Prediction of contraction channel scour depth: based on interpretability analysis and PCA-enhanced SVR

In river flow, both artificial and natural structures, such as bridge abutments and landslides, lead to the reduced fluid cross-section area. The reduced width significantly accelerates the flow and therefore leads to increased bed shear stress in the narrowed riverbed, creating a high risk of scour or erosion. Therefore, the accurate calculation of contraction scour depth is crucial for risk assessment and the safe operation of these structures.

Straub (1934) conducted the earliest documented study on contraction scour, deriving a correlation for predicting scour depth under dynamic conditions based on the continuity equations for water flow and sediment. Subsequent expansions and modifications of this correlation were made by researchers (Ashida 1963; Laursen 1963; Komura 1966; Gill 1981; Webby 1984; Lim 1993; Lim & Cheng 1998; Raikar 2004; Briaud et al. 2005; Oliveto & Marino 2019; Nowroozpour & Ettema 2021). Researchers have proposed various criteria for classifying contraction channels. Komura (1966) defined the long contraction channel when the ratio of the contraction section length to the uncontracted channel width (L/b₁) exceeds 1, whereas Raikar (2004) and Webby (1984) specified a threshold ratio of 2, above which the channel is regarded as a long contraction channel. Further research indicates that an increase in the length of contraction section reduces and stabilizes shear stress on riverbeds. Additionally, the L/b₁ ratio also affects scour morphology, influencing the depth of scour (Oliveto & Marino 2019).

In hydraulic engineering, scouring is generally classified into two types: clear water scour and live-bed scour. Extensive experimental studies using flume tests under controlled laboratory conditions have established a robust set of empirical formulas. Refined by Dey & Raikar (2005), these formulas offer highly accurate predictions for maximum equilibrium scour depth. Despite these advancements, previous studies have often oversimplified the complexity of the problem in their empirical models, limiting their applicability in intricate scenarios (Le et al. 2024). Recently, Lagasse et al. (2021) conducted extensive research and evaluation on the applicability of contraction and pier scour formulas and provided a relatively rare field case study (i.e., the U.S. Highway 287 crossing Spring Creek in Fort Collins, Colorado), marking a critical step toward the practical application of these formulas.

As research progresses, Najafzadeh et al. (2018) expanded the range of predictive methods by using gene expression programming (GEP), M5 Tree (MT), and evolutionary polynomial regression (EPR) to predict maximum equilibrium scour depth. Moreover, enhancing prediction accuracy while reducing the number of input variables has become central to their studies. Najafzadeh et al. (2016) conducted a variable sensitivity analysis using the ANFIS by sequentially removing variables and assessing their impact on model performance, assuming that significant decreases in prediction accuracy would indicate a high importance of the removed variables. More recently, absolute values of correlation coefficients have frequently adopted to assess the importance of variables, combining ML and ensemble techniques to generate numerous hybrid models and select the optimized model that best matches different variable input combinations. For instance, Sharafati et al. (2021) combined five metaheuristic algorithms with the ANFIS model to develop a reliable and robust prediction model that broadly reflects the relationship between input parameters and the target scour depth. The Dagging-Iterative Classifier Optimizer hybrid model by Khosravi et al. (2021) was developed by integrating the Dagging (DA) and Random Subspace (RS) algorithms with five distinct ML models. Recent advances have highlighted the significant role of ML in sediment scour prediction, particularly in estimating scour depth around bridge piers. Eini et al. (2023) provided a comprehensive summary of various ML models and proposed an XGBoost hybrid model enhanced by optimization algorithms, which markedly improved the accuracy of scour depth predictions around circular bridge piers. Furthermore, SHapley Additive exPlanation (SHAP) analysis was employed to interpret the variable importance in the model. In addition, Eini et al. (2024) advanced this approach by combining Bayesian optimization with SVM and XGBoost, achieving enhanced prediction accuracy for scour depth around different pier shapes and demonstrating the substantial potential of ML in scour prediction.

However, the inputs of the aforementioned models are based on traditional correlation measurement methods such as Pearson, Spearman, and Kendall. Constrained by the assumptions (data distribution) of these algorithms, which cause discrepancies in the ranking of variable importance and numerous models with limited interpretability. In recent years, to mitigate the ‘black-box effect’ of prediction models, researchers have adopted interpretable feature selection algorithms such as SHAP or Sobel Operator (SOBEL) analysis for ordering and combining input variables (Guo et al. 2024), given that SHAP results do not rely on any particular model. After identifying variable importance, developing models that can capture these importances becomes a new challenge. Studies on hybrid dimensionality reduction algorithms have confirmed that feature reduction significantly improves the generalization capability and computational efficiency of ML models (Laghrissi et al. 2021; Chang et al. 2022; Pandey et al. 2023). Jia et al. (2022) elaborated on common feature selection and extraction methods and their applications, including PCA, Kernel PCA (KPCA), linear discriminant analysis (LDA), non-negative matrix factorization (NMF), and t-distributed stochastic neighbor embedding (tSNE). They emphasized that during the identification and learning process, adopting appropriate, reliable, and practical feature reduction techniques is crucial for identifying data characteristics and improving accuracy (Cao et al. 2003; Anowar et al. 2021). PCA, as a linear dimensionality reduction technique, transforms high-dimensional data into a lower-dimensional space, effectively mitigating the ‘curse of dimensionality’ by reducing the number of features. Given that the experimental data are sample-based, they are expected to follow the central limit theorem and approximate a Gaussian distribution, aligning well with PCA's assumptions regarding data distribution. This makes PCA especially suitable for handling the high-dimensional, sparse data in this study, as it provides effective denoising, redundancy removal, and computational efficiency. While alternative methods like KPCA and tSNE can also perform dimensionality reduction, they have limitations in data distribution assumptions and feature interpretability, making them less ideal for this study (Jia et al. 2022). Additionally, PCA offers a scalable approach for handling additional data features that may emerge in future observational datasets. To further capture the nonlinear relationships in the data post-dimension reduction, we selected support vector regression (SVR) with a Gaussian kernel as the ML model. SVR leverages kernel functions to map data into a higher-dimensional feature space, enabling the accurate capture of nonlinear patterns with greater precision and robustness. With fewer hyperparameters, it is easier to optimize and, due to its inherent regularization, can avoid overfitting in limited samples, supporting improved generalizability when expanding the dataset in the future (Eini et al. 2024).

Therefore, we developed an interpretable scour depth prediction model based on PCA dimensionality reduction and SVR. The traditional variable importance rankings were unified using PCA algorithm, and the SVR model was optimized through Bayesian grid search and k-fold cross-validation to learn the features of data. Meanwhile, SHAP was used to reveal the patterns learned by the model from the dataset. Additionally, to comprehensively evaluate the predictive performance of the optimized model, it was compared with other literature models in terms of metrics and uncertainty analysis, and its accuracy and conservatism were compared with empirical formulas.

Table 1 summarizes the distribution of available datasets from contraction channel clear water scour experiments (Laursen 1963; Komura 1966; Gill 1981; Webby 1984; Lim 1993; Dey & Raikar 2005) in literature. The summary includes additional details not found in earlier literature, such as the contraction channel length (L) for each of the 182 experimental samples (Lagasse et al. 2021).

In addition to PCA, several other dimensionality reduction techniques are utilized to handle complex data structures. These include KPCA, tSNE, LDA, and NMF. (1) KPCA extends PCA to nonlinear data structures by using a Kernel trick to project data into a higher-dimensional space, enabling the capture of complex patterns (Schölkopf et al. 1997); (2) tSNE excels in visualizing high-dimensional data by preserving local relationships through probabilistic distributions, making it suitable for exploratory data analysis (Gisbrecht et al. 2015); (3) LDA differentiates itself by focusing on maximizing class separability, which is highly beneficial in supervised learning scenarios where class labels are known (Park & Park 2008); (4) NMF is particularly used in contexts like image processing and text mining, where it decomposes non-negative data into simpler, meaningful components, facilitating topic identification and feature extraction (Lee & Seung 1999).

The parameter settings for the five dimensionality reduction algorithms mentioned above are detailed in Table 2.

SHAP is an advanced model interpretation method used to address the ‘black-box effect’ in ML models. It provides an explanation for model decisions by calculating the average marginal contribution of each feature to the model prediction, known as the Shapley value. The sign of the Shapley value (positive or negative) indicates the direction in which the feature affects the prediction result, while the magnitude of the absolute value reflects the importance of the feature: the larger the absolute value, the more significant the influence of the feature, and thus the more important the feature is considered. The application of SHAP has expanded to various technical and engineering fields, including the scour domain (Kim et al. 2024), demonstrating its broad applicability and practical value.

To evaluate the model performances, eight evaluation metrics are used: coefficient of determination (R²), root mean square error (RMSE), mean absolute error (MAE), mean bias error (MBE), mean absolute percentage error (MAPE), Pearson's correlation coefficient (CC) (Kim et al. 2024; Kumar et al. 2024).

Table 3 summarizes the relative sensitivity of the scour depth for different input variables derived based on different criteria in the literature. Raikar et al. (2016) used PCA, selecting the factors with the largest proportion of principal components (PCs) as representative variables to explain the results of the model (e.g., PC1 = F_1ec). However, since each PC is composed of a linear combination of all original dimensionless quantities, it is difficult to distinguish the dominant variables. Najafzadeh et al. (2016) evaluated the sensitivity of the results to the input variables by sequentially excluding each variable from the ANFIS model ensemble. A highly important variable is recognized as its removal leads to a significant drop in model prediction accuracy. Although this method provides a preliminary assessment, the mechanisms by which variables influence the outcomes remain unexplained, and the interpretability is highly dependent on the selected ML model. Though not explicitly stated, it is inferred that Sharafati et al. (2021) used the Pearson correlation method (Table 3 – current investigation before PCA) and observed that the shrinkage ratio b₀ is the most important factor, exhibiting a negative correlation with the results. This observation is consistent with previous studies, i.e., the larger the b₀ = b₂/b₁, the less concentrated the flow and the smaller the scouring capacity. Khosravi et al. (2021) also found that b₀ is negatively correlated with the scour depth, but their results showed that σ_g is positively correlated with the scour depth, contrary to the known ‘shielding effect’ theory, which suggests that an increase in the sediment non-uniformity coefficient should reduce scour depth. Previous studies have consistently found that h₀ is always positively correlated with scour depth, in accordance with experiments suggesting that larger h₀ values are more likely to generate large-scale vortex structures, thereby increasing scour depth. However, the importance of h₀ varies across different studies, and no uniform significance has been assigned. This variation could be attributed to changes in h₀ during actual scour processes, as noted by Lagasse et al. (2021).

Given the discrepancy in ranking order and the lack of interpretability, which might increase the difficulty of matching with appropriate ML models, it is necessary to carefully review the consistency of different commonly used variable correlation algorithms in identifying variable importance before inputting them into ML models. In the ‘current investigation before PCA’ section of Table 3, the relative significances indicated by the correlation coefficients of different input variables derived from Pearson, Spearman, and Kendall algorithms are listed. Except for b₀, which is consistently observed to posses a negative correlation coefficient with the highest magnitude. The relative significances of the remaining variables vary among different models. This phenomenon may stem from the assumptions of traditional Pearson, Spearman, and Kendall algorithms, which, respectively, require linear relationships, monotonic relationships, and association consistency. These assumptions are difficult to satisfy for datasets with small sample sizes.

As shown in the ‘Current investigation after PCA’ section of Table 3, after PCA processing, the importance rankings of traditional correlation algorithms have been unified, with the importance of PCs decreasing in the order of PC3, PC5, PC4, PC8, PC7, PC2, PC6, and PC1. Although the SHAP rankings of PCs show slight inconsistencies, three of the top four rankings are consistent with traditional algorithms, which might be due to the use of the RBF kernel in SVR.

For h₀, its SHAP value nearly linearly increases from −0.04 to 0.02 as the feature values increase, which aligns with observed differences in wall shear stress contributions between shallow water effects at lower depths and the formation of large-scale vortices, such as horseshoe or necklace vortices, at greater depths. The development of these vortex structures is a significant factor in increasing scour depth (Eini et al. 2024), and current data indicate that this trend shows no signs of diminishing. The increase in particle gradation σ_g stabilizes its impact on scour depth, reducing scour depth, which aligns with actual physical processes. The impact of Fr₀ on scour depth is similar to that of h₀, with a linear increase, following the mechanism described by Raikar et al. (2016) using excess Froude number (F_1ec), where Froude number comprises the difference between the contraction flow velocity and the critical sediment entrainment velocity. When this value exceeds 0, it significantly increases scour depth; however, current data only involve clear water scour, and more data are required for live-bed scour. The slight decrease of d₀ near zero is consistent with Dey & Raikar's (2005) experimental findings, which are due to variations in the bed shear stress requirements caused by the transitional properties of the Shields curve. For gravel, in order to maintain the flow velocity U₁ close to U_C, U₁ was significantly increased (thus increasing U₁/U_C), greatly enhancing bed shear stress in the contraction zone and thereby increasing scour potential, leading d_s0 to increase at a constant rate. Although the sample size for density ratio S is limited, its negative correlation with scour depth aligns with experimental laws. Before PCA processing, SHAP visualization analysis reveals significant linear relationships between the top-ranked variables such as b₀ (CC = −0.933), h₀ (CC = 0.952), and σ_g (CC = −0.952) and the SHAP values of the dependent variable (Figure 4(a)), with the trend of influence consistent with the physical interpretation (Section 4.1). In contrast, lower-ranked variables S and L₀ did not show clear linear relationships. (Previous research indicates that high S values reduce sediment scour, but most natural sediments cluster around an S value of 2.65, suggesting a need for more diverse data.) Additionally, longer L values may alter flow control from downstream to upstream, a hypothesis that requires further validation with larger datasets (Lagasse et al. 2021). The above observations indicate a high correlation between the top-ranked variables identified by traditional correlation algorithms and their SHAP values in the ‘current investigation after PCA’ section of Table 3.

To further prove this point, when the data are processed by PCA (as shown in Figure 4(d)), SHAP visualization analysis shows that as variables such as PC3 (CC = 0.977) and PC5 (CC = 0.994) increased, the corresponding SHAP values exhibit significant linear growth. These variables consistently ranked high in traditional algorithms, indicating a strong linear contribution relationship between PC3, PC5, and scour depth. This indirectly confirms that the PCA method's effect of linearizing data features make it easier for SVR to identify these features. To directly verify this point, Section 4.2 will input variables based on correlation rankings to explore their specific impact on prediction accuracy.

The method of cumulatively inputting variables ranked by the conventional PCA principal component variance contribution rate did not yield ideal results (as shown in Figure 5(c)), reaching the maximum R² = 0.936 when five PCs were input. However, adding more PCs has a negligible effect on the accuracy. This is mainly due to that PC3, PC4, and PC5 have been included in the first five PCs.

If PCA is not performed and the dimensionless variables ranked by SHAP analysis are directly input in order of importance, an R² = 0.955 can be achieved with only b₀, h₀, and σ_g taken as inputs. However, as more variables are added, the accuracy slightly decreases, confirming the physical experimental laws that adding more physical variables does not necessarily lead to better results (e.g., ignoring the Reynolds number in dimensionless analysis because it is not considered in turbulent flows). Most studies also overlook d₀ and U₁/U_C when fitting data using empirical formulas, as outlined in Section 4.3. Nevertheless, it can at least enhance the stability of SVR predictions. Moreover, the SHAP interpretability analysis method's SHAP value distribution for scour depth is consistent with the conventional theoretical knowledge of scour depth influencing factors, indicating that the SVR model's predictions are reasonable (the Laursen formula in Section 4.3. also uses b₀ and h₀ variables).

Further uncertainty analysis (see Table 4) revealed that all models tend to underestimate (negative MBE). Additionally, the standard deviation is defined as where is the predicted value and is the observed value, and the 95% confidence interval (95% CI) is calculated to provide precise estimates (Ebtehaj et al. 2017). The results show that the CC-PCA-4 model has the smallest and narrowest uncertainty interval ([−0.01816, 0.01483]), further confirming its superior predictive performance.

To provide a more comprehensive evaluation of the model, we have added a sensitivity analysis. The sensitivity of the input variables to the output was analyzed using the best SVR model. Multiple training datasets were generated by removing each input variable one by one, and results were evaluated based on the R² metric. Findings in Table 5 indicate that b₀ plays a critical role in predicting scour depth compared to other input variables, as it yielded the lowest R² on the test set. For σ_g and h₀; however, the sensitivity ranking differs slightly from the importance derived through SHAP analysis. This is understandable, as both variables are similarly close in terms of sensitivity and importance. While SHAP values measure each feature's marginal contribution to the model output, eliminating a feature during hyperparameter optimization changes the interactions among the remaining features. These complex interactions may lead to different sensitivity analysis results. For the other variables, the sensitivity ranking is consistent with the SHAP analysis (Figure 4(b)).

Khosravi et al. (2021) evaluated five models – isotonic regression (ISOR), sequential minimal optimization (SMO), iterative classifier optimizer (ICO), locally weighted learning (LWL), and least median of squares regression (LMS) – combined with the DA and RS optimization algorithms, finding that the DA-ICO-4 model performed best. Sharafati et al. (2021) applied five optimization algorithms – ant colony optimization (ACO), biogeographic-based optimization (BBO), GA, invasive weeds optimization (IWO), and teaching–learning-based optimization (TLBO) – to optimize the ANFIS model, with ANFIS-BBO-M2 showing the best performance. Najafzadeh et al. (2016) used a base version of ANFIS, which outperformed a simple hyperparameter-tuned SVM (CC = 0.88); they subsequently applied GEP (CC = 0.89), MT (CC = 0.874), and EPR (CC = 0.903), with EPR yielding relatively better results. Raikar et al. (2016) tested ANN (BP) models with single and double hidden layers, as well as GA. Pursuing a lower MSE, the single-layer network (4-16-1 structure) achieved an R² of only 0.808, inferior to the GA's R² of 0.955.

This research proposes a PCA-enhanced SVR model to predict contraction scour depth. Around 182 laboratory samples were used to construct the model. SHAP analysis was used to explain the prediction mechanism of the model, and its effectiveness was evaluated by comparison with a priori knowledge. The use of PCA allows the ML model to achieve high accuracy without the need for largely extensive searches for input variable matching. Finally, the model's predictive performance was validated by comparing it with existing models and empirical formulas from the aspects of uncertainty, predictive distribution, accuracy, and conservatism. Overall, this study validates the feasibility of using the interpretable ML approach to predict scour depth and provides an explanation and comprehensive evaluation of its mechanism. The detailed conclusions of this study are as follows:

• 1. SHAP analysis shows that the SVR model tends to identify variables that are strongly linearly correlated with scour depth and SHAP values. The identified key variables, such as b₀, h₀, and σ_g, have influence trends consistent with the physical experimental laws.

• 2. The data after PCA dimensionality reduction enhanced the correlation between independent variables (PCs) and dependent variables (scour depth and SHAP values), making the importance rankings of PCA-processed variables more consistent according to traditional correlation algorithms.

• 3. The SVR model outperforms other models and empirical formulas in accuracy. With the top four PCA-processed correlated input parameters (CC-PCA-4 scheme), the R² value is 0.971, and MAPE is 7.54%. Its results show a narrower uncertainty interval ([–0.01816, 0.01483]). Conservatism analysis shows that when the safety factor k = 1.2, the model's prediction error (MAPE) is 22.1% while maintaining a high conservatism of 94.6%.

Despite the effectiveness of the model demonstrated in this study, there remain certain limitations regarding data scale and complexity. Data acquisition costs are exceedingly high, whether from physical sediment model experiments, numerical simulations, or field measurements. This constraint is one reason why current research predominantly relies on laboratory data. Given these conditions, a feasible approach for rapid dataset expansion is to establish an automated experimental and monitoring platform to address data gaps in density ratios and contraction segment lengths. Additionally, we aim to integrate more statistical metrics into the existing data types through this platform, such as time-varying turbulence energy, sediment concentration, and time-varying scour depth, thereby extending the data range to include live-bed scour and extreme flood conditions. We also plan to collect more field measurement cases and, based on this enriched dataset, develop a physics-based online support vector machine that can enable dynamic prediction, assessment, and early warning, providing a reliable tool for infrastructure safety and health monitoring.

The authors are consistent with the ethical requirements.

The authors all consent to participate in the paper editing.

The authors all consent to the publication of the paper.

The authors wish to acknowledge the financial support by the National Natural Science Foundation of China (Grant Nos. 52179060, 52209081, and 52479060).

Data cannot be made publicly available; readers should contact the corresponding author for details.

The authors declare there is no conflict.