0.95 |

All independent variables except θ | 4.67 | 5.87 |

All independent variables except h/p | 5.71 | 7.17 |

All independent variables | 0.80 | 1.00 |

Name . | Q . | Ht . | P . | W . | L . | N . | α . | Cdo . |
---|---|---|---|---|---|---|---|---|

Bartletts Ferry | 5,920.00 | 2.19 | 3.43 | 18.30 | 70.30 | 20.50 | 14.00 | 0.43 |

Boardman | 387.00 | 1.77 | 2.76 | 18.30 | 53.50 | 2.00 | 19.40 | 0.50 |

Ritchard | 1,555.00 | 2.74 | 3.05 | 83.80 | 411.00 | 9.00 | 11.50 | 0.28 |

Worona | 1,020.00 | 1.36 | 2.13 | 13.41 | 31.23 | 11.00 | 25.00 | 0.63 |

UTE | 15,570.00 | 5.79 | 9.14 | 18.30 | 73.70 | 14.00 | 15.20 | 0.37 |

Name . | Q . | Ht . | P . | W . | L . | N . | α . | Cdo . |
---|---|---|---|---|---|---|---|---|

Bartletts Ferry | 5,920.00 | 2.19 | 3.43 | 18.30 | 70.30 | 20.50 | 14.00 | 0.43 |

Boardman | 387.00 | 1.77 | 2.76 | 18.30 | 53.50 | 2.00 | 19.40 | 0.50 |

Ritchard | 1,555.00 | 2.74 | 3.05 | 83.80 | 411.00 | 9.00 | 11.50 | 0.28 |

Worona | 1,020.00 | 1.36 | 2.13 | 13.41 | 31.23 | 11.00 | 25.00 | 0.63 |

UTE | 15,570.00 | 5.79 | 9.14 | 18.30 | 73.70 | 14.00 | 15.20 | 0.37 |

Performance parameters . | Present Study . | Ghodsian (2009) . | Kumar et al. (2011)
. | Karami et al. (2017)
. | |
---|---|---|---|---|---|

RM1 . | RM2 . | ||||

MAPE | 415.81 | 17.00 | 36.19 | 20.59 | 82.84 |

RMSE | 1.93 | 0.09 | 0.162 | 0.098 | 0.381 |

Performance parameters . | Present Study . | Ghodsian (2009) . | Kumar et al. (2011)
. | Karami et al. (2017)
. | |
---|---|---|---|---|---|

RM1 . | RM2 . | ||||

MAPE | 415.81 | 17.00 | 36.19 | 20.59 | 82.84 |

RMSE | 1.93 | 0.09 | 0.162 | 0.098 | 0.381 |

### Accuracy assessment of various discharge coefficients prediction models for the prototype data

*α*≤ 25° have been selected for accuracy assessment of various discharge coefficients prediction models. The performance of different discharge coefficient prediction models for prototype LW has been evaluated, and results are provided in Table 17 and box-and-whisker plot (Figure 8). These results indicate that the performance of RM1 and models proposed by Karami

*et al.*(2017) are inaccurate; that of RM2 in the present study and Kumar

*et al.*(2011) are well predicted, and the performance of Ghodsian (2009) prediction models is reasonable on the basis of scale of the judgment of predicted accuracy (Lewis scale 1982). The inaccuracy of the present RM1 may be attributed to the involvement of insufficient size of cycle (N) of the LW in the proposed equation. A detailed investigation of the effect of N on the discharge coefficient is recommended for future research.

## CONCLUSION

SVR and NLR models for the discharge coefficient of a TL weir have been developed in the present study using a sizeable amount of laboratory data available in the literature. The performance of these models has been assessed based on statistical performance indices such as mean absolute percentage error (MAPE) and mean square error (MSE). It was found that the SVR model performs better than the NLR model. However, a simple discharge coefficient equation obtained from the NLR approach may be used for field application. The sensitivity analysis indicates that the parameters *h/p* have the most, *θ* moderate, and *W/p* the least influence on the discharge coefficient. However, the influence of N is also almost in the range of W/p. The results of the accuracy assessment of various discharge coefficients prediction models for the prototype data indicate that the performance of RM1 and models proposed by Karami *et al.* (2017) are inaccurate; that of RM2 in the present study and Kumar *et al.* (2011) are well predicted, and performance of Ghodsian (2009) prediction models is reasonable on the basis of scale of the judgment of predicted accuracy (Lewis scale 1982). The inaccuracy of the present RM1 may be attributed to the involvement of insufficient size of cycle (N) of the LW in the proposed equation. A detailed investigation of the effect of N on the discharge coefficient is recommended for future research.

## DATA AVAILABILITY STATEMENT

All relevant data are available from an online repository or repositories.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

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