## 3 Case Study
### Model Building and Sampling  
(1)Identification of field condition covariates  
Waterproofing designs for the deformation joints of utility tunnels are specific to construction methods. In China, utility tunnels are generally constructed by cast-in-situ concrete, which is also the case in this case study. For this reason, this paper will focus on the deformation joints of cast-in-situ utility tunnels, for which the waterproof facilities are typically designed as shown in Figure 3.  
It can be seen from Figure 3 that such a waterproofing system consists, from exterior to interior, of a waterproof membrane, a sealant (for tunnel top or sidewalls) or an external \begin{table}
\begin{tabular}{c c c} \hline \hline
**Factors** & **Failure Reasons** & **Influence Modes** \\ \hline Differential land settlement & Dislocations and rotations of deformation joints caused by differential land settlements lead to the failure of waterproof facilities. & Long-term influence that is strong in the early stage due to accelerating land settlement but weakens with time \\ \hline Construction defect & Improper construction methods lead to component installation defects. For example, the internal waterstops fall off during concrete casting, or do not bond well to the side structures due to insufficient vibration. & Early-stage influence \\ \hline Component defect & Inferior waterproof products reduce the service life of joint waterproof facilities. & Early-stage influence \\ \hline Natural disaster & Fire or earthquake disasters damage the waterproof system. & Contingent influence \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._  
Figure 3: Waterproofing designs for the deformation joints of cast-in-situ utility tunnels. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Covariates** & **Field Conditions** & **Values** \\ \hline \(X_{1}\) & Groundwater type & Unconfined: 0; Confined: 1 \\ \hline \(X_{2}\) & Repair & With: 0; Without: 1 \\ \hline \(X_{3}\) & Quality defect & With: 0; Without: 1 \\ \hline \(X_{4}\) & Geological condition & Rock or hard soil: 0; Soft soil: 1; Medium: 0.5 \\ \hline \(X_{5}\) & Foundation treatment & With: 0; Without: 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Field condition covariates and corresponding values. waterstop (for tunnel bottom), a foam board filler, an internal waterstop, a foam board filler and a sealant. The joint leakages in cast-in-situ utility tunnels indicate that the entire waterproofing system or some of its components are out of service. Based on the authors' field investigation, there are multiple factors as listed in Table 2 that can lead to the failure of joint waterproof facilities. As contingent influence factors, such as construction disturbances and natural disasters, are characterized with low occurrence probability, short effect time and less regularity in the long-term operation and maintenance process, their impacts on the reliability estimation for the joint waterproof facilities of utility tunnels depends more on the scenario-specific analysis. Therefore, this case study only selected the predicable long-term and early-stage failure factors to identify the field condition covariates of joint waterproof facilities. In this manner, the field condition covariates that are related to the long-term and early-stage failure factors of joint waterproof facilities were identified as groundwater type, repair, quality defects, geological conditions and foundation treatment, as listed in Table 3.  
Specification of Prior Distributions for Model Parameters  
Theoretically, as long as there are enough observation data, the Bayesian estimation can derive accurate posteriors from any prior distributions. However, in the case of limited samples of this case study, the specification prior distributions will have a great influence on the parameter estimation results.