Analytic network process for local scour formula ranking with parametric sensitivity analysis and soil class clustering Open Access

b

bridge pier width;

b_n

bridge pier width normal to flow;

L

bridge pier length;

V

flow velocity;

V₀′

incipient velocity at the pier;

V_c

critical flow velocity;

y

flow depth;

Fr

Froude number;

D₅₀

median sediment size;

K_sh

factor for the pier nose shape;

K_se

factor for the angle of attack of the flow;

K_an

factor for sediment condition.

Local scour is one of the main events that can threaten the stability of bridges built across rivers. This phenomenon appears as erosion of soil on which the bridge piers are supported (Pandey & Azamathulla 2021), eventually collapsing the bridge with overall erosion of soil (Annad & Lefkir 2022b; Devi & Kumar 2022). Owing to this phenomenon, significant losses in human life, property, and consequently the economy are continually reported everywhere around the world (Toth & Brandimarte 2011; Bestawy et al. 2020). In Europe, estimates show that the mitigation costs of this risk will reach 541 M€/year for the period 2040–2070 (Nemry & Demirel 2012).

Several researchers have proposed various empirical formulas for predicting scour at bridges (Annad & Lefkir 2022b). However, it should be noted that most of the formulas claimed to calculate field scour at bridge piers were developed using laboratory data, and these formulas have rarely been validated by field measurements. Indeed, until now, little data has been sufficiently available to validate these formulas (Ahmad et al. 2018). This prompted researchers to investigate the accuracy of these formulas. One of the earliest was Gaudio et al. (2010), who evaluated six scour calculation formulas in their study. The results showed that the predictions of these six formulas differed significantly from each other. The selected formulas were evaluated using a set of field data under uniform riverbed sediment, concluding that the formulas' ability to estimate the maximum scour depth under steady-state conditions was unsatisfactory. In a study published by Najafzadeh et al. (2015), group data processing method (GMDH) networks were used for predicting abutment scour depth. GMDH performances were then compared with several conventional equations. It was found that the scour depth estimated using the conventional formulas showed a significantly higher error than that obtained using the GMDH network presented in this study. In another study conducted by Qi et al. (2016), three types of widely used equations (the Chinese equation, the HEC-18 FHWA, and the Melville) were compared. These formulas were validated using laboratory and field data. The results showed that the influence of every parameter on the scour depths calculated by the three formulas was considerably varied. Wang et al. (2017) conducted eight local scour experiments around a group of piles and large-size piles. The scour depth was observed by experimental tests and then compared with empirical formulas. The authors concluded that although many empirical formulas have been proposed to calculate scour, there is no comprehensive formula that meets all the requirements.

As mentioned earlier, it appears to be a fact that many of the empirical formulas do not always provide an efficient assessment of local scour. Moreover, some of these formulas may be suitable for a specific condition but fail in another context (Annad & Lefkir 2022b). This deficiency is probably due to the development conditions of the formulas (i.e., the latter are based on laboratory data that are not able to reproduce a complex field situation) (Vonkeman & Basson 2019). Furthermore, due to the complexity of the scour phenomenon, many empirical equations fail to estimate the scour accurately (Najafzadeh & Kargar 2019). Indeed, scour is defined as being a highly complicated process (i.e., the result of fluid–structure–soil interaction, FSSI). However, for one reason or another, many formulas aim to reduce its complexity (Mohamed et al. 2006). Usually, in empirical formulas, the ‘fluid’ is considered to remain constant, and it is expected to initiate only clear water scour (Ahmad et al. 2018). For the second component of the FSSI interaction, which is the ‘structure’, empirical laboratory formulas are established on reduced scale models (Liang et al. 2020). Although the scour phenomenon is strongly influenced by sediment properties, the ‘soil’, is often assumed to be reduced, and with constant granulometry Pizarro et al. (2020). Such simplifications, disregarding the sediment size and type when estimating scour, may lead to computational inaccuracies within the model (Rustiati et al. 2017). Indeed, Pandey et al. (2019) performed 85 different experiments with different pier diameters and flow rates, conducted under a variety of riverbed conditions. They highlighted that the maximum scour depth does not only depend on structural and flow parameters but also on the particle size as well as the geometric standard deviation of the streambed material.

The present study aims at implementing a new multi-criteria decision support method intended to evaluate a given hydraulic phenomenon. Some multi-criteria decision studies have been proposed in the hydraulic field. Bruno et al. (2020) presented a comparative analysis of different clariflocculation treatment processes for removing pollutants from salt water using a frequent multi-criteria analysis. Dehshiri (2022) presented a new hybrid MCDM method for the location of offshore wind-farm sites. Fuzzy SWARA was used to weight the criteria and fuzzy WASPAS was used to rank the sites. Naraghi et al. (2019) developed a framework using fuzzy AHP to select the most suitable residue removal procedure. Using the analytical hierarchy process, Boukhari et al. (2018) developed a sustainability assessment tool for water supply and sanitation services (WSS).

Using multi-criteria decision methods, this study aims to compare scour formulas. To begin the process, a set of 18 distinct and different local scour formulas are selected. The proposed formulas are then subjected to a comparative framework designed to evaluate their performance under different soil classes. The comparative approach proposed in this study evaluates, as a first step, the efficiency of the formulas through a selection of statistical performance criteria. Afterward, to assess the variation impact of the different input parameters included in the scour equation on the calculation output, sensitivity and correlation analyses are performed. To complete the approach, the findings of these three analyses are integrated into the Analytical Network Process (ANP) to appreciate each formula's performance and thereby identify and classify the most appropriate ones for each soil type.

A large variety of formulas designed to predict bridge scour have been proposed. For each of these formulas, each author presented a calculation method based on his perception and interpretation of the phenomenon. Indeed, for the same objective, which is to calculate the scour in the area surrounding the bridge piers, each author used a particular design approach (e.g., either laboratory data or field data). The selection of the most influential parameters influencing the scouring process is a frequent matter of scientific research (Pandey et al. 2021; Asadi et al. 2022). Moreover, through the parameters involved in these formulas, one may argue that the scour phenomenon would be dependent on certain specific types of parameters rather than others.

In this study, a significant selection corresponding to 18 empirical local scour formulas was performed. To provide extensive coverage of these formulas and to ensure a more accurate and complete evaluation of the study, these 18 formulas were selected to encompass a wide variety of situations according to the different parameters included therein (structural, hydraulic, and geotechnical). Consequently, the selected formulas are categorized by their type, as follows (Table 1):

Table 1

Local scour formulas

Authors .	Equations .	Correction coefficients .
Formulas including only hydraulic parameters
Chitale (1962)	For both clear water and live bed (1)
Formulas including only structural parameters
Laursen (1962)	For both clear water and live bed (2)
Larras (1963)	For both clear water and live bed (3)	• Ksh = 1.5 for round or cylindrical shapes
Breusers (1965)	For both clear water and live bed (4)
Neill (1973)	For both clear water and live bed (5)	• Ksh = 1.5 for round and circular shapes, and 2 for a rectangular shape.
Chitale (1988)	For both clear water and live bed (6)
Formulas including hydraulic and structural parameters
‘Inglis-Poona I’ Inglis (1949)	For both clear water and live bed (7)
‘Inglis-Poona II’ Inglis (1949)	For both clear water and live bed (8)
‘Laursen I’ Neill (1964)	For both clear water and live bed (9)
Shen et al. (1969)	For both clear water and live bed (10)
‘Laursen-Callender’ Melville (1975)	For both clear water and live bed (11)
‘Mississippi’ Wilson (1995)	For both clear water and live bed (12)
Williams et al. (2018)	For both clear water and live bed (13)
Combined formulas (including different types of parameters)
Breusers et al. (1977)	For both clear water and live bed (14)
Froehlich (1988)	For both clear water and live bed (15)	• Ksh= 1.3 for a square nose, 1.0 for a round nose, and 0.7 for a sharp nose
The Chinese 65-1 Gao et al. (1992)	For clear water (16.a) For live bed (16b)	• K = • V0′ = 0.462(D50/b*)0.06Vc • Vc = 0.0246 (y/D50)0.14 • bn = (L−b) sin + b (if circular or sharp nose) • bn = L sin + b cos (if square nose) • n1 =
The Chinese 65-2 Gao et al. (1992)	For clear water (17a) For live bed (17b)	• Kse = • V0′ = 0.12(D50 + 0.5)0.55 • Vc = 0.28 • n2 =
HEC-18 (FHWA) Arneson et al. (2012)	For both clear water and live bed (18)	• Details in FHWA Publication Number: HIF-12-003

Authors .	Equations .	Correction coefficients .
Formulas including only hydraulic parameters
Chitale (1962)	For both clear water and live bed (1)
Formulas including only structural parameters
Laursen (1962)	For both clear water and live bed (2)
Larras (1963)	For both clear water and live bed (3)	• Ksh = 1.5 for round or cylindrical shapes
Breusers (1965)	For both clear water and live bed (4)
Neill (1973)	For both clear water and live bed (5)	• Ksh = 1.5 for round and circular shapes, and 2 for a rectangular shape.
Chitale (1988)	For both clear water and live bed (6)
Formulas including hydraulic and structural parameters
‘Inglis-Poona I’ Inglis (1949)	For both clear water and live bed (7)
‘Inglis-Poona II’ Inglis (1949)	For both clear water and live bed (8)
‘Laursen I’ Neill (1964)	For both clear water and live bed (9)
Shen et al. (1969)	For both clear water and live bed (10)
‘Laursen-Callender’ Melville (1975)	For both clear water and live bed (11)
‘Mississippi’ Wilson (1995)	For both clear water and live bed (12)
Williams et al. (2018)	For both clear water and live bed (13)
Combined formulas (including different types of parameters)
Breusers et al. (1977)	For both clear water and live bed (14)
Froehlich (1988)	For both clear water and live bed (15)	• Ksh= 1.3 for a square nose, 1.0 for a round nose, and 0.7 for a sharp nose
The Chinese 65-1 Gao et al. (1992)	For clear water (16.a) For live bed (16b)	• K = • V0′ = 0.462(D50/b*)0.06Vc • Vc = 0.0246 (y/D50)0.14 • bn = (L−b) sin + b (if circular or sharp nose) • bn = L sin + b cos (if square nose) • n1 =
The Chinese 65-2 Gao et al. (1992)	For clear water (17a) For live bed (17b)	• Kse = • V0′ = 0.12(D50 + 0.5)0.55 • Vc = 0.28 • n2 =
HEC-18 (FHWA) Arneson et al. (2012)	For both clear water and live bed (18)	• Details in FHWA Publication Number: HIF-12-003

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Formulas including hydraulic parameters: A very limited number of local scour formulas containing only hydraulic parameters are available in the literature. Chitale (1962) estimated that the scour can be calculated only through hydraulic parameters.

Formulas including structural parameters: The majority of these formulas consider essentially the shape and dimensions of the bridge pier.

Formulas including hydraulic and structural parameters: Since scour is mainly related to the flow acting upstream, some authors tried to propose formulas combining hydraulic and bridge pier parameters.

Combined formulas (including different types of parameters): Some authors prefer to propose formulas encompassing different types of parameters.

Although there is an abundance of scouring calculation formulas in the literature, some of them fail to accurately estimate the local scour (Brandimarte et al. 2012; Pandey et al. 2020). One of the failure reasons identified was that the majority of formulas were estimated using a simplified approach regarding the sedimentation conditions of the rivers (Pizarro et al. 2020). Indeed, several empirical formulas were derived based on small-scale laboratory models, considering only low-size uniform sediments (Wang et al. 2017). Some formulas may contain in their equations some characteristics related to the streambed, such as the grain size D₅₀, the critical velocity V_c, etc. Nevertheless, including D₅₀ in the scour estimation seems inaccurate, for the simple reason that D₅₀ has low values compared with the other parameters, and considering it in the calculation instead of its characteristics can skew the outcome result (Annad et al. 2021). Different erosion behaviors were identified by Briaud et al. (2001) for a variety of soil types with similar particle sizes. Based on this finding, and because the scour is mainly an erosion sediment process, this study aims to validate a wide selection of local scour formulas according to different soil types (classes).

In this study, an extensive scour database is used, the 2014 USGS Pier-Scour Database (PSDB-2014) (Benedict & Caldwell 2014). This database contains 1,858 field scour measurements taken from 32 publications collected in 23 US states, and six other countries.

Among the 1,858 data, 965 contained the necessary scour parameters (such as flow velocity, for example) to calculate the scour analytically using the formulas in Table 1. For the rest of the scour data, we opted for their exclusion since their parameters are not available, rather than completing the database by using data generation techniques, which is not the object of this study. Furthermore, the data generation techniques cannot reflect the field's real values.

The database was clustered by types of soil according to D₅₀ as the ISO 14688-1:2017 standard. This last one allows qualifying a soil by giving it a name (a class) according to its granulometry (the size of its sediment).

Table 2 presents the names and the number of measurements of the three soil classes considered in this study.

Table 3 can be used as a first step to describe the performance of each formula through its statistical results as mentioned above (Annad et al. 2021).

The first step in comparing and validating the formulas used in this study is to compare the scour depths calculated by the formulas with the scour depths recorded in the field by submitting each of them to statistical performance criteria as stated in Table 3.

As indicated above, the performance criteria are varied, making it possible to assess the similarity of the formulas through R², their dispersion, and variability through RSR, and finally, their tendency (i.e., the over- or under-estimation of scour) through PBIAS.

The remarkable results obtained for several formulas for the fine soil class increasingly deteriorate as the soil size increases (sand or gravel). This preliminary result supports the hypothesis raised in this study that the local scour is dependent on sediment size. Furthermore, since most of the formulas were developed in scaled-down laboratory models, in which the sediments involved are also scaled-down, this same result indicates that formulas under field conditions can be exclusively reliable only if the sediment conditions in the field are similar to the conditions in the laboratory environment. As previously mentioned, most of the formulas were established in the laboratory where the sediments are of small and constant size (i.e., fine soil, where the grain size is small). Indeed, on real bridge models, the granulometry of the soil is varied, non-uniform, and coarse-sized sediments are also present. Another result that is also noteworthy is that most formulas overestimate the local scour (PBIAS < 0).

As indicated in Figure 1, the first observation is that the more the granulometry of the soil increases, the more the formulas lose their efficiency. From the R² results, it is observed that most of the empirical formulas show a very good to good similarity of scour (50 < R² < 94) for soil classes corresponding to small grain sizes (fine soil). However, these results deteriorate more and more as the particle size of the soil increases. From the results of PBIAS for the sand and gravel classes, it is noteworthy that most empirical formulas tend not to estimate scour accurately (|PBIAS| > 25), such as Froehlich (1988) (PBIAS > 50 for the three soil classes), Inglis-Poona II (Inglis 1949), Williams et al. (2018) (respectively, PBIAS = 35% and 81%). Overestimation of scouring may be tolerable for safety reasons (i.e., when PBIAS < 0). However, the underestimation of the scour endangers the stability of the bridge, rendering it more vulnerable, and under no circumstances may this be allowed, regardless of the reasons (Chaudhuri et al. 2022).

The first part devoted to the statistical criteria described above cannot select the best formula since an appropriate statistical performance of each formula should be based on all three criteria simultaneously (multi-criteria evaluation). The performance as described in Figure 1 is incomplete because it is done for each criterion individually. Therefore, a multicriteria analysis is performed combining the results given by these three criteria (R², RSR, and PBIAS) as well as the impact of each parameter on the scour result to complete the analysis and decide on the best formula for each soil class.

Usually, scales in the discrete interval 1–9 are used in classical ANP to compare the components in each cluster. Scale 1 means equal importance between components, whereas scale 9 represents the extreme significance of an element in comparison with others (Farman et al. 2018). The scale for preferences and pairwise comparisons proposed by Saaty (1996) were defined empirically, leaving the judgment to the user to define which criterion or alternative is better than another and therefore will have greater weight. Being mainly dependent on the user, the empirical choice of scales can lead to uncertainties in the outcome, either by misinterpretation from the user or by lack of data allowing an adequate judgment.

The Pearson correlation coefficient (Pearson's r) is used to define the scales for assigning weights involved in the pairwise comparison of all criteria, and the pairwise comparison of all alternatives with each criterion is used (Tables 4,⁵–6). The Pearson correlation is a measured value between the local scour depth and the parameters. The weight assigned to one criterion (e.g., b) and another criterion (e.g., b_n) is the ratio of their correlation coefficients to the scour depth d_s. Consequently, the criterion highly correlated with the scour depth will be assigned a higher weight based on the ratio found.

Table 4

Sand class parameter correlation matrix

.	.	Scour depth .	b .	bn .	L .	V .	Vc .	y .	Fr .
	Scour depth	1.00	0.74	0.47	0.31	0.35	0.05	0.60	0.03
	b	0.74	1.00	0.60	0.39	0.21	0.02	0.42	0.02
	bn	0.47	0.60	1.00	0.51	0.03	0.12	0.21	0.09
	L	0.31	0.39	0.51	1.00	0.10	0.35	0.28	0.11
	V0	0.35	0.21	0.03	0.10	1.00	0.40	0.39	0.67
	Vc	0.05	0.02	0.12	0.35	0.40	1.00	0.04	0.28
	y	0.60	0.42	0.21	0.28	0.39	0.04	1.00	0.28
	Fr	0.03	0.02	0.09	0.11	0.67	0.28	0.28	1.00

.	.	Scour depth .	b .	bn .	L .	V .	Vc .	y .	Fr .
	Scour depth	1.00	0.74	0.47	0.31	0.35	0.05	0.60	0.03
	b	0.74	1.00	0.60	0.39	0.21	0.02	0.42	0.02
	bn	0.47	0.60	1.00	0.51	0.03	0.12	0.21	0.09
	L	0.31	0.39	0.51	1.00	0.10	0.35	0.28	0.11
	V0	0.35	0.21	0.03	0.10	1.00	0.40	0.39	0.67
	Vc	0.05	0.02	0.12	0.35	0.40	1.00	0.04	0.28
	y	0.60	0.42	0.21	0.28	0.39	0.04	1.00	0.28
	Fr	0.03	0.02	0.09	0.11	0.67	0.28	0.28	1.00

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Table 5

Sand class pairwise comparison matrix of all criteria

.	.	R2 .	RSR .	PBIAS .	b .	bn .	L .	V .	Vc .	y .	Fr .
= 1/	R2	1.00	1.00	1.00	1.36	2.12	3.26	2.88	18.82	1.68	35.43
	RSR	1.00	1.00	1.00	1.36	2.12	3.26	2.88	18.82	1.68	35.43
	PBIAS	1.00	1.00	1.00	1.36	2.12	3.26	2.88	18.82	1.68	35.43
	b	0.74	0.74	0.74	1.00	1.56	2.40	2.12	13.84	1.23	26.06
	bn	0.47	0.47	0.47	0.64	1.00	1.54	1.36	8.90	0.79	16.75
	L	0.31	0.31	0.31	0.42	0.65	1.00	0.88	5.77	0.51	10.87
	V0	0.35	0.35	0.35	0.47	0.74	1.13	1.00	6.54	0.58	12.31
	Vc	0.05	0.05	0.05	0.07	0.11	0.17	0.15	1.00	0.09	1.88
	y	0.60	0.60	0.60	0.81	1.26	1.94	1.71	11.21	1.00	21.12
	Fr	0.03	0.03	0.03	0.04	0.06	0.09	0.08	0.53	0.05	1.00
	Sum	5.54	5.54	5.54	7.53	11.72	18.05	15.94	104.25	9.30	196.30

.	.	R2 .	RSR .	PBIAS .	b .	bn .	L .	V .	Vc .	y .	Fr .
= 1/	R2	1.00	1.00	1.00	1.36	2.12	3.26	2.88	18.82	1.68	35.43
	RSR	1.00	1.00	1.00	1.36	2.12	3.26	2.88	18.82	1.68	35.43
	PBIAS	1.00	1.00	1.00	1.36	2.12	3.26	2.88	18.82	1.68	35.43
	b	0.74	0.74	0.74	1.00	1.56	2.40	2.12	13.84	1.23	26.06
	bn	0.47	0.47	0.47	0.64	1.00	1.54	1.36	8.90	0.79	16.75
	L	0.31	0.31	0.31	0.42	0.65	1.00	0.88	5.77	0.51	10.87
	V0	0.35	0.35	0.35	0.47	0.74	1.13	1.00	6.54	0.58	12.31
	Vc	0.05	0.05	0.05	0.07	0.11	0.17	0.15	1.00	0.09	1.88
	y	0.60	0.60	0.60	0.81	1.26	1.94	1.71	11.21	1.00	21.12
	Fr	0.03	0.03	0.03	0.04	0.06	0.09	0.08	0.53	0.05	1.00
	Sum	5.54	5.54	5.54	7.53	11.72	18.05	15.94	104.25	9.30	196.30

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Table 6

Sand class normalized pairwise matrix for all criteria

.	.	R2 .	RSR .	PBIAS .	b .	bn .	L .	V .	Vc .	y .	Fr .	Criteria weight .
=	R2	0.18	0.18	0.18	0.18	0.18	0.18	0.18	0.18	0.18	0.18
	RSR	0.18	0.18	0.18	0.18	0.18	0.18	0.18	0.18	0.18	0.18
	PBIAS	0.18	0.18	0.18	0.18	0.18	0.18	0.18	0.18	0.18	0.18
	b	0.13	0.13	0.13	0.13	0.13	0.13	0.13	0.13	0.13	0.13
	bn	0.09	0.09	0.09	0.09	0.09	0.09	0.09	0.09	0.09	0.09
	L	0.06	0.06	0.06	0.06	0.06	0.06	0.06	0.06	0.06	0.06
	V0	0.06	0.06	0.06	0.06	0.06	0.06	0.06	0.06	0.06	0.06
	Vc	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01
	y	0.11	0.11	0.11	0.11	0.11	0.11	0.11	0.11	0.11	0.11
	Fr	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01

.	.	R2 .	RSR .	PBIAS .	b .	bn .	L .	V .	Vc .	y .	Fr .	Criteria weight .
=	R2	0.18	0.18	0.18	0.18	0.18	0.18	0.18	0.18	0.18	0.18
	RSR	0.18	0.18	0.18	0.18	0.18	0.18	0.18	0.18	0.18	0.18
	PBIAS	0.18	0.18	0.18	0.18	0.18	0.18	0.18	0.18	0.18	0.18
	b	0.13	0.13	0.13	0.13	0.13	0.13	0.13	0.13	0.13	0.13
	bn	0.09	0.09	0.09	0.09	0.09	0.09	0.09	0.09	0.09	0.09
	L	0.06	0.06	0.06	0.06	0.06	0.06	0.06	0.06	0.06	0.06
	V0	0.06	0.06	0.06	0.06	0.06	0.06	0.06	0.06	0.06	0.06
	Vc	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01
	y	0.11	0.11	0.11	0.11	0.11	0.11	0.11	0.11	0.11	0.11
	Fr	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01

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The method of excluding the influence of a parameter (colored by the gray circle at each step in Figure 5) consists of assigning a nominal value of 1 if it is multiplied in the formula, or a nominal value of 0 if it is added or subtracted.

Sensitivity can then be measured by monitoring changes in the output through the statistical criteria (R², RSR, and PBIAS) given by the formulas as defined earlier. We then see the effect on the local scour at each time.

To obtain the weights to be assigned to all criteria, considering the alternative Inglis-Poona I for the fine soil class as an example (Table 7), as described in Tables 5 and 6, the same steps are followed.

Table 7

Fine soil normalized pairwise matrix of all criteria with Inglis-Poona I formula

Inglis-Poona I .	R2 .	RSR .	PBIAS .	b .	bn .	L .	V .	Vc .	y .	Fr .	Normalized weight .
R2	0.11	0.11	0.10	0.13	0.11	0.11	0.07	0.11	0.24	0.11
RSR	0.11	0.11	0.10	0.10	0.11	0.11	0.10	0.11	0.10	0.11
PBIAS	0.11	0.11	0.10	0.15	0.11	0.11	0.20	0.11	0.05	0.11
b	0.05	0.06	0.04	0.06	0.06	0.06	0.06	0.06	0.06	0.06
bn	0.11	0.11	0.10	0.10	0.11	0.11	0.10	0.11	0.10	0.11
L	0.11	0.11	0.10	0.10	0.11	0.11	0.10	0.11	0.10	0.11
V0	0.12	0.07	0.04	0.07	0.07	0.07	0.07	0.07	0.07	0.07
Vc	0.11	0.11	0.10	0.10	0.11	0.11	0.10	0.11	0.10	0.11
y	0.05	0.11	0.21	0.10	0.11	0.11	0.10	0.11	0.10	0.11
Fr	0.11	0.11	0.10	0.10	0.11	0.11	0.10	0.11	0.10	0.11

Inglis-Poona I .	R2 .	RSR .	PBIAS .	b .	bn .	L .	V .	Vc .	y .	Fr .	Normalized weight .
R2	0.11	0.11	0.10	0.13	0.11	0.11	0.07	0.11	0.24	0.11
RSR	0.11	0.11	0.10	0.10	0.11	0.11	0.10	0.11	0.10	0.11
PBIAS	0.11	0.11	0.10	0.15	0.11	0.11	0.20	0.11	0.05	0.11
b	0.05	0.06	0.04	0.06	0.06	0.06	0.06	0.06	0.06	0.06
bn	0.11	0.11	0.10	0.10	0.11	0.11	0.10	0.11	0.10	0.11
L	0.11	0.11	0.10	0.10	0.11	0.11	0.10	0.11	0.10	0.11
V0	0.12	0.07	0.04	0.07	0.07	0.07	0.07	0.07	0.07	0.07
Vc	0.11	0.11	0.10	0.10	0.11	0.11	0.10	0.11	0.10	0.11
y	0.05	0.11	0.21	0.10	0.11	0.11	0.10	0.11	0.10	0.11
Fr	0.11	0.11	0.10	0.10	0.11	0.11	0.10	0.11	0.10	0.11

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The ANP approach structures a model problem as a network of clusters in which each cluster has different connected components (Farman et al. 2018). The Hybrid Analytic Network Process (HANP) proposed in this study can be summarized in the following steps:

• (1)

  The first step is to select a significant number of the formulas.

• (2)

  The second step concerns the selection of criteria for the selection of the best formula. For this purpose, two clusters of criteria are defined. The first one is based on the statistical performance of the formulas (R², RSR, and PBIAS), while the second cluster is based on the parameters influencing the scour result as defined previously through the sensitivity and correlation analyses.

• (3)

  In each cluster, the components are compared as follows:

  • −   For the pairwise comparison of all criteria (Table 5), and for the pairwise comparison of all alternatives to each criterion (Table 8), the scale assigned to each element with another is based on the ratio of their correlation coefficients. Each element is weighted depending on its significance compared with the other components by assuming certain parameters. A matrix is created for each comparison.

  • −   For the pairwise comparison of all criteria in each cluster to each alternative (Table 9), the weights assigned to each criterion relative to another are based on the ratio of their statistical coefficients R², RSR, and PBIAS estimated from the sensitivity analysis.

• (4)

  In this step, the unweighted supermatrix (Figure 6) is constructed according to the network built in step (3). To obtain the weighted supermatrix wherein each column's sum equals 1, the unweighted supermatrix (Figure 6) is then normalized and scaled by dividing every cell by its column sum.

• (5)

  The limit matrix is computed to synthesize the approach by multiplying the weighted supermatrix by the power of k + 1.

• (6)

  The limit matrix finally leads to the determination of the best formula, where the weight of each formula is assigned.

After applying all the steps as stated above, the weights assigned to the formulas according to the limit supermatrix for each class of soil are as follows.

For the fine soil where (0.002 < D₅₀ ≤ 0.06 mm), the three best formulas are those of Larras (1963), Shen et al. (1969), and Breusers et al. (1977). These formulas contain mainly parameters such as the width and type of the pier b obstructing the flow and therefore subjected to scour, and the flow depth y upstream of the pier.

For the sand class where (0.063 < D₅₀ ≤ 2.00 mm), the three best formulas are in order: Larras (1963), Wilson (1995), and Inglis-Poona I (Inglis 1949). In these three formulas, besides the two parameters mentioned above, the flow velocity V_c is included.

For the gravel class where (2.00 < D₅₀ ≤ 63 mm), normally, the best formulas in order are Froehlich (1988), the Chinese 65-2 (Gao et al. 1992), Larras (1963), and HEC-18 (Arneson et al. 2012). These formulas are classified according to Table 1 as being formulas including different parameters.

The performance provided by a particular formula differs from one soil class to another. For example, if we consider the Chinese 65-2 formula (Gao et al. 1992), this formula is classified as being the least efficient for the two classes fine soil and sand (ranked last). However, it is classified as the second most efficient for the gravel class. The example is also true for other equations such as Inglis-Poona I, which only gives good results for the soil class of gravel (ranked 3) and provides unsatisfactory results for the other classes (ranked 14 for the fine soil class, and 17 for the gravel class). Therefore, the fact of deciding on the most suitable formula without considering the soil class seems to be a misleading decision.

Local scour is a very critical phenomenon threatening the stability of bridges, many of which have been destroyed due to its impact (Cook 2014). Several formulas exist for estimating scouring. Generally, these formulas neglect the soil class variation (Annad et al. 2021).

The purpose of this study is to develop a comparison framework involving several empirical formulas by soil class. The validation of the formulas clustered by soil type allows judging, more pertinently, the performance of each formula in estimating scour. For a more representative and accurate validation attempt, 18 different scour calculation formulas were tested by clustering the data according to the streambed size classes, following D₅₀. The field observations of PSDB-2014 (Benedict & Caldwell 2014) were exploited and three soil classes were selected (fine soil, sand, and gravel). This study aims to assess the scour formulas’ efficiency through a robust comparison system, making it possible to test their ability to calculate scour by soil class. The proposed system is a hybrid model of the Analytic Network Process (ANP) coupled with a sensitivity and correlation analysis. The new system allows, on the one hand, testing of the efficiency of the formulas by subjecting them to various statistical performance criteria (R², RSR, and PBIAS). The results obtained by these criteria allow us to qualify the performances of each formula and to determine the empirical formulas giving the best results according to the soil classes. On the other hand, studying the impact of variation of each parameter of the calculation equations and finally deciding on the best formula by soil class combines the outcome of these three analyses into the new approach proposed in this study, which is the Hybrid Analytic Network Process (HANP).

Our gratitude goes to the Algerian General Directorate of Scientific Research and Technological Development (DGRSDT) for supporting this study in the context of the MESRS-PRFU program.

All relevant data are available from: https://pubs.usgs.gov/ds/0845/

The authors declare there is no conflict.