Freezing depth prediction of surrounding rock in seasonally frozen tunnels based on bayes-optimized XGBoost

Abstract

During freeze-thaw cycles, frost heave forces induced by pore water phase transition in the surrounding rock of seasonally frozen tunnels act continuously on the lining structure, impairing its stability and service performance. The evolution of freezing depth is governed by the coupling effects of temperature, moisture and surrounding rock properties, and directly correlates with the distribution of frost heave forces. Thus, accurate prediction of freezing depth is critical for frost heave disaster control and structural design of cold-region tunnels. In this study, moisture migration and phase transition during surrounding rock freezing were considered to establish a hydrothermal coupling model and identify key influencing factors. On this basis, a dataset of tunnel freezing depth in seasonally frozen regions was constructed by combining Latin Hypercube Sampling (LHS) with numerical simulation methods. Subsequently, a Bayes-optimized XGBoost (BO-XGBoost) prediction model was constructed, where Bayesian optimization (BO) was employed for hyperparameter tuning. Distinguished from traditional manual or grid search methods, this algorithm realizes automatic and precise parameter optimization with higher efficiency, and effectively avoids the model from falling into local optimality during the tuning process. SHapley Additive exPlanations (SHAP) analysis was used to quantify feature contributions, and the Stefan solution verified prediction reliability. Results show that BO-XGBoost outperforms Random Forest (RF), Support Vector Regression (SVR) and Light Gradient Boosting Machine (LightGBM), with high sensitivity to temperature, moisture and thermal conductivity properties but low sensitivity to specific heat properties. Increased thermal conductivity of surrounding rock and lining deepens the predicted freezing depth, while higher specific heat capacity, initial moisture content, initial rock temperature and average temperature of the coldest month reduce it. The maximum relative error between model predictions and field monitoring data is only 7.9%, significantly lower than the 22.4% of the Stefan solution, providing a more accurate method for predicting the freezing depth of seasonally frozen tunnels.

Introduction

China’s transportation infrastructure is rapidly expanding into high-latitude and high-altitude regions, with cold-region tunnels increasingly dominating key transportation hubs. Under coupled climatic and hydrological effects, surrounding rock freezing depth exposes tunnel linings to high-intensity frost heave forces, severely endangering long-term tunnel stability and operational safety^1,2. Thus, accurate determination of surrounding rock freezing depth is fundamental for optimizing frost-resistant schemes and disaster mitigation in cold-region tunnels.​

Theoretical calculation is a primary method for determining freezing depth. Zhang et al.³ proposed an approximate analytical method for maximum freezing depth based on equivalent negative accumulated temperature and quasi-steady state theory, which outperformed the Stefan solution in case verifications. Xia et al.⁴ derived an analytical formula considering lining, insulation layer, and unfrozen water content via quasi-steady state assumption and integral method. Zhao et al.⁵ measured surrounding rock temperature of the Greater Khingan Mountains Tunnel, established a freezing depth equation using the Stefan method, and evaluated factor impacts. Liang et al.⁶ developed a modified prediction formula incorporating unfrozen water content with verified accuracy. Wu et al.⁷ derived a phase transition-included temperature field analytical formula using separation of variables and Fourier integral transform, validated by field monitoring data. However, these theories primarily rely on quasi-steady state assumptions and struggle to accurately capture temperature field redistribution under coupled moisture migration and phase transition, limiting their calculation accuracy.​

Freezing depth prediction in cold-region tunnels is a typical multi-field coupling problem, influenced by air temperature boundary conditions, surrounding rock thermophysical parameters, and initial stratum water content. Traditional analytical methods are unable to characterize the strong nonlinear interactions among these factors. In contrast, machine learning (ML) methods possess robust pattern recognition and function approximation capabilities. They can thus effectively capture the nonlinear coupling between characteristic factors and dependent variables, which facilitates the construction of high-precision surrogate models. Consequently, ML has been gradually applied to freezing depth prediction. Ma et al.⁸ proposed a temperature field-based hybrid prediction model, verified via engineering applications. Sun et al.⁹ constructed a BP neural network for efficient prediction of freezing depth and frost damage location. Yan et al.¹⁰ adopted RF and other ML methods to predict freezing depth under 7 working conditions. However, existing predictions focus solely on the temperature field, neglecting transient moisture migration and latent heat release during phase transition. Additionally, lack of systematic hyperparameter optimization and interpretability analysis results in low prediction accuracy and limited generalization ability.​

To address these issues, this study considers nonlinear temperature-moisture coupling in seasonally frozen tunnel surrounding rock during freezing periods and the impact of water-ice phase transition on freezing depth. Taking temperature, moisture content, surrounding rock properties, and lining characteristics as input features, a seasonally frozen tunnel freezing depth dataset was established via finite element numerical simulation combined with LHS. The BO algorithm was employed to automatically search for the optimal hyperparameter combination of the XGBoost prediction model. Stability and accuracy of the BO-XGBoost algorithm were verified via comparative analysis with RF, SVR, and LightGBM models. Furthermore, SHAP analysis was integrated to interpret feature contribution degrees and dependence relationships between each feature and freezing depth. Research results provide valuable references for cold-region tunnel structural design.

Establishment of a freezing depth dataset for seasonally frozen tunnels

Hydrothermal coupling governing equations

Based on Fourier’s law and considering moisture migration and phase transition in the surrounding rock, the governing equation for the temperature field of the surrounding rock can be derived as follows¹¹:

1

where: T denotes the temperature; ρc(θ) is the volumetric heat capacity of the surrounding rock; λ(θ) is the thermal conductivity of the surrounding rock; L represents the latent heat of water-ice phase transition, with a value of 334.56 kJ/kg. The expressions of ρc(θ) and λ(θ) are given as follows:

2

3

where: ρ_w, ρ_i and ρ_s denote the density of water, ice, and surrounding rock, respectively; θ_w, θ_i and θ_s​ represent the volumetric water content, volumetric ice content, and saturated water content of the soil, respectively; c_w, c_i and c_s are the specific heat capacity of water, ice, and surrounding rock, respectively; λ_w, λ_i and λ_s stand for the thermal conductivity of water, ice, and the skeleton of surrounding rock, respectively.

Assuming that there is no moisture infiltration in the tunnel lining and thus neglecting moisture migration and phase transition, the governing equation for the tunnel lining can be expressed as follows:

4

where: ρ_l, c_l and λ_l denote the density, specific heat capacity, and thermal conductivity of the lining structure, respectively.

Assuming that the moisture migration in the surrounding rock obeys Darcy’s Law, the governing equation for moisture migration can be expressed as follows:

5

where: D(θ_w) is the moisture diffusivity; k(θ_w) is the permeability coefficient. The expressions of these two parameters are given as follows:

6

7

8

where: S is the relative saturation; c(θ_w) is the specific water capacity; l and m are the constitutive parameters related to soil properties; k_s is the saturated permeability coefficient of unfrozen water. S can be expressed as:

9

where: θ_s is the saturated water content; θ_r is the residual water content.

In the numerical simulation process, to achieve the coupling of the temperature field and moisture field, a solid-liquid ratio function B(T) was introduced, which can be expressed as follows¹²:

10

where: T_f is the freezing temperature of the surrounding rock; B is the correlation coefficient associated with the solid-liquid ratio¹³.

Finite element model calculation

Model establishment and boundary condition setting

A certain seasonally frozen tunnel is located in the moderate-deep seasonal frozen soil region in the eastern Qinghai-Tibet plateau, with a total length of 3355 m and an annual average temperature of 5.9 °C. The surrounding rock is classified as grade V¹⁴. In this study, the partial differential equation (PDE) module of the COMSOL finite element software was adopted for numerical simulation. The boundary conditions of the model are set as follows¹⁵: the average temperature of the inner boundary of the lining is 5.9 °C with a temperature amplitude of 15 °C; the initial ground temperature of the surrounding rock is 2.9 °C; boundaries AB, CD and BD of the model are defined as adiabatic interfaces; the initial water content of the surrounding rock is 28%. It is assumed that there is no moisture migration between the lining and the surrounding rock, and the seepage energy at the contact interface is zero¹⁶. Meanwhile, the seepage flux across boundaries AB, CD and BD is set to zero. The finite element calculation model is illustrated in Fig. 1, and the calculation parameters are listed in Table 1^15,17.

Fig. 1.

Finite element model.

Table 1.

Hydro-thermal coupling calculation parameters of the seasonally frozen tunnel.

Parameters	Value	Parameters	Value
ρ l	2500kg/m3	λ s	1.38W/(m·℃)
ρ s	1700kg/m3	λ w	0.63W/(m·℃)
ρ w	1000kg/m3	λ i	2.31W/(m·℃)
ρ i	918kg/m3	θ s	45%
c l	1.046kJ/(kg·℃)	θ r	4%
c s	0.89kJ/(kg·℃)	θ 0	28%
c w	4.2kJ/(kg·℃)	T f	−0.18℃
c i	2.1kJ/(kg·℃)	L	334.56kJ/kg
λ l	1.85W/(m·℃)	k s	2e-8m/s

Simulation results and analysis

The contour plots of temperature, unfrozen water content, and volumetric ice content of the tunnel at a specific moment during the freezing period, derived from the numerical simulation, are illustrated in Fig. 2.

Fig. 2.

Temperature and moisture cloud maps of tunnels during the freezing period. (a) Temperature field (Unit: ℃), (b) Unfrozen water content (Unit: %), (c) Volumetric ice content (Unit: %).

As can be seen from the Fig. 2, in the tunnel freezing stage, the external cold energy conducts from the tunnel lining surface to the deep part of the surrounding rock, forming a significant radial temperature gradient. When the temperature of the surrounding rock zone adjacent to the lining drops below the freezing temperature, pore water migrates and freezes. This results in a reduction in the unfrozen water content and an increase in the volumetric ice content, thus forming a frozen ring of the surrounding rock.

To further investigate the variation law of surrounding rock freezing depth under the coupling effect of temperature and moisture migration, we extracted the surrounding rock freezing depth at different moments during the freezing period. The corresponding results are presented in Fig. 3. It can be observed that the maximum freezing depth of the tunnel during the freezing period is approximately 1.25 m, which is basically consistent with the maximum freezing depth of 1.22 m obtained from field monitoring¹⁴. In the first 36 days under negative temperature exposure, the cold energy only conducts from the lining to the surface layer of the surrounding rock, and no freezing front is formed. The temperature of the surrounding rock remains basically above the freezing temperature, and the surrounding rock is entirely in a thawed state. With the continuous application of negative temperature, the cold energy continuously transmits to the surrounding rock, leading to a rapid growth trend of the freezing depth, and the evolution of the surrounding rock temperature field is significantly dominated by cold energy conduction¹⁸. After 160 days of negative temperature exposure, the ambient temperature turns positive, and the thermal resistance effect of the deep geotemperature of the surrounding rock becomes increasingly pronounced. This results in a slowdown in the advancement of the freezing front, a decrease in the growth rate, and the gradual emergence of a thawed layer.

Fig. 3.

Variation of surrounding rock freezing depth with time under negative temperature exposure.

Establishment of a dataset for surrounding rock freezing depth

Based on numerical simulation results and relevant research findings^19–21, starting from the intrinsic mechanism of hydrothermal coupling during surrounding rock freezing: temperature directly affects cold energy input and serves as the energy basis for freezing front advancement, moisture conditions determine the total latent heat released during freezing, and thermophysical parameters of surrounding rock and lining govern heat conduction efficiency between the external environment and interior. To cover the entire physical process of freezing front propagation, the core influencing factors of surrounding rock freezing depth are identified as: average temperature of the coldest month, initial rock temperature, initial moisture content, and thermal conductivity/specific heat of surrounding rock and lining. To establish a scientifically controllable parameter variation gradient and fully capture factor fluctuation impacts, each characteristic factor was adjusted by ± 20%²² to quantify the response trend of freezing depth under different factor values. The value ranges of each factor are as follows: surrounding rock thermal conductivity (1.104, 1.656), surrounding rock specific heat (0.712, 1.068), lining thermal conductivity (1.48, 2.22), lining specific heat (0.8368, 1.2552), initial surrounding rock temperature (2.32, 3.48), initial moisture content (22.4, 33.6), and average temperature of the coldest month (−12.1, −6.1).

Given the characteristic variable ranges, a hybrid method combining LHS²³ and numerical simulation was adopted for sample generation. The 7-dimensional sample data affecting freezing depth were uniformly stratified in the parameter space, yielding 100 representative characteristic variable combinations. This ensures the reliability and stability of the subsequent freezing depth prediction model.

Table 2 presents the statistical results of LHS sampling, showing that the mean value of each characteristic variable is close to the interval midpoint with a small standard deviation, achieving uniform variable coverage and enhancing model training stability. Pearson correlation analysis of LHS results (Fig. 4) indicates uniform and independent sampling of characteristic variables, with absolute correlation coefficients ≤ 0.16 (weak correlation). This confirms that the influence of each variable on freezing depth is dominated by nonlinear coupling effects, avoiding multicollinearity from strong variable correlations and effectively mitigating model overfitting risks.

Table 2.

Feature variable statistics.

Feature	Min	Max	Mean	SD
λ s	1.1074	1.6508	1.3802	0.1606
c s	0.7129	1.0668	0.8899	0.1033
λ l	1.4841	2.2139	1.8693	0.2078
c l	0.8404	1.2539	1.0452	0.1165
T 0	2.3214	3.4766	2.9154	0.3128
θ 0	22.4791	33.5036	28.1322	3.1964
T c	−12.0783	−6.1355	−9.0998	1.6852

Fig. 4.

Pearson correlation heatmap.

Prediction of surrounding rock freezing depth in seasonally frozen tunnels based on the BO-XGBoost

Establishment process of the BO-XGBoost prediction model

The XGBoost algorithm belongs to the Boosting ensemble learning family, built on the core principle of Gradient Boosting Decision Trees (GBDT). It forms the final predictive model via ensemble optimization of multiple decision tree models²⁴. BO is a hyperparameter tuning method incorporating a prior process²⁵. In this study, surrounding rock freezing depth was designated as the model’s dependent variable, with other variables defined as characteristic variables. The dataset was randomly split into a training set and a test set at an 8:2 ratio. To eliminate feature dimensionality effects, Z-Score standardization was performed on the dataset, and the calculation formula is as follows:

11

where: Z is the standardized value; X is the original feature value; µ is the mean value of the original feature dataset; σ is the standard deviation of the original feature dataset.

During model training, preset hyperparameter ranges were adopted as prior information, with the optimization objective of minimizing the Mean Absolute Error (MAE) of freezing depth prediction. BO was then employed for global hyperparameter optimization of the XGBoost algorithm. To avoid model overfitting and ensure its generalization performance, this study referred to the universal hyperparameter ranges widely adopted in geotechnical engineering prediction studies for parameter setting. The hyperparameter ranges were set as follows: max_depth (3 ~ 10), learning_rate (0.01 ~ 0.3), n_estimators (50 ~ 200), gamma (0 ~ 1), min_child_weight (1 ~ 10), subsample (0.6 ~ 1.0), colsample_bytree (0.6 ~ 1.0), and reg_lambda (0 ~ 1). Model convergence was determined when the loss curves of both the training and validation sets stabilized (Fig. 5).​

Fig. 5.

Loss function curve of BO-XGBoost model training process.

To ensure model generalization, 5-fold cross-validation (CV) was applied to the training set. Specifically, the 80 training samples were equally divided into 5 subsets: 4 subsets for training and 1 for validation, with the process repeated 5 times. Optimal model parameters were determined based on the average prediction accuracy of 5-fold CV, yielding a model with higher stability and stronger generalization. The establishment process of the BO-XGBoost model is illustrated in Fig. 6, and the hyperparameter search results of the BO-XGBoost model are presented in Table 3.

Fig. 6.

BO-XGBoost prediction model construction.

Table 3.

Bayesian search for optimal hyperparameter combination.

Hyperparameter	Optimal value
max_depth	8
learning_rate	0.3
n_estimators	155
min_child_weight	10
subsample	1
colsample_bytree	0.6
gamma	0
reg_lambda	1

Prediction results and analysis

To verify the accuracy of the BO-XGBoost algorithm in predicting the freezing depth of surrounding rock in seasonally frozen tunnels, RF, SVR, and LightGBM were employed for comparative prediction. The actual and predicted values of the test set for each model are illustrated in Fig. 7. It can be observed that the prediction errors of all four models are within ± 0.12 m, which confirms the feasibility of applying machine learning algorithms for surrounding rock freezing depth prediction in frozen tunnels.

Fig. 7.

Different algorithms test set comparison (Unit: m). (a) BO-XGBoost, (b) RF, (c) SVR, (d) LightGBM.

Figure 8 presents the prediction performance plots of the test sets for the four models. The results demonstrate that all four machine learning prediction models can satisfactorily predict the freezing depth. However, the BO-XGBoost model exhibits a higher fitting degree between the measured and predicted value curves, along with a more uniform residual distribution. The residuals of the surrounding rock freezing depth obtained by BO-XGBoost, RF, SVR, and LightGBM are within the ranges of ± 0.1 m, ± 0.12 m, ± 0.12 m, and ± 0.11 m, respectively. It can be concluded that the BO-XGBoost model outperforms the other three models in terms of both prediction accuracy and error control.

Fig. 8.

Comparison of prediction performance and residuals of different algorithms on the test set. (a) BO-XGBoost, (b) RF, (c) SVR, (d) LightGBM.

To intuitively evaluate model accuracy, four metrics including mean absolute percentage error (MAPE), MAE, root mean square error (RMSE), and coefficient of determination (R²) were employed for quantitative assessment of prediction performance. Specifically, smaller values of the error metrics and a larger R² value indicate a higher prediction accuracy of the model²⁶. Their calculation formulas are as follows:

12

13

14

15

where: y_i is the actual value of the test set; ŷ_i is the predicted value of the model;is the mean value of the actual values.

The comparison of evaluation metrics of each model is illustrated in Fig. 9. It can be seen that the three error metrics (MAPE, MAE, RMSE) of the BO-XGBoost model are all smaller than those of the other three models, whereas its coefficient of determination (R²) is larger. This indicates that the model has a superior ability to capture nonlinear data patterns, with significantly better error control and prediction accuracy than the comparison models. Due to the lack of fine hyperparameter tuning, the RF and LightGBM models show slightly lower accuracy than BO-XGBoost, while the SVR model yields relatively unremarkable prediction results.

Fig. 9.

Comparison of prediction model evaluation metrics, (a) MAPE, (b) MAE, (c) RMSE, (d) R².

SHAP interpretability analysis and validation of surrounding rock freezing depth prediction model

SHAP interpretability analysis

Figure 10 shows the schematic diagram of the SHAP interpretability model. When conditioned on a specific feature, the variation in the model’s expected prediction result is attributed to that feature. The baseline value refers to the mean output of all samples. Red represents a positive promotion effect (e.g., Features 1, 2, and 3), while blue denotes a negative inhibitory effect (e.g., Feature 4). A single prediction result is derived from the superposition of multiple features with either positive or negative contributions²⁷.

Fig. 10.

SHAP interpretability schematic diagram.

The feature importance ranking based on SHAP analysis is illustrated in Fig. 11(a). Characteristic variables including temperature, initial moisture content, and thermal conductivity exhibit relatively high impacts on prediction results, while specific heat shows a lower influence. Figure 11(b) presents the SHAP summary plot. It can be observed that when the average temperature of the coldest month, initial moisture content, initial surrounding rock temperature, and surrounding rock skeleton specific heat decrease, the corresponding SHAP values are positive, and the predicted surrounding rock freezing depth tends to increase. Conversely, as the surrounding rock skeleton thermal conductivity and lining thermal conductivity increase, the SHAP values of these features show a positive growth trend, with the predicted freezing depth increasing synchronously. During the evolution of the surrounding rock freezing front, changes in ambient temperature, moisture migration, and thermal conductivity properties exert significant impacts on freezing depth, as reflected by the higher absolute magnitudes of their SHAP values. In contrast, during the water-ice phase transition, the regulatory effect of specific heat on freezing depth is relatively limited, resulting in significantly lower absolute magnitudes of SHAP values compared to other features and a lower ranking in feature importance.

Fig. 11.

SHAP analysis. (a) Feature contribution, (b) Summary plot.

To better understand the influence of each characteristic variable on freezing depth, single-factor SHAP value plots were generated as illustrated in Fig. 12. The key observations are as follows: (1) With the increase in the average temperature of the coldest month, the cold energy input from the external environment to the surrounding rock decreases, leading to a reduction in SHAP values and a corresponding decrease in the predicted freezing depth; (2) As the initial surrounding rock temperature rises, the internal heat of the surrounding rock during the early freezing stage increases, resulting in a downward trend of SHAP values—indicating that a lower initial surrounding rock temperature corresponds to a greater predicted freezing depth; (3) An increase in initial moisture content leads to more latent heat release from phase transition within the surrounding rock, which inhibits the advancement of the freezing front²⁸. Consequently, the SHAP values of this feature show a decreasing trend, meaning a lower initial moisture content results in a greater freezing depth; (4) Higher thermal conductivity of the surrounding rock and lining enhances the heat transfer capacity between the external environment and the surrounding rock, suggesting that increased thermal conductivity contributes to a larger predicted freezing depth; (5) With the increase in soil specific heat, more heat is released during the freezing and cooling process of the surrounding rock, consuming additional cold energy and leading to a decreasing trend of freezing depth. The influence of lining specific heat is relatively weak, with SHAP values fluctuating around 0, indicating it is a weakly influential feature.

Fig. 12.

SHAP dependence plot. (a) Mean temperature of coldest month (T_c), (b) Initial water content (θ₀), (c) Surrounding rock skeleton thermal conductivity (λ_s), (d) Lining thermal conductivity (λ_l), (e) Initial rock temperature (T₀), (f) Surrounding rock skeleton specific heat (c_s), (g) Lining specific heat (c_l).

Model validation

In this study, field monitoring data were collected from three tunnel sections, including ZK9 + 000 section of the Dongtianshan Tunnel, ZK33 + 970 section of the Qingshashan Tunnel, and RK7 + 340 section of the Alatan Tunne^3,15,17 as presented in Table 4. These data were compared with the freezing depth results calculated by the Stefan solution²⁹ and the predicted outputs of the BO-XGBoost model. This comparison was conducted to verify the accuracy and applicability of the prediction model proposed in this paper.

Table 4.

Basic data of the tunnels.

Tunnel section	Annual average air temperature/(℃)	Annual temperature amplitude/(℃)	Water content /(%)	Freezing depth /(m)
Dongtianshan Tunnel	10.75	25	30	1.36
Qingshashan Tunnel	5.9	15	28	1.25
Alatan Tunnel	0.7	13.2	15	1.65

The Stefan solution for the freezing depth of surrounding rock is expressed as follows:

16

where: T_f denotes the equivalent temperature of the tunnel inner surface, and Δt represents the equivalent duration of negative temperature.

Figure 13 presents the calculation results of the field monitoring data, BO-XGBoost predictions, and Stefan solution. As can be seen from Fig. 13, the BO-XGBoost prediction results for the three engineering cases are closer to the field monitoring data than those of the Stefan solution. The maximum relative error of the Stefan solution reaches 22.4%, while that of the BO-XGBoost model is only 7.9%. The errors mainly stem from the errors generated during the numerical simulation process and the inherent errors of the BO-XGBoost model prediction. The BO-XGBoost prediction model can control the freezing depth error within approximately 0.1 m, and its accuracy can meet the requirements of practical engineering⁴. In addition, it can be observed from the figure that the prediction error of freezing depth for the Alatan Tunnel is relatively large, which is mainly due to the fact that its extreme low temperature is outside the characteristic interval. The insufficient learning of the evolution characteristics of extreme working conditions by the model leads to a relatively high freezing depth prediction error.

Fig. 13.

Prediction model validation.

Conclusions

In this study, a freezing depth dataset of seasonally frozen tunnels was constructed via LHS combined with numerical simulation. BO-XGBoost model was employed to predict the surrounding rock freezing depth, with comparative analysis conducted against predictions from RF, SVR, and LightGBM models. Critical factors influencing surrounding rock freezing depth were identified using SHAP analysis. Furthermore, field monitoring results from three engineering cases, BO-XGBoost predictions, and Stefan solution calculations were compared to verify the applicability of the BO-XGBoost model for freezing depth prediction in seasonally frozen tunnels. The main research conclusions are as follows:​

(1) A hydrothermal coupling finite element model of seasonally frozen tunnels was established considering moisture migration and water-ice phase transition, and the time-dependent evolution law of the freezing front during the freezing process of tunnel surrounding rock was investigated. The simulated freezing depth was approximately 1.25 m, which is consistent with the field monitoring value of 1.22 m, with a relative error of only 2.46%, thus verifying the accuracy of the established model.​

(2) The original dataset was determined using LHS combined with numerical simulation. The BO algorithm was adopted to automatically search and optimize the hyperparameters of the XGBoost prediction model, which significantly improved the model performance. Comparative analysis with the prediction results of RF, SVR, and LightGBM models indicated that the BO-XGBoost model exhibits the optimal performance in both error control and prediction accuracy, providing a novel method for freezing depth prediction of seasonally frozen tunnels.​

(3) Feature importance ranking was performed using the SHAP interpretable model. The results indicated that temperature, moisture content, and thermal conductivity exert significant impacts on freezing depth, while the influence of specific heat is relatively weak. This conclusion is highly consistent with the basic theory of hydrothermal transfer in frozen soil, which verifies the physical rationality of the model prediction results.​

(4) SHAP dependence plots explained the influence of individual characteristic factors on freezing depth prediction. The findings demonstrated that: with the increase in the average temperature of the coldest month, initial moisture content, initial rock temperature, and surrounding rock specific heat, the SHAP values of the model gradually decrease to negative values, indicating an inhibitory effect on freezing depth advancement (negative correlation); the increase in thermal conductivity of surrounding rock and lining raises the SHAP values to positive values, promoting freezing depth advancement (positive correlation); the contribution of specific heat to freezing depth prediction is weak.​

(5) Comparative analysis of monitoring data and theoretical calculation results from typical tunnel projects in seasonally frozen regions reveals that the BO-XGBoost model can effectively predict freezing depth with an error of approximately 0.1 m. Additionally, the BO-XGBoost model exhibits high prediction accuracy within the characteristic variable range, while the error becomes relatively larger when extreme working conditions are considered.

Author contributions

J.J.: conceptualization, methodology, investigation, review the manuscript.J. L.: writing-original manuscript, simulation analysis, machine learning model establishment.

Funding

This research is supported by the National Natural Science Foundation of China (42261029) and Science and Technology Program of Gansu Province (26JRRA587).

Data availability

The datasets, models and codes generated in this study have been deposited in the database(https://doi.org/10.5281/zenodo.19397342).

Declarations

Competing interests

The authors declare no competing interests.