Modeling scour depth downstream of grade-control structures using data driven and empirical approaches

When a high velocity jet impinges a bed (which consists of sand, gravel, or weak and jointed rock), deep scour holes might be formed downstream of spillways and control structures. So, construction of a control structure downstream of spillways is necessary to prevent severe erosion of the bed and formation of deep scour holes. An important common feature in all local scouring problems is the existence of strong secondary flows, along with a vortex system induced by a hydraulic jump, that govern the scouring downstream of a spillway (Guan et al. 2014). Study of the scour phenomenon formed downstream of spillways and control structures is necessary because it crucially endangers the safety of the main structures as well as the adjacent structures, such as training walls and guide bunds. Various factors might govern the scour depth, including flow discharge, height of fall, bucket radius, lip angle, type of rock and the degree of its homogeneity, the time factor of the scouring process, and the mode of spillway operation, which makes it difficult to study this process (Bormann & Julien 1991; Azamathulla et al. 2005). The literature review by the authors showed that there are two varieties of studies focusing on the scour depth simulation: (I) studies focusing on the assessment of scour depth estimation formulas and (II) studies focusing on the temporal variations of scouring. Among group (I), Mason & Arumugam (1985), Chee & Yuen (1985), Yen (1987), and D'Agostino & Ferro (2004) proposed physical-based semi-empirical models for estimating scour depth, which will be discussed in the Methods section. On the other side and in the context of heuristic model applications (group II), data driven approaches (e.g. artificial neural networks (ANNs) and gene expression programming (GEP)) have been applied across wide aspects of water resources engineering in recent years. Among others, ANNs have been applied for predicting the friction factor of open channel flow (Yuhong & Wenxin 2009), prediction of water resource variables in river systems (Maier et al. 2010), estimating river suspended sediment (e.g. Lin & Namin 2005; Hamidi & Kayaalp 2008; Yang et al. 2009), predicting daily and monthly suspended sediment load (Nourani & Andalib 2015), and modeling dissolved oxygen (DO) concentration in lakes (Bertone et al. 2015).

Further, GEP has been applied for modeling river suspended sediment loads (Kisi & Shiri 2012; Kisi et al. 2012), modeling energy dissipation over spillways (Roushangar et al. 2014a), modeling river total bed material load discharge (Roushangar et al. 2014b), modeling the friction factor in alluvial channels (Roushangar et al. 2014c), modeling solid load discharge of an alluvial river (Roushangar & Alizadeh 2015), as well as estimating sediment transport in sewer pipes (Bonakdari & Ebtehaj 2015), all of which have confirmed the GEP capabilities for mapping the nonlinear behavior of the input-target data set.

Babovic & Abbott (1997) applied genetic programming (GP) for the evolution of equations in hydraulic issues. Azamathulla et al. (2005) estimated scour depth downstream of a ski-jump bucket using neural networks. Azamathulla et al. (2006) applied neural networks to estimate scour below spillways. Azamathulla et al. (2008) applied GP to predict ski-jump bucket spillway scour. Babovic (2000) studied sediment transport utilizing GP and evolutionary algorithms. Azamathulla & Ghani (2011) predicted the scour depth at culvert outlets by using an adaptive neuro-fuzzy inference system (ANFIS). Sheikhalipour & Hassanpour (2013) estimated suspended sediment loads using GEP. Azamathulla & Zakaria (2007) predicted the scour below flip-bucket spillways using ANFIS-based approaches. Dargahi (2003) evaluated the scour development downstream of spillways. Azamathulla et al. (2010) predicted sediment concentration for Malaysian rivers utilizing the GEP approach. Guven (2011) employed GEP in estimating scour geometry downstream of hydraulic structures. GEP was also utilized by the following: Azamathulla & Haque (2012) in predicting scour depth at culvert outlets; Azamathulla (2012) in predicting scour depth downstream of sills; and Guven & Gunal (2008) for predicting local scour downstream of hydraulic structures. Liao et al. (2012) used a probabilistic approach to create a preliminary inspection evaluation form for scour potential at bridge sites.

The present literature review shows that most studies have been focused on ANN and ANFIS applications for modeling scour depth (Azamathulla & Ghani 2011; Azamathulla 2012).

The present study aims at investigating the physically based traditional and data driven model (including ANN and GEP) capabilities for scour depth simulation, as well as comparison of the applied models. Therefore, modeling scour depth downstream of an inclined slope controlled structure using heuristic models has been carried out here for the first time. A sensitivity analysis was also performed to describe the most influential input parameters of the data driven models.

In this paper, three kinds of data corresponding to scouring downstream of (i) a ski-jump bucket (Figure 1(a)), (ii) a sharp-crested weir (Figure 1(b)), and (iii) an inclined slope controlled structure (Figure 1(c)), captured from published literature, are used.

Data corresponding to scouring downstream of a ski-jump bucket include 95 patterns that have been gathered at the Central Water and Power Research Station, India (Azamathulla et al. 2005).

The experimental data set used for simulating scour depth downstream of a sharp-crested weir includes 225 patterns captured from D'Agostino & Ferro (2004). The experiments carried out by Veronese (1937) include discharges varying from 0.001–0.083 (m²s⁻¹) and maximum scour depths of 0.055–0.22 m. The ratio between the weir width (b) and the channel width (B) was established to be unity (b = B = 0.5 m), and the water level difference between the upstream and downstream was set to be H = 1 m. Mossa (1998) carried out some experimental runs by modeling a sharp-crested weir. The experimental channel was 0.3 m in width, accompanied by a ratio of b/B as unity. The experimental discharges varied from 0.0045 to 0.0148 (m²s⁻¹), with corresponding maximum scour depths of 0.0352 and 0.145 m. D'Agostino (1994) carried out some experiments (114 run) in a channel of 0.5 m width. The runs were carried out through adopting two values of the ratio b/B (0.3 and 0.6), two values of the weir height (0.41 and 0.71 m) and two different gravel mixtures to reproduce the alluvial bed. The experimental discharges ranged from 0.0167 to 0.167 (m²s⁻¹), producing maximum scour depths of 0.045 and 0.285 m, respectively.

The experimental data set used for simulating scour depth downstream of slope control includes 88 patterns that have been carried out by Bormann & Julien (1991). These experiments have been conducted in a large outdoor flume (the ratio between weir channel width was unity, b = B = 0.91 m) at the Colorado State University Engineering Research Center. The elevation of the grade-control structure model was set at 2.13 m above the flume floor, with three downstream angles of the structure (λ = 1.57, 0.79 and 0.32 radian).

GEP was developed by Ferreira (2001) using fundamental principles of the genetic algorithms (GA) and GP (Koza 1992). GEP mimics biological evolution to create a computer program for simulating different phenomena. In GEP, a mathematical function is described as a chromosome with multi genes, and developed using the data presented to it. GEP performs the symbolic regression using most of the genetic operators of GA.

ANNs are learning systems, consisting of a number of interconnected simple processing elements called neurons or nodes with the attractive attribute of information processing characteristics such as nonlinearity, parallelism, noise tolerance, learning and a generalization capability (Haykin 1998).

Feed forward neural networks (FFNN) with a back-propagation (BP) algorithm are commonly used ANNs in engineering applications. It has been proved that three-layered BP network models are satisfactory in forecasting and simulating issues (e.g. Hornik 1989; ASCE 2000; Nourani et al. 2008). The standard BP algorithm is a gradient descent algorithm, in which the network weights are modified through a negative direction of the gradient of performance function.

In order to assess the performance of data driven models with respect to traditional formulas, the results of the GEP and FFNN models were compared with the corresponding selected semi-empirical formulas.

• 1. Veronese formula (Bureau of Indian Standards 1985):

  

  1

  where d_s is the equilibrium depth of scour below the tail water level, q is the unit discharge and H₁ is the head between the upper (reservoir) water level and the tail water level.
• 2. Azamathulla et al. (2005) suggested estimating the maximum scour depth downstream of a ski-jump bucket, as:

  

  2

  where d_s, q, R, ⁠, d_w, H₁, d₅₀, g denote the equivalent scour depth, unit discharge, radius of the bucket, lip angle of bucket, tail water depth, height of water on the control structure, bed material size and gravitational acceleration, respectively.
• 3. Mason & Arumugam (1985):

  

  3

  where d_s, q, d_w, H₁, d₅₀, g stand for equivalent scour depth, unit discharge, tail water depth, head between the upper (reservoir) water level and the tail water level, bed material size and gravitational acceleration, respectively.
• 4. Sofrelec (1980):

  

  4

where d_s, q, H₁ are the equivalent scour depth, unit discharge, and head between the upper (reservoir) water level and the tail water level, respectively.

• 1. Yen (1987):

  

  5

  with the equilibrium scour depth ⁠, unit discharge ⁠, total head (H), head of water falling from the weir and gravitational acceleration (g).
• 2. D'Agostino & Ferro (2004):

  

  6

  where d_s, H, d₅₀, d₉₀, b, B, h and z denote the equilibrium scour depth, total head, mean sediment size, diameter for which 90% of grains' size is smaller than, width of the weir, width of channel, head of water falling from the weir, and weir height, respectively.

• 1. Chee & Yuen (1985):

  

  7
• 2. Bormann & Julien (1991) 

  

  8

  where in both Equations (7) and (8), d_s, U₀, q, λ, d₅₀, d_w and z represent the equilibrium scour depth, jet velocity entering tail water, unit discharge, the face angle of the control structure, mean sediment size, tail water depth and weir height, respectively.

Input selection is a key aspect in data driven model development and applications. Several approaches have been suggested and reviewed by investigators for determining the most relevant input variables to feed data driven models (ASCE 2000; Bowden et al. 2005). Among others, the input selection methods based on a priori knowledge of the modeled system (Bowden et al. 2005) might be a successful alternative for modeling scour depth, taking into consideration the physical parameters affecting this phenomenon. Since there are multitudes of parameters inflecting local scour depth, some very important parameters (obtained through physical judgment) will be applied here as dimensionless input variables. A combination of this approach with sensitivity analysis was used here for better describing the most effective parameters. Once the optimum data driven approaches were selected, a sensitivity analysis was carried out to distinguish the irrelevant parameters.

Consequently, nine input configurations were built here for studying the effect of the non-dimensional parameters on the scour depth. Table 1 sums up the introduced input configurations.

Table 1

Input configurations of data driven models

Model .	Input variables .	Model .	Input variables .	Model .	Input variables .
Ski-jump bucket data
MB1		MB4		MB7
MB2		MB5		MB8
MB3		MB6		MB9
Sharp edge overflow data
MS1		MS3		MS5
MS2		MS4		MS6
Inclined slope controlled structure data
MI1		MI4		MI7
MI2		MI5		MI8
MI3		MI6		MI9

Model .	Input variables .	Model .	Input variables .	Model .	Input variables .
Ski-jump bucket data
MB1		MB4		MB7
MB2		MB5		MB8
MB3		MB6		MB9
Sharp edge overflow data
MS1		MS3		MS5
MS2		MS4		MS6
Inclined slope controlled structure data
MI1		MI4		MI7
MI2		MI5		MI8
MI3		MI6		MI9

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In this study, four basic arithmetic operators (+, −, ×, /) and four basic mathematical functions (√, exp, sin, cos) were utilized as the GEP function set. Further, each GEP model was evolved till the fitness function remained unchanged for 10,000 runs for each pre-defined number of genes (varying from 3 to 4), then the program was stopped. The model parameters and the size of the developed GEP models were then tuned (optimized) throughout refining (optimizing) the trained and fixed model as a starter. Table 2 lists the combination of all genetic operators used in this study. Further details about GEP applications in hydraulic structures studies might be found in e.g. Roushangar et al. (2014a, 2014b).

Multi-layer FFNN can have more than one hidden layer, although some studies have shown that a single hidden layer might be applied for approximating complex nonlinear functions (Tsukamoto 1979). Therefore, one hidden layer-FFNN was used for modeling scour depth. A difficult task with FFNN application involves choosing its architecture, e.g. the number of hidden nodes and the learning rate. Here, the number of hidden layer nodes is determined by trial and error (through examining various network structures). Consequently, several architectures were examined with variable hidden layer nodes (from 2 to 7). The tangent sigmoid and pure linear functions were found to be appropriate when used as the hidden and output node activation functions, respectively. Feed-forward–BP, TRAINLM (Levenverg–Marquardt algorithm) and LEARNGDM (gradient descent momentum) were applied for network type, training function and adoption learning function, respectively. The training of the FFNN models was stopped when the acceptable level of error was achieved. FFNN was implemented using MATLAB.

From 203 patterns of the sharp-crested weir, 132 patterns were selected for training and 71 patterns were reserved for testing. Table 4 summarizes the statistical criteria of all models. From the statistics, the MS5-based GEP and FFNN models have the highest accuracy. Figure 3(b) presents the observed vs. simulated scour depth values.

From the existing 80 patterns of inclined slope controlled structure data, 53 patterns were used for training and 27 patterns were considered for testing. Table 5 sums up the corresponding error statistics for the GEP and FFNN models. From the table, it is observed that the models corresponding to the MI3 configuration present the highest accuracies. Figure 3(c) illustrates the observed vs. simulated scour depth values of GEP and FFNN models.

The mathematical expression of these ETs reads as follows:

Based on the obtained results, the MB4, MS5 and MI3 models are selected as the best models for simulating scour depth for the ski-jump bucket, sharp-crested weir and inclined slope controlled structure cases, respectively. The MB4 model includes four dimensionless parameters, i.e. ⁠. Table 6 shows the results of the sensitivity analysis of the GEP and FFNN models. According to the table, is the most important parameter affecting the scour depth simulation downstream of the ski-jump bucket, while shows the lowest impact.

Table 6

Sensitivity analyses of the best data driven models

Omitted variable .	Model .	CC .	R2 .	RMSE .	MAE .
Ski-jump bucket (MB4 model)
	GEP	0.318	0.439	0.065	0.052
	FFNN	0.203	0.261	0.189	0.085
	GEP	0.411	0.457	0.064	0.045
	FFNN	0.535	0.581	0.060	0.039
	GEP	0.623	0.694	0.048	0.037
	FFNN	0.741	0.817	0.038	0.029
	GEP	0.598	0.651	0.051	0.036
	FFNN	0.477	0.511	0.081	0.046
Sharp edge overflow (MS5 model)
	GEP	0.224	0.304	0.273	0.159
	FFNN	0.345	0.567	0.207	0.146
	GEP	0.763	0.817	0.150	0.086
	FFNN	0.821	0.852	0.126	0.062
	GEP	0.613	0.709	0.174	0.097
	FFNN	0.736	0.826	0.142	0.074
	GEP	0.588	0.649	0.191	0.111
	FFNN	0.711	0.771	0.163	0.086
	GEP	0.491	0.519	0.219	0.128
	FFNN	0.628	0.679	0.192	0.090
Inclined slope controlled structure (MI3 model)
	GEP	0.411	0.498	0.282	0.246
	FFNN	0.398	0.439	0.187	0.095
	GEP	0.122	0.232	0.338	0.290
	FFNN	0.233	0.394	0.302	0.243
	GEP	0.321	0.413	0.298	0.232
	FFNN	0.285	0.382	0.306	0.237

Omitted variable .	Model .	CC .	R2 .	RMSE .	MAE .
Ski-jump bucket (MB4 model)
	GEP	0.318	0.439	0.065	0.052
	FFNN	0.203	0.261	0.189	0.085
	GEP	0.411	0.457	0.064	0.045
	FFNN	0.535	0.581	0.060	0.039
	GEP	0.623	0.694	0.048	0.037
	FFNN	0.741	0.817	0.038	0.029
	GEP	0.598	0.651	0.051	0.036
	FFNN	0.477	0.511	0.081	0.046
Sharp edge overflow (MS5 model)
	GEP	0.224	0.304	0.273	0.159
	FFNN	0.345	0.567	0.207	0.146
	GEP	0.763	0.817	0.150	0.086
	FFNN	0.821	0.852	0.126	0.062
	GEP	0.613	0.709	0.174	0.097
	FFNN	0.736	0.826	0.142	0.074
	GEP	0.588	0.649	0.191	0.111
	FFNN	0.711	0.771	0.163	0.086
	GEP	0.491	0.519	0.219	0.128
	FFNN	0.628	0.679	0.192	0.090
Inclined slope controlled structure (MI3 model)
	GEP	0.411	0.498	0.282	0.246
	FFNN	0.398	0.439	0.187	0.095
	GEP	0.122	0.232	0.338	0.290
	FFNN	0.233	0.394	0.302	0.243
	GEP	0.321	0.413	0.298	0.232
	FFNN	0.285	0.382	0.306	0.237

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In the case of the MS5 model, which includes four dimensionless parameters, i.e. ⁠, the term seems to be the most important parameter affecting the scour depth downstream of the sharp-crested weir, whereas shows the lowest impact (Table 6). Finally, for the slope controlled structure scour depth simulation, the MI3 model comprising 3 input parameters i.e. ⁠, according to Table 6, the omission of Fr₂ decreases the model accuracy to a great extent, so it seems that this parameter is the most influential factor in modeling scour depth downstream of an inclined slope controlled structure.

Comparisons of semi-empirical formulas and data driven model performances for the ski-jump bucket case (model: MB4), the sharp-crested wave case (model: MS5) and the inclined slope controlled structure case (model: M13) are shown in Figures 5, 6 and 7, respectively.

The aim of this study was to assess the capabilities of data driven approaches in the estimation of scour depth downstream of grade control structures. Consequently, various GEP and FFNN models were developed and validated for three kinds of data, including the ski-jump bucket, sharp-crested weir and inclined slope controlled structures. These new techniques predicted scour depth downstream of grade control structures with reasonable accuracy from desirable data sets. The use of the GEP in the present study is another momentous part of scour depth estimation procedures for grade control structures. Since the GEP method can determine the most influential parameters on scour depth for all three structures, the use of a parsimonious model would result in higher accuracies than semi-empirical formulas.

The best semi-empirical formula R² values related to the ski-jump bucket (Azamathulla et al. 2005), sharp-crested weir (Yen 1987), and inclined slope controlled structures (Bormann & Julien 1991) are equal to 0.582, 0.661, and 0.632, respectively, which are less than the GEP and FFNN-related values for all of them. The obtained results revealed the superiority of FFNN and GEP techniques in predicting scour depth downstream of grade control structures.

The present study showed that GEP can be applied successfully to formulate scour depth downstream of grade control structures where interrelationships among the relevant variables are poorly understood.

Once the optimum model was identified for each data set, the most influential parameters affecting the scour depth were investigated through a sensitivity analysis. The results showed that for the ski-jump bucket, the optimum model includes four dimensionless parameters ⁠, from which and showed the highest and least influence, respectively; for the sharp-crested weir, the parameters were found to be the most relevant factors on scour depth, where and showed the highest and lowest effect on scouring; and for the inclined slope controlled structure, was the most influential parameter, while the showed the lowest effectiveness.

The results presented here are based on laboratory data, which have been captured from the existing literature. Further studies are needed to support the conclusions by using field data. Therefore, as the available data are limited for this issue, a comparison among different data management scenarios for feeding the data driven models would be valuable.