## 2 Methodology
### Construction of a Coupled Risk Bayesian Network Model Structure  
Using the Bayesian network for estimating the probability of construction accidents in DFPs adjacent to existing subways requires adopting the association rules obtained in the previous step, utilizing directed edges to identify the causes of the outcomes, gradually constructing a Bayesian network directed acyclic graph and using Netica, version 5.18 (Norsys software Corp., Vancouver, BC, Canada), to construct a dynamic Bayesian network model containing time series. In addition, correcting the Bayesian network based on the coupled risk's influence degree on the overall risk is needed. Different risk factors interact with each other, forming coupled risks in the process of risk evolution while changing the intensity and frequency of the factors. The mechanism of accident coupling occurrence in DFP projects adjacent to existing underpass tunnels is shown in Figure 3. The types of risk coupling are classified according to the attributes and quantity of risk factors, as shown in Figure 4. In this study, the N-K model was used to analyze the construction safety's coupled risk of DFPs adjacent to existing tunnels. The basic principle herein is to calculate the mutual information value (\(T\)) between the risk of the pit itself, the risk of the existing tunnel, the risk of the surrounding environment, the risk of human factors, and other factors to measure the degree of influence of the coupling effect on the construction of the DFP adjacent to the existing underpass tunnel. The larger \(T\) indicates that the more these factors are coupled in a certain way, the greater the risk of an accident. The coupling degree \(T\) of the four-factor coupling risk is calculated as follows:  
\[T_{4}(FP,TU,SE,HU)=\sum_{h=1}^{H}\sum_{i=1}^{I}\sum_{j=1}^{I}\sum_{k=1}^{K}p_{ h,i,j,k}\times\log_{2}(p_{h,i,j,k}/(p_{h,m}.p_{i,}p_{\cdot j,}p_{\cdot\cdot k})) \tag{10}\]  
where \(FP\), \(TU\), \(SE\), and \(HU\) represent the four types of risk factors such as the stability of DFP, stability of existing tunnel, stability of the surrounding environment, and human factors,respectively; \(h=1,\ldots,H\); \(i=1,\ldots,I\); \(j=1,\ldots,J\); \(k=1,\ldots,K\); \(p_{h,i,j,k}\) is the probability of coupling between the stability of the pit in state \(h\), the stability of the existing tunnel in state \(i\), the stability of the surrounding environment in state \(j\), and the human factor in state \(k\). Most construction accidents in DFPs adjacent to existing underpass tunnels are caused by the partial coupling of risk factors, and the probability of complete coupling of risk factors leading to accidents is small. Based on the number of factors involved, the risk of safety coupling in constructing DFPs adjacent to existing underpass tunnels can be divided into single-factor coupling risk, two-factor coupling risk, and multi-factor coupling risk. Two-factor coupling risk refers to the risk caused by the interaction and influence between two risk factors affecting the safety of the construction of DFPs adjacent to existing subways, mainly including the foundation pit-tunnel coupling risk, the foundation pit-environment coupling risk, the foundation pit-human coupling risk, the tunnel-environment coupling risk, the tunnel-human coupling risk, and the environment-human coupling risk. The coupling degrees \(T\) are denoted as \(T_{21}\) (_FP_, _TU_), \(T_{22}\) (_FP_, _SE_), \(T_{23}\) (_FP_, _HU_), \(T_{24}\) (_TU_, _SE_), \(T_{25}\) (_TU_, _HU_), and \(T_{26}\) (_SE_, _HU_), and are calculated as follows:  
Figure 4: Types of risk coupling for accidents in deep foundation projects adjacent to existing underpass tunnels. Figure 3: Construction accident risk coupling mechanism of DFP adjacent to the existing underpass tunnel. \[\left\{