Prediction of hydro-suction dredging depth using data-driven methods

Abstract

In this study, data-driven methods (DDMs) including different kinds of group method of data handling (GMDH) hybrid models with particle swarm optimization (PSO) and Henry gas solubility optimization (HGSO) methods, and simple equations methods were applied to simulate the maximum hydro-suction dredging depth (h_s). Sixty-seven experiments were conducted under different hydraulic conditions to measure the h_s. Also, 33 data samples from three previous studies were used. The model input variables consisted of pipeline diameter (d), the distance between the pipe inlet and sediment level (Z), the velocity of flow passing through the pipeline (u₀), the water head (H), and the medium size of particles (D₅₀). Data-driven simulation results indicated that the HGSO algorithm accurately trains the GMDH methods better than the PSO algorithm, whereas the PSO algorithm trained simple simulation equations more precisely. Among all used DDMs, the integrative GMDH-HGSO algorithm provided the highest accuracy (RMSE = 7.086 mm). The results also showed that the integrative GMDHs enhance the accuracy of polynomial GMDHs by ~14.65% (based on the RMSE).

Graphical abstract

Keywords

sedimentation / water resources / dam engineering / machine learning / heuristic

1 Introduction

Rivers convey sediments from their basins to the lakes, seas, and oceans downstream. Storage or diversion dams or other hydraulic structures across the river valley attenuate the flow, causing sediments to deposit behind them. The accumulation of sediment deposits causes problems, such as reducing reservoirs’ storage [1] and consequently the reduction of flood control of the reservoirs [2]. Besides, reservoirs’ sedimentation may block irrigation and power-generation intakes [3–4], damage hydropower systems and turbines [5], and diminish the suitable water head in hydropower dams [6]. Flushing and hydro-suction are two common methods developed to extract the accumulated sediment without emptying the reservoir. However, installation and maintenance of the flushing equipment are costly and are useful for extracting sediments deposited near the bottom of outlets and conduits [7]. Thus, the hydro-suction method, as an attractive alternative, is an ambulant, economical, and environmentally friendly [8] dredging system. The hydro-suction system uses the available hydraulic head (the difference between water elevation in the upstream and downstream river water surface) to suck the particles (sediment) and transfer them downstream without needing an extra energy source (Fig. 1).

The successful application of hydro-suction systems has been severally reported [9–12]. Brahme and Herbich [13] studied the flow around a cutter head (hydro-suction head) by a physical model and indicated that the Reynolds number is vital in sediment movement modeling. Rehbinder [7] conducted some experiments and developed a theoretical analysis, concluding that the effect of the horizontal shear stress on the particles’ movement is low, whereas the seepage in the sediment surface greatly influences the lifting of the particles. Hotchkiss and Huang [8] designed and applied different inlet shapes of hydro-suction in a field test and declared that the applied system had the effectual ability to dredging the deposited sediments. Ullah et al. [14] conducted experiments on a hydro-suction system and evaluated the effect of the distance of the pipe inlet from the sediment level on the hydro-suction performance. They reported that in some experiments, a vortex flow around the pipe inlet was observed, which positively affected the lifting of particles. Chen et al. [15] designed some experiments and focused on the pipe inlet shape effect on the efficiency of the hydro-suction dredging method. The results indicated that the pipe inlet shape affected the efficiency of hydro-suction, and a 20° wedge-type inlet shape with three side holes was the best-designed pipe inlet. Tao et al. [16] designed some experiments in a clean and muddy water to analyze siphon resistance. The results showed that in the specific water head, pipe diameter directly influenced the pipe resistance, while the hump height and desilting concentration inversely affected it. Ke et al. [4] conducted some experiments to evaluate the effect of consolidating sediments (silt and clay). The results indicated that the performance of the hydro-suction system depends on the duration of self-weight consolidation, and the efficiency of hydro-suction decreases beyond 90% consolidation. Pishgar et al. [3] designed some experiments and evaluated the geometrical and mechanical parameters of burrowing-type suction pipe on the efficiency of sediment removal.

Successful applications of machine learning (ML) approaches and data-driven methods (DDMs) in engineering have been recently reported in several studies [17–23]. Hamdia et al. [24] used the genetic algorithm approach as an integrative ML model with deep neural networks and adaptive neuro-fuzzy inference system (ANFIS) to predict the fracture energy of polymer/nanoparticles composites. The results indicated the superiority of the integrative method to the individual ML methods. Guo et al. [25] applied a deep collocation method to analyze the bending of Kirchhoff plates. The results indicated that the proposed method can analyze the bending of different-shaped Kirchhoff plates. Samaniego et al. [26] successfully solved the energetic format of partial differential equations using DDMs. Thus, these findings approved the capabilities of the ML method in engineering fields.

Among several available DDMs, the group method of data handling (GMDH) is categorized as an inductive ML method that is employed in different sciences to model complex phenomena [27–30]. Lashteh Neshaei et al. [31] predicted the beach profile evolution with GMDH on beaches with seawalls. The results approved the accuracy of the GMDH network. Masoumi Shahr-Babak et al. [32] developed a hybrid method to predict the suction caisson uplift capacity. They used the harmony search algorithm (HS) to optimize the weights of the GMDH method. The results showed that the HS algorithm increased the efficiency of classic GMDH. Parsaie et al. [33] applied a particle swarm optimization (PSO) algorithm to train GMDH to model the discharge coefficient over weir-gate. The results confirmed the accuracy of the purposed model. Mahdavi-Meymand and Zounemat-Kermani [34] applied the firefly algorithm (FA) as an integrative algorithm embedded with GMDH to estimate spillways aerators' air demand. The results showed that the integrative GMDH-FA is more accurate than the standard GMDH model. Sayari et al. [20] applied teaching–learning based optimization (TLBO) as an integrative algorithm with GMDH to estimate the critical flow velocity of slurries. The results approved the higher accuracy of GMDH-TLBO compared to empirical equations. Qaderi et al. [35] developed several hybrid models based on the combination of shuffled complex evolutionary (SCE), HS, and GMDH to predict bridge pier scour depth. The results revealed that the HS and SCE algorithms promote the performance of GMDH.

The maximum scour depth is an important parameter in hydro-suction dredging method design. In this study, DDMs were applied to model the maximum hydro-suction scour depth (h_s). Based on the authors’ knowledge, DDMs have not been used to model the h_s yet.

The GMDH ability to model complex phenomena has been approved in previous studies, thus this method was chosen as the base simulation method. To optimize the GMDH weights, PSO and Henry gas solubility optimization (HGSO) meta-heuristic methods were used. Notably, the HGSO algorithm is a new swarm optimization approach, and the application of integrative GMDH combined with the HGSO algorithm (GMDH-HGSO) has not been reported elsewhere, which highlights the contribution of this study.

Another novelty facet of this study lies in the GMDH model establishment. The classic GMDH uses polynomial equations in neurons, while this study uses other complex nonlinear equations with more coefficients to apply in the GMDH structure.

2 Materials and methods

Herein, new data-driven approaches were used to simulate the maximum scour depth of a hydro-suction system. For the simulation, several databases including the data series gathered from three published resources, as well as observed data of a physical model measured by the authors of this study, were considered (Table 1). Figure 2 shows the flowchart of the process of this study.

2.1 Experimental procedure

The experiments were conducted at the Hydraulic Laboratory of the Water engineering Department of Shahid Bahonar University of Kerman, Iran. The model reservoir, which consists of a cubic tank of dimension 90 cm × 80 cm × 70 cm, was made of 16-mm thick glass. The variable parameters in tests include water head (H), pipeline diameter (d), and diving depth of pipe inlet to the sediment (Z) (Fig. 1). H and d values were 30, 50, and 70 cm; and 20, 16, and 13 mm, respectively. The Z parameter changed from 0 to the blockage depth with a 5-mm step. Twenty-cm sediment evenly leveled on the bed of the model. Particles’ specifications consist of medium diameter (D₅₀), uniformity coefficient (C_u), and relative density of 0.75, 2.51, and 2.65, respectively. Figure 3 shows the created model view. For all three pipeline diameters, the pipelines were 250-cm in length. Twenty-cm long polyethylene pipes, with the same pipeline diameters, connected to the pipelines were used as the tube mouth. At the beginning of the experiments, water entered the model until it was one-third full. Then, the pipeline was filled with water (to remove air) and the tube mouth was installed on the designed depth. The other side of the pipeline was blocked and installed on the desired head. After this stage, water flow continued again. When the model was filled and the water head was fixed, the hydro-suction was initiated. The experiment timeout was set to 2 h because the scour hole reached a stable state up to this time. After the last time out, the pipeline was removed from the sediments, and water was extracted from the tank by opening the valve under the tank. After water evacuation, the formed scour depth was measured with a point gauge of 0.02-mm accuracy.

2.2 Data-driven and optimization methods

In this study, DDMs were used to simulate the maximum scour depth of hydro-suction systems. The applied DDMs include the polynomial GMDH, nonlinear GMDH (which is a new method and will be introduced in this study), integrative GMDH embedded with PSO algorithm (GMDH-PSO), GMDH-HGSO algorithm (also a new approach), and regression methods. These methods are explained in the following sections.

2.2.1 Group method of data handling (GMDH)

The GMDH is a type of NN model proposed by Ivakhnenko [36]. GMDH is a self-organizing approach, and like other ML methods, enables simulating complex phenomena, modeling, image processing, and data mining. The GMDH is a network consisting of several layers and nodes and the connections between the nodes are approximated by Volterra functional series as follows:

y=C_{\mathrm{0}}+{\displaystyle \sum _{i=\mathrm{1}}^n{C}_i{x}_i+{\displaystyle \sum _{i=\mathrm{1}}^n{\displaystyle \sum _{j=\mathrm{1}}^n{C}_{ij}{x}_i{x}_j+{\displaystyle \sum _{i=\mathrm{1}}^n{\displaystyle \sum _{j=\mathrm{1}}^n{\displaystyle \sum _{k=\mathrm{1}}^n{C}_{ijk}{x}_i{x}_j{x}_k+\mathrm{...}}}}}}}\mathrm{.}

. (1)

The above equation is known as the Kolmogorov–Gabor polynomial [37]. Where n is the number of independent variables (the number of inputs to each node), C is the coefficients’ vector (weights of network), and x (x₁, x₂, …, x_n) and y are the input vector and output of a node, respectively. In most previous studies, the second-order polynomial form of the above equation has been used in the neurons as the transfer function [20,29,31]:

(2)y={{\displaystyle C}}_{\mathrm{0}}+{{\displaystyle C}}_{\mathrm{1}}{{\displaystyle x}}_{\mathrm{1}}+{{\displaystyle C}}_{\mathrm{2}}{{\displaystyle x}}_{\mathrm{2}}+{{\displaystyle C}}_{\mathrm{3}}{{\displaystyle x}}_{\mathrm{1}}^{\mathrm{2}}+{{\displaystyle C}}_{\mathrm{4}}{{\displaystyle x}}_{\mathrm{2}}^{\mathrm{2}}+{{\displaystyle C}}_{\mathrm{5}}{{\displaystyle x}}_{\mathrm{1}}{{\displaystyle x}}_{\mathrm{2}}.(\mathrm{G}\mathrm{M}\mathrm{D}\mathrm{H}-\mathrm{I})

Figure 4 shows example schemes of the GMDH structure, consisting of nodes, nodes’ connections, and layers (input, middle, and output layers). The GMDH consists of four inputs (x₁, x₂, x₃, x₄), one output (y), three middle layers, and maximum five neurons for each layer. In general, in the GMDH, the number of neurons for the next layer (N_{l+ 1}) is calculated as follows:

(3)N_{l+\mathrm{1}}=\left(\begin{array}{l}{N}_l\\ \hspace{0.17em}\mathrm{2}\end{array}\right)={\displaystyle \frac{N_l!}{\mathrm{2}!\hspace{0.17em}\times (N_l-\mathrm{2})!}}\mathrm{.}

In this study, several nodes with inappropriate results would be eliminated to prevent excessive network growth (black nodes in Fig. 4). The least-square optimization is the most used optimization method to train the GMDH network.

It is possible to use several different equations as transfer functions instead of polynomials. Hence, this study aims to approve other equations that may provide better results than polynomials for this specific problem. Thus, a NLE was used to connect the neurons as follows:

(4)y={{\displaystyle C}}_{\mathrm{0}}+{{\displaystyle C}}_{\mathrm{1}}{{\displaystyle x}}_{\mathrm{1}}^{C_{\mathrm{2}}}+{{\displaystyle C}}_{\mathrm{3}}{{\displaystyle x}}_{\mathrm{2}}^{C_{\mathrm{4}}}+{{\displaystyle C}}_{\mathrm{5}}{{\displaystyle x}}_{\mathrm{1}}^{C_{\mathrm{6}}}{{\displaystyle x}}_{\mathrm{2}}^{C_{\mathrm{7}}}.(\mathrm{G}\mathrm{M}\mathrm{D}\mathrm{H}-\mathrm{I}\mathrm{I})

The above equation has more constant coefficients than the previous equation (Eq. (2)), thus it is necessary to use the right optimization algorithms to determine these coefficients.

2.2.2 Particle swarm optimization (PSO)

PSO is a stochastic optimization method inspired by flock birds’ social behavior and belongs to the swarm intelligence optimization algorithms. Eberhart and Kennedy [38] introduced this algorithm, and its high abilities in simulating and modeling complex optimization problems have been approved ever since. In PSO, candidate solutions (particles) are randomly spread in the whole search space. Each particle is marked by velocity (v_i^t) and position (x_i^t) vectors. In each iteration, the objective function calculates the particles’ values and their new positions (x_i^{\mathrm{t+1}}) as follows:

(5)v_i^{t\mathrm{+1}}=\omega v_i^t+c_{\mathrm{1}}{r}_{\mathrm{1}}\left(x_{\mathrm{P}}^t-x_i^t\right)+c_{\mathrm{2}}{r}_{\mathrm{2}}\left(x_{\mathrm{G}}^t-x_i^t\right),

(6)x_i^{t\mathrm{+1}}=x_i^t+v_i^{t\mathrm{+1}},

where r₁ and r₂ are random numbers in the range [0,1], ω is the inertia weight, controlling the exploration and exploitation process, x_p is the ith particle best position, x_G is the global best position of all particles, and c₁ and c₂ are nonnegative constants.

2.2.3 Henry gas solubility optimization (HGSO)

HGSO is a new meta-heuristic algorithm inspired by Henry’s law, which was introduced by Hashim et al. [39]. Based on Henry’s law in a closed system with constant temperature, the amount of dissolved gas to the specific liquid depends on the partial pressure. Henry’s law suggests calculating the solubility of a gas as follows (C_g):

(7)C_{\mathrm{g}}=H_{\mathrm{g}}\times P_{\mathrm{g}},

where P_g is the gas partial pressure and H_g is the Henry’s constant. In the above equation, H_g changes as the system temperature changes. The H_g value in the next temperature can be calculated based on Van’t Hoff equation as follow:

(8){{\displaystyle H}}_{\mathrm{g}\mathrm{T}\mathrm{2}}=\exp \left(-{{\displaystyle C}}_H\times \left({\displaystyle \frac{\mathrm{1}}{T\mathrm{2}}}-{\displaystyle \frac{\mathrm{1}}{T\mathrm{1}}}\right)\right)\times {{\displaystyle H}}_{\mathrm{g}\mathrm{T}\mathrm{1}},

where C_H is a constant, {{\displaystyle H}}_{\mathrm{g}\mathrm{T}\mathrm{1}} is the reference H_g, T2 and T2 are the next and reference temperatures, respectively. The mathematical modeling process of the HGSO algorithm consists of seven main steps.

Step 1: Initialization process

The population in this algorithm is the number of gases (N). In this step, like other stochastic optimization algorithms, the positions of the population are randomly initialized in the space search. H_g, partial pressure, and C_H (all three are related to Eq. (7)) of j cluster are initialized in this step as follows:

(9){{\displaystyle H}}_{\mathrm{g}j}^t={{\displaystyle r}}_{\mathrm{1}}{{\displaystyle l}}_{\mathrm{1}},{{\displaystyle P}}_{\mathrm{g}j}^i={{\displaystyle r}}_{\mathrm{2}}{{\displaystyle l}}_{\mathrm{2}},{{\displaystyle C}}_{\mathrm{H}j}={{\displaystyle r}}_{\mathrm{3}}{{\displaystyle l}}_{\mathrm{3}},

where r₁, r_2, and r₃ are random numbers between 0 and 1, and l₁, l₂, and l₃ are constant values of 0.005, 100, and 0.01, respectively.

Step 2: Clustering

This algorithm considers the number of gas types (N_c). In this step, the population is divided into N_c equal clusters.

Step 3: Evaluation

The best cluster is recognized in this step as the gas that reaches the highest equilibrium.

Step 4 and 5: Update Henry’s coefficient and solubility

Eq. (8) is inspired to update Henry’s coefficient as follow:

(10){{\displaystyle H}}_{\mathrm{g}j}^{t+\mathrm{1}}=\exp \left(-{{\displaystyle C}}_{\mathrm{H}j}\times \left({\displaystyle \frac{\mathrm{1}}{T^{\mathrm{t}}}}-{\displaystyle \frac{\mathrm{1}}{T^{\mathrm{0}}}}\right)\right)\times {{\displaystyle H}}_{\mathrm{g}j}^t,T^t=\exp \left({\displaystyle \frac{-t}{MaxIter}}\right),

where T^t is the temperature at iteration t, T⁰ is a constant of 298.15, and MaxIter is the maximum number of iterations. The solubility of gas i in cluster j is updated based on the following equation:

(11){{\displaystyle C}}_{\mathrm{g}i,j}^t=K\times {{\displaystyle H}}_{\mathrm{g}\hspace{0.17em}j}^{t+\mathrm{1}}\times {{\displaystyle P}}_{\mathrm{g}i,j}^t,

where K is a constant coefficient and {{\displaystyle P}}_{\mathrm{g}i,j}^{\mathrm{t}} is the partial pressure on gas i in cluster j.

Step 6: Update position

The agents’ position is updated as follows:

(12){{\displaystyle x}}_{i,j}^{t+\mathrm{1}}={{\displaystyle x}}_{i,j}^t+F.{{\displaystyle r}}_{\mathrm{4}}.\gamma .({{\displaystyle x}}_{i,\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}^t-{{\displaystyle x}}_{i,j}^t)+F.{{\displaystyle r}}_{\mathrm{5}}.a.({{\displaystyle C}}_{\mathrm{g}i,j}^t.{{\displaystyle x}}_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}^t-{{\displaystyle x}}_{i,j}^t),

where x_{i,j}^t is the position of agent (gas) i in cluster j at iteration t, r₄ and r₅ are random numbers in the range [0,1], F is an index that changes the agent direction ( − or+ ), x_{i,\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}^t is the best agent in cluster j, x_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}^t is the best agent in all population, and γ is calculated as follows:

(13)\lambda =\beta .\exp \left({\displaystyle \frac{Cos{t}_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}^t+\epsilon }{Cos{t}_{i,j}^t+\epsilon }}\right),

where β is a constant, Cos{t}_{i,j}^{\mathrm{t}} is the fitness of agent i of cluster j, Cos{t}_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}^{\mathrm{t}} is the fitness of the best agent, and ε is a constant of 0.05.

Step 7: Escape from local optimum

To escape from local solutions, the number of worse agents (N_w) is re-initialized. N_w is calculated as below:

(14)N_w=\mathrm{Round}[N.(r_{\mathrm{6}}(B_{\mathrm{1}}-B_{\mathrm{2}})+B_{\mathrm{1}})],

where B₁ and B₂ are constant coefficients.

2.2.4 Integrative GMDH Methods

After creating the GMDH structure, the network coefficients (weights) must be optimized. In this study, two GMDHs (GMDH-I and GMDH-II) were used to model the h_s. GMDH-I uses Eq. (2), which has six constants that must be optimized, as the transfer function in internal neurons. GMDH-II, which has been novelly introduced in this study, uses Eq. (4), which has eight constants. PSO and HGSO algorithms were used to optimize these parameters.

2.2.5 Regression methods (simple methods)

Optimization methods can advantageously and efficiently optimize diverse equations with different constant coefficients. In this study, linear equation (LE) and NLE models were used for the process simulation as follows.

(15)y=C_{\mathrm{0}}+C_{\mathrm{1}}{x}_{\mathrm{1}}+C_{\mathrm{2}}{x}_{\mathrm{2}}+C_{\mathrm{3}}{x}_{\mathrm{3}}+\mathrm{...}+C_n{x}_n,(\mathrm{L}\mathrm{E})

(16)y={{\displaystyle C}}_{\mathrm{0}}+{{\displaystyle C}}_{\mathrm{1}}{{{\displaystyle x}}_{\mathrm{1}}}^{{{\displaystyle C}}_{\mathrm{2}}}+{{\displaystyle C}}_{\mathrm{3}}{{{\displaystyle x}}_{\mathrm{2}}}^{{{\displaystyle C}}_{\mathrm{4}}}+{{\displaystyle C}}_{\mathrm{5}}{{{\displaystyle x}}_{\mathrm{3}}}^{{{\displaystyle C}}_{\mathrm{6}}}\dots +{{\displaystyle C}}_{(\mathrm{2}n-\mathrm{1})}{{{\displaystyle x}}_n}^{{{\displaystyle C}}_{\mathrm{2}n}}{,}_{(\mathrm{N}\mathrm{L}\mathrm{E})}

where y is the output, x (x₁, x₂, …, x_n) is the input vector and C is the coefficient vector. The optimization processes of the regression equations are based on the PSO and HGSO algorithms.

2.3 Data preparation and modeling

Herein, 100 data sets were used to model the h_s. Sixty-seven tests were performed and registered, whereas 33 data series were extracted from three resources [14,40,41]. The independent parameters consist of d, Z, u₀, H, and D₅₀, and are shown in Fig. 1. Ullah et al. [14] used two cylindrical tanks of 1219 and 550 mm diameters, and 1219 and 705 mm depths. The smaller tank was placed in the larger one and was filled with sediments of 0.58-mm mean particle size. Scour holes of 9.7- and 20.4-mm diameters created by hydro-suction at different distances of the pipe inlet from the sediment level were analyzed. Moghbeli [40] built a model of 70 cm × 50 cm × 70 cm in dimension and tested three pipeline diameters on sediment dredging. The water head was also in three levels and the median sediment size was 0.23 mm. Forutan Eghlidi [41] designed and constructed a reservoir model of 100 cm × 100 cm × 70 cm in dimension and conducted experiments to evaluate the hydro-suction efficiency on different distances of the pipe head from the sediment level. In Fig. 5, boxplots of data sets are plotted, and more details regarding the geometric characteristics of the four physical models are presented in Table 1.

Figure 5 implies that in the observed maximum scour hole depth, no outliers exist, indicating the absence of any out-of-range-data in the data set. To determine the best subset of predictors, the best subsets regression (BSR) method was applied to the data set. The outcomes of the most effective input parameters on the output are tabulated in Table 2.

Table 2 suggest that the model with five inputs (d, Z, u₀, H, and D₅₀) is more precise than other models. Since the corresponding Mallows' Cp to the five independent parameters is the lowest value, this combination is the best subset for modeling the h_s. Also, the related R-squared value for this input configuration is higher than that of other models.

In this study, the whole data set was randomly split into training (70%), validation (15%), and testing (15%) sets. The training data set was used to train the models. The validation data were used to prevent overtraining of the models, and the testing data set was used to evaluate the implemented methods’ accuracy. Table 3 presents the ranges of data sets and the simple correlation coefficient (SCC) between the inputs and output (h_s).

The SCC values of Table 3 indicate that all considered inputs’ parameters positively affected the maximum scour hole of hydro-suction, except for the Z parameter. Table 3 also shows that the testing and validation data sets lie within the range of the training data set.

The applied PSO and HGSO algorithms have some initial parameters, which are presented in Table 4 and were chosen based on previous studies [19,35].

2.4 Evaluation of the Simulation Models

Here, four statistical parameters were used to assess the mathematical performance of the applied methods. These parameters include the root-mean square error (RMSE), coefficient of determination (R²), mean absolute error (MAE), and the mean index of agreement (IA), which can be determined as follows:

(17)RMSE=\sqrt{{\displaystyle \frac{\mathrm{1}}{{{\displaystyle N}}_{\mathrm{d}}}}\underset{i=\mathrm{1}}{\overset{{{\displaystyle N}}_{\mathrm{d}}}{\Sigma }}{({{{\displaystyle h}}_{\mathrm{s}}}^{\mathrm{m},i}-{{{\displaystyle h}}_{\mathrm{s}}}^{\mathrm{O},i})}^{\mathrm{2}}},

(18)R^{\mathrm{2}}={\displaystyle \frac{{{\displaystyle \left({\mathrm{\Sigma }}_{i=\mathrm{1}}^{{{\displaystyle N}}_{\mathrm{d}}}\left({{{\displaystyle h}}_{\mathrm{s}}}^{\mathrm{m},i}-{{\displaystyle \overline{{{\displaystyle h}}_s}}}^{\mathrm{m}}\right)\left({{{\displaystyle h}}_{\mathrm{s}}}^{\mathrm{O},i}-{{\displaystyle \overline{{{\displaystyle h}}_{\mathrm{s}}}}}^{\mathrm{O}}\right)\right)}}^{\mathrm{2}}}{{\mathrm{\Sigma }}_{i=\mathrm{1}}^{{{\displaystyle N}}_{\mathrm{d}}}{{\displaystyle \left({{{\displaystyle h}}_{\mathrm{s}}}^{\mathrm{m},i}-{{\displaystyle \overline{{{\displaystyle h}}_{\mathrm{s}}}}}^{\mathrm{m}}\right)}}^{\mathrm{2}}{\mathrm{\Sigma }}_{i=\mathrm{1}}^{{{\displaystyle N}}_{\mathrm{d}}}{{\displaystyle \left({{{\displaystyle h}}_{\mathrm{s}}}^{\mathrm{O},i}-{{\displaystyle \overline{{{\displaystyle h}}_{\mathrm{s}}}}}^{\mathrm{O}}\right)}}^{\mathrm{2}}}},

(19)MAE={\displaystyle \frac{\mathrm{1}}{{{\displaystyle N}}_{\mathrm{d}}}}\underset{i=\mathrm{1}}{\overset{{{\displaystyle N}}_{\mathrm{d}}}{\mathrm{\Sigma }}}\left|{{{\displaystyle h}}_{\mathrm{s}}}^{\mathrm{m},i}-{{{\displaystyle h}}_{\mathrm{s}}}^{\mathrm{O},i}\right|,

(20)IA=\mathrm{1}-{\displaystyle \frac{{{\displaystyle \Sigma }}_{i=\mathrm{1}}^{{{\displaystyle N}}_{\mathrm{d}}}{({{{\displaystyle h}}_{\mathrm{s}}}^{\mathrm{m},i}-{{{\displaystyle h}}_{\mathrm{s}}}^{\mathrm{O},i})}^{\mathrm{2}}}{{\mathrm{\Sigma }}_{i=\mathrm{1}}^{{{\displaystyle N}}_{\mathrm{d}}}{{\displaystyle \left(\left|{{{\displaystyle h}}_{\mathrm{s}}}^{\mathrm{m},i}-{{\displaystyle \overline{{{\displaystyle h}}_{\mathrm{s}}}}}^{\mathrm{m}}\right|+\left|{{{\displaystyle h}}_{\mathrm{s}}}^{\mathrm{O},i}-{{\displaystyle \overline{{{\displaystyle h}}_{\mathrm{s}}}}}^{\mathrm{O}}\right|\right)}}^{\mathrm{2}}}},

where N_d is the number of data, {{{\displaystyle h}}_{\mathrm{s}}}^{\mathrm{m}} is the simulated scour depth, {{{\displaystyle h}}_{\mathrm{s}}}^{\mathrm{O}} is the observed scour depth, {\overline{h_{\mathrm{s}}}}^{\mathrm{m}} is the average simulated scour depth, and{\overline{h_{\mathrm{s}}}}^{\mathrm{O}} is the average observed scour depth.

3 Results

3.1 Experimental observations

Generally, two distinct observable phases of sediment transport were distinguished in the hydro-suction system. In the first phase, vortex flow was not created around the pipe inlet, whereas in the second phase, vortex flow was generated (Fig. 6). Depending on the experimental condition, either the two phases or just the first phase may occur. The first phase starts from the initiation of flow and initially removes a huge amount of materials. This phase is short but may last until the end of the experiment, depending on the experimental condition. The results indicated that by dipping the pipe inlet to the sediments deeper, the performance of the hydro-suction increased. Also, the vortex flows positively affected the hydro-suction efficiency.

3.2 Soft computing simulation results

Here, eight DDMs were applied to model the h_s. To achieve this purpose, d, Z, u₀, H, and D₅₀ were considered as the input parameters. Table 5 presents the simulation results of the training and validation data sets.

From Table 5, the outcomes of all applied methods in the training and validation phases are acceptable, indicating that the R² and IA values of all methods are high (R² and IA>0.8). Generally, the PSO algorithm was more accurate than the HGSO algorithm in both the training and validation phases. The average training values of the RMSE, R², and IA of the hybrid GMDH with the PSO and HGSO algorithms are 5.951 and 6.733 mm, 0.923 and 0.909, and 0.979 and 0.971, respectively. The average validation RMSEs of the PSO and HGSO algorithms are 2.682 and 3.264 mm, respectively. The GMDH І and II performances are close to each other in the training phase. The average RMSE, R², and IA of the GMDH І are 6.357 mm, 0.913, and 0.922, while those for the GMDH II are 6.327 mm, 0919, and 0.974, respectively. In the validation phase, the GMDH II indicated better accuracy than the GMDH І. The average RMSE, R², and IA of GMDH II are 2.840 mm, 0.967, and 0.991, whereas those of the GMDH І are 3.110 mm, 0.959, and 0.966, respectively. In the training data sets, the GMDH II-PSO algorithm, with 10 layers and a maximum of five neurons in each layer, is the most accurate method.

In the validation data set, the GMDH І-PSO algorithm, with 10 layers and a maximum of 15 neurons in each layer, is the most accurate method. Interestingly, the results of simple methods are promising. The results of these methods in some cases are even better than that of the GMDH methods. Generally, the PSO algorithm provided better results in optimizing the coefficients of simple methods in training and validation data sets compared to the HGSO algorithm.

Analyzing the attained results of the models in the training and validation sets could reveal valuable information about the training ability of the models, which helps in avoiding model overtraining. The final assessment of the simulation models is done according to the results in the testing set (Table 6).

Table 6 shows that the GMDH II results are better than that of the GMDH І model. The average RMSE, R², MAE, R², and IA of the GMDH І are 10.520 mm, 0.835, 8.645 mm, 0.659, and 0.911, whereas those of the GMDH II are 9.340 mm, 0.833, 7.263 mm, 0.731, and 0.926, respectively. The best structure of the GMDH is a network with 10 layers and a maximum of 10 neurons in each layer. Overall, the GMDH II-HGSO method with 10 layers and a maximum of 10 neurons in each layer (lowest RMSE of 7.086 mm, highest IA of 0.956, and R² of 0.922) exhibits better performance than all applied models, indicating the HGSO algorithm as a new effective optimization algorithm. The GMDH II-PSO algorithm with the same network structure is the next most accurate model. Compared to the best models, using the HGSO algorithm instead of the PSO algorithm increases the accuracy of modeling by ~12.31% based on the RMSE criterion. The test results imply that the performance of integrative simple regression methods with optimization approaches is acceptable. Among these methods, the integrative PSO algorithm with an NLE has better performance than the others (RMSE of 8.829 mm).

The scatterplot trend is an effective diagnostic way to show the distribution of the attained results of different methods. Figure 7 illustrates the scatterplots of the applied GMDH models.

Figure 7 shows that the scatter points of the GMDH II-HGSO algorithm are closer to the agreement line (1:1 line) and are less scattered than those of the other models, indicating better simulation results. Most of the scatter points of the applied methods are located under the 1:1 line, revealing the underestimation of such models. The integrative GMDH with the HGSO algorithm show less scattered distribution than that with the PSO algorithm (Fig. 7), which indicates that the HGSO algorithm optimized the GMDH methods coefficients more accurately than the PSO algorithm.

Taylor's diagram is another graphical way to show and compare the performances of different models. The similarity between the results of the applied models is quantified based on their correlation, their centered root-mean-square, and standard deviations [42]. Figure 8 displays the Taylor diagram of the developed methods.

Figure 8 indicates that the performance points of all methods are close to each other, indicating the marginal performance of the applied methods. However, the GMDH points are closer to the observation point (the asterisk in Fig. 8), indicating the superiority of the GMDH models to the simple equations. Among all applied methods, the corresponding point of the GMDH-II-HGSO algorithm is the closest to the observation point, implying the highest model accuracy.

4 Discussion

Here, experimental and integrative DDMs were applied to model the h_s. Experimental results indicated that vortex flow, which is formed in some cases, directly influences the efficiency of the hydro-suction system. Ullah et al. (2005) also highlighted these vortexes. The results obtained here agree with those of previous studies, which reported that diving the pipe inlet through the sediments increases the efficiency of hydro-suction (Ullah et al., [14]; Forutan Eghlidi, [41]).

Herein, GMDH approaches including GMDH-I-PSO, GMDH-II-PSO, GMDH-I-HGSO, and GMDH-II-HGSO algorithms were used to model the h_s. GMDH-I, which has been used in several studies, uses polynomial equations in neurons. GMDH-II uses a nonlinear equation as the transfer function in neurons (a novel application). The use of sophisticated optimization algorithms such as PSO and HGSO is suitable for complex problems and was employed here to train the simulation models. The results showed that the new NL GMDH model improved the accuracy of the polynomial GMDH. Most optimization algorithms have initial parameters (like that in Table 4). However, it should be noted that the proposed initial values cannot always guarantee the best optimization process. The fewer initial values ease the application of optimization algorithms. The HGSO is a new optimization algorithm and the attained results revealed that the GMDH structure was trained more accurately than the PSO algorithm. However, from Table 4, compared to the PSO algorithm, the HGSO algorithm contains several initial parameters, indicating the need for a more complex process of setting the initial tuning values. Notably, the values of the initial tuning parameters were chosen based on the original reference study (Hashim et al., [39]). Thus, determining the HGSO algorithm’s initial parameters might be an interesting topic for future studies.

Herein, five input parameters including d, Z, u₀, H, and D₅₀ were considered in predicting the h_s. The results of the BSR method (Table 2) indicated that all input parameters significantly influence the modeling process. Tables 2 and 3 showed that Z is the most effective parameter (Mallows’ Cp is closest to 1; SCC = −0.822) and u₀ is the less effective parameter (the farthest distance from 1; SCC = 0.233).

5 Conclusions

Hydro-suction is a suitable method for dredging deposited sediments in reservoirs. In this study, DDMs were applied to model the induced maximum scour depth (h_s) created by the hydro-suction system. The pipeline diameter (d), water head (H), and depth of diving pipe inlet to de sediment (Z) were the variable parameters in the experiments. The experimental results indicated that, generally, the h_s increase as the Z parameter increases. In this study, a new integrative GMDH structure, using a nonlinear equation (NLE) as the transfer function in neurons, was presented. Two optimization algorithms, including PSO and HGSO were used to determine the coefficients of two GMDH models and two simple equations. The results indicated that the NL GMDH increased the accuracy of the classic GMDH by ~14.65% (based on RMSE criterion). The results also indicated that the HGSO algorithm, as a new optimization method, provided better accuracy than the PSO algorithm, while the general performance of the PSO algorithm in optimizing simple equations was better. Hence, the results of the regression equations implied that the NLE acted better than the applied LE (1.60% of RMSE improvement).

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