Abstract

Wave-induced scour depth below pipelines is a physically complex phenomenon, whose reliable prediction may be challenging for pipeline designers. This study shows the application of adaptive neuro-fuzzy inference system (ANFIS) incorporated with particle swarm optimization , ant colony (), differential evolution and genetic algorithm () and assesses the scour depth prediction performance and associated uncertainty in different scour conditions including live-bed and clear-water. To this end, the non-dimensional parameters Shields number (), Keulegan–Carpenter number () and embedded depth to diameter of pipe ratio () are considered as prediction variables. Results indicate that the model ( and ) is the most accurate predictive model in both scour conditions when all three mentioned non-dimensional input parameters are included. Besides, the model shows a better prediction performance than recently developed models. Based on the uncertainty analysis results, the prediction of scour depth is characterized by larger uncertainty in the clear-water condition, associated with both model structure and input variable combination, than in live-bed condition. Furthermore, the uncertainty in scour depth prediction for both live-bed and clear-water conditions is due more to the input variable combination than it is due to the model structure .

HIGHLIGHTS

  • Uncertainties of model structure and input variables through the prediction of wave-induced scour depth around pipelines are quantified using Monte Carlo simulation.

  • New hybrid data-intelligence models are developed to predict wave-induced scour depth around pipelines.

  • The proposed models show robust predictive techniques for wave-induced scour depth around pipelines.

  • The proposed models have better accuracy in the prediction of wave-induced scour depth around pipelines compared to other previously developed ones.

INTRODUCTION

Submarine pipelines are commonly utilized to carry gas and oil in offshore areas and usually lie on erodible seafloors. Wave action may wash out sediment around a pipeline due to a localized increase of bed shear stress, with consequent development of a scour hole that may undermine the stability of the pipeline and eventually lead to its collapse. In fact, the pipeline may become suspended in seawater as the scour develops, making its structure not able to withstand static and dynamic forces. Pipeline failure not only represents an economic loss but may also cause significant environmental consequences. Thus, consideration of the scour phenomenon beneath offshore pipelines is key during pipeline design (Fredsøe et al. 1988; Yasa & Etemad-Shahidi 2014).

Most of the available predictive scour depth formulas in this context are based on laboratory experiments (Lucassen 1984; Sumer & Fredsøe 1990; Çevik & Yüksel 1999). Such regression-based equations are of straightforward use, and they are commonly adopted to estimate scouring depth around pipelines and, generally, any river or marine structures; however, possible scale effects may lead to considerable inaccuracy in predicting scour for large-scale structures in the field (Tafarojnoruz 2012; Tafarojnoruz & Gaudio 2012). To overcome this limitation, numerical models may be developed for local erosion simulation (Zhao & Fernando 2007; Zhao et al. 2018); such studies, however, are still limited in number and are generally dependent on validation against laboratory observations. Furthermore, simulating scour phenomena with a numerical model is computationally burdensome: resolving a three-dimensional scour hole up to the point it reaches equilibrium may take weeks to months of machine time, which is often impractical for any project's purposes.

Soft computing (SC) techniques have been increasingly adopted to analyze and predict various hydraulic phenomena. For instance, the use of group method of data handling (GMDH), applied to the prediction of the longitudinal dispersion coefficient in rivers, offered more accurate estimations compared with the available empirical equations (Najafzadeh & Tafarojnoruz 2016); a study on the calculation of riprap stone size for protection of a steep slope revealed that evolutionary polynomial regression (EPR) is a robust alternative to the empirical mathematical formulations (Najafzadeh et al. 2018); numerous studies demonstrated the capability of artificial intelligence (AI) techniques in predicting local scour depth around hydraulic structures (Najafzadeh et al. 2017; Ebtehaj et al. 2018; Najafzadeh & Kargar 2019).

Scour development adjacent to submarine pipelines is generally caused by the shear stress on the seabed associated with waves, currents, or a combination of both. Previous investigations have used AI techniques to predict current-induced and wave-induced scour depth. For current-induced scouring, the earliest studies focused on artificial neural networks (ANNs)-based and Genetic Programming-based scour depth prediction around pipelines crossing rivers, producing an acceptable prediction performance (Azamathulla & Ghani 2010; Azamathulla & Zakaria 2011). Zanganeh et al. (2011) adopted an optimization-based methodology (i.e., PSO algorithm) to mitigate the shortcomings of an adaptive neuro-fuzzy inference system (ANFIS) model for current-induced scour prediction. Yasa & Etemad-Shahidi (2014) derived scour prediction formulations for live-bed and clear-water scour conditions by combining the model tree (MT) and regression model.

Utilizing some of the AI methods may lead to derive new prediction equations. These equations may generally have more complicated mathematical structures than those resulting from the conventional regression-based approaches, but at the same time may offer more accurate predictions. For instance, Najafzadeh & Sarkamaryan (2018) proposed gene-expression programming (GEP), EPR and MT algorithms to extract mathematical formulations for estimating current-induced scour depth below pipelines. Recent studies showed the capabilities of Multivariate Adaptive Regression Splines and Support Vector Machine techniques in predicting scour depth under pipelines in rivers (Haghiabi 2017, 2019; Parsaie et al. 2019).

For wave-induced scour, the prediction performance of an ANN approach, one of the most common AI models, was assessed in comparison with regression-based formulations (Kazeminezhad et al. 2010). Although this study reported accurate predictions using the ANN approach, its application is not easy to carry out by engineers. To overcome this limitation, an MT was later developed to derive more easily usable predictive equations (Etemad-Shahidi et al. 2011). These studies, as well as other applications of different AI approaches, e.g., GMDH (Najafzadeh et al. 2014a, 2014b), demonstrate the feasibility of SC models to estimate the scour depth caused by waves or currents around submarine pipelines. In particular, the combination of two AI techniques has been shown to enhance the prediction performance: for instance, a GMDH network programmed using a GEP technique provided excellent prediction results for scouring under pipelines (Najafzadeh & Saberi-Movahed 2018).

A common AI approach, ANFIS, combines ANN and fuzzy inference system (FIS) and is widely used to estimate scouring depth around hydraulic structures. Because of the flexibility provided by ANNs, ANFIS models can be trained following a complex mathematical mapping between inputs and outputs within a nonlinear framework. Moreover, the ‘IF and THEN’ rules, embedded in FIS, allow for forecasting the behavior of uncertain systems. In recent years, ANFIS models have been applied to estimate the scouring depth at bridge piers and abutments (Akib et al. 2014; Choi et al. 2017; Moradi et al. 2018), culvert outlets (Azamathulla & Ghani 2011) and long contractions along straight canals (Najafzadeh et al. 2016). It can be therefore expected that the ANFIS technique can satisfactorily be used for the prediction of scour around pipelines as well.

Recently, the nature-inspired algorithms, i.e., ant colony optimization (ACO), particle swarm optimization (PSO), differential evolution (DE) and genetic algorithm (GA) have been introduced for optimization purposes in various engineering problems. Specifically to water resources engineering, ACO algorithm has been used to analyze optimal groundwater long-term strategies (Li & Chan Hilton 2005, 2007); PSO algorithm has been adopted to optimize water distribution networks (Surco et al. 2018), rainfall-runoff forecasting models (Motahari & Mazandaranizadeh 2017) and scour depth estimations (Zanganeh et al. 2011; Najafzadeh 2015).

The progression of AI modeling in the field of hydraulic engineering indicates the limitations of the existed AI models and the enthusiasm for solving those limitations. The primary contribution of the present study is to address the internal tuning parameters that are associated with the ANFIS model. This has been scientifically evidenced over the recent literature, and thus the main motivation of the methodological phase is taken place.

The goal of this study is to enhance the capability of the ANFIS technique to estimate scouring depth under submarine pipelines by combining it with the aforementioned nature-inspired optimization algorithms. The prediction accuracy of the proposed methodology is then quantified and compared with the most recent formulations obtained with stochastic approaches for wave-induced pipeline scour depth (Etemad-Shahidi et al. 2011; Sharafati et al. 2018) using indices of prediction performance.

GOVERNING VARIABLES AND SCOUR DEPTH PREDICTION FORMULATIONS

Pipeline failure may occur because of various reasons. Inadequate cover or a low specific gravity of the pipeline may both result in pipeline deterioration and eventual failure; therefore, they must be considered when designing and laying a pipeline. The focus of this investigation is on bed scour, which is another possible cause of pipeline failure. The scouring process around any marine or river structure is a complex phenomenon, and various physical factors affect its development. Several governing variables regarding water, pipeline and seabed interaction are typically considered when predicting scour depth. Previous studies (Sumer & Fredsøe 1990; Kazeminezhad et al. 2010; Najafzadeh et al. 2014a) showed that the variables affecting the wave-induced scour under a pipeline are mainly related to the fluid (herein water), flow regime, seabed sediment and pipeline properties. In general, the maximum equilibrium scour depth at a submarine pipeline, S, may be expressed with the following unknown functional relationship, :
formula
(1)
where and are water and sediment mass densities, g is the acceleration of gravity, D is the pipe diameter, is the median diameter of the sediment particles; , and T are, respectively, the maximum magnitude of the undisturbed orbital fluid velocity at the bed, wave friction velocity and wave period, respectively; e, H and denote the gap between the pipeline and the initial bed, wave height and dynamic viscosity of water, respectively. Among the parameters within the above equation, and T represent the wave characteristics for any wave-induced scour condition around a pipeline.
Through dimensional analysis, the above relationship can be rewritten in terms of dimensionless parameters. For the case of a pipeline with a smooth outer surface (Najafzadeh et al. 2014a):
formula
(2)
where is an unknown function, R and represent Reynolds number and sediment grain Reynolds number, respectively, which are expressed as:
formula
(3)
formula
(4)
and the Shields number, and the Keulegan–Carpenter number, , are expressed as:
formula
(5)
formula
(6)
Both and R depend on the flow field close to the pipeline, whereas and also take into account the seabed characteristics. Because of the relationship of the Reynolds number parameters, i.e., R and , with the dimensionless variables and and the typical existence of turbulent flow field conditions around pipelines, the Reynolds number parameters can be neglected and Equation (2) is simplified as in the following functional relationship with unknown function (Kazeminezhad et al. 2010; Etemad-Shahidi et al. 2011; Najafzadeh et al. 2014a; Sharafati et al. 2018):
formula
(7)
Among the dimensionless parameters in the above equation, the magnitude of plays a critical role in the resulting maximum scour depth (Lucassen 1984; Sumer & Fredsøe 1990). The influence is so notable that Çevik & Yüksel (1999) proposed a simple scour depth prediction equation based on as the only effective parameter on , where no gap exists between the pipeline and the bed:
formula
(8)

Although for the estimation of maximum scour depth beneath a pipeline placed above the seabed level other researchers (Sumer & Fredsøe 2002; Mousavi et al. 2009) derived predictive formulas neglecting , a more accurate scour depth estimation is obtained if all the dimensionless parameters in Equation (7) are included. To this end, based on an MT approach, Etemad-Shahidi et al. (2011) derived a set of mathematical formulations to predict the scouring depth by considering all the parameters in Equation (7).

Recently, Sharafati et al. (2018) revised the coefficients and exponents of Etemad-Shahidi et al. (2011)’s formulations, assuming that the uncertainty associated with the dataset they utilized is the primary source of the variability of the parameters. The uncertainty analysis was performed using two stochastic approaches: generalized likelihood uncertainty estimation (GLUE) and sequential uncertainty fitting (SUFI). Based on the GLUE method, which provided more accurate results than SUFI, Sharafati et al. (2018) proposed the following formulae for clear-water (Equation (9)) and live-bed scour conditions (Equation (10)):
formula
(9)
formula
(10)
Equations (9) and (10) are the latest and most comprehensive equations for the estimation of wave-induced ultimate scour depth beneath pipelines. In the present investigation, predictions obtained with the new proposed methodology are compared with the results of these equations, as well as the equations by Etemad-Shahidi et al. (2011). The prediction improvement provided by the new model is discussed and the uncertainty due to the input parameters and the model structure is assessed in detail.

PROPOSED HYBRID ARTIFICIAL INTELLIGENCE MODELS

The recent years have seen a noticeable advancement of SC approaches (Sharafati et al. 2019). Such methods are suitable for solving complex problems characterized by a high level of non-linearity and non-stationarity (Yaseen et al. 2015). Hybridized AI-global optimization models have recently gained popularity (Ghorbani et al. 2017). Among them, the performance of nature bio-inspired models has been the best, owing to the AI models-associated hyper-parameters (Maier et al. 2014).

This study hybridized four nature bio-inspired algorithms, namely PSO, ACO, DE and GA, with an ANFIS model for wave-induced maximum scour depth prediction at pipelines. Figure 1 outlines the developed models in the form of flowcharts.

Figure 1

Proposed integrated ANFIS models with nature-inspired optimization algorithms (PSO, ACO, DE and GA).

Figure 1

Proposed integrated ANFIS models with nature-inspired optimization algorithms (PSO, ACO, DE and GA).

ANFIS model

ANFIS models are very well-established AI models based on fuzzy logic (Jang 1996) and their popularity is because they allow input variables (attributes) to execute numerical approximation of the internal mechanism relationships of a physical phenomenon (Yaseen et al. 2017). In essence, ANFIS models boost the learning capability of a classic ANN, which develops rules to map a set of inputs to an output value based on a set of fuzzy rules presented by Zadeh (1965). The fuzzy logic component is exceptionally beneficial as it aids in optimal solution generation (in terms of prediction performance) from imprecise/noisy input attributes.

The fuzzy logic approach is based on allowing each element of a dataset to fall in a particular class (set) partially, and its membership degree is described by a membership function. Achieving an accurate learning process based on knowledge and related experience requires optimal selection of the shape of the membership functions, and fuzzy rules (Kisi & Yaseen 2019).

The general five-layer structure of an ANFIS model is illustrated in Figure 2, describing the case of two input parameters and an output variable. The ANFIS rules are expressed through the following ‘if’ and ‘then’ functions:
formula
(11)
formula
(12)
where (A1 & A2) and (B1 & B2) are linguistic terms of the inputs and y, and ,, , ,, are coefficients of linear equations.
Figure 2

Conceptual structure of an ANFIS model.

Figure 2

Conceptual structure of an ANFIS model.

Within the first layer of the model, each node is considered as an adaptive node with the following node function:
formula
(13)
formula
(14)
where the subscript ‘1’ stands for ‘the first layer’, x or y represents the input value to the node i and and denote the membership functions representing the linguistic term (for input value ) and (for input value ), respectively. Various types of membership functions have earlier been reported in the literature, such as trapezoidal, Gaussian and triangular; however, this study adopted the Gaussian function, which is expressed thus:
formula
(15)
formula
(16)
where , , and are distribution variables.
The second model layer has each node fixed while the model output is computed as:
formula
(17)
where the subscript ‘2’ stands for ‘second layer’ and are the rules' weights.
Within the third layer of the model, each node computes the ratio of the corresponding rule weight and the sum of the weights as follows:
formula
(18)
In the fourth layer, each node is adaptive, and the output is computed as:
formula
(19)
where stands for the third layer output and the coefficients , and are updated during the training based on the training set.
In the fifth layer (the final one), the circulation of the nodes is carried out as follows:
formula
(20)

Optimization algorithms

The optimization of the rules that map the inputs to an output value is typically done with trial and error procedures, and the ANFIS model utilizing this type of optimization is referred to as ‘classic’ ANFIS model. Such trial and error procedures have the disadvantage of often being time-consuming and can lead to overfitting. These issues can be mitigated by using global optimization algorithms. The methodologies for optimization used in this investigation are outlined in the following sub-sections.

Particle swarm optimization algorithm

The PSO technique is a bio-based optimizer first introduced by Eberhart & Kennedy (1995); it was inspired by the pattern of movement of natural creatures, such as fish, insects and birds. This methodology models each candidate solution to an optimization problem as a particle flowing in the search domain of the optimization problem. The location (position) and speed of every single particle are adjusted following its own experience and the neighboring particles.

The formulation to update the location of particle i is:
formula
(21)
where is the position of the particle at time t + 1 and is the position of the particle at time t; is the velocity of the particle at time t + 1, which is updated thus:
formula
(22)
where is the particle velocity at time t, and are acceleration coefficients, and are random values varying between 0 and 1, and is the personal best position while is the global best position, in optimization terms, of the ith particle at time t. The parameters used for the application of the PSO optimization technique are summarized in Table 1.
Table 1

Parameters used for application of the various optimization techniques

Optimization techniqueDescription of the parameterParameter value
PSO Number of iterations 1,500 
Number of populations 50 
Inertia weight 
Inertia weight damping ratio 0.99 
Personal learning coefficient 0.9 
Global learning coefficient 
ACO Number of iterations 1,500 
Number of populations 50 
Intensification factor 0.5 
Deviation-distance ratio 
DE Lower bound of scaling factor 0.2 
Upper bound of scaling factor 0.8 
Crossover probability 0.15 
GA Number of iterations 1,500 
Number of populations 50 
Crossover percentage 0.7 
Number of offspring 80 (crossover percentage × number of populations) 
Mutation rate 0.15 
Mutation percentage 0.45 
Number of mutants 60 (mutation percentage × number of populations) 
Selection pressure 
ANFIS Train epochs 250 
Train-error goal 
Train-initial step size 0.015 
Train-step size decrease 0.95 
Train-step size increase 1.15 
Optimization techniqueDescription of the parameterParameter value
PSO Number of iterations 1,500 
Number of populations 50 
Inertia weight 
Inertia weight damping ratio 0.99 
Personal learning coefficient 0.9 
Global learning coefficient 
ACO Number of iterations 1,500 
Number of populations 50 
Intensification factor 0.5 
Deviation-distance ratio 
DE Lower bound of scaling factor 0.2 
Upper bound of scaling factor 0.8 
Crossover probability 0.15 
GA Number of iterations 1,500 
Number of populations 50 
Crossover percentage 0.7 
Number of offspring 80 (crossover percentage × number of populations) 
Mutation rate 0.15 
Mutation percentage 0.45 
Number of mutants 60 (mutation percentage × number of populations) 
Selection pressure 
ANFIS Train epochs 250 
Train-error goal 
Train-initial step size 0.015 
Train-step size decrease 0.95 
Train-step size increase 1.15 

Ant colony optimization algorithm

The ACO was first presented by Dorigo & Di Caro (1999) as an optimizer which has undergone several modifications to suit multiple engineering applications (Weise 2009; Afshar et al. 2015; Ajay Adithyan et al. 2018). The ACO algorithm is an optimization technique that is most effective in addressing both dynamic and static problems in the field of engineering (Dorigo et al. 1996; Blum 2005; Dorigo & Blum 2005; Dorigo & Socha 2007).

Although colonies of ants are composed of simple individuals, they are considered to have one of the most well-organized structures in nature (Guo & Zhu 2012). The stigmergy mechanism which facilitates self-organization controls activities like foraging, brood sorting, co-operative transport and division of labor (Dorigo & Di Caro 1999). Stigmergy in ant colonies is based on the pheromone track left by each ant, which affects the actions of all the other ants. The ACO algorithm, inspired by this concept, can find the best solution in an optimization problem through forward & backward movements, as well as a step-wise decision process. The parameters used for the application of the ACO optimization technique are summarized in Table 1.

Differential evolution algorithm

The DE methodology is a stochastic nature-inspired framework for solving problems that are considered highly nonlinear and multi-dimensional (Suribabu 2010; Chen & Chau 2016); it was first proposed by Storn & Price (1995). The optimization of a function with population size k and n real variables first requires the formulation of the vectors in the following manner:
formula
(23)
where G represents the number of generations. Each parameter has an upper and lower boundary defined as follows:
formula
(24)

Therefore, identity probability is invoked in setting the initial magnitudes of the variables. The parameters used for the application of the DE optimization technique are summarized in Table 1.

Genetic algorithm

GA algorithm is an evolution-based algorithm that was developed based on Darwin's Principle of Natural Selection for addressing numerous optimization problems (Koza 1994; Yang & Honavar 1998; Maulik & Bandyopadhyay 2000; Deb et al. 2002; Levasseur et al. 2008; Iba & Aranha 2012; Kubat 2017). The algorithm is initialized by generating an initial random population of individual solutions to the considered problem. The goodness of fit for each solution is then assessed using suitable metrics, and crossover and mutation operators are employed to generate the next generation of individual solutions and allow the population to evolve towards an optimal solution. The parameters used for the application of the GA optimization technique are summarized in Table 1.

Optimization of parameters

For live-bed conditions, two Gaussian and two linear functions are defined as the membership functions of the input and output variables, respectively. For clear-water conditions, one Gaussian and one linear function are considered as membership functions of the input and output variables, respectively. For instance, for a combination of three input variables, the total number of parameters to be optimized is 20 (12 antecedent, input-related parameters and 8 consequent, output-related parameters) for live-bed conditions and 10 (6 antecedent parameters and 4 consequent parameters) for clear-water conditions. The number of membership functions employed in this study is selected based on the available number of data for the training phase, and the number of parameters to optimize is less than the number of training data. Table 2 shows an example of optimized parameters for the ANFIS-PSO model with three input variables.

Table 2

Optimized parameters for the ANFIS-PSO model

ModelConditionAntecedent parameters
Consequent parameters
Membership function


Membership functionpqrs
ababab
ANFIS-PSO Live-bed Gaussian 76.97 10.48 0.026 0.0092 0.101 0.194 Linear 0.012 0.855 −0.179 0.06 
11.02 −29.23 0.069 −0.486 −2.18 0.28 0.032 −0.025 −0.109 −0.229 
Clear-water 1.65 11.73 0.039 0.12 −0.77 0.35 0.011 −2.08 1.16 −0.19 
ModelConditionAntecedent parameters
Consequent parameters
Membership function


Membership functionpqrs
ababab
ANFIS-PSO Live-bed Gaussian 76.97 10.48 0.026 0.0092 0.101 0.194 Linear 0.012 0.855 −0.179 0.06 
11.02 −29.23 0.069 −0.486 −2.18 0.28 0.032 −0.025 −0.109 −0.229 
Clear-water 1.65 11.73 0.039 0.12 −0.77 0.35 0.011 −2.08 1.16 −0.19 

Description of the proposed predictive models

Several combinations of input variables are considered to identify the optimal predictive model for wave-induced pipeline scour depth under live-bed, as well as clear-water conditions. Individually, seven input combinations, called to , are evaluated (Table 3). In total, 35 different predictive models are assessed, employing different predictive approaches (ANFIS, ANFIS-PSO, ANFIS-ACO, ANFIS-DE and and different input variable combinations ( to ).

Table 3

Combinations of input variables to predict wave-induced scour depth at pipelines

Input combinationPredictive variables
M1 ✓   
M2  ✓  
M3   ✓ 
M4 ✓ ✓  
M5 ✓  ✓ 
M6  ✓ ✓ 
M7 ✓ ✓ ✓ 
Input combinationPredictive variables
M1 ✓   
M2  ✓  
M3   ✓ 
M4 ✓ ✓  
M5 ✓  ✓ 
M6  ✓ ✓ 
M7 ✓ ✓ ✓ 

DATASET FOR THE ANALYSIS

To develop the mentioned AI models, laboratory experimental datasets with a total of 69 scour depth observations from four sources are collected (Lucassen 1984; Sumer & Fredsøe 1990; Pu et al. 2001; Mousavi et al. 2009). The datasets are deemed suitable for the present study because of the following: all the datasets were collected for conditions characterized by KC <100, which imply the pipelines were exposed to the wind wave, the pipe surface was hydraulically smooth, and the current induced by waves was perpendicular to the pipe; the experiments were performed under a wide range of Reynolds numbers, reproducing field conditions; and the channel width in all the tests was large enough to neglect the influence of sidewalls.

The analysis is performed separately for clear-water (21 observations) and live-bed (48 observations) scour conditions. The training-testing data are provided based on the 31–17 (in live-bed condition) and 13–8 (in clear-water condition) observations. The data division adopted between training and testing phases is the result of a trial and error search to attain the best performance. Table 4 summarizes the statistical characteristics of the dimensionless parameters e/D, θ, KC and S/D for the overall dataset considered; the range of variation of the parameters is wide enough to obtain robust results.

Table 4

Statistical characteristics of dimensionless parameters for the selected four datasets

ParameterInput
Output
Minimum 1.42 0.02 0.00 0.03 
Maximum 55.77 0.28 2.04 0.95 
Average 13.40 0.09 0.19 0.26 
Standard deviation 9.97 0.05 0.38 0.16 
Coefficient of variation 0.74 0.58 2.04 0.62 
ParameterInput
Output
Minimum 1.42 0.02 0.00 0.03 
Maximum 55.77 0.28 2.04 0.95 
Average 13.40 0.09 0.19 0.26 
Standard deviation 9.97 0.05 0.38 0.16 
Coefficient of variation 0.74 0.58 2.04 0.62 

INDICES OF PREDICTION PERFORMANCE

Four different indices, i.e. root mean square error () (Salih et al. 2019, 2020), mean absolute error () (Hai et al. 2020), correlation of determination () (Sharafati et al.2020a) and Willmott's Index (WI) (Malik et al. 2020; Mohammed et al. 2020), are computed to measure the adequacy of the predictive models for wave-induced pipeline scour depth. The indices are calculated as follows:
formula
(25)
formula
(26)
formula
(27)
formula
(28)
where and are observed and predicted non-dimensional scour depth magnitudes, respectively, while is the observed non-dimensional scour depth mean magnitude while is the predicted non-dimensional scour depth mean magnitude. NT stands for the number of considered datasets. The smaller or , or the closer to 1.0 or are, the better the prediction performance.
To quantify the performance improvement offered by the ANFIS models optimized using the nature-inspired algorithms compared with the classic ANFIS model, the Improvement Index (IM) is calculated in the testing stage (Sharafati et al. 2020b) as follows:
formula
(29)
where
formula
(30)
formula
(31)
formula
(32)
formula
(33)
where,, , and are, respectively, the computed , , and for the classic ANFIS model, while , , and are the computed indices for the models optimized using the nature-inspired algorithms, for the testing stage.

UNCERTAINTY ANALYSIS

In this study, two sources of uncertainty attributable to the model structure and the input variables are investigated. To quantify the uncertainty of the model structure, a set of five predicted scour depth values in the testing phase (i.e., predicted set by the aforementioned hybridized models) is assigned to each observed scour depth. For each predicted set, the mean and standard deviation are computed to describe a normal distribution function. Using this distribution, 1,000 scour depth values are generated through the ‘Monte Carlo’ simulation (MCS) technique. The MCS technique generally quantifies the uncertainty associated with random variables based on their probability density functions. A set of input variables is generated in each iteration to simulate a system (model). Then, the model outputs are generated randomly based on the obtained stochastic input variables. This process is repeated for an appropriate number of iterations to achieve a reliable description of the output variability due to the different predictive models adopted (Sharafati & Zahabiyoun 2014; Sharafati & Azamathulla 2018). Recent studies on scouring have used the MCS technique to quantify model output uncertainty, confirming this technique as a robust method to assess the uncertainty associated with scouring prediction (Khalid et al. 2019; Salamatian & Zarrati 2019; Homaei & Najafzadeh 2020; Wu & Luo 2020). To quantify the uncertainty of scour depth prediction, the 95% prediction confidence interval (i.e., the interval between the 97.5% and the 2.5% quantiles), called the ‘95 percent prediction uncertainty’ (95 PPU), is extracted using the generated scour depths for each observed scour depth. Individually, the uncertainty is measured using the index as follows (Sharafati & Azamathulla 2018):
formula
(34)
where denotes the standard deviation of the observed data and dx is computed as follows:
formula
(35)
where k stands for the number of observed data and and denote the value of upper quantile (i.e., 97.5%) and lower quantile (i.e., 2.5%) of the 95 PPU band, respectively. This procedure for computation of the index is carried out in this study for both live-bed and clear-water scour conditions.

To evaluate the uncertainty associated with the input variables, for each observed scour depth, the predicted scour depth is computed in the testing phase for a single model but multiple input combinations ( to ). Then, the uncertainty associated with the input variables is quantified using the same approach described above for the uncertainty related to the model structure. Again, this procedure is carried out for both live-bed and clear-water conditions.

RESULTS AND DISCUSSION

Assessment of the proposed predictive models

This study examines several ANFIS models hybridized with the different nature-inspired algorithms presented above. Each model uses a different tuning process to obtain the appropriate ANFIS model parameters. Hence, the models provide different prediction performances based on their tuning processes. Comparing the performance metrics of the models is a way to assess the impact of their tuning processes on prediction performance. Specifically, to analyze the prediction performance of the mentioned 35 different predictive models (ANFIS, ANFIS-PSO, ANFIS-ACO, ANFIS-DE and for the input combinations to ), the selected prediction performance indices () are computed for training, and testing phases and live-bed or clear-water scour conditions (Tables 59).

Table 5

Performance indices of the classic ANFIS model for both live-bed and clear-water conditions

Hydraulic conditionPhaseInput combinationRMSEMAEWI
Live bed Training M1 0.072 0.048 0.623 0.876 
M2 0.099 0.073 0.280 0.640 
M3 0.085 0.068 0.469 0.791 
M4 0.060 0.041 0.738 0.919 
M5 0.025 0.018 0.955 0.988 
M6 0.075 0.053 0.589 0.855 
M7 0.028 0.021 0.941 0.985 
Testing M1 0.077 0.054 0.043 0.552 
M2 0.134 0.103 0.015 0.337 
M3 0.082 0.064 0.275 0.672 
M4 0.049 0.034 0.466 0.794 
M5 0.058 0.043 0.568 0.837 
M6 0.073 0.055 0.334 0.717 
M7 0.077 0.061 0.478 0.775 
Clear water Training M1 0.042 0.028 0.978 0.994 
M2 0.105 0.065 0.858 0.960 
M3 0.268 0.205 0.079 0.332 
M4 0.013 0.010 0.998 0.999 
M5 0.039 0.027 0.981 0.995 
M6 0.070 0.053 0.938 0.983 
M7 0.006 0.004 0.997 0.998 
Testing M1 0.344 0.255 0.010 0.295 
M2 0.516 0.357 0.001 0.171 
M3 0.157 0.128 0.090 0.531 
M4 0.116 0.101 0.391 0.750 
M5 0.495 0.359 0.198 0.043 
M6 0.996 0.636 0.364 0.029 
M7 0.245 0.184 0.043 0.269 
Hydraulic conditionPhaseInput combinationRMSEMAEWI
Live bed Training M1 0.072 0.048 0.623 0.876 
M2 0.099 0.073 0.280 0.640 
M3 0.085 0.068 0.469 0.791 
M4 0.060 0.041 0.738 0.919 
M5 0.025 0.018 0.955 0.988 
M6 0.075 0.053 0.589 0.855 
M7 0.028 0.021 0.941 0.985 
Testing M1 0.077 0.054 0.043 0.552 
M2 0.134 0.103 0.015 0.337 
M3 0.082 0.064 0.275 0.672 
M4 0.049 0.034 0.466 0.794 
M5 0.058 0.043 0.568 0.837 
M6 0.073 0.055 0.334 0.717 
M7 0.077 0.061 0.478 0.775 
Clear water Training M1 0.042 0.028 0.978 0.994 
M2 0.105 0.065 0.858 0.960 
M3 0.268 0.205 0.079 0.332 
M4 0.013 0.010 0.998 0.999 
M5 0.039 0.027 0.981 0.995 
M6 0.070 0.053 0.938 0.983 
M7 0.006 0.004 0.997 0.998 
Testing M1 0.344 0.255 0.010 0.295 
M2 0.516 0.357 0.001 0.171 
M3 0.157 0.128 0.090 0.531 
M4 0.116 0.101 0.391 0.750 
M5 0.495 0.359 0.198 0.043 
M6 0.996 0.636 0.364 0.029 
M7 0.245 0.184 0.043 0.269 
Table 6

Performance indices of the ANFIS‐PSO model for both live-bed and clear-water conditions

Hydraulic conditionPhaseInput combinationRMSEMAEWI
Live bed Training M1 0.086 0.063 0.457 0.789 
M2 0.099 0.080 0.287 0.665 
M3 0.086 0.070 0.467 0.788 
M4 0.055 0.035 0.779 0.934 
M5 0.031 0.024 0.932 0.982 
M6 0.075 0.053 0.594 0.859 
M7 0.024 0.018 0.957 0.989 
Testing M1 0.075 0.055 0.002 0.448 
M2 0.088 0.070 0.044 0.468 
M3 0.081 0.061 0.286 0.680 
M4 0.047 0.033 0.511 0.787 
M5 0.036 0.029 0.761 0.930 
M6 0.087 0.065 0.250 0.658 
M7 0.032 0.026 0.832 0.923 
Clear water Training M1 0.040 0.029 0.979 0.995 
M2 0.101 0.057 0.933 0.964 
M3 0.268 0.205 0.079 0.332 
M4 0.026 0.018 0.991 0.998 
M5 0.017 0.011 0.996 0.999 
M6 0.076 0.046 0.939 0.978 
M7 0.0048 0.0029 0.9997 0.9999 
Testing M1 0.263 0.219 0.162 0.164 
M2 0.605 0.380 0.131 0.176 
M3 0.222 0.161 0.633 0.634 
M4 0.454 0.383 0.249 0.150 
M5 0.322 0.228 0.360 0.424 
M6 1.533 0.688 0.058 0.031 
M7 0.014 0.012 0.984 0.995 
Hydraulic conditionPhaseInput combinationRMSEMAEWI
Live bed Training M1 0.086 0.063 0.457 0.789 
M2 0.099 0.080 0.287 0.665 
M3 0.086 0.070 0.467 0.788 
M4 0.055 0.035 0.779 0.934 
M5 0.031 0.024 0.932 0.982 
M6 0.075 0.053 0.594 0.859 
M7 0.024 0.018 0.957 0.989 
Testing M1 0.075 0.055 0.002 0.448 
M2 0.088 0.070 0.044 0.468 
M3 0.081 0.061 0.286 0.680 
M4 0.047 0.033 0.511 0.787 
M5 0.036 0.029 0.761 0.930 
M6 0.087 0.065 0.250 0.658 
M7 0.032 0.026 0.832 0.923 
Clear water Training M1 0.040 0.029 0.979 0.995 
M2 0.101 0.057 0.933 0.964 
M3 0.268 0.205 0.079 0.332 
M4 0.026 0.018 0.991 0.998 
M5 0.017 0.011 0.996 0.999 
M6 0.076 0.046 0.939 0.978 
M7 0.0048 0.0029 0.9997 0.9999 
Testing M1 0.263 0.219 0.162 0.164 
M2 0.605 0.380 0.131 0.176 
M3 0.222 0.161 0.633 0.634 
M4 0.454 0.383 0.249 0.150 
M5 0.322 0.228 0.360 0.424 
M6 1.533 0.688 0.058 0.031 
M7 0.014 0.012 0.984 0.995 
Table 7

Performance indices of the ANFIS-ACO model for both live-bed and clear-water conditions

Hydraulic conditionPhaseInput combinationRMSEMAEWI
Live bed Training M1 0.102 0.072 0.255 0.669 
M2 0.115 0.095 0.032 0.251 
M3 0.102 0.084 0.244 0.607 
M4 0.106 0.088 0.184 0.558 
M5 0.065 0.048 0.697 0.904 
M6 0.101 0.080 0.268 0.565 
M7 0.063 0.049 0.715 0.903 
Testing M1 0.088 0.069 0.008 0.387 
M2 0.071 0.060 0.162 0.474 
M3 0.070 0.057 0.313 0.665 
M4 0.078 0.062 0.010 0.257 
M5 0.057 0.047 0.288 0.657 
M6 0.066 0.056 0.340 0.675 
M7 0.055 0.041 0.324 0.692 
Clear water Training M1 0.075 0.068 0.929 0.980 
M2 0.204 0.166 0.467 0.804 
M3 0.268 0.205 0.079 0.332 
M4 0.079 0.065 0.919 0.978 
M5 0.076 0.062 0.925 0.980 
M6 0.164 0.122 0.654 0.886 
M7 0.049 0.037 0.970 0.992 
Testing M1 0.160 0.126 0.124 0.259 
M2 0.348 0.284 0.004 0.333 
M3 0.222 0.161 0.633 0.634 
M4 0.204 0.146 0.031 0.360 
M5 0.053 0.045 0.71 0.909 
M6 0.313 0.258 0.849 0.595 
M7 0.116 0.093 0.811 0.837 
Hydraulic conditionPhaseInput combinationRMSEMAEWI
Live bed Training M1 0.102 0.072 0.255 0.669 
M2 0.115 0.095 0.032 0.251 
M3 0.102 0.084 0.244 0.607 
M4 0.106 0.088 0.184 0.558 
M5 0.065 0.048 0.697 0.904 
M6 0.101 0.080 0.268 0.565 
M7 0.063 0.049 0.715 0.903 
Testing M1 0.088 0.069 0.008 0.387 
M2 0.071 0.060 0.162 0.474 
M3 0.070 0.057 0.313 0.665 
M4 0.078 0.062 0.010 0.257 
M5 0.057 0.047 0.288 0.657 
M6 0.066 0.056 0.340 0.675 
M7 0.055 0.041 0.324 0.692 
Clear water Training M1 0.075 0.068 0.929 0.980 
M2 0.204 0.166 0.467 0.804 
M3 0.268 0.205 0.079 0.332 
M4 0.079 0.065 0.919 0.978 
M5 0.076 0.062 0.925 0.980 
M6 0.164 0.122 0.654 0.886 
M7 0.049 0.037 0.970 0.992 
Testing M1 0.160 0.126 0.124 0.259 
M2 0.348 0.284 0.004 0.333 
M3 0.222 0.161 0.633 0.634 
M4 0.204 0.146 0.031 0.360 
M5 0.053 0.045 0.71 0.909 
M6 0.313 0.258 0.849 0.595 
M7 0.116 0.093 0.811 0.837 
Table 8

Performance indices of the ANFIS-DE model for both live-bed and clear-water conditions

Hydraulic conditionPhaseInput combinationRMSEMAEWI
Live bed Training M1 0.097 0.074 0.376 0.726 
M2 0.114 0.092 0.054 0.323 
M3 0.098 0.078 0.311 0.648 
M4 0.099 0.080 0.297 0.701 
M5 0.057 0.042 0.766 0.930 
M6 0.098 0.075 0.331 0.707 
M7 0.060 0.047 0.744 0.915 
Testing M1 0.081 0.052 0.034 0.549 
M2 0.071 0.059 0.173 0.463 
M3 0.064 0.052 0.372 0.748 
M4 0.077 0.061 0.004 0.284 
M5 0.042 0.032 0.623 0.863 
M6 0.063 0.055 0.382 0.755 
M7 0.053 0.042 0.372 0.706 
Clear water Training M1 0.063 0.053 0.950 0.987 
M2 0.183 0.139 0.600 0.877 
M3 0.268 0.205 0.079 0.332 
M4 0.063 0.054 0.950 0.987 
M5 0.059 0.046 0.958 0.988 
M6 0.155 0.111 0.694 0.907 
M7 0.046 0.037 0.973 0.993 
Testing M1 0.204 0.164 0.221 0.156 
M2 0.600 0.429 0.002 0.199 
M3 0.222 0.161 0.633 0.634 
M4 0.109 0.086 0.002 0.418 
M5 0.277 0.200 0.069 0.212 
M6 1.745 0.941 0.420 0.121 
M7 0.090 0.068 0.692 0.846 
Hydraulic conditionPhaseInput combinationRMSEMAEWI
Live bed Training M1 0.097 0.074 0.376 0.726 
M2 0.114 0.092 0.054 0.323 
M3 0.098 0.078 0.311 0.648 
M4 0.099 0.080 0.297 0.701 
M5 0.057 0.042 0.766 0.930 
M6 0.098 0.075 0.331 0.707 
M7 0.060 0.047 0.744 0.915 
Testing M1 0.081 0.052 0.034 0.549 
M2 0.071 0.059 0.173 0.463 
M3 0.064 0.052 0.372 0.748 
M4 0.077 0.061 0.004 0.284 
M5 0.042 0.032 0.623 0.863 
M6 0.063 0.055 0.382 0.755 
M7 0.053 0.042 0.372 0.706 
Clear water Training M1 0.063 0.053 0.950 0.987 
M2 0.183 0.139 0.600 0.877 
M3 0.268 0.205 0.079 0.332 
M4 0.063 0.054 0.950 0.987 
M5 0.059 0.046 0.958 0.988 
M6 0.155 0.111 0.694 0.907 
M7 0.046 0.037 0.973 0.993 
Testing M1 0.204 0.164 0.221 0.156 
M2 0.600 0.429 0.002 0.199 
M3 0.222 0.161 0.633 0.634 
M4 0.109 0.086 0.002 0.418 
M5 0.277 0.200 0.069 0.212 
M6 1.745 0.941 0.420 0.121 
M7 0.090 0.068 0.692 0.846 
Table 9

Performance indices of ANFIS-GA model for both live-bed and clear-water conditions

Hydraulic conditionPhaseInput combinationRMSEMAEWI
Live bed Training M1 0.085 0.060 0.470 0.796 
M2 0.101 0.079 0.259 0.622 
M3 0.086 0.070 0.465 0.785 
M4 0.072 0.052 0.622 0.875 
M5 0.039 0.027 0.890 0.970 
M6 0.076 0.061 0.578 0.847 
M7 0.041 0.027 0.878 0.967 
Testing M1 0.072 0.049 0.013 0.477 
M2 0.082 0.070 0.015 0.420 
M3 0.081 0.063 0.280 0.674 
M4 0.066 0.048 0.133 0.575 
M5 0.042 0.032 0.599 0.870 
M6 0.065 0.053 0.269 0.683 
M7 0.040 0.034 0.648 0.887 
Clear water Training M1 0.045 0.035 0.974 0.993 
M2 0.107 0.068 0.853 0.959 
M3 0.268 0.205 0.079 0.332 
M4 0.024 0.017 0.993 0.998 
M5 0.046 0.031 0.973 0.993 
M6 0.095 0.066 0.885 0.968 
M7 0.010 0.007 0.999 0.999 
Testing M1 0.226 0.191 0.144 0.211 
M2 0.368 0.168 0.034 0.172 
M3 0.222 0.161 0.633 0.634 
M4 0.283 0.234 0.414 0.081 
M5 0.392 0.285 0.630 0.418 
M6 0.557 0.295 0.002 0.185 
M7 0.087 0.065 0.433 0.789 
Hydraulic conditionPhaseInput combinationRMSEMAEWI
Live bed Training M1 0.085 0.060 0.470 0.796 
M2 0.101 0.079 0.259 0.622 
M3 0.086 0.070 0.465 0.785 
M4 0.072 0.052 0.622 0.875 
M5 0.039 0.027 0.890 0.970 
M6 0.076 0.061 0.578 0.847 
M7 0.041 0.027 0.878 0.967 
Testing M1 0.072 0.049 0.013 0.477 
M2 0.082 0.070 0.015 0.420 
M3 0.081 0.063 0.280 0.674 
M4 0.066 0.048 0.133 0.575 
M5 0.042 0.032 0.599 0.870 
M6 0.065 0.053 0.269 0.683 
M7 0.040 0.034 0.648 0.887 
Clear water Training M1 0.045 0.035 0.974 0.993 
M2 0.107 0.068 0.853 0.959 
M3 0.268 0.205 0.079 0.332 
M4 0.024 0.017 0.993 0.998 
M5 0.046 0.031 0.973 0.993 
M6 0.095 0.066 0.885 0.968 
M7 0.010 0.007 0.999 0.999 
Testing M1 0.226 0.191 0.144 0.211 
M2 0.368 0.168 0.034 0.172 
M3 0.222 0.161 0.633 0.634 
M4 0.283 0.234 0.414 0.081 
M5 0.392 0.285 0.630 0.418 
M6 0.557 0.295 0.002 0.185 
M7 0.087 0.065 0.433 0.789 

Considering only the classic ANFIS model (see Table 5), exhibits the best prediction performance for live-bed conditions (, for training and , for testing) among all ANFIS prediction models. For the clear-water conditions, no clearly preferable ANFIS model emerges in training and testing phases, although (, for training and , for testing) shows the best performance. For both the clear-water and live-bed scour conditions, predictions with higher accuracy result from the training phase rather than the testing phase. Overall, the classic ANFIS model is not robust enough to predict wave-induced scour around the pipelines.

Table 6 indicates that the model offers more accurate predictions than the classic ANFIS model and the best prediction performance among all models considered. In particular, shows the best prediction performance for both live-bed conditions (, for training and , for testing) and clear-water conditions (, for training and , for testing).

Tables 79 (relative to ANFIS-ACO, ANFIS-DE and ANFIS-GA) show that the input variable combination that results in the best prediction performance is different for live-bed and clear-water conditions. The (), and models are the best performing models for live-bed conditions, whereas the , and ANFIS-GA-M7 models are the best performing models for clear-water conditions.

It must be noted that Tables 59 include both error indices (e.g., RMSE and MAE) and similarity indices (e.g., R2 and WI). For instance, the ANFIS‐M1 model in live-bed conditions results in 0.072, 0.048, 0.623 and 0.876 for RMSE, MAE, R2 and WI, respectively, in the training phase, and 0.077, 0.054, 0.043, 0.552, respectively, in the testing phase (Table 5). It can, therefore, be observed that the prediction performance is reduced in testing phase by 6.94, 12.50, 93.10 and 36.99%, judging respectively from RMSE, MAE, R2 and WI. This means that, for the case of the ANFIS-M1 model, the prediction performance based on the error indices is reduced from training to testing phase less significantly than the prediction performance based on the similarity indices. For the ANFIS-M5 and ANFIS-M7 models, instead, the error indices show a more significant prediction performance reduction than the similarity indices. This behavior excludes an issue of overfitting in our approach because overfitting would produce the same pattern of prediction performance reduction for the testing phase in both error and similarity indices. The low prediction performance indices, when observed in the testing phase, reflect instead of the limitations associated with the dataset sample size and the prediction capability of the model considered. For the best predictive model, ANFIS‐PSO‐M7, the prediction performance reduction from training to testing phase is quantified, using the RMSE and MAE indices, by a decrease in performance of 25 and 30.8%, respectively (in live-bed conditions), and a decrease of 65.7 and 75.8%, respectively (in clear-water conditions). Considering the R2 and WI indices, the decrease in performance is 15 and 7.2%, respectively, in live-bed conditions, and 1.6 and 0.49%, respectively, in clear-water conditions. Overall, the error metrics (RMSE and MAE) obtained in testing phase indicate a moderate prediction performance reduction (49.3%), while for the similarity metrics (R2 and WI) the reduction (6.1%) is negligible.

Our findings indicate that the classic ANFIS model is not accurate in predicting the wave-induced scour depth around pipelines, especially in clear-water conditions (R2 = 0.39 for testing phase) for which the models ANFIS‐PSO (R2 = 0.98), ANFIS‐ACO (R2 = 0.71), and ANFIS‐DE (R2 = 0.69) provide a significantly better prediction performance, as a result of the nature-inspired optimization algorithms introduced in this research for the ANFIS model.

The best prediction performance indices for each model in the testing phase is presented in Table 10.

Table 10

Best performance indices obtained for each ANFIS model type for the testing phase

Hydraulic conditionPhaseModelRMSEMAEWIIM
Live bed Testing ANFIS‐M5 0.058 0.043 0.568 0.837 ‐ 
ANFIS‐PSO‐M0.032 0.026 0.832 0.923 35.28 
ANFIS‐ACO‐M0.055 0.041 0.324 0.692 −12.61 
ANFIS‐DE‐M0.042 0.032 0.623 0.863 16.49 
ANFIS‐GA‐M0.042 0.032 0.599 0.870 15.64 
Clear water Testing ANFIS‐M4 0.116 0.101 0.391 0.750 ‐ 
ANFIS‐PSO‐M0.014 0.012 0.984 0.995 90.09 
ANFIS‐ACO‐M0.053 0.045 0.71 0.909 53.14 
ANFIS‐DE‐M0.090 0.068 0.692 0.846 36.22 
ANFIS‐GA‐M0.087 0.065 0.433 0.789 19.15 
Hydraulic conditionPhaseModelRMSEMAEWIIM
Live bed Testing ANFIS‐M5 0.058 0.043 0.568 0.837 ‐ 
ANFIS‐PSO‐M0.032 0.026 0.832 0.923 35.28 
ANFIS‐ACO‐M0.055 0.041 0.324 0.692 −12.61 
ANFIS‐DE‐M0.042 0.032 0.623 0.863 16.49 
ANFIS‐GA‐M0.042 0.032 0.599 0.870 15.64 
Clear water Testing ANFIS‐M4 0.116 0.101 0.391 0.750 ‐ 
ANFIS‐PSO‐M0.014 0.012 0.984 0.995 90.09 
ANFIS‐ACO‐M0.053 0.045 0.71 0.909 53.14 
ANFIS‐DE‐M0.090 0.068 0.692 0.846 36.22 
ANFIS‐GA‐M0.087 0.065 0.433 0.789 19.15 

As mentioned, the model with the input variable combination (all the three variables, , included) is the model resulting in the best prediction performance for both live-bed and clear-water conditions. Furthermore, the obtained values (also shown in Table 10) quantify the significant improvement in prediction provided by all the ANFIS models optimized using the nature-inspired algorithms compared with the classic ANFIS model for both live-bead and clear-water conditions in the testing phase, with the most significant improvement obtained with the ANFIS‐PSO model ( and , for testing phase).

A visual performance comparison between the different models is provided in the heat map in Figure 3, based on the standardized and performance indices. The RMSE and MAE indices are standardized using the formula , while the R2 and WI indices are standardized using the formula , where X is the index value. The resulting standardized values are therefore within the range 0 to 1 with the best index value having a standardized value of 1. As mentioned, the model (dark blue column) has the best performance indices for both live-bed and clear-water conditions. In contrast, and (red columns) offer the lowest performance for live-bed and clear-water conditions, respectively.

Figure 3

Heat map of scour depth model prediction performance, based on four standardized performance metrics, for testing phase in (a) live-bed and (b) clear-water conditions. Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/hydro.2020.184.

Figure 3

Heat map of scour depth model prediction performance, based on four standardized performance metrics, for testing phase in (a) live-bed and (b) clear-water conditions. Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/hydro.2020.184.

The prediction performance is also evaluated on two-dimensional scatter plots comparing the simulated and the observed values of scour depth (Figure 4), where the identity (1:1) line is a reference to visualize how close the simulated and observed values are. For live-bed conditions (Figure 4(a)), the points are generally the closest to the 1:1 line and show the most linear pattern (coefficient of determination ), whereas the points corresponding to the other models are noticeably more scattered (). Likewise, for clear-water conditions, the points are the nearest to the 1:1 line with , whereas most of the points for the other models suggest overestimation of the predicted scour depth.

Figure 4

Scatter plot of computed vs. observed scour depth in testing phase for (a) live-bed and (b) clear-water conditions.

Figure 4

Scatter plot of computed vs. observed scour depth in testing phase for (a) live-bed and (b) clear-water conditions.

Models are also comparatively assessed on a Taylor diagram (Figure 5), considering RMSE, R2 and normalized standard deviation (Taylor 2001). Again, in the diagram, the model with the best predictions (i.e., the closest points to the points labeled ‘observed’) is the for both live-bed and clear-water scour conditions.

Figure 5

Normalized Taylor diagrams of the predicted and the observed scour depth in testing phase for (a) live-bed conditions (b) clear-water conditions.

Figure 5

Normalized Taylor diagrams of the predicted and the observed scour depth in testing phase for (a) live-bed conditions (b) clear-water conditions.

The variability of the measured and predicted scour depth magnitudes is quantified and compared for the different models by computing and plotting the quantiles , and (Figure 6). The median scour depth predicted by the model is the closest to the median observed value for both live-bed conditions () and clear-water conditions (). From a comparison of the interquartile range ), which is the difference between Q75% and Q25%, it appears that the predictive models underestimate the variability of the observed data in live-bed conditions. In contrast, they either under- or overestimate the measured values in clear-water conditions.

Figure 6

Boxplot of observed and predicted scour depth in testing phase for (a) live-bed and (b) clear-water conditions.

Figure 6

Boxplot of observed and predicted scour depth in testing phase for (a) live-bed and (b) clear-water conditions.

Comparison of the proposed predictive models with the available models in the literature

Besides assessing the performance of the proposed models against observed values of scour depth, a comparison with other recently developed methodologies from literature is carried out. Specifically, two recent studies are considered (Etemad-Shahidi et al. 2011; Sharafati et al. 2018). Etemad-Shahidi et al. (2011) proposed several MT-based equations to predict wave-induced scour depth beneath pipelines in clear-water (Equation (36)) and live-bed (Equations (37) and (38)) conditions as follows:
formula
(36)
formula
(37)
formula
(38)

Sharafati et al. (2018) improved Etemad-Shahidi et al. (2011)’s equations using the stochastic GLUE and SUFI approaches and developed Equations (9) and (10). The best proposed model () was compared with the mentioned formulations from literature using prediction performance indices (Table 11) and several visual performance comparisons (Figure 7).

Table 11

Performance indices of the best proposed predictive model (ANFIS‐PSO‐M7) and the models from previous studies

Hydraulic conditionPhaseModelRMSEMAEWI
Live bed Training Etemad-Shahidi et al. (2011)  0.056 0.044 0.834 0.949 
Sharafati et al. (2018)  0.046 0.034 0.850 0.958 
Present study (ANFIS‐PSO‐M7) 0.024 0.018 0.957 0.989 
Testing Etemad-Shahidi et al. (2011)  0.039 0.034 0.688 0.894 
Sharafati et al. (2018)  0.038 0.033 0.798 0.894 
Present study (ANFIS‐PSO‐M7) 0.032 0.026 0.832 0.923 
Clear water Training Etemad-Shahidi et al. (2011)  0.121 0.083 0.947 0.931 
Sharafati et al. (2018)  0.071 0.049 0.951 0.982 
Present study (ANFIS‐PSO‐M7) 0.0048 0.0029 0.999 0.999 
Testing Etemad-Shahidi et al. (2011)  0.028 0.021 0.962 0.980 
Sharafati et al. (2018)  0.026 0.022 0.973 0.985 
Present study (ANFIS‐PSO‐M7) 0.014 0.012 0.984 0.995 
Hydraulic conditionPhaseModelRMSEMAEWI
Live bed Training Etemad-Shahidi et al. (2011)  0.056 0.044 0.834 0.949 
Sharafati et al. (2018)  0.046 0.034 0.850 0.958 
Present study (ANFIS‐PSO‐M7) 0.024 0.018 0.957 0.989 
Testing Etemad-Shahidi et al. (2011)  0.039 0.034 0.688 0.894 
Sharafati et al. (2018)  0.038 0.033 0.798 0.894 
Present study (ANFIS‐PSO‐M7) 0.032 0.026 0.832 0.923 
Clear water Training Etemad-Shahidi et al. (2011)  0.121 0.083 0.947 0.931 
Sharafati et al. (2018)  0.071 0.049 0.951 0.982 
Present study (ANFIS‐PSO‐M7) 0.0048 0.0029 0.999 0.999 
Testing Etemad-Shahidi et al. (2011)  0.028 0.021 0.962 0.980 
Sharafati et al. (2018)  0.026 0.022 0.973 0.985 
Present study (ANFIS‐PSO‐M7) 0.014 0.012 0.984 0.995 
Figure 7

Comparison between the best proposed predictive model (ANFIS-PSO-M7) from this study and recently developed models for testing phase. (a) Scatter plot for live-bed conditions, (b) scatter plot for clear-water conditions, (c) heat map of standardized performance metrics for live bed conditions, (d) heat map of standardized performance metrics for clear-water conditions, (e) Taylor diagram for live-bed conditions, (f) Taylor diagram for clear-water conditions.

Figure 7

Comparison between the best proposed predictive model (ANFIS-PSO-M7) from this study and recently developed models for testing phase. (a) Scatter plot for live-bed conditions, (b) scatter plot for clear-water conditions, (c) heat map of standardized performance metrics for live bed conditions, (d) heat map of standardized performance metrics for clear-water conditions, (e) Taylor diagram for live-bed conditions, (f) Taylor diagram for clear-water conditions.

In the two-dimensional scatter plots (Figure 7(a) and 7(b)) the proposed model points are closer to the 1:1 line and show a more linear pattern (coefficient of determination closer to unity) compared to Etemad-Shahidi et al. (2011)’s and Sharafati et al. (2018)’s models for both live-bed and clear-water scour conditions. The better performance of the model is also confirmed by the heat maps (Figure 7(c) and 7(d)) and the Taylor diagrams (Figure 7(e) and 7(f)).

Although the ANFIS‐PSO model provides a better prediction performance compared to the formulas obtained by Etemad-Shahidi et al. (2011) and Sharafati et al. (2018), its additional structure complexity may potentially hinder its application by some practitioners in the field of scouring. Indeed, the ANFIS model comprises several unknown parameters which needs tuning with an optimization algorithm such as PSO. Hence, we especially advise to use the model proposed in the present study in high-stakes pipeline projects that require a very accurate prediction of scour depth.

To verify that the best predictive model (ANFIS‐PSO‐M7) is consistent with the physics of the pipeline scour phenomenon, Figure 8 shows how the normalized scour depth S/D is predicted to vary with varying e/D, θ and KC in live-bed conditions. The ANFIS‐PSO‐M7 correctly predicts S/D to increase for decreasing e/D or increasing θ and KC, as observed in physical investigations (Sumer & Fredsøe 2002).

Figure 8

Normalized scour depth in live-bed as a function of (a) normalized distance between pipeline and sea bed, (b) Shields number and (c) Keulegan–Carpenter number.

Figure 8

Normalized scour depth in live-bed as a function of (a) normalized distance between pipeline and sea bed, (b) Shields number and (c) Keulegan–Carpenter number.

Uncertainty analysis of the proposed predictive models

The uncertainty associated with the model structure is evaluated considering the considered five models (ANFIS, ANFIS‐PSO, ANFIS‐ACO, ANFIS‐DE and ) with input variable combination (the best performing combination as shown earlier). The uncertainty associated with the input variables is assessed for the model (the best performing model, as explained earlier) and different input variable combinations (). Figures 9 and 10 show the generated 95 PPU band for model structure and input variable uncertainty, respectively. The figures also show the corresponding observed values and are provided for both live-bed and clear-water scour conditions.

Figure 9

Generated 95 PPU band accounting for model structure uncertainty for (a) live-bed and (b) clear-water conditions.

Figure 9

Generated 95 PPU band accounting for model structure uncertainty for (a) live-bed and (b) clear-water conditions.

Figure 10

Generated 95 PPU band accounting for the input variable combination uncertainty for (a) live-bed and (b) clear-water conditions.

Figure 10

Generated 95 PPU band accounting for the input variable combination uncertainty for (a) live-bed and (b) clear-water conditions.

From Figure 9, the uncertainty in predicted scour depth associated with the model structure in clear-water () is higher than in live-bed conditions (). From Figure 10, the uncertainty in predicted scour depth associated with the input variables is also higher in clear water () compared with live-bed conditions (). It can generally be concluded that prediction of scour depth caused by waves at pipelines in clear-water conditions is characterized by more considerable uncertainty, due to both model structure and input variables, than in live-bed conditions. Furthermore, the uncertainty associated with the input variables is larger than the one associated with the model structure .

SUMMARY AND CONCLUSION

This study proposed and assessed the application of nature-inspired optimization algorithms to enhance the ANFIS model performance in predicting wave-induced pipeline scour depth. The considered algorithms (PSO, ACO, DE and GA) are alternatives to the common trial and error methods for optimization, which are not time-efficient and may lead to overfitting.

The proposed models were trained and tested using four datasets (Lucassen 1984; Sumer & Fredsøe 1990; Pu et al. 2001; Mousavi et al. 2009), considering different combinations of input variables () derived from dimensional analysis. The prediction accuracy of the various proposed models was assessed based on indices of prediction performance (RMSE, MAE, R2, WI) and visual comparison (heatmap of standardized performance metrics, scatter plot, normalized Taylor diagram and boxplot of the predicted and observed scour depth). From the comparison results, it emerged that the ANFIS model including all the three input variables and optimized using a PSO algorithm provides the most accurate wave-induced pipeline scour depth predictions, for both live-bed and clear-water scour conditions.

This paper also evaluated two sources of uncertainty associated with the scour depth prediction, disaggregating the uncertainty of the model structure (type of optimization algorithm) and the one due to combination of the input variables (selection of input variables for the model). To evaluate uncertainty, a Monte Carlo simulation technique is used, and the 95% prediction uncertainty is quantified through the index. From the results, the model structure and the input parameters selection both lead to more considerable uncertainty in scour depth prediction for clear-water conditions than for live-bed conditions. Also, the uncertainty due to the input variable combination is larger than the model structure-associated uncertainty.

This study shows that a relatively simple improvement in the optimization of an ANFIS model, based on the PSO algorithm, may lead to significant improvement in prediction performance not only in comparison with a classic ANFIS model optimized through trial and error procedure but also in comparison with recently developed models based on the MT approach (Etemad-Shahidi et al. 2011) or GLUE and SUFI stochastic approaches (Sharafati et al. 2018). The added complexity of the ANFIS‐PSO model compared to simpler formulations (still suitable for most projects) is counterbalanced by a higher accuracy.

This paper provides new insight into scouring depth prediction for the design of submarine pipelines. Although the use of conventional equations is straightforward for practical purposes, their prediction is not always accurate. This study shows that an ANFIS‐PSO model can be trained and applied for more accurate scour depth predictions that can support a more robust and safer design.

DATA AVAILABILITY STATEMENT

Data cannot be made publicly available; readers should contact the corresponding author for details.

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