A comparison between advanced hybrid machine learning algorithms and empirical equations applied to abutment scour depth prediction

Highlights

• 5 new hybrid machine learning models applied to abutment scour depth prediction.

• A standalone machine learning model and 2 empirical methods served as benchmarks.

• 4 different input scenarios are considered for the machine learning models.

• Same input variable has a different effectiveness on different abutment shape.

• The machine learning models substantially outperform empirical methods.

Abstract

Complex vortex flow patterns around bridge piers, especially during floods, cause scour process that can result in the failure of foundations. Abutment scour is a complex three-dimensional phenomenon that is difficult to predict especially with traditional formulas obtained using empirical approaches such as regressions. This paper presents a test of a standalone Kstar model with five novel hybrid algorithm of bagging (BA-Kstar), dagging (DA-Kstar), random committee (RC-Kstar), random subspace (RS-Kstar), and weighted instance handler wrapper (WIHW-Kstar) to predict scour depth (d_s) for clear water condition. The dataset consists of 99 scour depth data from flume experiments (Dey and Barbhuiya, 2005) using abutment shapes such as vertical, semicircular and 45° wing. Four dimensionless parameter of relative flow depth (h/l), excess abutment Froude number (Fe), relative sediment size (d₅₀/l) and relative submergence (d₅₀/h) were considered for the prediction of relative scour depth (d_s/l). A portion of the dataset was used for the calibration (70%), and the remaining used for model validation. Pearson correlation coefficients helped deciding relevance of the input parameters combination and finally four different combinations of input parameters were used. The performance of the models was assessed visually and with quantitative metrics. Overall, the best input combination for vertical abutment shape is the combination of F_e, d₅₀/l and h/l, while for semicircular and 45° wing the combination of the Fe and d₅₀/l is the most effective input parameter combination. Our results show that incorporating Fe, d₅₀/l and h/l lead to higher performance while involving d₅₀/h reduced the models prediction power for vertical abutment shape and for semicircular and 45° wing involving h/l and d₅₀/h lead to more error. The WIHW-Kstar provided the highest performance in scour depth prediction around vertical abutment shape while RC-Kstar model outperform of other models for scour depth prediction around semicircular and 45° wing.

Introduction

Bridges are critical infrastructures, and the failure of their piers can lead to severe economical and social consequences. The most common failure mode for bridges over rivers is generally due to intense local scouring around their piers. Therefore, a reliable estimation of abutment scour and its disruptive effects are crucially important to design these infrastructures as overestimating or underestimating scour can result in higher construction cost and abutment failure, respectively (Azamathulla et al., 2009, Cardoso and Bettess, 1999). Bridge abutments change the local flow pattern and generally cause the formation of a three-part separation zone around a bridge pier. The pressure gradient due to the presence of the pier forces a down-flow that causes the scouring in front of the pier. This leads to the generation of a so-called horseshoe vortex which facilitate further the scouring in front of pier (Coleman et al., 2003, Török et al., 2014). The shear stresses at the upstream face of an abutment due to the principal vortices also facilitate secondary vortices. In addition, unsteady shear layers that are generated at the pier rotate in vertical axes (wake vortices) as small eddies. Furthermore, bow-waves can also contribute to the scouring process. A combination of these vortexes eventually lead to scour holes around piers of bridges (Laursen and Toch, 1956; Liu et al. 1961; Kwan, 1988, Hosseini et al., 2016), and detailed descriptions on abutment scour depth process are widely available in the literature (e.g., Dey, 1997, Dey and Barbhuiya, 2005, Gazi et al., 2019, Kothyari et al., 1992, Melville and Raudkivi, 1977, Melville and Sutherland, 1988, Moonen and Allegrini, 2015, Namaee and Sui, 2019, Raudkivi and Ettema, 1983).

A local scour can be created in conditions in which no sediments are proceeding from upstream reaches (i.e., clear water, or no sediment feeding from upstream) or in more natural conditions in which the flow approach the pier with sediments (i.e., sediment is fed from upstream reaches). A wide range of experimental and field studies investigated the process of scour depth around bridges under clear water conditions, due to the simplicity of this condition. Dey and Lambert (2005) conducted experiments under clear water conditions and investigated the evolution over time and the equilibrium conditions of scour depth for three different shapes of short abutments (vertical wall, semicircular, and 45° wing wall) using both uniform and nonuniform sediments. They applied the concept of mass conservation of sediment to derive numerical equations to calculate the scour depth evolution over time. Oliveto and Hager (2002) conducted a further set of experiments and proposed an equation that allows to calculate the scour depth around both piers and abutments that worked reasonably well when applied to other experimental datasets too, especially for rectangular cross section and uniform distribution of roughness. Amini et al. (2012) further revealed that the pile spacing, diameter, and the submerge ratio are three important parameter which can affect the scour depth. Although many flume experiments have been carried out to support scour depth modeling (Ataie-Ashtiani et al., 2010, Ataie-Ashtiani and Beheshti, 2006, Singh et al., 2020, Yang et al., 2020), this approach suffers from scale effect issues which can have an impact on the applicability of the results. Also, the experimental approach is costly and time-consuming. Flume experimental data are generally used to derive empirical equations based on regressions but this approach, albeit practical, are too simplistic to represent the complexity of flows around a bridge piers (Azamathulla et al., 2009). Numerical investigations of scour depth have been attempted using SSIIM models (Hamidi and Siadatmousavi, 2018, Jahangirzadeh et al., 2014), Smagorinsky subgrid model combined with a ghost-cell immersed boundary method (Kim et al., 2014), Virtual Flow Simulator (VFS-Geophysics) (Khosronejad et al., 2020), FLUENT (Yang et al., 2005), and FLOW-3D (Omara et al., 2019), but applications of these models are restricted due to the paucity of large experimentally and field measurement dataset for their calibration and validation. Although these models consider the physics of the scouring processes, their implementation is difficult, time consuming, and needs large and accurate datasets.

An alternative to traditional approaches is provided by the use of Artificial Intelligence (AI) as it is user-friendly, easy to perform, requires less data, is robust to missing data, and provides high accuracy to predict complex phenomena especially in engineering and geoscience fields. Artificial intelligence has the ability to train complex and hidden relationships between inputs and outputs without a detailed knowledge of the physics of the problem. Employing AI for predicting scour depth around different hydraulic structures has indeed been attempted in literature in the past decade (Ebtehaj et al., 2018, Guven and Azamathulla, 2012, Guven and Gunal, 2008, Najafzadeh et al., 2013a, Najafzadeh and Lim, 2015).

Artificial Neural Networks (ANN) is the traditional and most widely used algorithm for scour depth prediction (Amini et al., 2012, Kaya, 2010, Yazdandoost and Birgani, 2011). Three different ANN techniques as the Feed Forward Back Propagation (FFBP), Feed Forward Cascade Correlation (FFCC) and Radial Basis Function (RBF) were applied by Muzzammil (2008) to estimate scour depth in clear-water condition for vertical wall abutments. In his study the input and output data were normalized (0 and 1) and the impact of dimensionless and dimensional inputs in modeling the scour depth was investigated. There were only a few later application of ANN models due to many critical disadvantages as the low speed convergence and poor generalization power (Choubin et al., 2018, Hooshyaripor et al., 2014). Also, the performances of the ANN model strongly depend of the extension of the dataset (Hooshyaripor and Tahershamsi, 2013). To overcome this issue, adaptive Neuro-Fuzzy Inference System (ANFIS) was developed as an ensemble of ANN and fuzzy logic. Bateni and Jeng (2007) employed an ANFIS method to simulate the scour depth. Hosseini et al. (2016) compared the prediction power of ANFIS, ANN and multiple nonlinear regression (MNLR) for scour depth perdition and finally stated that the ANFIS model has a higher prediction capability than ANN and MNLR models. Still, the ANFIS model suffers from determining the weights in a membership function, which affect significantly the result. Abd El-Hady Rady (2020) reported on the superiority of genetic programming (GP) over ANFIS algorithm for scour depth around bridge pier. Support Vector Machine (SVM) is another type of neuron-based model which was successfully applied in scour depth prediction. Parsaie et al. (2019) observed that the SVM model has a higher prediction capability for scour depth prediction than ANN and ANFIS algorithm. Ahmad et al. (2018) revealed that SVM is sensitive to hyper-parameter selection, and Najafzadeh et al. (2016) reported that ANFIS performed better than SVM and traditional existing equations. Further, the Group Method of Data Handling (GMDH) is a model which can automatically select the number of neurons and the network layers and allows to obtain a mathematical model in terms of polynomials for the target parameter. However, being a kind of neuron-based model, GMDH is sensitive to the extension of the dataset. Najafzadeh et al., 2013a, Najafzadeh et al., 2013b applied both GMDH and SVM approaches to a set of experimental data to predict scouring depth in four different shapes of abutments in both clear water and sediment feeding conditions. They used a backward path (BP) algorithm to design topology of the GMDH model in order to improve the performance of model and discovered that the BP-GMDH performed better than the SVM in both conditions. Scour depth around abutments was also predicted by using the pareto evolutionary structure of ANFIS network by Azimi et al., 2017, Azimi et al., 2019. Using the dataset from Dey and Lambert (2005) they used a sensitivity analysis to rank the role of eleven different dimensional input variables which effect scour depth. Also, they compared their best developed model (i.e., ANFIS-GA/SVD 7) with other techniques employed previously (i.e., Azamathulla, 2012, Moradi et al., 2019, Muzzammil, 2010, Najafzadeh et al., 2013b) and revealed that the ANFIS-GA/SVD model could provide more accurate results of scour depth in comparison with GEP, ANFIS–SC, ANFIS and GMDH models. Extreme Learning Machine (ELM) is another neuron-based algorithm with faster training phase and has been successfully applied in a different field of study. Ebtehaj et al. (2018) reported that the ELM model has a higher performance than ANN and SVM models to predict scour depth. In a recent study, Bonakdari et al. (2020) used the ELM technique to predict scour depth in clear-water condition considering four different nondimensional input parameters to estimate scour depth. About 11 different input combinations were tested to find the best one and they finally concluded that the model which contains all input variables allowed to obtain a better performance. They extended a matrix-based equation to calculate the scour depth, but their equation is highly complex and need large mathematic calculations. Because ELM is a version of the ANN model, its performance strongly depends on the extent of the dataset and is hampered by low performances with small datasets.

To overcome the shortcomings of the aforementioned traditional machine learning algorithm, different kind of data mining algorithms have been recently developed. These are tree-based [random forest (RF), random tree (RT), M5 prime (M5P), reduced error pruning tree (REPT)], rule-based [M5 Rule (M5R)], lazy-learn-base [Kstar, instance-based K-nearest neighbors (IBK), locally-weighted learning (LWL)], regression-based [sequential minimal optimization regression algorithm (SMO)] and ensemble-based [bagging (BA), random committee (RC), random subspace (RS)] algorithms. Some of these techniques operate for classification as well as for regression, based on the learner. The superiority of the RF algorithm over ANN and SVM in infiltration process prediction was reported by Sihag et al. (2020). Also, Yan et al. (2012) found that M5P algorithm had a higher prediction capability than ANN model for daily suspended sediment load prediction. Different metaheuristic algorithms were applied to solve the weakness of neuron-based models which suffer from determination of weights in membership function and operators. Khosravi et al. (2019a) compared predictive modeling of standalone new algorithms of M5P, RT, RF, REPT, Kstar with standalone and optimized ANFIS model using metaheuristic algorithms for reference evaporation prediction. They found that new a decision-tree based standalone model and Kstar models have higher performance than the ANFSI model, while optimized ANFIS models performed slightly better than standalone new models. Except for better prediction power and more flexibility, new algorithms require fewer parametric settings, making them more practical for real applications.

Sheikh Khozani et al. (2019) employed different standalone and a hybrid model to predict apparent shear stress in compound channels. They found that the BA-M5P model predicted the apparent shear stress with higher accuracy than standalone models. Khosravi et al. (2020) implemented four standalone algorithm of decision tree and four ensemble-based model using BA algorithm for bedload transport rate prediction, and found that ensemble-based models predicted bedload with higher prediction accuracy. Similar observations have been reported by Bui et al., 2020, Khosravi et al., 2018.

The main objective of the present study is to predict abutment scour depth using a suite of new standalone and ensemble-based models. To meet the aim, standalone KStar algorithm are applied as a base model along with five novel ensemble-based models of BA-KStar, dagging (DA-KStar), RC-Kstar, RS-Kstar and Weighted Instance Handler Wrapper model (WIHW-Kstar). Finally, the results are compared with two traditional empirical equations (Dey and Barbhuiya, 2005, Muzzammil, 2010) as a benchmark.