Uncertainty Analysis of Flow Resistance Modeling In Sewer Pipes With Different Bed Conditions Using KELM Method


 An accurate prediction of flow resistance in channels or pipes which is used for carrying rain water or sewage is an important issue in hydraulic engineering. In this study, the capability of two artificial intelligence methods namely; Feed Forward Neural Network (FFNN) and Kernel Extreme Learning Machine (KELM) was utilized for the Manning roughness coefficient modeling in circular sewer pipes with smooth and rough beds. During model developing process, at first the most effective variables related to hydraulic and sediment properties were selected using Factorial Analysis (FA) method, then, various models were developed based on the selected parameters. The models were tested using some available experimental datasets. The obtained results showed that there resulted high efficiency with FFNN and KELM methods in Manning coefficient modeling. It was found that both hydraulic and sediment properties were effective in modeling process. The sensitivity analysis results showed that d50/R variable (the ratio of median diameter of sediment size to the hydraulic radius) was the most significant parameter in modeling process. Finally, an uncertainty analysis was conducted to assess the dependability of the best applied ‎model, and the results revealed that the KELM model possesses an acceptable level of ‎uncertainty in roughness coefficient modeling.


Introduction
Precise prediction of roughness coe cient is of utmost importance in circular channels such as sewer pipes, which are used for carrying rain water and sewage. Sewer system is one of the essential hydraulic structures. The presence of sediment always in uences the movement of waste water in sewer pipe and also reduces the capacity of the pipes with its deposition. Reduced ability of sediment movement in sewer pipes sometime causes surcharging, blockage, and over ow of sewage water. Therefore, exact information of roughness coe cient is necessary for the design and proper functioning of sewer system. In sewer system exact information of roughness coe cient of pipes depends on several factors like bed forms and bed material etc. In the past few decades several researchers investigated the phenomena of ow resistance in open and closed channels [Meyer- Peter and Mueller (1948); Einstein and Barbarossa (1952); Ackers (1961); Richardson and Simons (1967); Henderson (1984); Ashley et al. (1992); Yang et al. (2005); and Van der Mark et al. (2008)] and a large number of conventional models has been proposed for the ow resistance in channels but these models were very complex. May (1989) investigated the limit of sediment deposition in pipe and this investigations were carried out on 77 and 158 mm diameter smooth pipes and ow conditions, which were full or partially full. Ackers et al. (1964) found presence of permanent deposition of sediments in utilized sewer pipes. Nalluri and Ota (2000) developed a model for estimation of sediment transport capacity at the highest limit of deposition. But these models do not provide satisfactory results with the larger channels. Thus, Ota and Nalluri (2003) made an extension of their previous work and developed a model for the estimation of sediment transport in large diameter channels. Henderson (1984) has investigated the performance of sewer system and suggested the sewer roughness has a vital role in the performance of sewer system capacity. Sewer hydraulic roughness is affected by various factors: sewer material, connections of pipes, sliming and existence of sediment deposits. Novak and Nalluri (1972) investigated the channel roughness. Ojos (1978) extended the work of Novak and Nalluri (1972) by using 305 mm broad rectangular channel for measuring the effects of particle groupings, spacing and roughness of the channels. Kleijwegt (1992) carried out experiments on 152 mm diameter pipes running fully and partial fully. Also, May (2003) carried out experiments using horizontal pipe for studying sediment transport capacity and roughness coe cient of pipes and proposed a new design method. However, the outcomes of conventional models are not generalised due to the roughness coe cient complexity and uncertainty. Therefore, it is necessary to adopt or develop new methods for the accurate estimation of roughness coe cient in pipes considering different hydraulic or bed conditions.  (Ebtehaj et al., 2018) are some of the examples. In general, the task of a machine learning algorithm can be described as follows: (1) with given a set of input variables and the associated output variable(s), (2) the objective is learning a functional relationship for the input-output variables set.
Due to the complexity of the ow resistance process, in the past few decades, accurate prediction of this parameter remains one of the most important tasks for scienti c planning and management of water resources system. Due to the nonlinearity of the roughness coe cient in sewer pipes, the existing regression methods often don't show desired accuracy. Consequently, the application of many of these methods is limited to special cases of their development. This issue causes an uncertainty in the estimation of the roughness coe cient in channel pipes. With the advantages of easy implementation and high exibility, arti cial intelligence methods have been widely employed to address the complex hydraulic and hydrological prediction problems. Among arti cial models, kernel based approaches such as KELM are relatively new important methods based on the different kernels type which are based on statistical learning theory initiated. Such models are capable of adapting itself to predict any variable of interest via su cient inputs. The aim of this study was to assess the most signi cant variables in modeling the roughness coe cient of smooth and rough bed pipes using nonlinear approaches. Therefore, two AI models (i.e. FFNN, KELM) were applied to predict roughness coe cient in channels with smooth and rough beds and also investigate the best input models and effective parameters for each state. In this regard, the most important ow and sediment parameters were selected using Factorial Analysis (FA) method. FA method can be useful where there is a nonlinear relationship between inputs and output variables. In fact, it tries to recognize underlying factors, which explain the correlations pattern within a set of observed variables. Also, based on Tezcan et al. (2015), FA can be used for data reduction to select a small number of factors which explain most of the variance that is observed in a much larger number of manifest variables. Also, uncertainty analysis was performed to investigate the applied models dependability.

Materials And Methods
The datasets In this study, to evaluate the roughness coe cient in circular channels, the experimental datasets of Ghani (1993) and May et al. (1989) were used. Ghani (1993) studied sediment transport and ow resistance in smooth and rough beds under part-full ow conditions. Two hundred and fty four experiments on sediment transport of non-cohesive sediments in the non-deposition state were done in sewer pipes with diameters (D) of 154, 305, and 450 mm and a length of 20.5 m. All the three sizes were used for smooth rigid beds and the 305 mm pipe also was used for a rough rigid bed.
For transport experiments with a rough rigid boundary, two sizes of sand (d 50 = 0.5 and 1.0 mm), where d 50 is the median particle size were used to roughen the pipe arti cially. The rigid boundary tests were done with a proportional ow depth of 0.15 < y 0 /D < 0.80, where y 0 is the depth of uniform ow. May et al. (1989) carried out thirty eight tests which were under part-full ow conditions on a pipe with diameter of 300 mm and length of 20 m. The non-cohesive sediments with median diameter of 0.72 mm, ow velocity in the range of 0.082 < V < 1.5 m/s, and speci c gravity of 2.62 were used during the experiments. Table 1 shows the ranges of some parameters used in these experiments. In this table S 0 , y/D, d 50 , D, V, Cv, k 0 , and Re are slope of pipe, proportional ow depth, particles median diameter, pipes diameter, ow velocity, sediment concentration, clear-water equivalent sand roughness, and ow Reynolds number, respectively. Two AI techniques including FFNN and Kernel extreme KELM were used to predict roughness coe cient in circular channels with smooth and rigid beds, whose brief description are provided below.
-Feed Forward Neural Network (FFNN) The aim of ANN as a Meta model approach is to achieve a nonlinear relationship between inputs and output data series (Naja et al., 2018). ANN is based on a collection of connected nodes called neurons, which are linked to certain of their neighbors with varying connective coe cients which show the connections strengths. FFNN is the most common algorithm of ANN, with having three layers of input, hidden and target. In this method the transfer of information is done in uni-direction: from the input nodes to the hidden and output nodes. Figure 1, shows a three-layer FFNN model.
The hidden neuron sums up the input connections weights. For selecting the information which should be moved to the next neuron, in the hidden layer the weighted summation should be passed through an activation function (Tayfur, 2012).

-Kernel extreme Learning Machine (KELM)
Among data driven techniques, Kernel-based methods such as Kernel Extreme Learning Machine (KELM) are considered as relatively innovative and signi cant techniques in terms of various kernels types and the statistical learning theory. These models can adapt themselves for predicting any parameter of interest by adequate inputs. Furthermore, they can model non-linear decision boundaries, and numerous kernels exist in this regard.
These methods are also objectively strong against over tting, particularly in high-dimensional spaces. Nevertheless, proper selection of the kernel kind is the most essential step in the KELM method because of its direct effect on classi cation precision and training. There are various kernels functions such as linear, radial basis, and polynomial kernels. Where, and ND are the observed, estimated, mean observed, mean estimated values, and number of experiments, respectively. Also, all input variables were scaled between 0-0.8 in order to eliminate the input and output variables dimensions. where D gr is dimensionless particle number and F rm is Modi ed Froude number. The ow resistance in sediment transport sever pipes can be expressed as a function of different sets of input variables. In this study for selecting the most effective variables in modeling the roughness coe cient in circular channels the FA was used. Factorial Analysis is originated from experimental design to identify the interaction effects of several factors on a response variable (Tezcan et al., 2015). The results of FA are listed in Table 2 and Fig. 2(a). In this table variables Fr and A are ow Froude number and ow cross sectional area, respectively. Based on the results listed in Table 2, the most effective parameters were selected and several models were developed via combining these parameters as inputs for AI methods. Table 3 shows the considered models in assessing the roughness coe cient in circular channels with different bed conditions. According to the Fig. 2(b), two states were considered in models preparing. In the state 1, only hydraulic properties were used for Manning roughness coe cient modeling and in the state 2, the combination of both hydraulic and sediment properties were used as inputs. In this study, the Manning coe cient (n) was selected as output parameter. In the next steps, the most essential parameters in the prediction procedure were determined by utilizing sensitivity analysis, followed by employing Monte Carlo uncertainty analysis (UA) to evaluate the dependability of the applied models. should be determined. For kernel based approaches designing, the appropriate kind of kernel function selection should be done. In this research, the roughness coe cient in smooth bed channel was predicted using different kernel types. In this regard, the model HS(VI) for the smooth bed was run via KELM and according to Fig. 3(a), the RBF kernel function [ in which γ is kernel parameter] was ned as the best kernel function. Figure 3 shows the RMSE via γ values for assessing the impact of RBF kernel parameter of γ on employed algorithm performance for testing set of model HS(VI) in the smooth bed channel. In this study, optimization of γ was performed by a systematic grid search of the parameter using crossvalidation. On the other hand, in ANN modeling the network topology has direct effects on its computational complexity and generalization capability; therefore, the appropriate structure of ANN should be selected. Various networks were tested to determine the hidden layer node numbers.
Different numbers of neurons (i.e. 2, 3, 5, and 7) in hidden layer were tested. Also, it was found that the tangent sigmoid and pure linear functions are suitable for the hidden and output node activation functions, respectively.

Results And Discussion
The results obtained for the state 1 (the use of only hydraulic parameters) For assessing the impact of hydraulic parameters on roughness modeling, two models were de ned using y/D and Re parameters. These two parameters were selected based on FA results listed in Table 2. Modeling based on hydraulic parameters can be useful in the cases where only data related to ow characteristics are available. Table 4 and Fig. 4 show the results of FFNN and KELM models. According to the results, it could be seen that when only hydraulic parameters were used for roughness coe cient modeling, the predicted and observed values led to poor agreement. For the state 1, the model H(II) with input combination of Re and y/D yielded in more accurate outcome and it seems that adding y/D variable to inputs increased the models e ciency. As it could be seen from Table 4, the applied models represented more desirable results compared with rough channel. Also, the KELM model was slightly accurate than the FFNN model. However, according to the results, it could be induced that the models containing only hydraulic characteristics were not so accurate. The scatter plots of the KELM-best model for smooth and rough channels are shown in Fig. 4. In this section for roughness coe cient predicting in circular channels with different bed condition, both hydraulic and sediment parameters were used for preparing the models. The results obtained for this state are presented in Table 5 and Fig. 5. Based on the statistical performance criteria, it could be seen that for both smooth and rough bed channels the model HS(VI) with input parameters Cv, F rm , D gr , d 50 /R, D 2 /A showed the most accurate results. It could be inferred that the use of d 50 /R and D 2 /A variables increased the models accuracy. By comparing the results of the states 1 and 2, it could be indicated that in roughness coe cient modeling, the use of both hydraulic and sediment parameters performed more successfully than when only hydraulic parameters were used. In state 2, the rst three models were developed without considering Cv as input and the last three models were developed considering Cv as input. From the results of these two sets, it seems that in modeling the roughness coe cient, using parameter Cv increased the models e ciency. Figure 5(a) displays the scatter plots of the KLEM-best model for two datasets. Also, for evaluating the impact of each independent parameter sensitivity analysis was performed for the model HS(VI). Therefore, in the HS(VI) model the input variables were omitted one by one and the KLEM model was rerun. The sensitivity analysis results for test series are presented in Fig. 5(c). The RMSE error criterion was used for showing the impact of each variable. It was found that variable d 50 /R in both circular channels was the most signi cant parameter. In this section, the performance of applied methods was assessed for a wider range of data. In this regard, applied datasets were mixed; then, two developed models in the state 2 were selected and reanalyzed for the new datasets. The obtained results are listed in Fig. 6. From the results, it was found that the model HS(VI) led to better prediction than the model HS(V) and using parameter D 2 /A caused an improvement in models' e ciency. However, comparison between the results of Tables 4 and 5 and Fig. 6 showed that FFNN and KELM models for this state lead to undesired accuracy and separate datasets yielded better predictions. However it should be noted that the mixed datasets results is capable to cover wider range of data and in this case roughness coe cient can be studied without regarding the pipe bed condition (i.e. smooth or rough states).

The results of uncertainty analysis
The uncertainty of the best KELM model was determined by the uncertainty analysis (UA). In the current study, the Monte Carlo UA method was utilized as well. In the UA method, two elements are applied to measure the robustness and evaluate model uncertainty. The rst one is the percentage of the investigated outputs within the range of 95PPU, and the next one represents the average distance between the lower (X L ) and upper (X U ) uncertainty bands (Noori et al., 2015). Accordingly, the intended model needs to be run several times (1000 times in this study), and the experimental cumulative distribution probability of the models should be determined as well. The lower and upper bands are regarded as the probabilities of 97.5 and 2.5% for the cumulative distribution, respectively. Two important indices should be taken into account at the appropriate con dence level. First, the 95PPU band brackets most observations. In addition, the average distance between the upper and lower parts of the 95PPU (d-Factor) should be smaller than the standard deviation of the observed data (Abbaspour et al., 2007). The indicated indices were used for estimating input uncertainties. According to Abbaspour et al. (2007), the average width of the con dence interval band can be computed by Eq. (2): where 95PPU, k, and X reg represent the predicted uncertainty of 95%, the observed data number, and the currently registered data, respectively. Figure   7 presents the obtained results for the UA. According to the d-Factor values and 95% PPU, for both smooth and rough bed channels the predicted and observed values were within the 95% PPU band in most cases. Additionally, the ndings revealed that the rate of d-Factors for training and testing datasets was lower compared to the standard deviation of the observed data. Therefore, based on the results, it could be induced that the roughness coe cient modeling via KELM model resulted in the allowable uncertainty level.

Conclusions
An accurate prediction of the roughness coe cient in channels with different bed condition is an important issue due to its impact on such structures performances. This study aimed to assess the e ciency of the FFNN and KELM methods in roughness coe cient modeling in smooth and rough bed circular channels. In the model developing process, rst of all, the most effective variables were determined via FA, then, considering the various combinations of hydraulic and sediment parameters different models were developed. The results indicated that the models which took advantages of both hydraulic and sediment characteristics performed much better than developed models based on only hydraulic variables. The results showed that the model HS(VI) with Cv, F rm , D gr , d 50 /R, D 2 /A variables had superior prediction ability in roughness coe cient modeling. Also, the use of y/D, Cv and d 50 /R variables increased the models accuracy. It was observed that the obtained RMSE values for smooth bed channel were less than those obtained for rough bed channel. Also, the KELM method was more accurate than FFNN method. The results of sensitivity analysis indicated that the impact of variable d 50 /R on obtaining a model with higher accuracy was more than other used parameters. Moreover, the dependability of the superior applied models was assessed through UA, and it was revealed that the KELM model possessed an acceptable uncertainty degree in the Manning roughness coe cient modeling in channels with different bed condition. Funding The authors did not receive support from any organization for the submitted work.

Declarations
Availability of Data and Material Some or all data, models, or code that support the ndings of this study are available from the corresponding author upon reasonable request.

Ethics Approval
Not applicable.

Consent to Participate
Not applicable.

Consent for Publication
Not applicable.

Con ict of Interest
The authors declare no con icts of interest/competing interests. 32. Yu X., Liong S.Y., Babovic V. 2004 EC-SVM approach for real-time hydrologic forecasting. Journal of Hydroinformatics, 6(3), 209-223. 33. Zhu S., Luo X., Xu Z. and Ye L. 2018 Seasonal stream ow forecasts using mixture-kernel GPR and advanced methods of input variable selection. Hydrology Research, 50(1), 200-214. Figure 1 Typical three layered FFNNs with a backpropagation training algorithm.       Uncertainty analysis for the best model of the KELM method.