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The process involved in the local scour below pipelines is so complex as to make it difficult to establish a general empirical model to provide accurate estimation for scour. This paper describes the use of an adaptive neuro-fuzzy inference system (ANFIS) and a Gamma Test (GT) to estimate the submerged pipeline scour depth. The data sets of laboratory measurements were collected from published literature and used to train the network or evolve the program. The developed networks were validated by using the observations that were not involved in training. The performance of ANFIS was found to be more effective when compared with the results of regression equations and GT Network modelling in predicting the scour depth of pipelines.

Scour is a major cause for the failure of underwater pipeline. Interactions between the pipeline and its erodible bed under strong current and/or wave conditions may cause scouring around the pipelines. This process involves the complexities of both the three-dimensional flow pattern and the sediment movement. Scouring underneath the pipeline may expose a section of the pipe, causing it to become unsupported in the stream. If the free span of the pipe is long enough, the pipe may experience resonant flow-induced oscillations, leading to settlement and subsequently structural failure. Accurate estimate of the scour depth is important in the design of submarine pipelines [

A number of empirical formulas have been developed in the past to estimate equilibrium scour depth below pipeline, including [

Predictive approaches such as artificial neural networks (ANNs) and adaptive neuro-fuzzy inference systems (ANFIS) have been recently shown to yield effective estimates of scour around hydraulic structures. ANNs have been reported to provide reasonably good solutions for hydraulic engineering problems, in cases of highly nonlinear and complex relationship among the input-output pairs in corresponding data.

The objective of this study is to develop a predictive model for scour depth, and in particular 1) to develop an ANFIS model with the aid of Gamma Test, 2) to evaluate the uncertainty inherent by using ANFIS and Gamma Test models for scour depth estimation in clear-water condition and (3) to compare the results obtained the ANFIS model with the empirical methods.

The variables influencing the equilibrium scour depth

where

Author | Equation |
---|---|

Chao and Hennessy (1972) | |

Kjeldsen et al. (1973) | |

Ibrahim and Nalluri (1986) | |

Dutch research group | |

Moncada-M. and Aguirre-Pe(1999) |

flow depth,

The nine independent variables in Equation (1) can be reduced to a set of six non-dimensional parameters. The Buckingham theorem applied to Equation (1), selecting

where

Reynolds number is considered negligible under a fully turbulent flow over a rough bed ([

During the last two decades, researchers have noticed that the use of soft computing techniques as an alternative to conventional statistical methods based on controlled laboratory or field data has provided significantly better results. Neural network (NN) and ANFIS are the most widely used branches of soft computing in hydraulic engineering. Within the larger field of hydraulics, a few researchers have dealt with the scour around and downstream of hydraulic structures using NN [

Therefore, the present study is based on a new soft computing technique, combined neural and fuzzy networks.

The most important characteristics of these methods are the ability to implement human knowledge by tongue labels and fuzzy rules, nonlinearity of these systems and their adaptability [

Parameters | Unit | Data Rang | Mean | Std Dev |
---|---|---|---|---|

a) Range of different input-output parameters used for the estimation of scour depth | ||||

Velocity (u) | cm/s | 21.8 - 73.6 | 44.67 | 12.46 |

Flow depth (Y) | cm | 3.8 - 28 | 13.43 | 6.21 |

Particle mean diameter (d_{50}) | mm | 0.48 - 7.6 | 1.92 | 1.61 |

Diameter of the pier (D) | cm | 2.34 - 7 | 4.37 | 1.44 |

Equilibrium scour depth (d_{s}) | cm | 0.02 - 11.3 | 4.75 | 2.39 |

b) Range of different non-dimensional input-output parameters used for the estimation of scour depth | ||||

dimensionless Shields parameter (τ_{*}) | 0.038 - 0.70 | 0.23 | 0.17 | |

normalized flow depth (Y/D) | 1.06 - 7 | 3.14 | 1.2 | |

pipeline diameter cross section of sediment size (D/d_{50}) | 3.28 - 145.8 | 38.17 | 31.41 | |

Froude number (Fr) | 0.2 - 0.83 | 0.46 | 0.15 | |

Non-dimensional equilibrium scour depth (d_{s}/D) | 0.008 - 1.66 | 1.04 | 0.32 |

Based on the above statements, combining of fuzzy systems, which work on logical rules, with artificial neural networks, which extract knowledge from numerical information, we can develop models that simultaneously use numerical formation and tongue statements to model any phenomenon. This combined method of artificial neural network and fuzzy systems is named the adaptive neuro-fuzzy inference system ([

where A_{1 }, A_{2} and ××× A_{n} are the fuzzy sets. In this system, if the section of the rule is a fuzzy value and the result section of the rule is a real function of the input values and usually is a linear statement such as: [

In order to begin the training an FIS structure is needed first. The FIS structure specifies the parameters of FIS system for learning. Genfis2 function meets these requirements because it generates a Sugeno-type FIS structure using subtractive clustering and requires separate sets of input and output data as input arguments. When there is only one output, Genfis2 may be used to generate an initial FIS for ANFIS training. Genfis2 accomplishes this by extracting a set of rules that models the data behavior. In order to find optimum cluster centers, several cluster radii were examined and the radius of 0.5 for clusters, proved to yield the best results [

Another Fuzzy interface system using for training of the ANFIS model is Genfis3. It generates an FIS using fuzzy c-means (FCM) clustering by extracting a set of rules that models the data behavior. The function requires separate sets of input and output data as input arguments. When there is only one output, you can use genfis3 to generate an initial FIS for ANFIS training. The rule extraction method first uses the FCM function to determine the number of rules and membership functions for the antecedents and consequents. The functional set and op-

erational parameters for dimensional analysis in best combination used in the ANFIS modeling in this study are listed in

In this study, attempts were made to improve the GT to illustrate the best combination of nonlinear model inputs prior to model construction and evaluation for selection of the best input parameter for ANFIS network and to yield a better prediction of scour depth below an underwater pipeline.

How the data are presented for training is one of the most important aspects of ANN and ANFIS models. Often this can be done in more than one way [

A code was developed in MATLAB to perform the analysis. Before applying the ANFIS algorithm, all data must be normalized. In order to normalize the data, they are transformed into the range of [0.05, 0.95]. The following formula is used for the normalization of data:

where X_{n} and X_{r} are the normalized and the original inputs; and X_{min} and X_{max} are the minimum and maximum of input ranges, respectively.

Out of the total of 215 input-output pairs, about 75% (161 sets), are randomly selected, were used for training, whereas the remaining 25% (54 sets) were employed for testing. Five of nine parameters in (1) namely fluid density, the buoyant sediment density, fluid dynamic viscosity, gravitational acceleration and the slope of energy line are constant in all experiments. Therefore, the first combination involves just four of the nine parameters in Equation (1) as the input pattern and the equilibrium scour depth

been used for the ANFIS model.

The selection of model input variables is a complex issue, even for linear multivariable regression analysis and nonlinear models such as ANN. Choosing the optimum inputs to arrive at good predictions is important in nonlinear modeling. This paper demonstrates a new model “Gamma Test” to identify the best combination and number of input data to make a prediction with the best possible accuracy. The Gamma Test is a non-linear analysis tool which allows quantification of the extent to which a smooth relationship exists within a numerical input/output

Rule Parameters | ANFIS | |
---|---|---|

Training of the ANFIS model using Genfis2 (subtractive clustering) | Training of the ANFIS model using Genfis3 (fuzzy c-means clustering) | |

Rule 1 Rule 2 Rule 3 Rule 4 Rule 5 Rule 6 Rule 7 Rule 8 Rule 9 | [0.1255 −0.6761 0.5241 −0.7729 0.4106] [0.2284 −0.3918 0.9777 0.173 0.1064] [1.13 0.1192 0.4859 −1.985 0.1239] [−0.3304 0.6991 −0.1687 1.279 0.09621] [0.6013 0.9184 0.3647 −0.2692 −0.4788] [1.124 0.2302 0.256 −0.2427 −0.01926] [−0.8196 0.3539 0.8058 0.332 0.4059] [2.059 0.257 −0.05244 −3.504 0.3668] [−0.1408 −1.259 0.7326 −121.7 116.1] | [0.07086 0.3199 0 0] [0.06754 0.4445 0 0] [0.06815 0.353 0 0] [0.08313 0.5967 0 0] [0.09327 0.2105 0 0] [0.06589 0.4481 0 0] [0.1432 0.7998 0 0] [0.08337 0.5635 0 0] [0.07432 0.2955 0 0] |

data set. The Gamma Test was first reported by [

The Gamma Test estimates the minimum mean square error (MSE) that can be achieved when modeling the unseen data with any continuous nonlinear models. The basic idea is quite distinct from the earlier attempts with nonlinear analysis. Suppose we have a set of data observations of the form

in which input vectors are confined to some closed bounded set

The domain of a possible model is now restricted to the class of smooth functions which have bounded first partial derivatives. The Gamma statistic is an estimate of the model’s output variance that cannot be accounted for by a smooth data model.

The Gamma Test can estimate Var. (r) directly from the data, even though function f is unknown. This estimate is calculated by computing the following equations called a delta function:

where |…| denotes Euclidean distance, ^{th} nearest neighbor to^{th} nearest neighbor and

It can be shown that

Calculating the regression line gradient can also provide helpful information on the complexity of the system under investigation. First, it is remarkable that the vertical intercept Γ of the y- (or Gamma) axis offers an estimate of the best MSE achievable, utilizing a modeling technique for unknown smooth functions of continuous variables [

The Gamma Test is a non-parametric method and the results apply regardless of the particular techniques used to subsequently build a model of f . We can standardize the result by considering another term V_{ratio}, which returns a scale invariant noise estimate between zero and one. V_{ratio} is defined as:

where, _{ratio} close to zero indicates that there is a high degree of predictability of the given output y. The reliability of Γ statistic can be determined by running a series of Gamma Tests for a definite number of unique data points (M), to establish the size of data set required to produce a stable asymptote. This is known as an M-test. The M-test also helps us to decide how much data we are likely to need to obtain a model of a given quality, in the sense of predicting with mean square error around the noise level. The M-test result would avoid the over fitting of a model beyond the stage where the MSE on the training data is smaller than Var(r) and help us to decide the required data length to build a meaningful model. For model identification is used from full embedding section to determine which combination yields the smallest absolute Gamma value. The Gamma Test analysis can be performed using winGamma^{TM} software implementation [

In order to measure and compare the uncertainty related to the results of ANFIS models, there needs to compare some objective criteria. In this study we used coefficient of determination (R^{2}), root mean squared error (RMSE) and mean absolute error (MAE):

where

Gamma Test is used to measure uncertainty by Gamma value, gradient and V_{ratio}_{ }.This paper demonstrates all combinations of input data that affect the pipeline scour depth by using full embedding. A full embedding tries every combination of inputs to determine which combination yields the smallest absolute Gamma value. It returns the number of results requested. If there are m scalar inputs then there are

Parameter type | Model | RMSE | MAE | R^{2} | ANFIS Rule |
---|---|---|---|---|---|

dimensional | d_{s} = f(u, Y, D, d_{50}) | 0.0742 | 0.39 | 0.96 | 9 |

d_{s} = f(u, Y, D) | 0.1013 | 0.473 | 0.91 | 6 | |

d_{s} = f(Y, D, d_{50}) | 0.0803 | 0.394 | 0.95 | 7 | |

d_{s} = f(u, D, d_{50}) | 0.116 | 0.5124 | 0.89 | 4 | |

Non-dimensional | d_{s}/D= f(τ_{*}, Y/D, D/d_{50} , Fr) | 0.0137 | 0.0711 | 0.83 | 18 |

d_{s}/D= f(τ_{*}, D/d_{50}, Fr) | 0.0212 | 0.1128 | 0.72 | 8 | |

d_{s}/D= f(τ_{*}, Y/D, D/d_{50}) | 0.0159 | 0.790 | 0.82 | 14 | |

d_{s}/D= f(τ_{*}, Y/D, Fr) | 0.0179 | 0.0903 | 0.77 | 19 |

model complexity. V_{ratio} is the measure of predictability of given outputs using available inputs. An input data set with low values of MSE, gradient and V_{ratio}_{ }is considered as the best scenario for the modeling. Four representative combinations for each data set (include the best one) are tabulated in _{50} and τ_{*} > D/d_{50} > Fr > Y/D for first and second combinations, respectively.

The quantity of sufficient input data to predict the desirable output was analyzed using the Gamma Test.

In this study, different combinations of input data (non-dimensional data sets) were explored to assess their influence on the scour depth modeling (_{50}); and the Froude number. Each parameter (except energy slope) in Equations (1) and (2) was considered in turn in the ANFIS for the sensitivity analysis. The results show that, of the parameters in Equation (1), the mean particle size (d_{50}) has the most significant effect on the scour depth and the flow discharge has the least effect on it.

Similarly, for non-dimensional parameters in Equation (2), sensitivity analysis shows that dimensionless Shields parameter (τ_{*}) and Y/D have respectively the most and the least effect on normalized scour depth. The quantity of input data required to predict the desirable output was analyzed using the M-Test with various data lengths for two combinations. This shows that a training data length of 173 and 158 is sufficient for the Gamma statistics to become stable and low respectively for dimensional (original data) and non-dimensional combinations. Statistical results of different combination are in

Parameter combination | Training Model | R2 | RMSE | MAE | |||
---|---|---|---|---|---|---|---|

Training | Validation | Training | Validation | Training | Validation | ||

Dimensional | subtractive Clustring | 0.95 | 0.96 | 0.0365 | 0.0742 | 0.38 | 0.39 |

Fuzzy C means | 0.63 | 0.81 | 0.1139 | 0.1567 | 0.89 | 0.72 | |

Non-dimensional | subtractive Clustring | 0.87 | 0.83 | 0.008 | 0.013 | 0.083 | 0.071 |

Fuzzy C means | 0.70 | 0.53 | 0.0136 | 0.0318 | 0.123 | 0.176 |

Parameter | Dimensional | Non-dimensional | ||||||
---|---|---|---|---|---|---|---|---|

u, Y, D, d_{50} | U, Y, D | Y, D, d_{50} | u, D, d_{50} | τ_{*}, Y/D, D/d_{50}, Fr | τ_{*}, D/d_{50}, Fr | τ_{*}, Y/D, D/d_{50} | τ_{*}, Y/D, Fr | |

Gamma (Γ) Gradient (A) Standard error V-ratio Near neighbors M Mask | 0.013 0.124 0.0027 0.053 10 215 1111 | 0.011 0.215 0.0027 0.043 10 215 1110 | 0.014 0.227 0.0035 0.057 10 215 0111 | 0.039 0.147 0.0133 0.158 10 215 1011 | 0.047 0.54 0.0089 0.191 10 215 1111 | 0.057 1.105 0.0099 0.231 10 215 1011 | 0.063 0.56 0.014 0.254 10 215 1110 | 0.076 0.78 0.016 0.304 10 215 1101 |

combination (original data) has better ability to predict scour depth. The result for the original data show a high coefficient of determination (R^{2}), also the RMSE in the second combination is better than the first combination in both training and validation periods but this variation is low compare with R^{2} variation. This study is useful for applications of pipeline scour for field conditions because the ANFIS model was developed with wide range of data, which could be deemed as the closest to field conditions, particularly helping to identify parameters that most likely define scour processes and explain scour variability, and ANFIS model is shown to agree well with actual measurements.

The application of a relatively new soft computing approach of ANFIS to predict the local pipeline scour depth was described. An ANFIS and GT model was developed to predict the values of the relative scour depth from laboratory measurements. A new approach was presented to estimate the equilibrium depth scour below underwater pipelines from optimum data sets with the ANFIS and GT modeling techniques. The application of the ANFIS in this study is another important contribution to scour depth estimation methodologies for pipes. The present study indicates that employing the original data set yielded a network that can predict measured pipeline scour depth more accurately than standard regression analysis based formulas [