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Group Method of Data Handling to Predict Scour at Downstream of a Ski-Jump Bucket Spillway

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Abstract

This study introduces a method to predict scour hole geometry at downstream of a ski-jump bucket by group method of data handling (GMDH). The GMDH network was developed using evolutionary and iterative algorithms including those of genetic programming (GP), particle swarm optimization (PSO), and back propagation (BP) algorithms. Results of alternative GMDH networks were compared with those obtained using artificial neural networks, genetic programming, ANFIS, empirical equations, and regression-based equations. Performances indicated that proposed GMDH-BP produced more accurate results in comparison with other methods. Moreover, the most effective independent parameters on scour hole geometry were determined using sensitivity analysis. Finally, combination of evolutionary and iterative algorithms has been confirmed that the GMDH network is a useful soft computing tool for prediction of scour hole geometry at downstream of a ski-jump bucket.

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Author information

Authors and Affiliations

  1. Department of Civil Engineering, Shahid Bahonar University of Kerman, P.O. BOX 76169133, Kerman, Iran

    Mohammad Najafzadeh, Gholam-Abbas Barani & Masoud-Reza Hessami-Kermani

Authors
  1. Mohammad Najafzadeh
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  2. Gholam-Abbas Barani
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  3. Masoud-Reza Hessami-Kermani
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Corresponding author

Correspondence to Mohammad Najafzadeh.

Additional information

Communicated by: H. A. Babaie

Appendices

Appendix A. Selective polynomials of the GMDH-PSO and GMDH-GP networks for predicting the d s /d w

Selective polynomial neurons of the GMDH-PSO were represented as follows:

$$ \begin{array}{l}{\left({d}_s/{d}_w\right)}_1^1=1+0.5296\left(q/\sqrt{g.}{d}_w^3\right)+0.2929\left(H/{d}_w\right)+0.31627\left(q/\sqrt{g.}{d}_w^3\right)\left(H/{d}_w\right)\hfill \\ {}+0.2775{\left(H/{d}_w\right)}^2-0.03919{\left(q/\sqrt{q.}{d}_w^3\right)}^2\hfill \end{array} $$
(41)
$$ \begin{array}{l}{\left({d}_s/{d}_w\right)}_5^1=-0.734+0.4496\left(q/\sqrt{g.}{d}_w^3\right)+0.896\left(R/{d}_w\right)-0.056\left(q/\sqrt{g.}{d}_w^3\right)\left(R/{d}_w\right)\hfill \\ {}+0.061{\left(q/\sqrt{g.}{d}_w^3\right)}^2-0.0238{\left(R/{d}_w\right)}^2\hfill \end{array} $$
(42)
$$ \begin{array}{l}{\left({d}_s/{d}_w\right)}_8^1=1+\left(q/\sqrt{g.}{d}_w^3\right)+\left({d}_{50}/{d}_w\right)+0.9903\left(q/\sqrt{g.}{d}_w^3\right)\left({d}_{50}/{d}_w\right)\hfill \\ {}-0.0925{\left({d}_{50}/{d}_w\right)}^2+0.47166{\left(q/\sqrt{g.}{d}_w^3\right)}^2\hfill \end{array} $$
(43)
$$ \begin{array}{l}{\left({d}_s/{d}_w\right)}_{10}^1=1+0.1841\left(q/\sqrt{g.}{d}_w^3\right)+0.11955\left(\varphi \right)-0.7924\left(q/\sqrt{g.}{d}_w^3\right)\left(\varphi \right)\hfill \\ {}+0.36708{\left(q/\sqrt{g.}{d}_w^3\right)}^2+0.2586{\left(\varphi \right)}^2\hfill \end{array} $$
(44)
$$ \begin{array}{l}{\left({d}_s/{d}_w\right)}_2^2=0.299+{\left({d}_s/{d}_w\right)}_1^1-0.0607{\left({d}_s/{d}_w\right)}_8^1-0.2573{\left({d}_s/{d}_w\right)}_1^1{\left({d}_s/{d}_w\right)}_8^1\hfill \\ {}+0.12091{\left({\left({d}_s/{d}_w\right)}_1^1\right)}^2+0.10617{\left({\left({d}_s/{d}_w\right)}_8^1\right)}^2\hfill \end{array} $$
(45)
$$ \begin{array}{l}{\left({d}_s/{d}_w\right)}_3^2=-0.0899+0.59299{\left({d}_s/{d}_w\right)}_3^1+0.4347{\left({d}_s/{d}_w\right)}_9^1+0.0839{\left({d}_s/{d}_w\right)}_3^1{\left({d}_s/{d}_w\right)}_9^1\hfill \\ {}-0.03295{\left({\left({d}_s/{d}_w\right)}_3^1\right)}^2-0.0472{\left({\left({d}_s/{d}_w\right)}_9^1\right)}^2\hfill \end{array} $$
(46)
$$ \begin{array}{l}{\left({d}_s/{d}_w\right)}_6^2=-0.5+0.24131{\left({d}_s/{d}_w\right)}_8^1+0.93{\left({d}_s/{d}_w\right)}_{10}^1+0.1845{\left({d}_s/{d}_w\right)}_8^1{\left({d}_s/{d}_w\right)}_{10}^1\hfill \\ {}-0.241{\left({\left({d}_s/{d}_w\right)}_8^1\right)}^2+0.06898{\left({\left({d}_s/{d}_w\right)}_{10}^1\right)}^2\hfill \end{array} $$
(47)
$$ \begin{array}{l}{\left({d}_s/{d}_w\right)}_1^3=0.4067+0.408{\left({d}_s/{d}_w\right)}_2^2+0.7757{\left({d}_s/{d}_w\right)}_3^2-0.1232{\left({d}_s/{d}_w\right)}_2^2{\left({d}_s/{d}_w\right)}_3^2\hfill \\ {}+0.07767{\left({\left({d}_s/{d}_w\right)}_2^2\right)}^2+0.0112{\left({\left({d}_s/{d}_w\right)}_3^2\right)}^2\hfill \end{array} $$
(48)
$$ \begin{array}{l}{\left({d}_s/{d}_w\right)}_2^3=-0.7601+0.0238{\left({d}_s/{d}_w\right)}_2^2+{\left({d}_s/{d}_w\right)}_6^2+0.3503{\left({d}_s/{d}_w\right)}_2^2{\left({d}_s/{d}_w\right)}_6^2\hfill \\ {}+0.1167{\left({\left({d}_s/{d}_w\right)}_2^2\right)}^2-0.2033{\left({\left({d}_s/{d}_w\right)}_6^2\right)}^2\hfill \end{array} $$
(49)
$$ \begin{array}{l}{\left({d}_s/{d}_w\right)}_1^4=0.3317-0.0681{\left({d}_s/{d}_w\right)}_1^3+0.7588{\left({d}_s/{d}_w\right)}_2^3-0.61847{\left({d}_s/{d}_w\right)}_1^3{\left({d}_s/{d}_w\right)}_2^3\hfill \\ {}-0.2927{\left({\left({d}_s/{d}_w\right)}_1^3\right)}^2+0.35601{\left({\left({d}_s/{d}_w\right)}_2^3\right)}^2\hfill \end{array} $$
(50)

Structure of the proposed GMDH-PSO network for predicting the d s /d w was represented as:

figure a

Fig. 8 Improved structure of the GMDH-BP network for predicting the d s /d w

and for the GMDH-GP:

$$ \begin{array}{l}{\left({d}_s/{d}_w\right)}_4^1= GP\left(q/\sqrt{g.{d}_w^3},{d}_{50}/{d}_w\right)=0.3313-9344\left(-4.289810+q/\sqrt{g.{d}_w^3}\right)\left(q/\sqrt{g.{d}_w^3}\right)\hfill \\ {}-3.166\left(q/\sqrt{g.{d}_w^3}\right)\left({d}_{50}/{d}_w\right)\left(-\left(q/\sqrt{g.{d}_w^3}\right)+{d}_{50}/{d}_w\right)\hfill \end{array} $$
(51)
$$ {\left({d}_s/{d}_w\right)}_5^1= GP\left(q/\sqrt{g.{d}_w^3},R/{d}_w\right)=0.7-0.5304\left(R/{d}_w\right)-0.5304{\left(q/\sqrt{g.{d}_w^3}\right)}^2+5.685\left(q/\sqrt{g.{d}_w^3}\right) $$
(52)
$$ \begin{array}{l}{\left({d}_s/{d}_w\right)}_{10}^1= GP\left(q/\sqrt{g.}{d}_w^3,\varphi \right)=-0.17-1.614{\left(q/\sqrt{g.}{d}_w^3\right)}^2{\left(\varphi \right)}^2\hfill \\ {}+\left(0.7924-\left(\varphi \right)\right)\left(q/\sqrt{g.}{d}_w^3\right)\hfill \end{array} $$
(53)
$$ \begin{array}{l}{\left({d}_s/{d}_w\right)}_1^2= GP\left({\left({d}_s/{d}_w\right)}_4^1,{\left({d}_s/{d}_w\right)}_5^1\right)=0.422+0.1305{\left({d}_s/{d}_w\right)}_4^1{\left({d}_s/{d}_w\right)}_5^1-0.08619{\left({\left({d}_s/{d}_w\right)}_4^1\right)}^2\hfill \\ {}+0.4232{\left({d}_s/{d}_w\right)}_4^1+0.2532{\left({d}_s/{d}_w\right)}_5^1\hfill \end{array} $$
(54)
$$ \begin{array}{l}{\left({d}_s/{d}_w\right)}_2^2= GP\left({\left({d}_s/{d}_w\right)}_4^1,{\left({d}_s/{d}_w\right)}_{10}^1\right)=1.277-0.2366{\left({d}_s/{d}_w\right)}_4^1+0.2366{\left({d}_s/{d}_w\right)}_{10}^1\hfill \\ {}+0.186\left({\left({d}_s/{d}_w\right)}_4^1{\left({d}_s/{d}_w\right)}_{10}^1-0.0087{\left({\left({d}_s/{d}_w\right)}_4^1\right)}^2{\left({d}_s/{d}_w\right)}_{10}^1\right)\hfill \end{array} $$
(55)
$$ \begin{array}{l}{\left({d}_s/{d}_w\right)}_1^3= GP\left({\left({d}_s/{d}_w\right)}_1^2,{\left({d}_s/{d}_w\right)}_2^2\right)=-0.451+0.4635{\left({d}_s/{d}_w\right)}_1^2{\left({d}_s/{d}_w\right)}_2^2\hfill \\ {}+1.28{\left({d}_s/{d}_w\right)}_2^2-0.4913{\left({\left({d}_s/{d}_w\right)}_2^2\right)}^2\hfill \end{array} $$
(56)

Structure of the proposed GMDH-GP network for predicting the d s /d w was represented as:

figure b

Fig. 9 Improved structure of the GMDH-GP network for predicting the d s /d w

Appendix. B. Selective polynomials of the alternative GMDH networks for predicting the l s /d w

Selective polynomial neurons of the GMDH-BP were proposed as follows:

$$ \begin{array}{l}{\left({l}_s/{d}_w\right)}_1^1=0.2663+0.7866\left(q/\sqrt{g.}{d}_w^3\right)+0.181\left(H/{d}_w\right)+0.2064\left(q/\sqrt{g.}{d}_w^3\right)\left(H/{d}_w\right)\hfill \\ {}-0.111{\left(q/\sqrt{g.}{d}_w^3\right)}^2-0.0939{\left(H/{d}_w\right)}^2\hfill \end{array} $$
(57)
$$ \begin{array}{l}{\left({l}_s/{d}_w\right)}_5^1=1.2928+1.579\left(q/\sqrt{g.}{d}_w^3\right)-0.7217\left(R/{d}_w\right)+0.893\left(q/\sqrt{g.}{d}_w^3\right)\left(R/{d}_w\right)\hfill \\ {}-0.4553{\left(q/\sqrt{g.}{d}_w^3\right)}^2-0.4338{\left(R/{d}_w\right)}^2\hfill \end{array} $$
(58)
$$ \begin{array}{l}{\left({l}_s/{d}_w\right)}_8^1=4.789+10.6187\left(q/\sqrt{g.}{d}_w^3\right)-24.796\left({d}_{50}/{d}_w\right)-4.884\left(q/\sqrt{g.}{d}_w^3\right)\left({d}_{50}/{d}_w\right)\hfill \\ {}-0.8731{\left(q/\sqrt{g.}{d}_w^3\right)}^2+127.013{\left({d}_{50}/{d}_w\right)}^2\hfill \end{array} $$
(59)
$$ \begin{array}{l}{\left({l}_s/{d}_w\right)}_{10}^1=0.8662+1.494\left(q/\sqrt{g.}{d}_w^3\right)-0.5758\left(\varphi \right)-0.7969\left(q/\sqrt{g.}{d}_w^3\right)\left(\varphi \right)\hfill \\ {}-0.4048{\left(q/\sqrt{g.}{d}_w^3\right)}^2-0.3898{\left(\varphi \right)}^2\hfill \end{array} $$
(60)
$$ \begin{array}{l}{\left({l}_s/{d}_w\right)}_1^2=0.2663+0.7865{\left({l}_s/{d}_w\right)}_1^1+0.18098{\left({l}_s/{d}_w\right)}_5^1+0.2064{\left({l}_s/{d}_w\right)}_1^1{\left({l}_s/{d}_w\right)}_5^1\hfill \\ {}-0.11136{\left({\left({l}_s/{d}_w\right)}_1^1\right)}^2-0.09398{\left({\left({l}_s/{d}_w\right)}_5^1\right)}^2\hfill \end{array} $$
(61)
$$ \begin{array}{l}{\left({l}_s/{d}_w\right)}_2^2=1.2928+1.5797{\left({l}_s/{d}_w\right)}_1^1-0.72179{\left({l}_s/{d}_w\right)}_8^1+0.8929{\left({l}_s/{d}_w\right)}_3^1{\left({l}_s/{d}_w\right)}_8^1\hfill \\ {}-0.45531{\left({\left({l}_s/{d}_w\right)}_1^1\right)}^2-0.43386{\left({\left({l}_s/{d}_w\right)}_8^1\right)}^2\hfill \end{array} $$
(62)
$$ \begin{array}{l}{\left({l}_s/{d}_w\right)}_3^2=0.86624+1.4941{\left({l}_s/{d}_w\right)}_1^1-0.578{\left({l}_s/{d}_w\right)}_{10}^1+0.7969{\left({l}_s/{d}_w\right)}_1^1{\left({l}_s/{d}_w\right)}_{10}^1\hfill \\ {}-0.40489{\left({\left({l}_s/{d}_w\right)}_1^1\right)}^2-0.3898{\left({\left({l}_s/{d}_w\right)}_{10}^1\right)}^2\hfill \end{array} $$
(63)
$$ \begin{array}{l}{\left({l}_s/{d}_w\right)}_1^3=0.14835+0.50484{\left({l}_s/{d}_w\right)}_1^2+0.49762{\left({l}_s/{d}_w\right)}_2^2+1.03938{\left({l}_s/{d}_w\right)}_1^2{\left({l}_s/{d}_w\right)}_2^2\hfill \\ {}-0.52389{\left({\left({l}_s/{d}_w\right)}_1^2\right)}^2-0.5155{\left({\left({l}_s/{d}_w\right)}_2^2\right)}^2\hfill \end{array} $$
(64)
$$ \begin{array}{l}{\left({l}_s/{d}_w\right)}_2^3=0.04938+1.0437{\left({l}_s/{d}_w\right)}_1^2-0.035138{\left({l}_s/{d}_w\right)}_3^2+0.87513{\left({l}_s/{d}_w\right)}_1^2{\left({l}_s/{d}_w\right)}_3^2\hfill \\ {}-0.45887{\left({\left({l}_s/{d}_w\right)}_1^2\right)}^2-0.41646{\left({\left({l}_s/{d}_w\right)}_3^2\right)}^2\hfill \end{array} $$
(65)
$$ \begin{array}{l}{\left({l}_s/{d}_w\right)}_1^4=-0.0517-1.0069{\left({l}_s/{d}_w\right)}_1^3+2.0254{\left({l}_s/{d}_w\right)}_2^3+2.552{\left({l}_s/{d}_w\right)}_1^3{\left({l}_s/{d}_w\right)}_2^3\hfill \\ {}-1.239{\left({\left({l}_s/{d}_w\right)}_1^3\right)}^2-1.31311{\left({\left({l}_s/{d}_w\right)}_2^3\right)}^2\hfill \end{array} $$
(66)

Furthermore, structure of the proposed GMDH-PSO network for predicting the l s /d w was represented as:

figure c

Fig. 10 Improved structure of the GMDH-BP network for predicting the l s /d w

Selective polynomial neurons of the GMDH-PSO were given as follows:

$$ \begin{array}{l}{\left({l}_s/{d}_w\right)}_1^1=0.9403+\left(q/\sqrt{g.}{d}_w^3\right)+0.3255\left(H/{d}_w\right)+0.4097\left(q/\sqrt{g.}{d}_w^3\right)\left(H/{d}_w\right)\hfill \\ {}-0.01{\left(q/\sqrt{g.}{d}_w^3\right)}^2-0.01{\left(H/{d}_w\right)}^2\hfill \end{array} $$
(67)
$$ \begin{array}{l}{\left({l}_s/{d}_w\right)}_6^1=0.9869+\left(H/{d}_w\right)+\left({d}_{50}/{d}_w\right)+0.9996\left(H/{d}_w\right)\left({d}_{50}/{d}_w\right)\hfill \\ {}-0.000631{\left(H/{d}_w\right)}^2+0.47166{\left({d}_{50}/{d}_w\right)}^2\hfill \end{array} $$
(68)
$$ \begin{array}{l}{\left({l}_s/{d}_w\right)}_9^1=1+0.94703\left(H/{d}_w\right)+0.9348\left(\varphi \right)+0.79233\left(H/{d}_w\right)\left(\varphi \right)\\ {}-0.01{\left(H/{d}_w\right)}^2+0.82577{\left(\varphi \right)}^2\end{array} $$
(69)
$$ \begin{array}{l}{\left({l}_s/{d}_w\right)}_2^2=0.8006+0.43532{\left({l}_s/{d}_w\right)}_1^1+0.11991{\left({l}_s/{d}_w\right)}_6^1-0.01{\left({l}_s/{d}_w\right)}_1^1{\left({l}_s/{d}_w\right)}_6^1\\ {}-0.01{\left({\left({l}_s/{d}_w\right)}_1^1\right)}^2+0.0503{\left({\left({l}_s/{d}_w\right)}_6^1\right)}^2\end{array} $$
(70)
$$ \begin{array}{l}{\left({l}_s/{d}_w\right)}_3^2=0.3843+0.9528{\left({l}_s/{d}_w\right)}_1^1+0.70082{\left({l}_s/{d}_w\right)}_9^1-0.01{\left({l}_s/{d}_w\right)}_1^1{\left({l}_s/{d}_w\right)}_9^1\hfill \\ {}-0.00457{\left({\left({l}_s/{d}_w\right)}_1^1\right)}^2-0.01{\left({\left({l}_s/{d}_w\right)}_9^1\right)}^2\hfill \end{array} $$
(71)
$$ \begin{array}{l}{\left({l}_s/{d}_w\right)}_6^2=1-0.01{\left({l}_s/{d}_w\right)}_6^1+0.86967{\left({l}_s/{d}_w\right)}_9^1-0.01{\left({l}_s/{d}_w\right)}_6^1{\left({l}_s/{d}_w\right)}_9^1\hfill \\ {}0.02381{\left({\left({l}_s/{d}_w\right)}_6^1\right)}^2-0.00943{\left({\left({l}_s/{d}_w\right)}_9^1\right)}^2\hfill \end{array} $$
(72)
$$ \begin{array}{l}{\left({l}_s/{d}_w\right)}_1^3=0.080334+0.03444{\left({l}_s/{d}_w\right)}_2^2+{\left({l}_s/{d}_w\right)}_3^2+0.02277{\left({l}_s/{d}_w\right)}_2^2{\left({l}_s/{d}_w\right)}_3^2\hfill \\ {}-0.01{\left({\left({l}_s/{d}_w\right)}_2^2\right)}^2-0.01{\left({\left({l}_s/{d}_w\right)}_3^2\right)}^2\hfill \end{array} $$
(73)
$$ \begin{array}{l}{\left({l}_s/{d}_w\right)}_2^3=-0.01+{\left({l}_s/{d}_w\right)}_2^2+0.471835{\left({l}_s/{d}_w\right)}_6^2-0.01{\left({l}_s/{d}_w\right)}_2^2{\left({l}_s/{d}_w\right)}_6^2\hfill \\ {}-0.01{\left({\left({l}_s/{d}_w\right)}_2^2\right)}^2+0.004188{\left({\left({l}_s/{d}_w\right)}_6^2\right)}^2\hfill \end{array} $$
(74)
$$ \begin{array}{l}{\left({l}_s/{d}_w\right)}_1^4=0.40802+0.4203{\left({l}_s/{d}_w\right)}_1^3+0.38145{\left({l}_s/{d}_w\right)}_2^3-0.01{\left({l}_s/{d}_w\right)}_1^3{\left({l}_s/{d}_w\right)}_2^3\hfill \\ {}-0.02779{\left({\left({l}_s/{d}_w\right)}_1^3\right)}^2-0.01{\left({\left({l}_s/{d}_w\right)}_2^3\right)}^2\hfill \end{array} $$
(75)

Structure of the proposed GMDH-BP network for predicting the l s /d w was represented as:

figure d

Fig. 11 Improved structure of the GMDH-BP network for predicting the l s /d w

and for the proposed GMDH-GP model:

$$ {\left({l}_s/{d}_w\right)}_1^1= GP\left(q/\sqrt{g.{d}_w^3},H/{d}_w\right)=3.7-0.9857{\left(q/\sqrt{g.{d}_w^3}\right)}^2+10.9\left(q/\sqrt{g.{d}_w^3}\right) $$
(76)
$$ \begin{array}{l}{\left({l}_s/{d}_w\right)}_8^1= GP\left(q/\sqrt{g.{d}_w^3},{d}_{50}/{d}_w\right)=4+10.77\left(q/\sqrt{g.{d}_w^3}\right)+10.77{\left({d}_{50}/{d}_w\right)}^2-0.9738{\left(q/\sqrt{g.{d}_w^3}\right)}^2\hfill \\ {}-0.01{\left({d}_{50}/{d}_w\right)}^2\hfill \end{array} $$
(77)
$$ \begin{array}{l}{\left({l}_s/{d}_w\right)}_{10}^1= GP\left(q/\sqrt{g.{d}_w^3},\varphi \right)=1.928+11.63\left(q/\sqrt{g.{d}_w^3}\right)+5.518{\left(\varphi \right)}^2-1.004\left(\varphi \right)\left(q/\sqrt{g.{d}_w^3}\right)\hfill \\ {}-1.004{\left(q/\sqrt{g.{d}_w^3}\right)}^2\hfill \end{array} $$
(78)
$$ {\left({l}_s/{d}_w\right)}_2^2= GP\left({\left({l}_s/{d}_w\right)}_1^1,{\left({l}_s/{d}_w\right)}_{10}^1\right)=-1.201-0.0504{\left({l}_s/{d}_w\right)}_1^1{\left({l}_s/{d}_w\right)}_{10}^1+0.04463{\left({\left({l}_s/{d}_w\right)}_{10}^1\right)}^2+1.188{\left({l}_s/{d}_w\right)}_{10}^1 $$
(79)
$$ {\left({l}_s/{d}_w\right)}_3^2= GP\left({\left({l}_s/{d}_w\right)}_8^1,{\left({l}_s/{d}_w\right)}_{10}^1\right)=-0.99+1.198{\left({l}_s/{d}_w\right)}_{10}^1-0.004518{\left({\left({l}_s/{d}_w\right)}_{10}^1\right)}^2-0.04441{\left({l}_s/{d}_w\right)}_8^1 $$
(80)
$$ {\left({l}_s/{d}_w\right)}_1^3= GP\left({\left({l}_s/{d}_w\right)}_2^2,{\left({l}_s/{d}_w\right)}_3^2\right)=0.07471+1.497{\left({l}_s/{d}_w\right)}_2^2-0.4991{\left({l}_s/{d}_w\right)}_3^2+1.826{\left({\left({l}_s/{d}_w\right)}_2^2-{\left({l}_s/{d}_w\right)}_3^2\right)}^2 $$
(81)

Structure of the proposed GMDH-GP network for predicting the l s /d w was represented as:

figure e

Fig. 12 Improved structure of the GMDH-GP network for predicting the l s /d w

Appendix C. Selective polynomials of the alternative GMDH networks for predicting the w s /d w

Selective polynomial neurons of the GMDH-BP can be expressed as follows:

$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_1^1=3.0059+7.4598\left(q/\sqrt{g.}{d}_w^3\right)+0.24244\left(H/{d}_w\right)-0.07705\left(q/\sqrt{g.}{d}_w^3\right)\left(H/{d}_w\right)\hfill \\ {}-0.6064{\left(q/\sqrt{g.}{d}_w^3\right)}^2+0.02356{\left(H/{d}_w\right)}^2\hfill \end{array} $$
(82)
$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_3^1=0.59015+2.518\left(R/{d}_w\right)+32.0963\left({d}_{50}/{d}_w\right)-10.2989\left(R/{d}_w\right)\left({d}_{50}/{d}_w\right)\hfill \\ {}+0.005464{\left(R/{d}_w\right)}^2+164.6615{\left({d}_{50}/{d}_w\right)}^2\hfill \end{array} $$
(83)
$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_6^1=2.0569+0.5275\left(H/{d}_w\right)+49.523\left({d}_{50}/{d}_w\right)+8.3201\left(H/{d}_w\right)\left({d}_{50}/{d}_w\right)\hfill \\ {}+0.007203{\left(H/{d}_w\right)}^2-399.941{\left({d}_{50}/{d}_w\right)}^2\hfill \end{array} $$
(84)
$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_9^1=-52.8145+5.5315\left(H/{d}_w\right)+168.467\left(\varphi \right)-7.996\left(H/{d}_w\right)\left(\varphi \right)\hfill \\ {}+0.01805{\left(H/{d}_w\right)}^2-121.9274{\left(\varphi \right)}^2\hfill \end{array} $$
(85)
$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_1^2=-1.09644+0.368036{\left({w}_s/{d}_w\right)}_1^1+0.74573{\left({w}_s/{d}_w\right)}_3^1-0.07171{\left({w}_s/{d}_w\right)}_1^1{\left({w}_s/{d}_w\right)}_3^1\hfill \\ {}+0.037116{\left({\left({w}_s/{d}_w\right)}_1^1\right)}^2+0.031346{\left({\left({w}_s/{d}_w\right)}_3^1\right)}^2\hfill \end{array} $$
(86)
$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_2^2=-0.23395-0.08297{\left({w}_s/{d}_w\right)}_1^1+1.0433{\left({w}_s/{d}_w\right)}_6^1-0.3297{\left({w}_s/{d}_w\right)}_1^1{\left({w}_s/{d}_w\right)}_6^1\hfill \\ {}+0.16358{\left({\left({w}_s/{d}_w\right)}_1^1\right)}^2+0.16695{\left({\left({w}_s/{d}_w\right)}_6^1\right)}^2\hfill \end{array} $$
(87)
$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_5^2=0.6848+0.53005{\left({d}_s/{d}_w\right)}_3^1+0.37741{\left({d}_s/{d}_w\right)}_9^1+0.09447{\left({d}_s/{d}_w\right)}_3^1{\left({d}_s/{d}_w\right)}_9^1\hfill \\ {}-0.055287{\left({\left({d}_s/{d}_w\right)}_3^1\right)}^2-0.034938{\left({\left({d}_s/{d}_w\right)}_9^1\right)}^2\hfill \end{array} $$
(88)
$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_2^3=0.51324-0.15495{\left({w}_s/{d}_w\right)}_1^2+1.160389{\left({w}_s/{d}_w\right)}_5^2+0.347713{\left({w}_s/{d}_w\right)}_1^2{\left({w}_s/{d}_w\right)}_5^2\hfill \\ {}-0.15086{\left({\left({w}_s/{d}_w\right)}_1^2\right)}^2-0.197052{\left({\left({w}_s/{d}_w\right)}_5^2\right)}^2\hfill \end{array} $$
(89)
$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_3^3=0.15692+0.086079{\left({w}_s/{d}_w\right)}_2^2+0.930205{\left({w}_s/{d}_w\right)}_5^2+0.25362{\left({w}_s/{d}_w\right)}_2^2{\left({w}_s/{d}_w\right)}_5^2\hfill \\ {}-0.104692{\left({\left({w}_s/{d}_w\right)}_2^2\right)}^2-0.14929{\left({\left({w}_s/{d}_w\right)}_5^2\right)}^2\hfill \end{array} $$
(90)
$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_1^4=-0.88519+1.06243{\left({w}_s/{d}_w\right)}_2^3+0.043505{\left({w}_s/{d}_w\right)}_3^3-1.09344{\left({w}_s/{d}_w\right)}_2^3{\left({w}_s/{d}_w\right)}_3^3\hfill \\ {}+0.53076{\left({\left({w}_s/{d}_w\right)}_2^3\right)}^2+0.558807{\left({\left({w}_s/{d}_w\right)}_3^3\right)}^2\hfill \end{array} $$
(100)

Also, structure of the proposed GMDH-BP network for predicting the w s /d w was represented as:

figure f

Fig. 13 Improved structure of the GMDH-BP network for predicting the w s /d w

Selective polynomial neurons of the GMDH-BP can be expressed as follows:

$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_1^1=-0.75-0.50385\left(q/\sqrt{g.}{d}_w^3\right)+1.7409\left(H/{d}_w\right)+0.33616\left(q/\sqrt{g.}{d}_w^3\right)\left(H/{d}_w\right)\hfill \\ {}-0.17057{\left(q/\sqrt{g.}{d}_w^3\right)}^2-0.04425{\left(H/{d}_w\right)}^2\hfill \end{array} $$
(101)
$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_8^1=2+2\left(q/\sqrt{g.}{d}_w^3\right)+1.3846\left({d}_{50}/{d}_w\right)+2\left(q/\sqrt{g.}{d}_w^3\right)\left({d}_{50}/{d}_w\right)\hfill \\ {}+0.126398{\left(q/\sqrt{g.}{d}_w^3\right)}^2-0.124435{\left({d}_{50}/{d}_w\right)}^2\hfill \end{array} $$
(102)
$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_{10}^1=2+2\left(q/\sqrt{g.}{d}_w^3\right)+2\left(\varphi \right)+1.2215\left(q/\sqrt{g.}{d}_w^3\right)\left(\varphi \right)\hfill \\ {}+0.08696{\left(q/\sqrt{g.}{d}_w^3\right)}^2+0.62547{\left(\varphi \right)}^2\hfill \end{array} $$
(103)
$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_2^2=0.16324+1.290411{\left({w}_s/{d}_w\right)}_1^1+0.783622{\left({w}_s/{d}_w\right)}_8^1+0.313924{\left({w}_s/{d}_w\right)}_1^1{\left({w}_s/{d}_w\right)}_8^1\hfill \\ {}-0.09234{\left({\left({w}_s/{d}_w\right)}_1^1\right)}^2-0.13886{\left({\left({w}_s/{d}_w\right)}_8^1\right)}^2\hfill \end{array} $$
(104)
$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_3^2=1.13187+0.55885{\left({w}_s/{d}_w\right)}_3^1+1.6008{\left({w}_s/{d}_w\right)}_9^1+0.4028{\left({w}_s/{d}_w\right)}_3^1{\left({w}_s/{d}_w\right)}_9^1\hfill \\ {}-0.12526{\left({\left({w}_s/{d}_w\right)}_3^1\right)}^2-0.19329{\left({\left({w}_s/{d}_w\right)}_9^1\right)}^2\hfill \end{array} $$
(105)
$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_6^2=0.48605+0.16088{\left({d}_s/{d}_w\right)}_8^1+0.17249{\left({d}_s/{d}_w\right)}_{10}^1+0.85044{\left({d}_s/{d}_w\right)}_8^1{\left({d}_s/{d}_w\right)}_{10}^1\hfill \\ {}-0.75{\left({\left({d}_s/{d}_w\right)}_8^1\right)}^2-0.083213{\left({\left({d}_s/{d}_w\right)}_{10}^1\right)}^2\hfill \end{array} $$
(106)
$$ \begin{array}{l}{\left({d}_s/{d}_w\right)}_1^3=0.13941+0.35861{\left({d}_s/{d}_w\right)}_2^2+10668{\left({d}_s/{d}_w\right)}_3^2+0.22018{\left({d}_s/{d}_w\right)}_2^2{\left({d}_s/{d}_w\right)}_3^2\hfill \\ {}+0.05174{\left({\left({d}_s/{d}_w\right)}_2^2\right)}^2-0.31643{\left({\left({d}_s/{d}_w\right)}_3^2\right)}^2\hfill \end{array} $$
(107)
$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_2^3=-0.74774+1.41079{\left({w}_s/{d}_w\right)}_2^2+0.64573{\left({w}_s/{d}_w\right)}_6^2+0.028514{\left({w}_s/{d}_w\right)}_2^2{\left({w}_s/{d}_w\right)}_6^2\hfill \\ {}-0.02972{\left({\left({w}_s/{d}_w\right)}_2^2\right)}^2-0.015677{\left({\left({w}_s/{d}_w\right)}_6^2\right)}^2\hfill \end{array} $$
(108)
$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_1^4=-0.390901-0.38199{\left({w}_s/{d}_w\right)}_1^3+0.575127{\left({w}_s/{d}_w\right)}_2^3+0.015529{\left({w}_s/{d}_w\right)}_1^3{\left({w}_s/{d}_w\right)}_2^3\hfill \\ {}+0.003{\left({\left({w}_s/{d}_w\right)}_1^3\right)}^2-0.0006342{\left({\left({w}_s/{d}_w\right)}_2^3\right)}^2\hfill \end{array} $$
(109)

Structure of the proposed GMDH-PSO network for predicting the w s /d w was represented as:

figure g

Fig. 14 Improved structure of the GMDH-PSO network for predicting the w s /d w

and for the proposed GMDH-GP model:

$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_1^1= GP\left(q/\sqrt{g.{d}_w^3},H/{d}_w\right)=5.976+0.1422{\left(q/\sqrt{g.{d}_w^3}\right)}^3+\left(-0.9703-\left(H/{d}_w\right)\right){\left(q/\sqrt{g.{d}_w^3}\right)}^2\hfill \\ {}+1.034\left(R/{d}_w\right)\left(q/\sqrt{g.{d}_w^3}\right)\hfill \end{array} $$
(110)
$$ {\left({w}_s/{d}_w\right)}_6^1= GP\left(H/{d}_w,{d}_{50}/{d}_w\right)=2.207+0.4485\left(H/{d}_w\right)+0.02278{\left(H/{d}_w\right)}^2+36.86\left({d}_{50}/{d}_w\right) $$
(111)
$$ {\left({w}_s/{d}_w\right)}_9^1= GP\left(H/{d}_w,\varphi \right)=4.084+\left(0.7502\varphi -1.178{\varphi}^2\right){\left(H/{d}_w\right)}^2+1.072\left(H/{d}_w\right)\varphi $$
(112)
$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_1^2= GP\left({\left({w}_s/{d}_w\right)}_1^1,{\left({w}_s/{d}_w\right)}_6^1\right)=-0.1871+0.7347{\left({w}_s/{d}_w\right)}_6^1+0.2449{\left({w}_s/{d}_w\right)}_1^1\hfill \\ {}+0.1298{\left({\left({w}_s/{d}_w\right)}_1^1-{\left({w}_s/{d}_w\right)}_6^1\right)}^2\hfill \end{array} $$
(113)
$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_3^2= GP\left({\left({w}_s/{d}_w\right)}_6^1,{\left({w}_s/{d}_w\right)}_9^1\right)=0.04935+0.05274\left({\left({w}_s/{d}_w\right)}_6^1-{\left({w}_s/{d}_w\right)}_9^1\right)\left({\left({w}_s/{d}_w\right)}_9^1\right)\hfill \\ {}+0.91{\left({w}_s/{d}_w\right)}_9^1+0.09064{\left({w}_s/{d}_w\right)}_6^1\hfill \end{array} $$
(114)
$$ \begin{array}{l}{\left({w}_s/{d}_w\right)}_1^3= GP\left({\left({w}_s/{d}_w\right)}_1^2,{\left({w}_s/{d}_w\right)}_3^2\right)=0.059+0.9439{\left({w}_s/{d}_w\right)}_3^2+0.05653{\left({w}_s/{d}_w\right)}_1^2\hfill \\ {}+0.05534{\left({w}_s/{d}_w\right)}_2^2{\left({w}_s/{d}_w\right)}_3^2-0.05534{\left({\left({w}_s/{d}_w\right)}_3^2\right)}^2\hfill \end{array} $$
(115)

Structure of the proposed GMDH-GP network for predicting the w s /d w was represented as:

figure h

Fig. 15 Improved structure of the GMDH-GP network for predicting the w s /d w

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Najafzadeh, M., Barani, GA. & Hessami-Kermani, MR. Group Method of Data Handling to Predict Scour at Downstream of a Ski-Jump Bucket Spillway. Earth Sci Inform 7, 231–248 (2014). https://doi.org/10.1007/s12145-013-0140-4

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