ANFIS- and GEP-based model for prediction of scour depth around bridge pier in clear-water scouring and live-bed scouring conditions

μ

fluid dynamic viscosity

b

pier diameter

d₅₀

median diameter of sediment

d_se

scour depth

F_d50

particle densiometric Froude number

Fr

Froude number

F_rc

critical Froude number

g

acceleration due to gravity

U

approach mean velocity

U_c

critical velocity

y

flow depth

ρ

fluid density

σ

geometric standard deviation of bed sediment size

ANFIS

adaptive network-based fuzzy inference system

ANN

artificial neural network

CWS

clear-water scouring

FFBP

feed-forward back-propagation

FIS

fuzzy interference system

GA

genetic algorithm

GEP

gene expression programming

GP

grid partition

GT

Gamma test

IS

insertion sequence

LBS

live-bed scouring

LR

linear regression

MAPE

mean absolute percentage error

MFs

membership functions

MLP

multi-layer perceptron

MLR

multivariable linear regression

MSE

mean square error

R²

coefficient of determination

RBF

radial basis function

RIS

root insertion sequence

RMSE

root mean square error

SDR

scour depth ratio

Many river bridges have collapsed because of heavy local scouring during high floods. Local scour depth refers to the erosion of dirt and rock particles located around the abutment or pier of a highway bridge that crosses over a water body to a certain depth. The scour depth is a significant factor in the bridge design as the base collapses due to improper scour depth estimation. Among two types of local scouring, clear-water scouring (CWS) is caused when the bed material upstream of the scour area is at rest and the shear stress of flow exceeds the critical shear stress, however, the live-bed scouring (LBS) takes place when there is general bed load movement and the upstream shear stress is greater than the threshold value. Laursen (1963) investigated the behavior of clear-water scours with geometry parameters, flow parameters, and sediment size and developed relationships for the scour in a long contraction, at an abutment, and around a pier. Ettema (1980) defined the influence of sediment size on scour depth at the circular shape of bridge piers for uniform sediments. Their experimental results showed that larger depths of clear-water scour can occur at piers placed in beds composed of uniform sediment armored by a layer of coarser sediment than that of the pier in a bed of uniform sediment. Raudkivi & Ettema (1983) reported that with smaller values of the flow depth to pier diameter ratio (y/b), the equilibrium scour depth reduces at a faster rate. They found that the decrease in the equilibrium depth is due to the interference of the water surface roller formed around the pier with the downward flow into the scour hole. Kothyari et al. (1992) proposed a semi-analytical method for predicting maximum scour depth at the bridge pier for LBS conditions. They considered the primary vortex in front of the pier to be the prime agent causing scour at bridge piers. Sheppard et al. (2004) used a large-size model to explore clear-water local bridge pier scour and found that the concentration of suspended fine silt (wash load) has a major effect on equilibrium scouring depths through experiments. According to Raikar & Dey (2005), equilibrium scour depth increases as gravel size decreases and they found that the depth of scour in gravel beds is generally higher compared to the sand beds. Through experimentation, they found that the variation of equilibrium scour depth with gravel size represents a considerable deviation from the variation of this depth with sand size. Kothyari et al. (1992) proposed a scour depth model by considering actual and entrainment densiometric particle Froude numbers. They also developed a relationship for the temporal scour evolution at bridge foundation elements based on the similitude of Froude by relating the scour depth to the difference between the actual and the entrainment densiometric particle Froude numbers. From their experimental findings, Aksoy et al. (2017) reported that the scour depth was increased with pier diameter and flow velocity and developed a scour depth predictive model using flow velocity and the flow depth to pier diameter ratio (y/b). Pandey et al. (2018) reported the scour hole is created through the formation of an armor layer in a non-uniform gravel bed condition. Overall, the scour depth is significantly affected by the geometry of the pier, flow depth, approach flow velocity, and bed sediment size as it is evident from the lab and field experiments by different researchers. Hamidifar et al. (2021) investigated and verified different equations to determine the critical flow velocity ratio which was further used in estimating scour depth. They proposed better combinations of scour depth predictive model for CWS conditions.

Previously, many artificial intelligence (AI) techniques have been used for scour depth prediction, however, all these methods are not providing accurate results for different flow, geometry, and roughness conditions as well as for different types of scouring (Khan et al. 2012; Sharafati et al. 2021; Nil & Das 2023). The literature review of different AI techniques used in bridge pier scour depth modeling is presented in Table 1.

The scour depth at a bridge pier is determined by the fluid flow, bed sediments, and pier characteristics.

The maximum scour depth to flow depth ratio was chosen as an output, and the five dimensionless parameters were chosen as inputs during the model development in this study. Different bridge pier scours depth predictive equations for clear-water and LBS are presented in Table 2, Equations (5)–(12).

Table 2

Different scour depth predictive equations used in the present study

Authors and years .	Equation .	Scouring condition .	Eq. no. .
Colemon (1972)		LBS condition	5
Jain & Fischer (1979)	, for,	Both CWS and LBS conditions For 0 < ≤ 0.2, the larger of the two scour depths computed from Equation (4) and (5) is used	6
	, for		7
Johnson (1995)		LBS condition	8
Ettema et al. (1998)		CWS condition	9
Yanmaz (2001)		LBS condition	10
Kim et al. (2015)		CWS condition	11
Sheppard et al. (2004)	where , , and	CWS condition	12

Authors and years .	Equation .	Scouring condition .	Eq. no. .
Colemon (1972)		LBS condition	5
Jain & Fischer (1979)	, for,	Both CWS and LBS conditions For 0 < ≤ 0.2, the larger of the two scour depths computed from Equation (4) and (5) is used	6
	, for		7
Johnson (1995)		LBS condition	8
Ettema et al. (1998)		CWS condition	9
Yanmaz (2001)		LBS condition	10
Kim et al. (2015)		CWS condition	11
Sheppard et al. (2004)	where , , and	CWS condition	12

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In this paper, an attempt has been made to use two AI techniques such as adaptive network-based fuzzy inference system (ANFIS) and gene expression programming (GEP) in predicting the CWS depth ratio and LBS depth ratio (d_s/y), where d_s is scour depth and y is approach flow depth. Also, the most efficient model has been proposed for estimating local scour depth around the bridge pier.

This section provides an in-depth discussion of the methods used in the present study to develop the ANFIS and GEP models for predicting scour depth. By reading this section, readers will gain a comprehensive understanding of the background and context of the model development process.

The adaptive neuro-fuzzy inference system (neuro-FIS) uses a set of artificial neural networks (ANNs) and fuzzy systems to build fuzzy IF-THEN rules with suitable membership functions (MFs) obtained from the training pair, culminating in an inference. They eliminate the requirement for manual fuzzy system parameter optimization and instead rely on ANN to modify the system parameters. When ANN and FIS work together, they improve the system operation without requiring administrators to intervene.

ANFIS is a rule-based fuzzy rationale model whose rules are applied during the model's training process. This inference framework is built using five layers, as shown in Figure 2. In this network layout, the input (layer 0) and yield (layer 5) hubs reflect the input and yield origins, respectively. In the hidden layers, a few fixed and adjustable hubs serve as the MFs and rules. Two knowledge factors, x, and y have been considered, and a yield variable z is used to explain the methodology of the ANFIS model. If-then fuzzy rules are used in the ANFIS model to express the relationship between knowledge and yield and the model contains the two fuzzy rules that can be expressed as follows in terms of Takagi and Sugeno's form:

Rule 1: If x is A₁ and y is B₁ then

The semantic levels are A₁, B₁, and A₂, B₂, and the ensuing parameters are p₁, q₁, r₁ and p₂, q₂, r₂. We have a zero-order TSK fuzzy-model if z₁ and z₂ are constants rather than linear equations. The ANFIS structure is comprised of five layers (Figure 2). The following are descriptions of the various layers: Layer 1 is the Fuzzification layer where every node is the flexible node with a node feature. Layer 2 is the Rule layer where every node is a fixed node that acts as the simple multiplier. Layer 3 is the Normalization layer and every node is an adaptive node with the letter N. The ith node determines the ratio of the terminating quality of the ith rule to the total of all terminating strengths. The fourth layer is the defuzzification layer and in this layer, every node has a purpose and is versatile. Layer 5 is the output layer, where a single node is a settled node that calculates the overall yield by adding all approaching signals.

This methodology builds a Sugeno-type FIS structure from training datasets. Grid partition (GP) divides the input dataset into different neighboring fuzzy localities using a pivot-paralleled partition depending on a predetermined number of MFs. The eight types of MFs in the GP strategy are trimf, trapmf, gbellmf, gaussmf, gauss2mf, pmif, dsigmf, and psigmf. Each item of data can be expressed as a connection with the number of MFs (Muzzammil & Ayyub 2010). Only one yield can be used in a Sugeno-type. Either a linear or a constant yield function can be used for the selection of FIS type. This methodology generates the same number of rules as the number of yield MFs. The elective models in this subsection are made up of various FIS structures created using various MF features like (types and numbers) as the input MFs parameter.

GEP was created by Candida Ferreira in 1999 as a new evolutionary AI-based approach. This method is based on Koza's genetic programming (GP) methodology (1992). Similar to genetic algorithms (GAs), the genome is encoded as fixed-length linear chromosomes, which are subsequently represented as phenotypes by GEP in the form of expression trees. GEP combines the advantages of both of its predecessors, GA and GP, by avoiding some of their disadvantages. GEP is a full-featured genotype/phenotype approach that separates all elements. As a result, the fully functional genotype/phenotype system developed by GEP beats the conventional GP system (Ferreira 2002).

The mechanism in GEP, like other evolutionary approaches, begins with the random generation of the initial population. It consists of individual chromosomes of fixed length. One or more genes can be found on a chromosome. The fitness of each chromosome in the initial population is then calculated using a fitness function based on MSE. The fitness value of these chromosomes is then used to pick them by using the roulette wheel selection procedure. The better the chromosomes match; the more likely they are to be selected in the next generation. After selection, these are replicated with certain genetic operator modifications. Genetic operators such as mutation, inversion, transposition, and recombination are utilized to change GEP. Mutation has been discovered to be the most powerful genetic operator and, in many circumstances, the only one available for chromosome modifications. The same modification approach is then applied to the new individuals, and the process is repeated until the maximum number of generations has been reached or the required precision has been obtained (Ferreira 2002). Several of these genetic operators used for chromosome genetic alteration are explained in the GEP scheme as follows (Ferreira 2006):

Mutation: Among all the operators, mutation is the most significant and powerful. In GEP modeling, mutations can occur anywhere in the genome. However, the structural composition of the chromosome should remain the same. In other words, the leading of a gene can replace the function with one of the other functions or ends, but only the end of the gene can be changed. When connected to the other terminal, the tail has no function. Thus, all the new individuals created by mutations are structurally correct programs.

Inversion: Inside the head of the gene, this operator inverts a sequence. It selects the chromosome, the gene to change, and the start and end points of the inverted head section.

Insertion sequence (IS) transposition: The IS elements are small parts of the genome with first-position function or termination. This operator randomly selects the chromosome or gene, as well as the beginning and end of the IS variable, and transposes them to the beginning of the gene immediately after the base.

Root insertion sequence (RIS) transposition: It is a small genomic fragment similar to the IS element, with the exception that starting point is always a function. It picks a chromosome, a gene to change, and starting and end points of the RIS element at random, and then transposes it to the gene's starting point.

Gene transposition: It occurs when the whole gene acts as transposing and transposes to the chromosome's beginning. Gene transposition differs from other types of transposition in that the transposon (gene) is deleted at the point of origin.

Single or double crossover/recombination: The parent chromosomes are coupled and the same spot is chosen in a single crossover. The gene is then swapped between the two chromosomes downstream of the crossover location. Two parent chromosomes are coupled in a double crossover, and two crossover locations are selected at random. The material that exists between the crossover locations is subsequently exchanged between parent chromosomes, resulting in the formation of two new offspring chromosomes (Guven & Aytek 2009).

To create a predictive model for scour depth using AI techniques, it is crucial to establish input and output data sets for the ANFIS and GEP models to be trained and tested. These data sets have been sourced from published literature and will be further elaborated upon in the following section.

For the present study, many clear-water and LBS data have been taken from the literature and are presented in Tables 3 and 4 for circular/cylindrical bridge piers. The detailed data set are available in Supplementary file, Appendix A and Appendix B for CWS and LBS, respectively. Scour in the bridge depends on hydraulic parameters and sediment properties like pier diameter (b), flow depth (y), approach mean velocity (U), critical velocity (U_c), median diameter of sediment (d₅₀), Froude number (Fr) and geometric standard deviation of bed particle size (σ). In the present study, a total of 213 datasets for LBS modeling and 442 datasets are used for CWS modeling. The ranges of different parameters of CSW and LBS collected from the literature are depicted in Tables 3 and 4. Out of 213 LBS data, about 75% (159 sets) were selected for training and 25% (54 sets) were for testing. Table 3 shows the range of non-dimensional parameters used for clear-water scour modeling. Out of 442, about 75% (332 sets) were selected for training and 25% (110 sets) were selected for testing.

In the present study, 50 different combinations of input parameters have been made in the winGamma software for clear-water and LBS separately. Out of 50, nine different combinations including the best input combinations are presented in Tables 5 and 6 for CWS and LBS, respectively. From Table 5, it is observed that the combination of the five parameters with mask [11111] can make a proper model for CWS as compared with other combinations due to its lesser values of Gamma and V-ratio which are very close to zero as compared to the other combination. From Table 6, it is observed that the combination of four parameters with mask [11011] shows the minimum Gamma and V-ratio values. In this blend, the U/U_c ratio is not included, however, this ratio is the most important parameter which influences the scour depth (Kothyari et al. 1992; Sheppard et al. 2004). The combination having five non-dimensional parameters is used for the ANFIS and GEP model development for clear-water and LBS.

To check the performance of the present model different statistical error indices like mean absolute percentage error (MAPE) and coefficient of determination (R²) are utilized as mentioned in Khuntia et al. (2019), Das et al. (2020), Devi et al. (2021) and Kumar et al. (2023). In Table 7, there are 16 combinations made with different MFs and different MF types for CWS using normalized data. Out of 16 combinations, the membership function [22222] and MF type (gbellmf) of the fifth number combination shows R² = 0.95 which is better than other combinations. The performance of various models of clear-water is presented in Table 7.

In Table 8, there are 16 combinations made with different MFs and different membership function types for LBS using normalized data. Out of 16 combinations, the membership function [33222] and MF type (gaussmf) of the tenth number combination shows R² = 0.95 which is better than other combinations. The performance of various models of CWS is presented in Table 8.

In this section, the scour depth of a bridge pier is modeled using the GEP approach is described. The GEP model is developed for the prediction of the scour depth by incorporating five independent input parameters. GeneXpro Tools 5.0 software package is used for the present study.

Table 9 shows the performance of the GEP model for CWS using normalized data. Using these data, four attempts have been done with the variation of chromosome number, fitness function, and number of runs for modeling the CWS and LBS. Table 9 demonstrates the performance of the GEP model for CWS using normalized data and Table 10 shows the corresponding parameters of the optimized GEP model. In Table 9, the coefficient of determination (R²) and MAPE values are depicted for the training and testing stage. Here, trial number 2 has been selected as less error is found in terms of R² and MAPE.

Table 11 depicts the performance of the GEP model for LBS using normalized data and Table 12 shows the parameters of the optimized GEP model for LBS. The results are presented in Table 11 for LBS. In Table 11, trial number 1 shows a low MAPE value and high R² value but the expression tree of trial number 1 is complex. In trial number 2, the R² and MAPE values are more than the trial number, however, it has been selected for the present study as the expression tree of trial number 2 is not very complex as compared to the expression of trial no 1.

Scour depth ratio (SDR) (=d_s/y) value computed using Equation (17) gives a MAPE value of 12.9% but using Equation (16), it is found to be 14.2% for the testing case and 15.30% for the total dataset (Table 11). However, the expression of Equation (16) is simpler than Equation (17) as it contains one logarithmic operator only. Therefore, for this present research, Equation (16) is suggested to be used in predicting the d_s/y value for LBS conditions.

In Figure 9, the SDR values obtained from the developed ANFIS and GEP lie very close to the best-fit line as compared to other scour depth predictive models (such as Colemon 1971; Jain & Fischer 1979, Johnson 1995; Yanmaz 2001) for LBS condition. The predicted SDR values from all the models are found to be very close to the best-fit line, for low observed values (i.e., 0.4–0.6) except for Jain & Fischer's (1979) scour depth predictive model. For the LBS data set used in the present research, it is found that the values predicted from the Jain & Fischer (1979) and Johnson (1995) models are higher than most of the observed SDR values and for Colemon (1972) model, it is lower than the observed values.

The statistical error indices such as MAPE, root mean square error (RMSE), and coefficient of determination (R²) are computed for both the present models i.e., ANFIS and GEP models along with four scour depth predictive equations such as Jain & Fischer (1979), Ettema et al. (1998), Sheppard et al. (2004), Kim et al. (2015), and was compared to different conditions. By examining the error for different ranges of b/y, U/U_c, and Fr, it is possible to determine the strength of present models and other existing equations, which are most useful under various conditions.

In CWS, the following ranges are selected for b/y, U/U_c, and Fr:

• (a)

  b/y < 0.25, 0.25 ≤ b/y < 0.5, 0.5 ≤ b/y < 1.0, 1 ≤ b/y < 1.5 and b/y ≥ 1.5

• (b)

  U/U_c < 0.5, 0.5 ≤ U/U_c < 0.7, 0.7 ≤ U/U_c <0.8, 0.8 ≤ U/U_c < 0.9, and 0.9 ≤ U/U_c <1.0

• (c)

  Fr < 0.20, 0.20 ≤ Fr < 0.40, 0.40 ≤ Fr <0.6, 0.60 ≤b/y < 0.80, and 0.8 ≤ Fr < 1.0

In LBS, the following ranges are selected for b/y, U/U_c, and Fr:

• (a)

  b/y < 0.25, 0.25 ≤b/y < 0.5, 0.5 ≤b/y < 1.0, and 1.0 ≤ b/y < 1.5

• (b)

  1.0 < U/U_c <1.2, 1.2 ≤ U/U_c < 1.5, 1.5 ≤ U/U_c < 2.5, 2.5 ≤U/U_c <3.5 and U/U_c ≥ 3.5

• (c)

  Fr < 0.25, 0.25 ≤ Fr <0.50, 0.50 ≤Fr <0.75, 0.75 ≤ b/y < 1.0, and Fr ≥ 1.0

Table 13(a)–13(c) provides the MAPE, RMSE, and R² values of two present models and four scour depth predictive equations for selected ranges of b/y, U/U_c, and Fr, respectively. For b/y < 0.25, MAPE is high for Ettema et al. (1998), Kim et al. (2015), GEP (present model), and Sheppard et al. (2004). In addition, RMSE is high for Jain & Fischer (1979), GEP (present model), Kim et al. (2015) and Ettema et al. (1998) and R² is low for Kim et al. (2015) and Jain & Fischer (1979). For both 0.5 ≤ b/y < 1.0 and 1.0 ≤ b/y < 1.5, MAPE and RMSE are high for Sheppard et al. (2004), Kim et al. (2015), Ettema et al. (1998) and Jain & Fischer (1979). For b/y ≥ 1.5, MAPE is high for Jain & Fischer (1979), Sheppard et al. (2004), and Kim et al. (2015). In addition, RMSE is high for Sheppard et al. (2004), Kim et al. (2015) and Ettema et al. (1998) and R² is low for GEP, Kim et al. (2015) and Ettema et al. (1998). At all the ranges of b/y, ANFIS (present model) performs well with minimum MAPE and RMSE and high R² value. Table 13(b) provides the MAPE, RMSE, and R² values for two present models and four scour depth predictive equations for selected ranges of U/U_c. For U/U_c < 0.5, MAPE is high for all the approaches. In addition, RMSE is high for Sheppard et al. (2004), Jain & Fischer (1979), GEP (present model), Kim et al. (2015), and Ettema et al. (1998), and R² is low for Kim et al. (2015), Jain & Fischer (1979), however, the ANFIS model provides R² value close to 0.90. For both 0.5 ≤U/U_c < 0.7 and 0.7 ≤U/U_c <0.8, Jain & Fischer (1979), Ettema et al. (1998), Sheppard et al. (2004) and Kim et al. (2015) show R² value in the range of 0.44–0.68. MAPE and RMSE are high for Ettema et al. (1998). For 0.8 ≤U/U_c < 0.9, Jain & Fischer (1979) and Ettema et al. (1998) shows a very high MAPE which considers the (b/d₅₀) value in their equation. Sheppard et al. (2004) show an R² value greater than 0.70 with an RMSE value of 0.52 for 0.9 ≤ U/U_c < 1.0. The developed ANFIS model provides MAPE of less than 15% and an R² value of more than 0.90 for all ranges except U/U_c < 0.5. In Table 13(c), for Fr < 0.2, the least MAPE and RMSE are found from the ANFIS model with a high R² value. Further, the errors are gradually reduced for high values of Fr. However, the previous models are providing unsatisfactory results with high errors and low R² values.

Table 14(a)–14(c) provides the MAPE, RMSE, and R² values found from the present approaches and four scour depth predictive equations, for selected ranges of b/y, U/U_c, and Fr, respectively. In Table 14(a), it is observed that the Johnson (1995) model renders less values of MAPE for the low range of b/y (<0.25), and present approaches failed to provide satisfactory results, due to fewer numbers of input data sets in this range. For b/y > 1.5, it has been observed that the present ANFIS model shows an R² value of 0.67 indicating the poor performance of the model in this domain as there are only 12 input data sets in the training and testing stage. However, the present GEP model shows less error and high R² values for all other ranges of b/y value. Table 14(b) demonstrates that the present GEP model and Colemon (1972) render less values of MAPE and RMSE for all the ranges of U/U_c. However, the present GEP model shows the least error for all the ranges of U/U_c value. In Table 14(c), it is observed from the error analysis that for Fr < 0.25, the present GEP model and Yanmaz (2001) render less values of MAPE and RMSE. For 0.25 ≤ Fr < 0.50 and 0.75 ≤ Fr < 1.0, the present ANFIS model renders less error than other models. However, the present GEP model shows the least error for other ranges of Fr value.

Overall, the proposed ANFIS model shows better accuracy for both CWS and LBS and the GEP model shows better accuracy for certain definite input ranges of b/y, U/U_c, and Fr in the case of LBS and performed well over the other existing empirical models.

In the present study, 442 and 213 data have been used for the modeling of clear-water and live-bed scour depth, respectively. The Gamma test reveals that pier diameter, flow depth, Froude number, approach flow velocity, critical flow velocity, geometric standard deviation, and median grain size are the most critical parameters to predict the scour depth at the bridge pier. In this study, for the prediction of clear-water and LBS depths through the ANFIS model, generalized bell membership function (gbellmf) of size [22222] and Gaussian membership function (gaussmf) of size [33222] have been recommended, respectively. The coefficient of determination (R²) value using normalized data was found to be 0.95 and 0.95 for clear-water and LBS, respectively. The MAPE value using normalized data was found to be less than 10 and 6% for clear-water and LBS, respectively.

In the GEP model for the prediction of scour depth in both clear-water and live-bed conditions, MSE fitness function and 30 chromosomes have been recommended. The coefficient of determination (R²) value using normalized data was found to be 0.88 and 0.90 for clear-water and LBS, respectively. The MAPE value using normalized data was found to be less than 22 and 16% for clear-water and LBS, respectively. Comparison of the performance of the ANFIS model with a GEP model for the prediction of clear-water and LBS depth demonstrates that the ANFIS model is more accurate for CWS and GEP model is more accurate as evident from the error indices.

For different input ranges of b/y, U/U_c, and Fr, the ANFIS model was found to provide less error in terms of MAPE and RMSE and high R² value for CWS condition, however, in LBS condition, the GEP model was found to provide less error and high R² value for certain ranges of different input parameter.

The limitation of the present approach is the range of data sets used in modeling of CWS and LBS. The model will provide better results if the input parameter value lies in the same range as mentioned in Tables 3 and 4 respectively for CWS and LBS. Further research is needed to improve the accuracy of the model by incorporating the impact of bridge pier shapes.

All relevant data are available from an online repository or repositories. (https://drive.google.com/drive/folders/1fk1spzZP-2m4BsWG2R2jlERm0EvZMY1C?usp=share_link)

The authors declare there is no conflict.