A multi-output descriptive neural network for estimation of scour

geometry downstream of grade-control structures

Aytac GUVEN

University of Gaziantep, Department of Civil Engineering,

27310 Gaziantep -TURKEY

e-mail: aguven@gantep.edu.tr

ABSTRACT

Several researches have been attempted to predict the maximum depth and location of

local scour, particularly, based on conventional regression analysis. Some of these equations

in the literature failed to predict the scour depths satisfactorily. This study presents explicit

formulation extracted from a multi-output descriptive neural network (DNN), which predicts

both the depth and location of maximum scour. Present method extracts rules (information)

conveyed from input layer to output layer of a NN consisting two outputs. The present DNN

is compared to non-linear and linear regression equations derived by the author and other

empirical equations in the literature. The results show that the proposed DNN predict the

maximum scour depth and its location in strict agreement with the measured ones, and

dominantly better than the other equations. This study shows that the explicit formulation

extracted from DNN can replace the conventional regression equations with much more

accuracy.

Key words: Scouring, descriptive neural networks, modeling, regression, river hydraulics

1. IntroductionLocal scour is the erosion of bed surface and the hydraulic structures due to the impact

effect of flowing water. Grade-control structures are built in order to prevent excessive

channel-bed degradation in alluvial channels. However, local scour downstream of grade-

control structures occurs due to erosive action of the weir overflow and this action may

undermine these structures [11]. Thus, the structural design of the grade control structures

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must include sufficient protective provisions against local scour and comprehensive

understanding of the mechanics, location and extent of the downstream [16].

Local scour downstream of the grade-control structures is a specific case considered in

this study. Although these structures are built to prevent excessive bed degradation in alluvial

channels, they also cause scouring due to the erosive action of overflowing water. Borrmann

and Julien [11] investigated the scour hole development downstream of sharp-crested grade-

control structures, having a weir width bw and height z, as given in figure 1. They observed

that, as the water flowed over the weir and enters the tailwater yt, the flow separated from the

structure at crest of structure and a vortex was formed in the separation zone. The diffused

flow velocity in the vicinity of point impingement point exerted a shear stress on bed

sediment particles. When the applied shear stress exceeded the critical shear stress, sediment

was dislodged and transported beyond the impingement zone (Point B in figure 1) [11]. The

process continued until the rate of scour went to zero. This state is generally characterized by

maximum scour depth, dm and its location xm. In fact, almost all experimental studies on local

scour aims to characterize the local scour by dm and predict it based on incoming flow

parameters and scour geometry characteristics [27]. Due to the complex structure of scouring

process together with difficulties raised by the various turbulent flow conditions, most of the

studies dealing with local scour have been experimental. A Detailed literature review on these

experimental studies can be found in Mason and Arumugam [25], Bormann and Julien [11]

and Sarkar and Dey [27]. In fact, the majority of these experimental studies aimed to derive

empirical relation among the scour parameters (depth and location of maximum scour) based

on conventional regression analysis. The existing formulae performed satisfactorily on

corresponding experimental data. However, their major drawback is that they involve

assuming ideal conditions, rough approximation and averaging of widely varying prototype

conditions [7]. Another issue is the scale problem, namely, almost all the existing formulae

have been derived based on regression analysis on experimental data of small or medium

scale laboratory researches. The scale problem arises from the limitations of the laboratory

conditions, since large-scale physical modelling of scouring around hydraulic structures needs

larger working space than available. This means that the existing formulae are not capable of

extrapolating larger scale scour parameters. Additionally, the scouring process is assumed as

two-dimensional or quasi two-dimensional in these laboratory studies, neglecting the side-

wall and secondary flow effects [19]. However, the scouring process in nature is much more

complex and containing many hydraulic factors, which are neglected or simplified in

laboratory studies. Accordingly, the resulting formulae derived based on these experimental

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data suffer from generalization on larger scale scour estimations and they generally ignore

some important hydraulic parameters influencing on scouring [16-18].

In recent decade, new techniques have been proposed as an alternative to conventional

regression analysis and numerical methods. Soft computing techniques have been widely used

and well-validated in prediction of water resources variables [2,9,15,24,27]. Neural networks

(NNs), a branch of soft computing technique, have been successively applied in prediction of

hydraulic data in last two decades. NNs are relatively stable with respect to noise in data and

have a good generalization potential to represent input-output relationships [14]. NNs involve

a structure where non-linear functions are present and a parameter identification process is

based on techniques which search for global maximums in the space of feasible parameter

values. Thus, NNs can represent the non-linear effects present in the scouring process. All

these features of NN make it an intelligent tool in formulation of maximum depth of scour

downstream hydraulic structures. On the other hand, NNs have problem of inability to

extrapolate.

Due to the black box nature of neural networks, they sometimes are not classified as data

mining tools for discover interesting and understandable data mining definition. However,

once an NN model is trained for its generalization properties, it can be assumed that the

trained model represents the physical process of the system. The knowledge acquired for the

problem domain during the training process is encoded within the NN in two forms: (a) in the

network architecture itself (through number of hidden units) and (b) in a set of constants or

weights [32]. Lange [22] states that ANNs are black-box models that only develop the

relation between input and output variables without the modeling of any physical processes,

however, it must be realized that the data that are employed in developing black-box models

contain important information about the physical processes being modeled, and this

information gets embedded or captured inside the model [8]. Although there are several

attempts in other scientific branches that have shown that useful information could be

obtained from trained neural networks (see Yao [32] for relevant references), only a few

researches have been recorded in water engineering [3,8,14,16,17].

In the literature a few studies are encountered, which use NNs in prediction of local scour

downstream of hydraulic structures. Liriano and Day [23] predicted the scour depth at culvert

outlets, Azinfar et al. [4] predicted the scour depth downstream of a sluice gate. Azamathulla

et al. [5-7] estimated the scour downstream of a ski jump and below spillways using NNs.

Guven and Gunal [16] proposed a NN model for prediction of local scour downstream of

grade-control structures, based on wide-spread experimental data of others. Most of the

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above-mentioned studies are developed based on small- or medium-scaled experimental data,

which could later show scale problems while applying to prototype cases. This study differs

from all above studies in that it uses both large-scale experimental and field data in training

and testing stages. More importantly, it focuses on modelling two important scour parameters

in a single NN model. Namely, a practical mathematical formulation is presented using a two-

output NN model. Although Azamathulla et al. [5,6] mentioned about the inner structure of

NN and its basic model equations, they did not give the explicit form of their proposed

models. This study directly aims to derive the knowledge (explicit formulation) from a well

trained and tested NN model. The proposed formulation can further be utilised by any

researcher who is even not familiar with NNs technique. The author would like to inform the

readers about most recent study on prediction of local scour downstream of grade-control

structures by a new soft computing technique: Gene-Expression Programming presented by

Guven and Gunal [17].

In this study, dimensional analysis and experimental scour data are used to establish a

non-dimensional relationship describing the geometrical pattern of the scour profile. These

are the normalized maximum scour depth, dm/z, and the normalized location of maximum

scour depth, xm/z. The experimental measurements were taken from a large scale model

research carried out by Bormann and Julien [11]. The non-dimensional parameters derived by

dimensional analysis: densimetric froude number (Frd), normalized tailwater depth (yt/z) and

sediment uniformity ratio d90/d50 (d50 and d90 represent bed grain size for which 50% and 90%

of sampled particles are finer (m), respectively) were used as inputs in training phase of

proposed DNN model, and the corresponding dm/z and xm/z are used as the outputs. The

derivation of DNN is also explained in this study.

2. Overview of descriptive neural networksDNN technique involves three basic stages. First the neural network forecasting is built by

training a neural network by available data. This stage contains all manual procedure

components involved in construction of neural networks such as data preprocessing,

input/output selection, sensitivity analysis, data organization, model construction, post

analysis and model recommendation. The second stage is to extract rules (formulation) from

wel trained NN. The NN architecture (number of hidden neurons) and model weights are

decoded to get some rules that govern the forecasting. By extracting hidden information from

previously constructed NN, we will be able to the mechanism of a forecating NN model. In

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the third stage, the formulations extrcated from the previous stage is incorporated to the

network generated by first stage to form a descriptive neural network (DNN). Researchers

genarally extrcat if-then type rules association rules [32]. In this study, the information hidden

in well trained NN is extracted as linear regression equations embedded in input layer and

logistic functions embedded output layer of the NN architecture. By this, it is aimed to present

the proposed DNN as a regression-based explicit formulation alternative to conventional

regression equations.

Although the basics of NNs have been sufficiently given in previous studies, the author

finds it necessary to remind, especially non-specialised reader, of most important elements of

NNs in order to comprehend the explicit neural networks formulation.

NNs technique is a data processing tool that mimics the function of the human brain and

nerves built on the so-called neurons – processing elements – connected to each other.

Artificial neurons are organized in such a way that the structure resembles a network. This

technique differs from the traditional data processing; it learns the relationship between the

input and output data [21].

Multilayer neural network model, which is considered in this study, usually consists of

three layers: input, hidden and output layers. The input layer constitutes input nodes

representing input variables. The output of the input nodes are normalized and transferred to

the hidden layer in which they are processed through a transfer function. The output layer

consists of output variables.

The basic element of NNs is an artificial neuron, which consists of three main

components; weights, bias, and an activation function. Each neuron receives inputs xi (i = 1, 2,

…, n) attached with a weight wij (j 1) which shows the connection strength for a particular

input for each connection. Every input is then multiplied by the corresponding weight of the

neuron connection and summed as

(1)

A bias bi, a type of correction weight with a constant non-zero value, is added to the

summation in Eq. (1) as

(2)

In other word, Wi in Eq. (1) is the weighted sum of the ith neuron for the input received

from the preceding layer with n neurons, wij is the weight between the ith neuron in the hidden

layer and the jth neuron in the preceding (input) layer, and xj is the output of the jth neuron in

the input layer. After being corrected by a bias as in Eq. (2), the summation is transferred

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using a scalar-to-scalar function called an “activation or transfer function”, f(Ui), to yield a

value called the unit’s “activation”, given as

(3)

Activation functions serve to introduce nonlinearity into NNs which makes it more

powerful than linear transformation. In his study, most commonly used sigmoid transfer

function (y=1/(1+exp(-x)) is utilised; although there are other types available, like tangent-

hyperbolic, Gaussian and linear. Interested reader who is looking for more information can

consult any textbook in neural computing, e.g. [20,21] or can refer to other previously

published works in related journals [24].

3. Scour data usedExperimental measurements of Bormann and Julien [11] (82 data sets) are used as Train,

Cross-Validation and Test sets of the proposed NNs model. They worked with a large-scale

flume. Experimental discharges, Q, range between 0.3 to 2.5 m3/s, producing maximum scour

depths reaching 1.40 m. The tests of Bormann and Julien [11] are characterized by a sediment

size range of 1.5<d90/d50<5.3. The other details of the experimental study can be found in

Bormann and Julien [11].

The physical relationship of scour downstream of sharp crested weir due to erosive action

of impinging water (see Fig. 1) is given in Eq.4, and the functional relationship obtained by

dimensional analysis on physical parameters in Eq.4 is given in Eq.5. In the present study, the

dimensionless parameters given in Eq.5 were used as input and output parameters in NN

modeling. Table 1 reports the ranges of input and output parameters i.e. maximum and

minimum values of dimensionless groups, which are used in this study.

(4)

(5)

Table 2 shows the correlation matrix representing the results of multiple linear regression

analysis in possible pair of groups, considering dm/z and xm/z as a dependent variable,

respectively. The numbers given in the table shows the degree of correlation between the

input and the dependent parameters. Table 2 shows that yt/z and d90/d50 have considerable

correlation with dm/z and xm/z, but Frd has relatively poor correlation. However, when

developed a NN model considering Frd as single input, correlations of R=0.805 and 0.831

were obtained for the dm/z and xm/z outputs, respectively.

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4. Sensitivity analysis on scour parametersIn order to ascertain how dm/z and xm/z depend upon the input parameters (Frd, yt/z, d90/d50), a

sensitivity analysis on these parameters was carried out. Firstly, a NN model was developed

based on the three inputs and then for the next runs, each one input was removed from input

set and new models were built. The results of the sensitivity analysis are given in Table 3. It is

clear that removing any of the three input parameters worsened the performance of the

corresponding NN model based on the coefficient of determination (R2) and the mean squared

error (MSE) criteria. Especially, the NN model without Frd gave the worst performance

(R2=619, MSE=10.858 for dm/z and R2=0.483, MSE=385.534 for xm/z) when 2-input set was

used. NN models with unique input set can be said to fail to model the two outputs meanwhile

as it seen from the table, especially revealing very high MSE values and also some negative

predictions which are not acceptable. It can be deduced from the overall performance of the

developed NN models that each input parameter has a significant influence on the scour

parameters dm/z and xm/z.

5. Results of training and testing of NNs modelThe data obtained from Bormann and Julien [11] were randomly split into Train, Cross-

Validation and Test sets. As it is widely known, one of the main issues of NNs modelling is

the generalisation capacity of the model. Increasing the number of hidden neurons lead to

over-generalisation of the model on Train set and poor performance on Test Set. For this

reason, a portion of the data set out of the Train set is used as Cross-Validation set in order to

control and avoid the over-generalisation of the model. In this perspective, 21 sets among 82

data sets (20% of total) were reserved for the Cross-Validation and Test sets and the

remaining data (40 sets) were perceived in the training.

Multi layer feed forward neural networks with back-propagation learning algorithm were

used in this study. The model parameters were optimised by one of the most common and

successful back-propagation algorithm, Levenberg&Marquardt. Another important issue is to

find the optimal architecture of the NNs model. Most studies in the literature used trial

approach, which generally leads to local maxima or minima. In this study, this problem is

eliminated by using a Genetic Algorithm in order to find the optimal architecture of the

proposed NNs model. Namely, a fitness function was chosen based on MSE of the Cross-

Validation Set and the program searched for optimal architecture with least MSE for Cross-

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Validation Set. The optimal architecture of the proposed NNs was found to be 3-7-2 (no. of

inputs-no. of hidden neurons-no. of outputs), as shown in Fig. 2.

The overall performance of the tree sets was evaluated by MSE and the R2. The training

results of the proposed models showed that the NNs learned the highly non-linear relationship

between local scour parameters and the outputs with high correlations (R2=0.990 for dm/z and

R2=0.993 for xm/z) and low errors (MSE=0.327 for dm/z and MSE=3.251 for xm/z). Validation

of the trained model proved the high generalization capacity of the proposed model with a

high correlation and low error (R2=.0.964, MSE=0.637 for dm/z and R2=0.974, MSE=11.724

for xm/z) (see Figs. 3 and 4).

6. Derivation of DNNThe model parameters of DNN were obtained from the trained NN, and the explicit

formulation is derived using the weights of the trained NN model. As mentioned earlier, the

proposed NNs model involves three input nodes, seven hidden nodes and two output nodes.

First of all, inputs and outputs are normalized before the learning process of the NN. In input

layer, each input is multiplied by a connection weight, namely products and biases are simply

summed (Eqs.(8a)-(8g)). In hidden layer, these equations are transformed through a transfer

function (sigmoid) to generate two results. Finally, in output layer, the outputs are obtained by

de-normalising the result obtained in hidden layer.

As mentioned earlier, the goal is to obtain a unified formulation that predicts both dm/z and

xm/z using unique NNs architecture, in a functional form in terms of measured dimensionless

variables as:

(6)

(7)

where values for Ui (i=1,…,7) are given as

(8a)

(8b)

(8c)

(8d)

(8e)

8

(8f)

(8g)

It should be noted that the above formulation is valid between the maximum and minimum

values of the input and output parameters given in Table 1.

6. Regression Analysis on scour dataMost of the studies on prediction of local sour parameters have applied nonlinear regression

analysis on experimental data, and optimised the parameters of their pre-defined equations

based on the correspondent data [11,13,26,28]. These empirical equations are generally in

power equation form (y=xa).

In this section, multiple non-linear regression (MNLR) and multiple linear regression

(MLR) techniques are applied on the same data sets used in NNs modelling. Firstly, a power

type nonlinear relation was used among the local sour parameters, and the model parameters

were optimised by Levenberg&Marquardt algorithm, which revealed following equations:

(9)

(10)

Secondly, simple linear relation was used between the input and output parameters and least-

square method was used for model calibration. The fit of linear regression equation to the

experimental data set yielded following equations:

(11)

(12)

The statistical performance of Eqs.(9)-(12) are discussed in further sections.

7. Other local scour equationsMason and Arumugam [26] made a detailed review on empirical equations predicting

maximum scour depth downstream of grade-control structures. They also studied on

performance of past empirical studies and proposed a general power form for these scour

equations as:

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(13)

where a, b, c, d, e, f , i and K are exponents of scour equation, ΔH is drop across the structure

(m), U0 is jet velocity impinging on the tailwater (m/s), g is gravitational acceleration (m/s2),

is jet angle near bed (rad.) and ds is representative bed grain size (m).

Based on the relationship in Eq. 13, Mason and Arumugam [26] proposed Eq.14, given in

Table 4. They observed that their equation predicted the model data is much better by using ds

equal to mean bed grain size, dm (median bed grain size).

Bormann and Julien [11] carried out a semi-numerical analysis on their measured data and

calibrated their semi-theoretical equation that predicts maximum scour depth under a wide

range of conditions: wall jets, vertical jets, free jets, submerged jets and flow over large-scale

grade control structures. Their proposed equation is remarkably similar to the regression

based equations proposed in the literature:

(18)

The angle (Fig. 1) is experimentally inferred by Bormann and Julien (1991) as:

(19)

where is downstream face angle of the grade-control structure (radian), y0 is water depth at

the crest (m).

It can be clearly deduced from Eq.18 that it is applicable to the cases for which the drop

height z is less than the term of Eq.18 within the square brackets.

Borrmann and Julien [11] also compared their equation (Eq.18) to some previous studies

in the literature. Table 4 shows some of these studies and their equations predicting dm

downstream of grade-control structures, which are also compared to present NNs model in

further section of this study.

D’Agostino and Ferro [13] updated the formulas for maximum scour depth given in the

literature and improved the estimation of maximum scour depth by introducing variables

representative of both the jet and contraction and the bed particle grain size distribution. The

authors used incomplete self similarity theory for deducing the physically dimensionless

groups controlling the geometrical pattern of scour profile:

(20)

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(21)

where B is channel width (m), A50 and A90 are dimensionless groups derived by dimensionless

analysis:

(22)

(23)

8. Comparison of DNN with other equations8.1 Performance Criteria

It is important to define the criteria by which the performance of the model and its prediction

accuracy will be evaluated in model development process. Various statistical measures have

been developed and used to assess the model performance. However, in our case we should

consider some specific error measure that would evaluate the performance of the compared

models with respect to model size as well as its accuracy in predicting the test data, since the

number of fitting data in a NNs model is naturally many more than that in regression-based

ones. Thus, we employed an effective measure, Akaike’s Information Criterion (AIC)

proposed by Akaike [1], coefficient of determination (R2) and mean absolute error (MAE), in

comparison of the proposed DNN with other regression equations based on the Test Set. AIC

(Eq.24) is used to measure the exchange between testing performance and model size (no. of

model weights). The goal is to minimize AIC to obtain a model with the best generalization.

(24)

(25)

where N is the number of input-output pairs in the testing set, k is the number of model

weights, ei is the experimental value and pi is the predicted value.

8.2 Maximum scour depth, dm

In this section the performance of DNN (Eq.6) and the other equations are compared based on

the Test Set, as shown in Table 5. The table shows the overall best performance of DNN with

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the lowest values of error measures (AIC=-61.921, MAE=0.341) and highest correlation

(R2=0.819). On the other hand, MLNR (Eqn.9) and MLR (Eqn.11) predicted dm much better

than the other regression-based equations based on AIC and MAE criteria, but Bormann [10]

predicted with higher R2 (0.688), which only indicated relatively weak tendency of

predictions of MNLR and MLR to linearly co-vary with the experimental values. Almost all

predictions of Chee and Yuen [12] (Eq.15), MNLR (Eq.9) and D’Agostino and Ferro [13]

(Eq.20) are observed to be below the experimental values, and also negative values of Chee

and Yuen [12] are seen in Table 5, which are physically unacceptable.

8.3 Location of maximum scour depth, xm

All of the studies in Table 4 only predicted dm and did not consider xm, so DNN of xm (Eq.7)

will be compared to MNLR (Eq.10), MLR (Eq.12) and D’Agostino and Ferro [13]’s empirical

model (Eq.21). Table 6 indicates the same performance of DNN in prediction of dm. Namely,

DNN (Eq.7) predicted xm values with lowest errors (AIC=-9.232, MAE=0.109) and highest

correlation (R2=0.907). MNLR and MLR derived by the author showed overall better

performance compared to Eq.21, except R2, which is better than that of MLR. It should be

noted that MNLR, MLR and Eq.21 under-predicted all xm values with considerable deviation,

while those of DNN are very close to experimental values.

9. Further applicationIn this section, the robustness of the proposed DNN (Eq.6), MNLR (Eqn.9), MLR (Eq.11) and

the other equations is evaluated in estimation of large-scale field data corresponding to

erosion downstream of dams, which were not used in training stage of the proposed NNs

model (see Table 7). The field data was taken from D’Agostino and Ferro [13], which

collected three field measurements of Veronese [30], Scimemi [29] and Whittaker and

Schleiss [31].

Veronese [30] presented field observations on the ‘‘Rocchetta’’ dam built on Noce River,

Trento, Italy. Scimemi [29] observed the bottom erosion downstream of the Conowingo dam

on the Susquehanna River. Whittaker and Schleiss [31] compared various formulas for the

Cabora-Bassa dam in Mozambique in Africa.

The statistical results of DNN estimations are very promising, since almost perfect

agreement is observed between the estimated and the field dm values (R2=0.997, MAE=0.408),

as shown in Table 7. The predictions of MLR (Eq.11) and some of the other regression-based

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equations, given in Table 4, are not given because they predicted negative values, which are

physically unacceptable.

Additionally, the xm values of these field studies, which are not available, are estimated

based on the developed unified NNs model. It should be noted that the values of field

measurements are very near to those of Borrmann and Julien [11]. The estimated xm values are

also observed to fall in the range of experimental values of Borrmann and Julien [11]. In this

perspective, the estimated xm values can be said to be realistic.

The underlying reason for the overall superiority of the proposed NN model to the

existing and the proposed regression-based empirical formulas is in that, regression analysis

applies a pre-defined linear or non-linear model on the widely-scattered relation among

scouring parameters, which suffers from inappropriate predictions. In fact, the correlation

matrix among the input and output parameters given in Table 2 proves that linear regression

fails to capture the relation among the input and output parameters, which assures to be highly

non-linear and complex. On the other hand, the proposed NN model uses parallel processors

incorporated with strong learning (back-propagation) and optimization tools that are able to

capture the highly non-linear relation among these parameters even though multiple

constraints such as multiple outputs in present case. It should be stressed that the existing and

the proposed regression equations, mentioned above, are capable of predicting only one

output, either dm/z or xm/z, but the proposed NN model predict both of them at once when the

values of input parameters are provided.

10. ConclusionsThe depth and location of maximum scour downstream of grade-control structures is

predicted by a two-output neural network model. The knowledge from the developed model is

extracted in order to build explicit formulation of the proposed descriptive NN model (DNN).

DNN predicted the depth and location of the maximum scour with very good agreement with

measured data and superior to nonlinear and linear equations developed by the author.

The significance of this study is in that, the proposed DNN model predicts both the depth

and the location of the maximum scour (dm/z and xm/z), and more significantly, this study

focuses on extracting the explicit formulation from the developed NN model. By this, the

proposed DNN model can be used to re-evaluate based on further experimental or field

observations, and also in comparison with conventional analytical methods.

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The procedure of DNN is longer than conventional regression analysis technique but not

more complex. DNN is such simple that it can be used, by anyone who is even not familiar

with NNs. It can be used in a spreadsheet on a very simple PC or even in a hand calculator

conditional that input variables are measured.

DNN was also compared to conventional empirical models in the literature, and the results

for Testing set exhibited overall superior performance of DNN against those equations with

the lowest error (MAE=0.341 and AIC=-61.921) and highest correlation (R2=0.819). Eq.15

proposed by Chee and Yuen [12] failed to predict dm with negative values, which are not

acceptable. The best performance among the regression equations belong to Eq.9 proposed by

the author with R2=0.677 and MAE=0.488 for dm/z and R2=0.706 and MAE=0.307 for xm/z,

respectively.

The robustness of DNN is well-validated in prediction and estimation of field data

obtained from other experimental studies, and the results are very promising. DNN predicted

the observed dm/z with quite accuracy (R2=0.997, MAE=0.408). MLR (Eq.11) and some of the

other regression-based equations, given in Table 4 predicted negative values for almost all

observed scour depths, which are physically unacceptable. These results prove the efficiency

of DNN in predicting complex local scour problem and encourage the use of DNN in

preliminary design of hydraulic structures.

The results of this study showed that conventional regression-based equations of

predicting maximum scour depth downstream of hydraulic structures could better be replaced

by DNN. Also, the performance of the nonlinear regression equations developed by the author

(Eqs.9-10) showed that they can be alternatively used in early design of hydraulic structures.

AcknowledgementThe author wishes to thanks to Scientific Research Projects Unit of Gaziantep University for

providing support during this study.

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17

Table 1. Minimum and maximum values of input and output variables

Dimensionless variable Minimum value Maximum value

Frd 1.026 1.921

yt/z 2.395 25.60

d90/d50 3.800 5.267

dm/z 0.368 25.20

xm/z 1.605 158.6

18

Table 2. Correlation matrix for Bormann and Julien (1991) experiment (82 data)

Parameter  Frd yt/z d90/d50 dm/z xm/z

Frd 1

yt/z -0.381 1

d90/d50 -0.259 0.470 1

dm/z 0.097 0.695 0.556 1

xm/z 0.048 0.765 0.581 0.931 1

19

Table 3. Results of the sensitivity analysis based on different input sets

Input set

Output= dm/z Output= xm/z

R2 MSE R2 MSE

Frd, yt/z, d90/d50 0.964 0.637 0.974 11.724

Frd, yt/z 0.858 3.976 0.919 27.136

Frd, d90/d50 0.888 3.210 0.952 35.367

yt/z, d90/d50 0.619 10.858 0.483 385.534

Frd 0.641 23.641 0.691 664.733

yt/z 0.001 52.894 0.624 1668.49

d90/d50 0.421 15.996 0.396 447.174

20

Table 4. Local scour equations

ResearchEquation

Eq. no

Martins (1975) (14)

Chee and Yuen (1985) (15)

Mason and Arumugam (1985)(16)

Bormann (1988) (17)

21

Tablo 1. Experimental and predicted dm/z for Test Set

Run Experimental dm (m)

Predicted dm (m)

Martins (1975)

Chee and Yuen (1985)

Mason and Arumugam (1985)

Bormann (1988)

Bormann and Julien (1991) (Eqn. 14)

D'Agostino and Ferro (2004) (Eqn. 16)

MNLR (Eqn.9)

MLR (Eqn.11)

DNN (Eqn. 6)

1 1.12 2.50 0.32 3.11 0.37 0.27 0.17 0.31 1.24 1.152 1.40 2.52 0.25 2.94 0.43 0.32 0.18 0.18 1.33 1.253 1.16 2.13 0.11 2.47 0.29 0.21 0.16 0.25 1.28 1.194 0.72 2.05 -0.05 2.67 0.15 0.10 0.14 0.68 1.33 0.625 1.32 2.48 0.39 2.94 0.63 0.48 0.17 0.64 1.89 1.266 0.97 2.41 0.27 2.79 0.35 0.26 0.20 0.10 0.91 0.977 1.39 2.51 0.14 2.82 0.37 0.27 0.21 1.35 0.86 1.198 0.27 1.00 -0.15 1.15 0.04 0.02 0.10 1.48 0.12 0.219 0.10 0.65 -0.14 0.65 0.05 0.03 0.08 2.62 0.12 0.2010 0.58 2.29 -0.09 2.94 0.21 0.15 0.17 1.04 0.19 0.4911 1.05 1.70 -0.10 1.89 0.26 0.18 0.15 1.85 0.99 0.8712 0.64 0.83 -0.21 0.84 0.14 0.09 0.10 1.16 1.33 0.7813 0.18 0.45 -0.21 0.46 0.11 0.07 0.07 0.73 1.30 0.7914 0.36 0.72 0.03 0.67 0.07 0.04 0.09 0.32 0.28 0.3415 0.53 1.15 0.16 1.11 0.15 0.11 0.13 0.11 0.36 0.5916 0.21 0.65 -0.04 0.63 0.11 0.07 0.08 0.65 0.63 0.3417 0.52 0.99 0.04 0.99 0.18 0.12 0.11 0.62 0.71 0.7918 0.47 2.55 0.26 3.05 0.22 0.16 0.22 0.18 0.81 0.4819 0.59 2.52 0.19 3.12 0.28 0.21 0.19 0.54 0.54 0.5520 0.57 1.01 0.03 1.06 0.16 0.11 0.11 0.47 0.51 0.5121 0.97 0.57 -0.04 0.51 0.11 0.07 0.08 0.52 0.87 0.91

AIC 8.900 -1.58 26.326 -11.228 -6.634 -3.657 -47.987 -25.694 -61.921R2 0.456 0.350 0.400 0.687 0.674 0.388 0.677 0.415 0.819MAE 1.687 1.067 2.015 0.666 0.771 0.740 0.488 0.840 0.341

22

Table 6. Experimental and predicted xm values for Test Set

RunExperimental

xm (m)

Predicted xm (m)

D'Agostino and

Ferro (2004)

(Eqn. 17)

MNLR

(Eqn.9)

MLR

(Eqn.11)

DNN

(Eqn. 7)

1 6.10 1.40 3.86 7.30 5.64

2 7.93 1.47 4.43 7.04 6.37

3 5.49 1.32 4.12 6.75 6.01

4 4.27 1.15 3.19 8.45 4.47

5 6.71 1.41 4.90 9.62 6.56

6 4.88 1.58 4.29 5.00 4.88

7 7.93 1.40 3.07 4.42 6.75

8 2.44 0.76 1.08 0.77 1.76

9 1.83 0.64 1.04 0.06 1.69

10 4.27 1.19 2.76 2.91 3.70

11 5.49 1.09 2.83 4.58 5.49

12 3.05 0.77 2.04 5.41 3.54

13 3.05 0.57 2.14 5.36 3.59

14 1.83 0.72 0.99 1.42 1.99

15 3.05 0.96 1.29 1.85 3.18

16 1.83 0.66 1.46 2.76 1.99

17 3.66 0.83 1.74 3.18 4.12

18 3.05 1.46 2.05 5.03 3.00

19 4.27 1.31 2.59 4.11 3.60

20 2.44 0.81 1.91 2.60 2.30

21 3.66 0.64 1.23 3.52 2.57

AIC 64.444 35.787 32.748 -9.232

R2 0.588 0.706 0.468 0.907

MAE 0.724 0.387 0.387 0.109

23

Table 7. Large-scale field data and dm predictions

Research z (m) yt (m) q (m2/s) d50 (m) bw=B (m) dm (m)

Predicted dm (m)

Mason and

Arumugam

(1985)

D'Agostino

and Ferro

(2004)

MNLR

(Eqn.9)

DNN

(Eqn.7)

Veronese (1937) 12.9 5 4.57 0.1 25 3 0.54 4.84 2.72 4.99

Scimemi (1939) 57 40 275 2.1 58 28 45.42 23.56 18.86 28.95

Whittaker and Schleiss

(1984)19.2 7 38.97 0.75 11.6 6.2 12.16 4.59 3.53 9.48

24

Figure legends

Figure 1. Sketch of local scour downstream of sharp-crested grade-control structure

Figure 2. Optimal architecture of the proposed NN model

Figure 3. Predicted and experimental dm/z for Train, Cross-Validation and Test sets

Figure 4. Predicted and experimental xm/z for Train, Cross-Validation and Test sets

25

Fig.1

26

Fig.2

27

Fig.3

28

Fig.4

29