Table 9 provides the statistical indicators of the employed kernel-depend models during the testing phase. It can be seen that under free-flow conditions, using the ratio of the downstream flow depth to the gate opening (y3/w) as the only input parameter provided the desired prediction accuracy while using only the ratio of the downstream flow depth to the radial gate radius (y3/r) as input did not yield the appropriate prediction accuracy. This can show the importance of the gate opening on the prediction process of the discharge coefficient under free-flow conditions. According to SI values, when comparing F(II) and F(III), adding the ratio of the gate opening to the radial gate radius (w/r) improves the average prediction accuracy of employed KELM-GWO and GPR approaches by 4 and 7%, respectively. It can be seen that the parameter y1/w is the most influential parameter for the prediction of the discharge coefficient of radial gates under free-flow conditions. Taking into consideration the statistical indices, KELM-GWO-Polynomial and GPR-Exponential models generated the most accurate results with model F(IV), where y1/w, w/r, and α-w/r were utilized as the input parameters. As mentioned in the previous section, under free-flow conditions, the comparison results of the employed kernel-depend methods revealed that the hybridized KELM-GWO-Polynomial with model F(IV) as input combination outperformed the SVM and GPR standalone models with R = 0.927, SI = 0.023, BIAS = −0.005946, and RMSE = 0.018. It should be noticed that the prediction of discharge coefficient under field conditions for multiple gates with different openings is complicated. It may become more intractable if one gate is free-flowing and another in the transition zone.

Table 9

Global statistical indices of the employed kernel-depend methods

Flow conditionMethodModelPerformance criteria
Testing
RSIBIASRMSE
Free flow KELM-GWO F(I) 0.305 0.057 −0.005410 0.044 
F(II) 0.866 0.028 0.002324 0.022 
F(III) 0.907 0.026 0.008926 0.021 
F(IV) 0.927 0.023 0.005946 0.018 
GPR F(I) 0.549 0.050 0.000421 0.038 
F(II) 0.896 0.025 −0.003127 0.019 
F(III) 0.908 0.024 −0.002365 0.018 
F(IV) 0.911 0.023 0.002162 0.018 
SVM F(I) 0.442 0.052 −0.004215 0.040 
F(II) 0.870 0.028 0.002492 0.022 
F(III) 0.880 0.028 −0.004524 0.021 
F(IV) 0.896 0.027 0.001713 0.021 
Submerged flow KELM-GWO S(I) 0.657 0.276 0.080632 0.214 
S(II) 0.680 0.254 0.031145 0.197 
S(III) 0.767 0.255 0.035795 0.197 
S(IV) 0.839 0.185 0.027117 0.144 
GPR S(I) 0.727 0.213 −0.000803 0.165 
S(II) 0.828 0.176 0.012242 0.136 
S(III) 0.880 0.148 0.006356 0.115 
S(IV) 0.944 0.102 0.009989 0.079 
SVM S(I) 0.864 0.179 −0.017431 0.139 
S(II) 0.822 0.183 0.010703 0.141 
S(III) 0.897 0.162 −0.015473 0.125 
S(IV) 0.940 0.106 0.000572 0.022 
Flow conditionMethodModelPerformance criteria
Testing
RSIBIASRMSE
Free flow KELM-GWO F(I) 0.305 0.057 −0.005410 0.044 
F(II) 0.866 0.028 0.002324 0.022 
F(III) 0.907 0.026 0.008926 0.021 
F(IV) 0.927 0.023 0.005946 0.018 
GPR F(I) 0.549 0.050 0.000421 0.038 
F(II) 0.896 0.025 −0.003127 0.019 
F(III) 0.908 0.024 −0.002365 0.018 
F(IV) 0.911 0.023 0.002162 0.018 
SVM F(I) 0.442 0.052 −0.004215 0.040 
F(II) 0.870 0.028 0.002492 0.022 
F(III) 0.880 0.028 −0.004524 0.021 
F(IV) 0.896 0.027 0.001713 0.021 
Submerged flow KELM-GWO S(I) 0.657 0.276 0.080632 0.214 
S(II) 0.680 0.254 0.031145 0.197 
S(III) 0.767 0.255 0.035795 0.197 
S(IV) 0.839 0.185 0.027117 0.144 
GPR S(I) 0.727 0.213 −0.000803 0.165 
S(II) 0.828 0.176 0.012242 0.136 
S(III) 0.880 0.148 0.006356 0.115 
S(IV) 0.944 0.102 0.009989 0.079 
SVM S(I) 0.864 0.179 −0.017431 0.139 
S(II) 0.822 0.183 0.010703 0.141 
S(III) 0.897 0.162 −0.015473 0.125 
S(IV) 0.940 0.106 0.000572 0.022 

Italic values indicate superior results.

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