## 2 Methodology
### An Improved Weibull Model Incorporating Field Conditions  
In the classical Weibull model, supposing we have independently identically distributed lifetime \(t\), the failure probability density function of the two-parameter Weibull distribution for joint waterproof facilities is:  
\[f(t)=\frac{m}{\eta}(\frac{t}{\eta})^{m-1}\exp\biggl{[}-(\frac{t}{\eta})^{m} \biggr{]} \tag{1}\]  
where \(m\) is a unitless shape parameter and \(\eta\) is a scale parameter. The Weibull distribution cumulative distribution function \(F(t)\) is often written as:  
\[F(t)=\int_{0}^{t}\!\!f(t)dt=1-\exp\biggl{[}-(\frac{t}{\eta})^{m}\biggr{]} \tag{2}\]  
as well as the reliability function \(R(t)\) is:  
\[R(t)=1-F(t)=\exp\biggl{[}-(\frac{t}{\eta})^{m}\biggr{]} \tag{3}\]  
The failure rate function (or hazard function) \(\lambda(t)\) can then be derived as:  
\[\lambda(t)=\frac{f(t)}{R(t)}=\frac{m}{\eta}(\frac{t}{\eta})^{m-1} \tag{4}\]  
Figure 2: Methodological flow chart of reliability estimation for critical utility tunnel facilities based on an improved Bayesian Weibull model. From Equation (4), it can be seen that the shape parameter \(m\) enables the Weibull distribution to be applied to the aforementioned three life stages of the bathtub curve. That is, the shape parameter \(m\) models the three stages of early failure, useful life and wear-out with the corresponding values less than one, equal to one and greater than one. The proposed method that incorporates the explanatory variables of field conditions into the Weibull model follows the Cox proportional hazard model [26], which allows a simultaneous exploration of the effects of several covariates on reliability [27]. With the Cox proportional hazard model, the failure rate of the joint waterproof facilities of utility tunnels can be extended to incorporate field condition factors as follows:  
\[\lambda(t,X)=\lambda(t)\exp(\beta_{0}+\sum\limits_{i=1}^{n}\beta_{i}X_{i}) \tag{5}\]  
where \(\lambda(t)\) is the basic failure rate of the joint waterproof facilities of utility tunnels that only depends on lifetime \(t\), \(X_{i}\) denotes the covariates of the \(i\)th principal field condition factor that influences the reliability distribution and the regression coefficient \(\beta_{i}\) represents the degree of influence of the \(i\)th covariate. Hence the key indicators for the Weibull model, i.e., the failure rate function \(\lambda(t)\), the reliability function \(R(t)\) and the failure probability density function \(f(t)\), can be expressed as follows:  
\[\lambda(t,X)=\frac{m}{\eta}(\frac{t}{\eta})^{m-1}\exp(\beta_{0}+\sum\limits_{ i=1}^{n}\beta_{i}X_{i}) \tag{6}\]  
\[R(t,X)=\exp\left[-(\frac{t}{\eta})^{m}\exp(\beta_{0}+\sum\limits_{i=1}^{n} \beta_{i}X_{i})\right] \tag{7}\]  
\[f(t,X)=\frac{m}{\eta}(\frac{t}{\eta})^{m-1}\exp(Z)\exp\left[-(\frac{t}{\eta}) ^{m}\exp(Z)\right] \tag{8}\]  
where \(Z=\beta_{0}+\sum\limits_{i=1}^{n}\beta_{i}X_{i}\). From the improved Weibull model, two key indicators in the reliability analysis for joint waterproof facilities can be derived. The first key indicator is the failure warning time. It is defined as the time (unit: days) when the reliability decreases to 80% of the original state, which is deduced from Equation (7) as follows:  
\[T_{0.8}=\eta\left[-\frac{\ln 0.8}{\exp(\beta_{0}+\sum\limits_{i=1}^{n}\beta_{i} X_{i})}\right]^{1/m} \tag{9}\]  
The other key indicator is expected failure time (unit: days) which is calculated as:  
\[F(t)=\int_{0}^{t}\!\!f(t)dt=1-\exp\biggl{[}-(\frac{t}{\eta})^{m}\biggr{]} \tag{2}\]0  
From the aforementioned analysis process, it is evident that the dynamic reliability estimation and subsequent early warning of failure risks can be delivered if the model parameters of \(m\), \(\eta\) and \(\beta_{i}\) can be obtained at any momentary position. However, as the improved Weibull model increases the number of model parameters, the derivation of these parameters remains a challenge owing to the deficiencies in empirical research and the complexities in the varying bathtub stages.