## Abstract

Grade-control structures (GCS) are commonly used to protect fish habitat by preventing excessive river-bed degradation in mountain streams. However, flow over the GCS can cause localized scour immediately downstream of the weir. This paper aims to develop more accurate models for prediction of the maximum scour depth downstream of GCS, using a more extensive dataset and evolutionary gene expression programming (GEP). Three GEP models are developed relating maximum scour depth and various control variables. The developed models had the lowest error compared to available models. A parametric analysis is performed for further verification of the developed GEP model. The results indicate that the proposed relations are simple and can more accurately predict the scour depth downstream GCS.

## INTRODUCTION

Grade-control structures (GCS) (Figure 1) are commonly prescribed to improve aquatic habitats in mountain streams by preventing excessive river-bed degradation (Veronese 1937; Mason & Arumugam 1985; Lenzi *et al.* 2003a; Muzzammil & Siddiqui 2009). GCS simulate alluvial river bed forms (riffles), do not require additional fish passage consideration, and naturally aerate flowing water over the weir (Farhoudi & Smith 1985; Hassan & Narayanan 1985; Bormann & Julien 1991).

However, the flow over GCS can cause localized scour downstream of the weir, potentially leading to failure of the structure (e.g., Guven & Gunal 2008; Abou-Seida *et al.* 2012). The key factors influencing the scour process include upstream water head, weir height, tail water level, bed particle grain-size distribution, and particle density (Bormann & Julien 1991; Lenzi & Comiti 2003; Marion *et al.* 2004; Pagliara 2007; Goel & Pal 2009; Azamathulla 2012; Atieh *et al.* 2017). To ensure safe design of the GCS to withstand expected flood events would require models for site-specific accurate prediction of the maximum depth of scour. Such models have been historically developed based on a combination of typically small set of laboratory-scale experiment data and limited field surveys (e.g., Bormann & Julien 1991; Mossa 1998; D'Agostino & Ferro 2003; Lenzi & Comiti 2003; Marion *et al.* 2004; Pagliara *et al.* 2004; Meftah & Mossa 2006; Pagliara 2007; Pagliara & Palermo 2008; Guven 2011; Radecki-Pawlik 2013; Guan *et al.* 2016). However, Scurlock *et al.* (2012) showed that for most models, the available experimental data – used for model development – were not sufficient to yield reliable scour depths and that prediction errors can reach 300%.

Another major disadvantage in most of the previously developed GCS scour depth models is that they are based on regression analysis, which may not be able to capture the complex relationship between key factors. Machine learning techniques, including gene expression programming (GEP), however, have been recently used by many researchers for developing complex models as an efficient alternative to traditional regression and other machine learning methods (Sattar 2014a, 2014b, 2016a, 2016b; Gazendam *et al.* 2016; Sabouri *et al.* 2016; Sattar *et al.* 2016, 2017; Gharabaghi & Sattar 2017; Thompson *et al.* 2016).

Thus, this work aims to overcome many of the previous studies' shortcomings by collecting large-scale field scour data downstream GCS in Polish mountain streams that can complement the existing database of experimental and field measurements. Moreover, this study presents the novel application of GEP for development of a scour depth prediction model that is capable of predicting scour downstream structures with higher accuracy than previous regression-based models. In this study, we conducted field surveys of 17 scour holes downstream of GCS in two Polish streams to augment 248 existing experimental and field measurements from several published papers from studies around the globe to build a comprehensive database for model development using GEP. The control variables used as predictors include flow rate, GCS height and width, and bed material representative particle sizes. GEP is used to find optimum prediction models with the least error and the best fit. The prediction uncertainty of the developed GEP models is quantified and compared with those of existing equations, and a parametric analysis is performed for further verification of the developed GEP models.

## MODELS FOR SCOUR DEPTH PREDICTION

Many historic studies used to predict the scour depth downstream GCS have used semi-analytical approaches; however, none of these models has proven to be sufficient for full description of the complex scour process and the related turbulent hydrodynamic factors. Bormann & Julien (1991) analyzed the jet diffusion and corresponding sediment incipient motion and developed a model with a simple equation that predicted the maximum scour depth. Their model has been modified by Stein & Julien (1994) to account for additional factors and provide more accurate predictions. Another similar model has been developed by Hoffmans (1998) based on the jet momentum dynamic approach. Chen & Hong (2001) performed complex analysis for the scour hole shape and proposed equations for maximum scour depth for uniform and graded sediment.

*et al.*(2003b) presented a summary of these historic models and highlighted major shortcomings, including insufficient representations for the complex scour process, dependency on various coefficients that are difficult to calculate, requirement for field calibration for model before application on a specific site, and presence of high degree of uncertainty in their results. One of the first empirical models for maximum scour depth downstream of GCS was developed by Mason & Arumugam (1985) and suggested a general form for a scour equation as: where

*a, b, c, d, e, f*are exponents with different values (review of exponent values for different studies is available in Bormann & Julien 1991),

*k*is a coefficient based on experimental results and a function of the tail water level,

*g*is the gravitational acceleration,

*d*is the effective grain diameter and taken as

_{s}*d*

_{90},

*q*is the unit discharge over the structure,

*U*is the jet velocity across the structure,

*D*is the head drop according to the structure, and is the jet angle at bed impact (see Figure 2).

_{g}*b*is the grade control structure width,

*H*is the approaching flow depth,

_{o}*B*is the channel width,

*d*

_{90},

*d*

_{50}are the 90-th and 50-th percent finer bed sediment diameters, and

*A*

_{50}is a dimensionless number defined as: where

*Q*is the flow through the GCS, is the sediment relative density.

*H*is the tail water depth downstream of the GCS. Equations for scour depth prediction from Equations (2)–(10) have some advantages over the equation form in Equation (1). They account for non-uniform bed sediment through the term , and the 3D flow owing to channel contraction through the term . Moreover, these equations can be used to predict scour depth even if it is larger than the structure height, which is a major limitation of Equation (1).

_{t}## MATERIALS AND METHODS

### Site selection

Many rivers in Poland are regulated using dams, bed sills, and low-head GCSs. These structures lead to channel bed stabilization and prevent successive erosion, where a dynamic equilibrium forms between bed scouring and aggradation. In this study, a detailed field survey for 17 scour hole downstream GCS was carried on two mountainous streams in Poland during the period from 2009 to 2015. The Porębianka River is 15.4 km long with a catchment area of 72 km^{2} and the Poniczanka River is 10.2 km long with a catchment area of 10.3 km^{2}, both located in the southern part of Poland in the Gorce Mountains (Figure 3). These mountainous rivers begin at the Obidowa peak (1,000 m a.s.l.) and rapidly drop elevations to below 500 m a.s.l. at the outlet.

### Grade-control structures

A total of 25 GCS have been constructed on the Porębianka River to stabilize the channel bed from Mszana Dolna town to Niedźwiedź village. These GCS are similar in construction and consist of two sheet pilings spread a distance of 12 to 24 m apart with elevation drops ranging from 1 to 2 m resulting in angles of inclination of 3° and 5°. The area between the sheet pilings is filled with boulders of 0.9 m average diameter, also known as the sloping rock apron (Figure 2). In the central part of the upper and lower sheet pilings, the crown elevation of the sheet pile at the centerline of the river is reduced about 20 cm on a 4 m length in order to concentrate the flow of water during the low flows for ease of fish migration.

Out of a total of 25 scour holes, 17 were chosen after careful inspection, such that the bottoms of all the holes were armored by coarse material with an average size equivalent to the largest found in the stream (Figure 4). The boulders and cobbles at each hole's bottom were tight and adjoined to each other with no clasts on their surface or fine sediment. This ensures that the scour hole has reached the dynamic equilibrium between water-sediment discharges and its dimensions. For each scour hole, the GCS height and the maximum scour depth were measured from surveyed profiles (Figure 5).

### GEP for model development

The functional relationships between the scour depth and related parameters can be determined by regression analysis or machine learning methods (e.g., artificial neural networks (ANNs), GEP). Although ANNs are successfully used in hydrology and water resources models, many studies have proven that GEP can be useful in many hydraulic engineering applications (e.g., Guven & Gunal 2008; El Hakeem & Sattar 2015; Najafzadeh & Sattar 2015; Sattar & Gharabaghi 2015; Sattar 2016a, 2016b; Sattar *et al.* 2016; Thompson *et al.* 2016).

*f*, is defined in the following equation where the error used is the root relative squared error (RRSE). Equation (11) is used to determine the root relative square error RRSE of an individual program

_{i}*i*(

*i*-the offspring). where

*P*

_{(ij)}is the value predicted by the program

*i*for fitness case

*j*,

*T*is the target value for fitness case

_{j}*j*, , and

*n*is the number of samples. The RRSE ranges from 0 to infinity, with zero corresponding to a perfect fit (ideal). Genetic mutations are the main essence of genetic evolution in GEP and can be described in seven forms (Ferreira 2001). The GEP code developed by Radi & Poli (2002) has been utilized in this study.

The following procedure was used to develop the GEP models:

- 1.
An initial set of control variables is chosen as terminals for GEP.

- 2.
The initial work environment is set for GEP by defining the chromosome architecture (number of genes, head size, functions) and mutation rates.

- 3.
GEP randomly formulates the chromosomes of the parent program and implements genetic operators to yield many first-generation offspring.

- 4.
GEP uses the fitness criteria to find the fittest offspring. This offspring represents the solution to the problem in the first generation.

- 5.
GEP considers the selected offspring the new parent and implements genetic operators to yield many second-generation offspring.

- 6.
GEP evolution continues per steps 3, 4, and 5 until the specified program fitness is met.

- 7.
The final GEP model (the fittest offspring of generation

*i*) is scored on a set of performance indicators. These indicators are the square of the Pearson product moment correlation coefficient (*R*^{2}), the relative absolute error (RAE), coefficient of efficiency (*E*), and index of agreement (_{sn}*D*).

### Validation of developed GEP models

*k*), or for the observed versus predicted values (

*k′*), at least one should be close to 1. Additionally, the coefficient of determination for the regression lines through the origin

*m*and

*n*should be less than 0.1. Moreover, the cross-validation coefficient

*R*should satisfy: where the squared correlation coefficients through the origin between the predicted and observed values and between the observed and predicted values are calculated from:

_{m}### Uncertainty analysis of GEP models

*S*, a confidence band can be defined around the predicted values of an error using Wilson score method without continuity correction (Newcombe 1998); the use of ±1.96

_{e}*S*yields an approximately 95% confidence band.

_{e}### Sensitivity analysis of GEP models

*et al.*2013; Robertson

*et al.*2015): where

*S*= the marginal sensitivity coefficient, = the change in GEP prediction, and = the change in the parameter. Normalizing the parameters indicates the percent change in the objective function for a 10% change in each individual parameter. The marginal sensitivity coefficients were normalized to establish a basis of comparison: where is the normalized sensitivity coefficient, = the expected value of the perturbed parameter, and = the expected solution. Higher normalized sensitivity coefficient implies higher sensitivity of the model to a specific parameter and vice versa. Moreover, the sign of the normalized sensitivity coefficient determines the gradient of change of solution with the increase in a specific parameter, i.e., when there is a positive sign, this means that the model prediction for scour depth increases with the increase in this parameter.

_{c}## RESULTS AND DISCUSSION

### Scour depth database

In this study, a comprehensive database of 265 scour depth measurements are compiled. These measurements include small GCS (Veronese 1937; D'Agostino & Ferro 2004), medium GCS (Bormann & Julien 1991) and large GCS on actual streams (Falciai & Giacomin 1978; D'Agostino 1994). The database includes 17 field measurements conducted in this study for large GCS in Polish streams. The database includes measurements of scour hole maximum depth (*D _{m}*) and various related flow, sediment and structure geometry parameters, i.e., the flow discharge (

*Q*), the jet thickness (

*H*), and the jet velocity (

_{o}*U*), tail water level (

_{o}*H*), median bed sediment size (

_{t}*d*

_{50}), length of the GCS (

*L*), height of the GCS (

_{g}*z*), and angle of inclination of the GCS downstream face (

_{g}*α*

_{g}). Falciai & Giacomin (1978) carried out 29 field measurements (Table 1) for scour depth downstream GCS in several rivers in the Tuscan Apennines in Italy with width from 5 to 20 m.

Variable category | Variables | Veronese (1937) | Falciai & Giacomin (1978) | Bormann & Julien (1991) | This study (2016) | D'Agostino (1994) | D'Agostino & Ferro (2004) |
---|---|---|---|---|---|---|---|

Data points | 265 | 33 | 26 | 66 | 17 | 11 | 112 |

Flow | Q (m^{3}/s) | 0.01–0.04 | 6.60–182.70 | 0.26–2.25 | 20.0–65.0 | 3.98–3.98 | 0.01–0.04 |

U_{o} (m/s) | 0.34–0.65 | 0.50–2.80 | 0.88–4.65 | 1.25–3.20 | 1.00–1.20 | 0.38–0.85 | |

H_{o} (m) | 0.04–0.13 | 0.79–3.96 | 0.09–1.19 | 0.67–.12 | 0.40–0.50 | 0.04–0.21 | |

H_{t} (m) | 0.05–0.25 | 1.13–4.90 | 0.24–1.65 | 0.86–1.56 | 0.50–0.70 | 0.08–0.44 | |

Sediment | d_{50} (mm) | 9.1–36.2 | 19.0–100.0 | 0.30–0.45 | 40.0–50.0 | 60 | 4.1–11.5 |

d_{90} (mm) | 9.1–36.2 | 37.0–117.0 | 1.58–1.71 | 58.0–69.0 | 60 | 7.0–17.6 | |

GCS | L_{g} (m) | − | − | − | 12.0–24.0 | − | − |

z_{g} (m) | 1.0–1.1 | 0.3–6.2 | 0.0–0.4 | 1.0–2.0 | 0.6–1.4 | − | |

α_{g} (^{o}) | − | − | 18–90 | 3.0–5.0 | − | − | |

Scour hole | D_{m} (m) | 0.1–0.2 | 0.40–3.50 | 0.10–1.52 | 0.15–2.30 | 0.3–0.7 | 0.045–0.3 |

Variable category | Variables | Veronese (1937) | Falciai & Giacomin (1978) | Bormann & Julien (1991) | This study (2016) | D'Agostino (1994) | D'Agostino & Ferro (2004) |
---|---|---|---|---|---|---|---|

Data points | 265 | 33 | 26 | 66 | 17 | 11 | 112 |

Flow | Q (m^{3}/s) | 0.01–0.04 | 6.60–182.70 | 0.26–2.25 | 20.0–65.0 | 3.98–3.98 | 0.01–0.04 |

U_{o} (m/s) | 0.34–0.65 | 0.50–2.80 | 0.88–4.65 | 1.25–3.20 | 1.00–1.20 | 0.38–0.85 | |

H_{o} (m) | 0.04–0.13 | 0.79–3.96 | 0.09–1.19 | 0.67–.12 | 0.40–0.50 | 0.04–0.21 | |

H_{t} (m) | 0.05–0.25 | 1.13–4.90 | 0.24–1.65 | 0.86–1.56 | 0.50–0.70 | 0.08–0.44 | |

Sediment | d_{50} (mm) | 9.1–36.2 | 19.0–100.0 | 0.30–0.45 | 40.0–50.0 | 60 | 4.1–11.5 |

d_{90} (mm) | 9.1–36.2 | 37.0–117.0 | 1.58–1.71 | 58.0–69.0 | 60 | 7.0–17.6 | |

GCS | L_{g} (m) | − | − | − | 12.0–24.0 | − | − |

z_{g} (m) | 1.0–1.1 | 0.3–6.2 | 0.0–0.4 | 1.0–2.0 | 0.6–1.4 | − | |

α_{g} (^{o}) | − | − | 18–90 | 3.0–5.0 | − | − | |

Scour hole | D_{m} (m) | 0.1–0.2 | 0.40–3.50 | 0.10–1.52 | 0.15–2.30 | 0.3–0.7 | 0.045–0.3 |

The unit discharges had a range from 1.2 to 13.4 m^{2}/s, the mean sediment diameter *d*_{50} from 19 to 100 mm, and the height of the GCS had a range from 0.3 to 6.2 m. The measured scour depth had a minimum value of 0.4 m and a maximum value of 3.5 m. The maximum scour depth data measured in the current study are from Porebianka and Poniczanka Rivers in Poland. The width of the rivers ranges from 8 to 12 m. The unit discharge has a range of 1 to 4.75 m^{2}/s, the jet velocity has a minimum value of 1.25 m/s and a maximum value of 3.2 m/s. The maximum measured scour depth ranges from 0.15 to 2.3 m. Table 2 presents the statistical parameters used in the study, which include maxima, minimum, average, and standard deviation.

Variable | Flow | Sediment | GCS | Scour hole | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Q (m^{3}/s) | q (m^{3}/s/m) | U_{o} (m/s) | H_{o} (m) | H_{t} (m) | d_{50} (mm) | d_{90} (mm) | L_{g} (m) | z_{g} (m) | α_{g} (^{o}) | D_{m} (m) | |

Minimum | 0.26 | 0.29 | 0.57 | 0.09 | 0.14 | 0.30 | 1.58 | 12 | 0.05 | 4.77 | 0.05 |

Average | 9.90 | 1.10 | 2.29 | 0.50 | 0.70 | 14.58 | 18.84 | 18 | 1.02 | 52.57 | 0.50 |

Maximum | 182.70 | 13.40 | 4.45 | 3.96 | 5.00 | 100.0 | 109.0 | 24 | 6.20 | 89.95 | 3.50 |

Standard deviation | 26 | 1.82 | 1.04 | 0.66 | 0.87 | 1.82 | 23.30 | − | 1.43 | 32.42 | 0.55 |

Variable | Flow | Sediment | GCS | Scour hole | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Q (m^{3}/s) | q (m^{3}/s/m) | U_{o} (m/s) | H_{o} (m) | H_{t} (m) | d_{50} (mm) | d_{90} (mm) | L_{g} (m) | z_{g} (m) | α_{g} (^{o}) | D_{m} (m) | |

Minimum | 0.26 | 0.29 | 0.57 | 0.09 | 0.14 | 0.30 | 1.58 | 12 | 0.05 | 4.77 | 0.05 |

Average | 9.90 | 1.10 | 2.29 | 0.50 | 0.70 | 14.58 | 18.84 | 18 | 1.02 | 52.57 | 0.50 |

Maximum | 182.70 | 13.40 | 4.45 | 3.96 | 5.00 | 100.0 | 109.0 | 24 | 6.20 | 89.95 | 3.50 |

Standard deviation | 26 | 1.82 | 1.04 | 0.66 | 0.87 | 1.82 | 23.30 | − | 1.43 | 32.42 | 0.55 |

### Key factors affecting scour depth

The equation above shows that the dimensionless scour downstream GCS depends on the GCS parameters , channel parameters , , and sediment non-uniform coefficient . Three GEP-based models were developed using different combinations of the above predictive parameters, as shown in Table 3. The effect of bed sediment non-uniformity was tested by excluding the parameter , while the effect of the GCS was tested by excluding the parameter . Using each parameter combination, the GEP model runs with an evolutionary mode until the fittest model is developed.

Model | Equation number | Input parameters to predict scour | Data type |
---|---|---|---|

Pagliara & Palermo (2008) | (5) | Laboratory data | |

D'Agostino & Ferro (2004) | (2) | Laboratory data | |

(3) | |||

Laucelli & Giustolisi (2011) | (6) | Field and laboratory data | |

(7) | |||

(8) | |||

(9) | |||

Guven (2011) | (10) | Laboratory data | |

GEP (this study) | (26) | Field and laboratory data | |

(27) | |||

(28) |

Model | Equation number | Input parameters to predict scour | Data type |
---|---|---|---|

Pagliara & Palermo (2008) | (5) | Laboratory data | |

D'Agostino & Ferro (2004) | (2) | Laboratory data | |

(3) | |||

Laucelli & Giustolisi (2011) | (6) | Field and laboratory data | |

(7) | |||

(8) | |||

(9) | |||

Guven (2011) | (10) | Laboratory data | |

GEP (this study) | (26) | Field and laboratory data | |

(27) | |||

(28) |

No strong correlation or interdependency was found among the predictor variables, thus eliminating the problems that could arise in analysis from exaggerating the strength of the relations between variables. The experimental database was divided into training and testing subsets. Several combinations were considered and the selection was made such that the main statistical descriptive parameters (mean and standard deviation) of both the training and testing subsets were consistent. Of the 265 values, 198 (75%) were used to develop the models, and 67 (25%) were used to test the models developed by using the GEP.

### Developed GEP models

The best GEP models are obtained using the procedure specified in Sattar (2016c) with the optimal parameter settings shown in Table 4. It is to be noted that the optimal GEP parameters for this study might not be the optimum for other problems depending on the complexity of the input data. The following GEP models for the prediction of scour downstream GCS had high scores and low prediction errors.

Parameter | Optimal setting |
---|---|

Number of generations | 15,000 |

Number of chromosomes | 40 |

Number of genes | 3 |

Head size | 3 |

Linking function | Multiplication |

Fitness function error type | RRSE |

Mutation rate | 0.005 |

Inversion rate | 0.1 |

One point recombination rate | 0.3 |

Two point recombination rate | 0.5 |

Gene recombination rate | 0.1 |

Gene transposition rate | 0.1 |

Function set | x, /, power |

Random numerical constants | 3 |

RNC mutation | 0.001 |

Parameter | Optimal setting |
---|---|

Number of generations | 15,000 |

Number of chromosomes | 40 |

Number of genes | 3 |

Head size | 3 |

Linking function | Multiplication |

Fitness function error type | RRSE |

Mutation rate | 0.005 |

Inversion rate | 0.1 |

One point recombination rate | 0.3 |

Two point recombination rate | 0.5 |

Gene recombination rate | 0.1 |

Gene transposition rate | 0.1 |

Function set | x, /, power |

Random numerical constants | 3 |

RNC mutation | 0.001 |

*R*

^{2}of 0.95 for training and of 0.90 for testing datasets. The error measures, RMSE and RAE show low values for training and testing datasets. The RMSE for GEP-01 and GEP-02 are 0.75 and 0.88, respectively, for training and testing sets. For GEP-03, error values for training and testing were lowest among the GEP models with values of 0.60 and 0.62 for training and testing sets, respectively. Similar trends are observed in the RAE calculated values. On the other hand, the

*E*and

_{sn}*D*show very good values for training and testing subsets with average values of 0.96, 0.99 for training subset and 0.90, 0.98 for testing subset.

Model | Equation number | Data partitioning | R^{2} | RMSE | RAE | E_{sn} | D |
---|---|---|---|---|---|---|---|

Pagliara & Palermo (2008) | (5) | NA | 0.48 | 2.75 | 0.62 | 0.30 | 0.82 |

D'Agostino & Ferro (2004) | (2) | NA | 0.63 | 2.33 | 0.46 | 0.49 | 0.78 |

(3) | 0.75 | 2.26 | 0.59 | 0.53 | 0.81 | ||

Laucelli & Giustolisi (2011) | (6) | NA | 0.68 | 2.41 | 0.79 | 0.46 | 0.70 |

(7) | 0.53 | 2.33 | 0.79 | 0.49 | 0.76 | ||

(8) | 0.76 | 2.71 | 0.75 | 0.32 | 0.53 | ||

(9) | 0.68 | 2.07 | 0.73 | 0.60 | 0.82 | ||

Guven (2011) | (10) | NA | 0.71 | 33.54 | 3.71 | 103.43 | 0.27 |

GEP-01 | (26) | Train | 0.95 | 0.75 | 0.21 | 0.95 | 0.99 |

Test | 0.88 | 0.89 | 0.29 | 0.85 | 0.97 | ||

GEP-02 | (27) | Train | 0.95 | 0.77 | 0.25 | 0.94 | 0.99 |

Test | 0.88 | 0.88 | 0.30 | 0.86 | 0.97 | ||

GEP-03 | (28) | Train | 0.97 | 0.60 | 0.17 | 0.97 | 0.99 |

Test | 0.93 | 0.62 | 0.19 | 0.93 | 0.98 |

Model | Equation number | Data partitioning | R^{2} | RMSE | RAE | E_{sn} | D |
---|---|---|---|---|---|---|---|

Pagliara & Palermo (2008) | (5) | NA | 0.48 | 2.75 | 0.62 | 0.30 | 0.82 |

D'Agostino & Ferro (2004) | (2) | NA | 0.63 | 2.33 | 0.46 | 0.49 | 0.78 |

(3) | 0.75 | 2.26 | 0.59 | 0.53 | 0.81 | ||

Laucelli & Giustolisi (2011) | (6) | NA | 0.68 | 2.41 | 0.79 | 0.46 | 0.70 |

(7) | 0.53 | 2.33 | 0.79 | 0.49 | 0.76 | ||

(8) | 0.76 | 2.71 | 0.75 | 0.32 | 0.53 | ||

(9) | 0.68 | 2.07 | 0.73 | 0.60 | 0.82 | ||

Guven (2011) | (10) | NA | 0.71 | 33.54 | 3.71 | 103.43 | 0.27 |

GEP-01 | (26) | Train | 0.95 | 0.75 | 0.21 | 0.95 | 0.99 |

Test | 0.88 | 0.89 | 0.29 | 0.85 | 0.97 | ||

GEP-02 | (27) | Train | 0.95 | 0.77 | 0.25 | 0.94 | 0.99 |

Test | 0.88 | 0.88 | 0.30 | 0.86 | 0.97 | ||

GEP-03 | (28) | Train | 0.97 | 0.60 | 0.17 | 0.97 | 0.99 |

Test | 0.93 | 0.62 | 0.19 | 0.93 | 0.98 |

The Q-Q plot for the predicted versus the measured for the three GEP models is shown in Figure 6. The figure shows the quality of prediction of the GEP models as they are applied on training and testing datasets. It is clear that the GEP-01 and GEP-02 models overestimated the scour depth for , while this is less noticeable in the GEP-03 model. Moreover, the terms , , appeared as ‘important predictors’ in the developed GEP models. This is in agreement with the findings of most of the previous research, e.g., D'Agostino & Ferro (2004), Laucelli & Giustolisi (2011), and Guven (2011); while the term appeared only in the GEP-03 model, similar to Laucelli & Giustolisi (2011).

### Error analysis

Three different formulae for prediction of scour depth downstream large GCS have been developed. Table 5 shows the statistical measures of the developed GEP model prediction errors as compared to the historic models. The model developed by D'Agostino & Ferro (2004) and Laucelli & Giustolisi (2011) yielded the highest *R*^{2} among all available models with value of 0.76, which is lower than that of all three developed GEP models. Pagliara & Palermo's (2008) model yielded the least *R*^{2} with the capacity to represent only 50% of the scour data. The RMSE for the models of D'Agostino & Ferro (2004), Laucelli & Giustolisi (2011), and Pagliara & Palermo (2008) were close with an average value of 2.5, which is higher than that of the developed GEP models. On the other hand, Guven (2011) yielded the highest RMSE with a value of 33.5. Moreover, the results of *E _{sn}* and D indices for all available models were low and unacceptable ranging from 0.3 to 0.8 as compared to the GEP models which yielded high values of 0.95 on average. It is shown from the above discussion that the GEP models outperformed all available models on the scour dataset including large GCS. The GEP models can predict the target scour values with acceptable accuracy and less error than the available models.

### Validation tests

As discussed before, the performance of developed models on testing data subsets was used to evaluate the prediction capability of the developed GEP models. Table 6 shows the validation criteria used and corresponding model performance, as calculated from Equations (17)–(20). Models have to satisfy some or all of the validation criteria. The gradients of the regression line for the predicted versus observed scour was close to 1 and within the recommended ranges of 0.85 to 1.15. This was accompanied by good values for the coefficient of determination for the regression line with average values of −0.10. The condition of cross-validity was also satisfied for all developed models. This shows a good and accepted performance of the developed GEP models against test dataset.

Model | R (R > 0.8) | K (0.85 < K < 1.15) | K′ (0.85 < K′ < 1.15) | M (m < 0.1) | N (n < 0.1) |
---|---|---|---|---|---|

GEP-01 | 0.95 | 0.92 | 0.88 | −0.11 | −0.10 |

GEP-02 | 0.97 | 1.11 | 0.72 | −0.07 | −0.03 |

GEP-03 | 0.94 | 0.83 | 0.96 | −0.11 | −0.13 |

Model | R (R > 0.8) | K (0.85 < K < 1.15) | K′ (0.85 < K′ < 1.15) | M (m < 0.1) | N (n < 0.1) |
---|---|---|---|---|---|

GEP-01 | 0.95 | 0.92 | 0.88 | −0.11 | −0.10 |

GEP-02 | 0.97 | 1.11 | 0.72 | −0.07 | −0.03 |

GEP-03 | 0.94 | 0.83 | 0.96 | −0.11 | −0.13 |

### Uncertainty analysis

The uncertainties in the predictions of the scour downstream large GCS are presented in Table 7 for the developed GEP models as well as the available empirical models. The uncertainty analysis is applied to the complete scour dataset used in this study. Table 7 shows the mean prediction errors of the various models, the width of the uncertainty band and the 95% prediction interval error. All three GEP models have absolute mean prediction errors for scour depth downstream large GCS much less than those of the presented empirical models. The absolute mean prediction error for D'Agostino & Ferro (2004) and Laucelli & Giustolisi (2011) was four orders of magnitude larger than the GEP-01 and GEP-03 models. The deviation of prediction error for the GEP models was half that of the available empirical models with values of 0.25 and 0.6, respectively. According to the mean prediction error, GEP-01 and GEP-03 slightly overpredicted scour depth. This was the same observed behavior for the three empirical models of Laucelli & Giustolisi (2011), which showed overprediction, while the models of D'Agostino & Ferro (2004), Pagliara & Palermo (2008), and Guven (2011) showed underprediction for the scour depth. The uncertainty band width for the three GEP models is similar and ranged from ±0.44 to ±0.52. This range is half the range calculated for the other empirical models, which were from ±0.90 for the Pagliara & Palermo (2008) model to ±2.59 for Guven's (2011) model. Similarly, the lowest 95% confidence prediction error interval was observed for the GEP models. The GEP models had the lowest mean prediction error and the smallest uncertainty bands of all the compared models.

Model | Equation number | Mean prediction error | Deviation of prediction error | Width of uncertainty band | 95% prediction error interval |
---|---|---|---|---|---|

Pagliara & Palermo (2008) | (5) | −0.10 | 0.46 | ±0.90 | 0 to + 79 |

D'Agostino & Ferro (2004) | (2) | −0.04 | 0.31 | ±0.62 | 0 to + 18 |

(3) | −1.05 | 0.59 | ±1.16 | 0 to + 234 | |

Laucelli & Giustolisi (2011) | (6) | +0.40 | 0.53 | ±1.03 | 0 to + 46 |

(7) | +0.37 | 0.53 | ±1.04 | 0 to + 51 | |

(8) | +0.30 | 0.55 | ±1.07 | 0 to + 70 | |

(9) | +0.41 | 0.50 | ±0.97 | 0 to + 35 | |

Guven (2011) | (10) | −0.86 | 1.32 | ±2.59 | 0 to + 111 |

GEP-01 | (26) | +0.01 | 0.27 | ±0.51 | 0 to + 10 |

GEP-02 | (27) | +0.14 | 0.23 | ±0.44 | 0 to + 9 |

GEP-03 | (28) | +0.04 | 0.27 | ±0.52 | 0 to + 5 |

Model | Equation number | Mean prediction error | Deviation of prediction error | Width of uncertainty band | 95% prediction error interval |
---|---|---|---|---|---|

Pagliara & Palermo (2008) | (5) | −0.10 | 0.46 | ±0.90 | 0 to + 79 |

D'Agostino & Ferro (2004) | (2) | −0.04 | 0.31 | ±0.62 | 0 to + 18 |

(3) | −1.05 | 0.59 | ±1.16 | 0 to + 234 | |

Laucelli & Giustolisi (2011) | (6) | +0.40 | 0.53 | ±1.03 | 0 to + 46 |

(7) | +0.37 | 0.53 | ±1.04 | 0 to + 51 | |

(8) | +0.30 | 0.55 | ±1.07 | 0 to + 70 | |

(9) | +0.41 | 0.50 | ±0.97 | 0 to + 35 | |

Guven (2011) | (10) | −0.86 | 1.32 | ±2.59 | 0 to + 111 |

GEP-01 | (26) | +0.01 | 0.27 | ±0.51 | 0 to + 10 |

GEP-02 | (27) | +0.14 | 0.23 | ±0.44 | 0 to + 9 |

GEP-03 | (28) | +0.04 | 0.27 | ±0.52 | 0 to + 5 |

The results of the uncertainty associated with the predictions of the GEP models are shown in Table 8. The mean prediction for the scour depth was similar for GEP-01 and GEP-03 models with an average value of 1.25 m. The expected uncertainties in prediction of GEP-01 and GEP-02 models were similar with values of 41%, while the GEP-03 model yielded a lower prediction uncertainty of 24%. Uncertainties in the order or magnitude of 25–40% have been reported in the literature to be acceptable ranges for reliable models (Sattar 2014b). These uncertainty bands can be used together with GEP model predictions to provide an estimate for the maximum scour depth downstream large CGS.

Model | Median | Uncertainty % |
---|---|---|

GEP-01 | 1.33 | 44 |

GEP-02 | 1.67 | 41 |

GEP-03 | 1.22 | 24 |

Model | Median | Uncertainty % |
---|---|---|

GEP-01 | 1.33 | 44 |

GEP-02 | 1.67 | 41 |

GEP-03 | 1.22 | 24 |

### Parametric analysis

In this section, a parametric analysis is performed on the developed GEP models to test their prediction behavior compared to the physics of the scour downstream GCS and how it is influenced by various input parameters. Beside being simple models, the developed GEP models show a clear combination of parameters that give them an advantage over existing grey-box machine learning methods. The average values of all parameters are used in prediction models and only one parameter (test parameter) is varied from a minimum value to a maximum value. Both the chosen average and minimum/maximum values are from the parameter's range specified in Table 2.

Figure 7 shows the versus and as calculated by the three GEP models and those of D'Agostino & Ferro (2004), Guven (2011), and Laucelli & Giustolisi (2011). The scour depth is directly proportional to the bed sediment gradation represented by the term , where it increases with the increase in the term .

This is consistent with the trends produced by the models of Guven (2011) and Laucelli & Giustolisi (2011) plotted in the same graph and is supported by the results of the sensitivity analysis shown in Table 9, where the is one of the lower ranked influencing parameters on the scour depth.

Attribute | S_{n} | ||
---|---|---|---|

GEP-01 | GEP-02 | GEP-03 | |

A_{50} | +0.79 | +0.58 | +0.59 |

b/z_{g} | 0.0 | 0.0 | +0.11 |

H_{t}/z_{g} | −0.08 | −0.13 | −0.08 |

d_{90}/d_{50} | +0.25 | 0.0 | +0.15 |

Attribute | S_{n} | ||
---|---|---|---|

GEP-01 | GEP-02 | GEP-03 | |

A_{50} | +0.79 | +0.58 | +0.59 |

b/z_{g} | 0.0 | 0.0 | +0.11 |

H_{t}/z_{g} | −0.08 | −0.13 | −0.08 |

d_{90}/d_{50} | +0.25 | 0.0 | +0.15 |

This observation is consistent with previous experimental findings, where Guan *et al.* (2016) confirmed that experimental and field measurements of scour depth downstream GCS did not reveal pronounced effects for the sediment size in the scour process in rivers, and that its effect is much less than the effect of other parameters. Figure 7 also shows the change of the normalized scour depth versus the normalized tail water depth . The developed GEP models show the same trend of change as the models of D'Agostino & Ferro (2004) and Laucelli & Giustolisi (2011). The increase in results in a corresponding decrease in the normalized scour depth. This has been discussed and shown by Bormann & Julien (1991), who related the increase in the downstream water level to the increase in the flow diffusion length in the scour hole resulting in reducing the hydrodynamic forces acting on the stream bed. Again, the change in the scour depth is very small compared to a corresponding large change in the tail water depth. The normalized tail water depth had only 8 to 13% influence on the scour depth as calculated in Table 9. This behavior is also consistent with the findings of experimental and field measurements of Laucelli & Giustolisi (2011) and Scurlock *et al.* (2012), who found and confirmed that the effect of the tail water on the scour depth is minimal compared to other parameters.

According to results in Table 9, the parameter *A*_{50} showed the highest influence on the scour depth with 58% influence on GEP-02 and GEP-03 models, and increased to 80% in the GEP-01 model. The impact is clearly shown in Figure 8, where the increase in the term caused a similar increase in normalized scour depth for the three GEP models in addition to D'Agostino & Ferro (2004) and Guven (2011) models.

Pagliara *et al.*’s (2004) experimental measurements showed that the high scour depths downstream GCS are related to high values of the *A*_{50} parameter and low sediment concentrations. Following the *A*_{50} parameter, the normalized GCS width was ranked the second most influential parameter on the scour depth, as shown in Table 9.

The term appeared only in the GEP-03 model with influence value of 11%. The normalized scour depth increased with the increase in , as shown in Figure 8 for the GEP-03 model, which showed a similar trend to D'Agostino & Ferro (2004), Guven (2011), and Laucelli & Giustolisi (2011) models. This term accounts for the change in the stream width relative to the GCS. This change can trigger water circulation leading to a deflection and concentration for flow exiting the GCS ramp (Pagliara & Palermo 2008) resulting in an increase in the erosive water capacity and thus the increase in the scour depth.

## CONCLUSIONS

The main goal of this paper was to develop more accurate models for prediction of the scour depth downstream of GCS to ensure safety and stability during flood flows. This is the first study of its kind to compile a comprehensive database of 265 scour depth measurements, consisting of both laboratory experiments and field surveys, to develop a more accurate model using GEP. The collected extensive dataset covers a wide range of the key factors influencing the scour process, including upstream water level, weir height, tail water level, and bed particle size distribution and particle density, therefore allowing the development of a superior model, compared to existing regression-based models, using GEP.

Selection criteria based on statistical measures and on external validation measures and the output of uncertainty analyses were used to select the best GEP models with the highest prediction accuracy and the least uncertainty. The prediction errors and uncertainties associated with the developed GEP models were significantly smaller than those associated with all of the existing empirical models. The root mean squared scour depth prediction error of the new GEP models dropped by 50% compared to the best of the existing empirical equations. The main reason for the superior performance of the new GEP models, compared with the existing empirical equations, is the use of the much larger/multi-scale dataset for the training of the new models and also the inclusion of the channel expansion parameter to enhance the performance of the new models.

Sensitivity analysis of the new GEP models revealed that the unit discharge has the highest influence on the scour depth, while the tail water depth and bed sediment size were found to have the least effect. The scour depth is directly proportional to the bed sediment gradation represented by the term and indirectly proportional to . The parameter *A*_{50} has the highest influence on the scour depth followed by the normalized GCS width as the second most influential factor.

## ACKNOWLEDGEMENTS

The authors would like to acknowledge the funding and support of the Natural Sciences and Engineering Research Council of Canada (NSERC) and the University of Agriculture in Krakow, Department of Land Reclamation and Environmental Development (BM/4374/KIWiG/2016 – Project: ‘The Researches for Young Scientists’).

## REFERENCES

*Experimental and Computational Solutions of Hydraulic Problems*(P. Rowinski, ed.). GeoPlanet: Earth and Planetary Sciences, Springer Netherlands, pp. 101–115