Performance criteria for semi-empirical formulas corresponding to the ski-jump bucket, sharp-crested weir and inclined slope, are listed in Table 7. The table also summarizes the results of the best data driven models. From the table, it is seen that the GEP and FFNN models are superior to semi-empirical models in estimating scour depth. Figures 5–7 compare the performance of the applied data driven and semi-empirical models in scour depth estimation for all three cases. Comparing the results, it may be stated that the GEP and FFNN methods produce more accurate results in the estimation of scour depth.
Table 7

Error statistics of the semi-empirical and data driven models of scour depth estimation

ModelCCR2RMSEMAE
Ski-jump bucket 
 Veronese (1937)  0.393 0.425 0.221 0.155 
 Azamathulla (2012)  0.503 0.582 0.059 0.041 
 Sofrelec (1980)  0.399 0.457 0.233 0.172 
 Mason & Arumugam (1985)  0.413 0.506 0.142 0.096 
 GEP-MB4 0.841 0.858 0.034 0.027 
 FFNN-MB4 0.863 0.876 0.032 0.024 
Sharp edge over flow 
 Yen (1987)  0.612 0.661 0.576 0.337 
 D'Agostino & Ferro (2004)  0.598 0.658 0.356 0.184 
 GEP-MS5 0.814 0.874 0.112 0.079 
 FFNN-MS5 0.862 0.885 0.104 0.059 
Inclined slope controlled structure 
 Chee & Yuen (1985)  0.501 0.598 0.489 0.413 
 Bormann & Julien (1991)  0.557 0.632 0.717 0.570 
 GEP-MI3 0.803 0.769 0.198 0.164 
 FFNN-MI3 0.833 0.835 0.175 0.131 
ModelCCR2RMSEMAE
Ski-jump bucket 
 Veronese (1937)  0.393 0.425 0.221 0.155 
 Azamathulla (2012)  0.503 0.582 0.059 0.041 
 Sofrelec (1980)  0.399 0.457 0.233 0.172 
 Mason & Arumugam (1985)  0.413 0.506 0.142 0.096 
 GEP-MB4 0.841 0.858 0.034 0.027 
 FFNN-MB4 0.863 0.876 0.032 0.024 
Sharp edge over flow 
 Yen (1987)  0.612 0.661 0.576 0.337 
 D'Agostino & Ferro (2004)  0.598 0.658 0.356 0.184 
 GEP-MS5 0.814 0.874 0.112 0.079 
 FFNN-MS5 0.862 0.885 0.104 0.059 
Inclined slope controlled structure 
 Chee & Yuen (1985)  0.501 0.598 0.489 0.413 
 Bormann & Julien (1991)  0.557 0.632 0.717 0.570 
 GEP-MI3 0.803 0.769 0.198 0.164 
 FFNN-MI3 0.833 0.835 0.175 0.131 
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Figure 5
Figure 5. Comparison of semi-empirical formulas and data driven model performances for the ski-jump bucket case (model: MB4).
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Comparison of semi-empirical formulas and data driven model performances for the ski-jump bucket case (model: MB4).

Figure 5
Figure 5. Comparison of semi-empirical formulas and data driven model performances for the ski-jump bucket case (model: MB4).
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Comparison of semi-empirical formulas and data driven model performances for the ski-jump bucket case (model: MB4).

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Figure 6
Figure 6. Comparison of semi-empirical formulas and data driven model performances for the sharp-crested weir case (model: MS5).
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Comparison of semi-empirical formulas and data driven model performances for the sharp-crested weir case (model: MS5).

Figure 6
Figure 6. Comparison of semi-empirical formulas and data driven model performances for the sharp-crested weir case (model: MS5).
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Comparison of semi-empirical formulas and data driven model performances for the sharp-crested weir case (model: MS5).

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Figure 7
Figure 7. Comparison of semi-empirical formulas and data driven model performances for the inclined slope controlled structure case (model: MI3).
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Comparison of semi-empirical formulas and data driven model performances for the inclined slope controlled structure case (model: MI3).

Figure 7
Figure 7. Comparison of semi-empirical formulas and data driven model performances for the inclined slope controlled structure case (model: MI3).
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Comparison of semi-empirical formulas and data driven model performances for the inclined slope controlled structure case (model: MI3).

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