## 3 First-Passage Reliability Problem  
### Mathematical Basis  
The classical first-passage reliability method was originally developed for time-dependent reliability analysis based on the stochastic process theory. The basic idea of the method is to assess the probability that the structural response does not exceed the safety limit for any time point within a reference period of \([0,T]\). [PERSON] (1958) proposed the first-passage calculation formula, which was used in structural dynamic reliability analysis. [PERSON] (1958) developed an approach for the first-crossing probability based on the continuous Markov process. [PERSON] and [PERSON] and [PERSON] (1958) derived the analytical solution of the first-passage time using the Markov's envelope process. [PERSON] (1958) proposed a Poisson approximation approach to calculate the crossing frequency. The Poisson approximation establishes the relationship between the crossing rate and the structural time-dependent reliability. Figure 1: Illustration of deformed retaining structures in foundation pits. (**a**) Vertical deformation. (**b**) Failure occurring at unmonitored locations. However, the computational accuracy depends on the assumption that the structural response exceeds the safety limit independently with a small probability. [PERSON] [43] showed that the solution to time-dependent reliability based on the Poisson process assumption is asymptotically accurate when the safety limit tends to infinity. Later on, the first-crossing reliability approach was extensively applied in the estimate of structural time-dependent safety level [31; 44; 45; 46; 32]. In this paper, the classical time-dependent reliability approach is extended, with some modifications, to estimate the deformation reliability of retaining structures, by noting that the spatial distribution of the deformation along the vertical direction is also a one-dimensional problem. Mathematically, within a length of \([0,X]\) of interest, the limit state function \(Z(x)\) takes the form of  
\[Z(x)=R(x)-S(x), \tag{1}\]  
where \(x\) denotes the location, \(S(x)\) is the structural deformation at \(x\), and \(R(x)\) is the deformation limit at \(x\). If \(Z(x)\geq 0\) holds for all applicable \(x\in[0,X]\), then the retaining structure is deemed as satisfactory. Typically, \(R(x)\) can be treated as deterministic (or simply a constant in some occasions, e.g., \(y_{\text{lim}}\), as discussed in Section 2). Taking into account the uncertainty associated with the process \(S(x)\), the probability of \(Z(x)\geq 0,\forall x\in[0,X]\) is the deformation reliability, denoted by  
\[\mathbb{L}(X)=\mathbb{P}(Z(x)\geq 0,\forall x\in[0,X]), \tag{2}\]  
where \(\mathbb{P}(\,)\) is the probability of the event in the bracket. Recall that the traditional \"point control\" approach (c.f. Section 2) simply gives  
\[\mathbb{L}_{\text{PC}}(X)=\mathbb{P}(Z(x)\geq 0,\forall x\in[0,X])=\prod_{i=1}^ {n}\mathbb{I}(y_{i}\geq y_{\text{lim}}), \tag{3}\]  
where \(\mathbb{I}(\,)\) is an indicative function, which returns \(1\) if the statement in the bracket is true and \(0\) otherwise. Now we apply the first-passage reliability approach to estimate Equation (2). We assume that \(S(x)\) is a weakly stationary Gaussian process. For the purpose of further derivation, we let  
\[\Omega(x)=R(x)-\mathbb{E}(S(x)),\quad Y(x)=S(x)-\mathbb{E}(S(x)), \tag{4}\]  
where \(\mathbb{E}(\,)\) denotes the mean value of the variable in the bracket. With this, \(Y(x)\) is a zero-mean stationary Gaussian process, as schematically shown in Figure 2. Thus, Equation (2) becomes  
\[\mathbb{L}(X)=\mathbb{P}(\Omega(x)-Y(x)\geq 0,\forall x\in[0,X]). \tag{5}\]  
The up-crossing rate of \(Y(x)\) with respect to \(\Omega(x)\) at location \(x\), \(\
u^{+}(x)\), is estimated according to  
\[