Uncertainty assessment of kernel based approaches on scour depth modeling in downstream of ski-jump bucket spillways Open Access

Control structures are provided for dams to release flood water in excess of reservoir capacity. Ski-jump bucket spillways are one of the most commonly used structures in this regard. In ski-jumps, the whole jet of flow is thrown into the air using a bucket. Part of the energy in the jet is dissipated in the air because of friction and some other part is dissipated at the point of impact with the riverbed downstream as a result of excavating a large scour hole (Dargahi 2003). The scouring continues up to the point at which the rate of bed erosion is balanced by the rate of deposition of material brought back into the scour hole by the return flow (Chang et al. 2004). In the past decades, accurately prediction of scour dimensions has been of much interest among many investigators to prevent scour damages. The depth of scour is governed by various parameters, such as discharge intensity, height of fall, bucket radius, bucket lip angle, type and size of rock, degree of rock homogeneity, time, and mode of operation of spillway. In order to estimate the scour depth downstream ski-jump buckets, several empirical formulas have been developed (Damle et al. 1966; Martins 1975; Hoffmans 1998; Lopardo et al. 2002; Dargahi 2003).

The outcomes of conventional models are not general due to the scour depth complexity and uncertainty. Therefore, it is necessary to adopt or develop new methods for the accurate estimation of scour depth downstream of ski-jump bucket spillways. Over the past few decades several artificial intelligence (AI) methods [e.g., Artificial Neural Networks (ANNs), Neuro-Fuzzy models (NF), Genetic Programming (GP), Gene Expression Programming (GEP), Support Vector Machine (SVM), and Kernel Extreme Learning Machine (KELM)] have been developed and applied for assessing the complex hydraulic and hydrologic phenomena efficiency. Daily dewpoint temperature prediction (Al-Shammari et al. 2016), relative energy dissipation prediction (Saghebian 2019), longitudinal dispersion coefficients computing in natural streams (Azamathulla & Wu 2011), side weir discharge coefficient modeling (Azamathulla et al. 2017), monthly streamflow modeling (Zhu et al. 2018; Pandhiani et al. 2020), roughness coefficient modeling in sewer pipes (Roushangar et al. 2020), and forecasting long-term evapotranspiration rates (Ashrafzadeh et al. 2020) are some examples. In this study, the capability of two kernel based (KB) models (i.e. SVM, KELM) was investigated for scour depth assessing downstream of ski-jump bucket spillways. In this regard, two states were considered and, using dimensional parameters (sate 1) and non-dimensional parameters (state 2), several models were developed and tested. The capability of applied methods was compared with some available semi-empirical equations. In addition, the most important parameters were determined using sensitivity analysis. Also, Monte Carlo uncertainty analysis was applied to investigate the dependability of the applied models. In addition, the capability of SVM and KELM approaches was compared with Hybrid Multilayer Perceptron Firefly Algorithm (MLP-FFA) as new artificial intelligence approach.

For determining the accuracy of the developed models laboratory investigations carried out at the Central Water and Power Research Station (CWPRS), India were used. Several experiments were performed considering different amounts for discharges and reservoir levels. The spillway gates were fully and partially open. The standing wave flume was used for hydraulic model discharge measurement. The sectional models were scaled to the range of 1:40–1:60, whereas comprehensive models had their scales varying from 1:50 to 1:100. A look at these observations revealed that additional measurements were necessary to make them more comprehensive; especially with respect to pattern of scour including width and distance of maximum scour depth from the spillway bucket lip (length). New hydraulic model studies were therefore conducted on three different bucket designs. The three hydraulic models simulated the dams across rivers Subarnarekha, Ranganadi, and Parbati Rivers in India.

The first dam was 52 m high and 720 m long. Its spillway consisted of 13 spans of 15 m wide each with crest at elevation 177 m. Radial gates of size 15 m16 m regulated the flow over this spillway. The design outflow flood was 26,150 m³/s. This corresponded to a maximum water level at an elevation of 192.37 m. The ski-jump bucket with bucket radius of 25 m and lip angle of 32.5° was provided at the toe for energy dissipation. The second (Ranganadi River) dam was 60 m high, made up of concrete with a rockfill portion on its right side. It had an overflow spillway with seven spans of 10 m width and 12 m height. The spillway catered to a maximum outflow flood of 12,500 m³/s. This corresponded to the maximum water level of 568.3 m and the full reservoir level of 567 m with the crest level of the spillway at 544 m. The ski-jump bucket modeled by a 1:60 scale model served as an energy dissipator at the toe of the spillway. It had a bucket radius of 18 m with 35° as the lip angle. The dam corresponding to the third spillway was 85 m high. It was designed to pass a maximum discharge of 1,850 m³/s at the full reservoir level of 2,198 m elevation. It had three spans, 6 m wide and 9 m high, separated by 6 m thick piers, and fitted with radial gates. An apron and a plunge pool along the downstream side fronted the bucket, which had a bucket radius of 28 m with the lip angle of 30°. This model based on Froude's law had a scale of 1:50. The downstream bed was made up of 2 mm diameter cohesionless sand particles. The riverbanks in this portion were assumed to be nonerodible and rigid. The various depths such as tail water depth, head over crest and other parameters were measured by using a point gauge having a graduation of 0.1 mm. The depth of scour was observed in a free formed plunge pool which was subsequently filled with sand having d50 size of 2 mm. Observations were made with four discharge passes (25, 50, 75, 100% of the maximum discharge) each with fully open as well as partially open gates. Experience showed that the equilibrium scour depth would be reached within this period, although the evolution of progressive scour depth is a function of time. In the end, 95 input-output pairs were compiled. Table 1 shows the range of data used in these experiments. The parameters mentioned in this table are Ys: scour depth, H₁: total head, Ls: scour length, q: unit discharge of spillway, φ: bucket lip angle, Yt: tail water depth, R: bucket radius, and d₅₀ mean sediment size. Also, Ranganadi dam and the schematic view of spillway and scour hole notations are shown in Figure 1.

Kernel based approaches are new methods which are used for classification and regression purposes. Kernel based approaches are based on statistical learning theory initiated and can be used for modeling the complex and non-linear phenomenon. Two important KB approaches are KELM and SVM which work based on different kernal types such as linear, polynomial, radial basis function (RBF) and sigmoid functions in SVM and linear, polynomial, and RBF in KELM (Saghebian et al. 2020).

Support Vector Machines as structural risk minimization (SRM) methods minimize an upper boundary on the expected risk (Vapnik 1995). According to Ji et al. (2017), this approach is applied for information categorization and dataset classification and regression (see Figure 2). The SVMs are based on the concept of the optimal hyper plane that separates samples of two classes by considering the widest gap between two classes. Support Vector Regression (SVR) is an extension of SVM regression. In the use of SVMs for regression aims, we tried to obtain a function with the most deviation from the actual target vectors for all given training data. For the non-linear SVR the kernel function concept is used (see Vapnik 1995 for more details).

In Equation (14), the attraction effect is shown using the first term, and randomization is shown using the second term. χ is the randomization coefficient. The χ value varies from 0 to 1 (for the current study χ = 0.5). ɛ_i is the random number vector. This parameter is obtained from a Gaussian distribution (ɛ_i is 0.96 in this study).

The aim of a model uncertainty analysis is to determine the statistical characteristics of the outputs of that model as a function of the uncertainty of the input parameters (Noori et al. 2015). Uncertainty is a factor associated with the estimation result which determines the estimation values range. Its value indicates the level of confidence in which the actual measured value falls, within the specified range (Noori et al. 2015). In the current research, the Monte Carlo method proposed by Abbaspour et al. (2007) was used to evaluate the uncertainty of the AI models in SPEIs series modeling. To verify model results uncertainties, 95% confidence interval (95PPU) and bandwidth factor (d-factor) which is the average distance between the upper (XU) and lower (XL) uncertainty bands should be used (Noori et al. 2015). In this regard, the considered model should be developed many times (1000 in this research), and the empirical cumulative distribution probability of the models should be calculated.

Each artificial intelligence method has its own settings and parameters and, to achieve the desired results, the optimized amount of these parameters should be determined. In SVM and KELM, designing the selection of appropriate type of kernel function is important. In this section of the paper, SVM and KELM methods were evaluated using model D(I) in order to select the best kernel functions of each model. Figure 4 indicates the percentage error of statistical RMSE parameter for model D(I) with different kernel functions. From the results, it was found that, among all kernel functions, the RBF yielded more accurate outcomes. Therefore, RBF was selected as a core tool of SVM and KELM methods and used in the rest of the models.

In the first state, several dimensional parameters were used for model development. These models were tested via SVM and KELM methods. The obtained results are shown in Table 3 and Figure 5. According to the performance criteria results, it could be deduced that the model D(I), with a combination of q, H₁, R, φ, Y_t, and d₅₀ as inputs, performed more successfully. Considering the four developed models results, it was found that using the combination of Y_t and q parameters, the models efficiency increased. Also, it was observed that the effect of the q variable on increasing the accuracy of the models was more than the H₁ variable. Based on Table 3 results, the efficiency of the KELM model in predicting the downstream scour depth of ski-jump bucket spillways was higher than for the SVM approach.

In the second state, several non-dimensional parameters were applied for model development. The results obtained for this state are indicated in Table 4 and Figure 6. The results showed that the model N(IV) with a combination of parameters had the superior performance. However, the model N(V) with parameters q²/[gY_t³], d₅₀/Y_t, and φ showed approximately the same results, therefore, this model with only three inputs could be selected as the optimum model. Based on the results, it could be seen that the use of q²/[gY_t³], d₅₀/Y_t, R/Y_t, and φ parameters as inputs increased the applied method capabilities in the scour depth modeling.

In this section, the most effective variables in predicting the scours depth of ski-jump bucket spillways was assessed using sensitivity analysis. In this regard, for each state, the superior models of KELM method were selected and re-run by removing each input variable. Table 5 and Figure 7 show the sensitivity analysis results. Based on Table 5, it could be induced that q in the state 1 and q²/[gY_t³] in the state 2 were the most important variables in estimating the scour depth downstream of ski-jump bucket spillways.

The experimental data were used to evaluate the performance of proposed best applied models compared with other data driven models. In this regard, for each state (i.e. modeling base on dimensional and non-dimensional parameters) the superior model was run using the MLP-FFA model and the results were compared with SVM and KELM. Table 6 shows the results of this comparison. As seen from Table 5, the MLP-FFA model led to desired accuracy and the efficiency of this model was more than the SVM. However, the KELM model slightly yielded better results in comparison with SVM and MLP-FFA models.

The capability of several existing scour depth equations was evaluated and the results were compared with KB models. The RMSE criterion was used as an indication of the accuracy of the equations. The results are shown on Figure 8. Based on the results, among all used formulas, Chee & Kung's (1974) equation led to reasonable accuracy. The correlation between models and observed values was rather good for the Y_s smaller values. However, Figure 8 showed that in general, the applied formulas were not so successful in scour depth modeling, while the SVM and KELM results were close to the observed data. These models had the lowest RMSE and this proved that the KB methods are efficient approaches in scour depth modeling of ski-jump bucket spillways.

In this part of the study the uncertainty analysis was done in order to find the uncertainty of the superior KELM model. In the proper confidence level, two important indices should be considered. First: the 95PPU band brackets most of the observations, and second: the d-Factor is smaller than the standard deviation of the observed data. The two mentioned indices were applied to account for input uncertainties. The obtained results for the uncertainty analysis are shown in Figure 9. Based on the values obtained for the d-Factor and 95%PPU, it could be indicated that for both considered states the observed and predicted values were within the 95PPU band in most of the cases. Also, it was found that the amount of d-Factors for train and test datasets was smaller than the standard deviation of the observed data. Therefore, it could be induced that scour depth modeling downstream of the ski-jump bucket spillway via KELM model led to an allowable degree of uncertainty.

Scour depth accurate prediction downstream of spillways is an important issue due to its impact on such structures performances. In this study, the efficiency of two KB methods was investigated in scour depth modeling of ski-jump bucket spillways. In the model developement process, based on dimensional (state 1) and non-dimensional (state 2) parameters, two states were tested. The results showed that in the state 1 model with input parameters q, H₁, R, φ, Y_t, and d₅₀ and in the state 2 model with input parameters of q²/[gY_t³], d₅₀/Y_t, R/Y_t, H₁/Y_t, and φ performed more successful. It was observed that using Y_t, q, q²/[gY_t³], d₅₀/ Y_t³, R/ Y_t, and φ as inputs enhanced the model efficiency. Also, comparison among applied kernel based models and some available equations revealed that the SVM and KELM models were more reliable in modeling the scour depth downstream of spillways. The results of sensitivity analysis indicated that the impact of variable q in the state 1 and variable q²/[gY_t³] in the state 2 on obtaining a model with higher accuracy were more than other used parameters. A comparison was performed between the SVM and KELM results and the MLP-FFA method. The results showed that KELM is more accurate than the other two intelligence approaches. Also, the superior applied models dependability was investigated using uncertainty analysis. The results showed that the KELM model had an allowable degree of uncertainty in scour depth modeling downstream of the ski-jump bucket spillway.

All relevant data are available from an online repository or repositories at http://cwprs.gov.in/.