## 3 Methodology for Dynamic Risk Evaluation and Early Warning of Crest Cracking
### Computation of Crest Cracking Probability and Estimation of Cracking Time  
As mentioned previously, the duration \(T_{AS}\) of abnormal state before crest cracking is uncertain and the uncertainty in \(T_{AS}\) can be quantitatively depicted with Probability Density Function (PDF). The exponential distribution can be used to describe the probability distribution of the time intervals of independent random events which follow Poisson distribution [20]. Thus, the exponential distribution is adopted to represent the probability distribution of the duration \(T_{AS}\) of abnormal state before crest cracking and its PDF is as follows:  
\[f(T_{AS})=\frac{1}{\lambda}\exp\left(-\frac{T_{AS}}{\lambda}\right)\ T_{AS}\geq 0 \tag{2}\]  
where \(\lambda\) is the parameter in the PDF of \(T_{AS}\), \(\lambda=E(T_{AS})\), i.e., \(\lambda\) is the expected value of the duration of abnormal state before crest cracking. When the abnormal state of dam crest deformation (i.e., \(\gamma_{S}>\gamma_{C}\)) occurs and continues, the probability of crest cracking in a given time \(\overline{t}\) can be computed as:  
\[P\left(T_{AS}\leq\overline{t}\right)\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\  
\[\pi_{i}(\lambda|E_{i})=\frac{L(E_{i}|\lambda)\pi_{i-1}(\lambda)}{\int_{\Theta}L(E_ {i}|\lambda)\pi_{i-1}(\lambda)d\lambda} \tag{5}\]  
where \(\pi_{i}(\lambda|E_{i})\) is the updated probability distribution of \(\lambda\) given that the \(i\)th information is obtained (i.e., the posterior probability distribution of \(\lambda\)), \(\pi_{i-1}(\lambda)\) is the probability distribution of \(\lambda\) before obtaining new information (i.e., the prior probability distribution of \(\lambda\)), \(L(E_{i}|\lambda)\) is likelihood function (i.e., given a value of \(\lambda\), the conditional probability of observing the \(i\)th information or a number proportional to the conditional probability), and \(\Theta\) is the possible range of \(\lambda\). How to establish the likelihood function is the critical step in updating of the probability distribution of \(\lambda\) using Equation (5). In the service life of an earth-rockfill dam, one may observe two types of information directly related to \(\lambda\) estimation. In terms of the two types of information, likelihood functions can be constructed as follows:  
(a) The first type of information: the observed time interval \(\overline{t_{i}}\) from the emergence of abnormal state until the \(i\)th crest cracking, and the corresponding likelihood function is expressed as:  
\[L(E_{i}|\lambda)=\frac{1}{\lambda}\exp\!\left(-\frac{\overline{t_{i}}}{ \lambda}\right) \tag{6}\]  
(b) The second type of information: the observed \(i\)th duration \(\overline{t^{\prime}_{i}}\) of abnormal state without crest cracking, and the corresponding likelihood function is formulated as:  
\[L(E_{i}|\lambda)=\int_{\overline{t_{2}}}^{+\infty}\frac{1}{\lambda}\exp\!\left( -\frac{t}{\lambda}\right)\!dt=\exp\!\left(-\frac{\overline{t^{\prime}_{i}}}{ \lambda}\right) \tag{7}\]  
According to the type of newly observed information, substituting the corresponding likelihood function and the prior probability distribution of \(\lambda\) into Equation (5), the probability distribution of \(\lambda\) can be updated. When the probability distribution of \(\lambda\) is estimated for the first time, the prior probability density function of \(\lambda\) without any observed information, \steps. At each load step, one layer of elements is activated for subsequent computation. 218 load steps are applied to simulate the water level fluctuation in the reservoir after the end of dam filling. The in-situ monitoring data of the seepage pressure measuring point in the upstream rockfill shows that the pore pressure in the rockfill is completely consistent with the current reservoir water level due to the strong water permeability of rockfill, and there is no hysteresis. Therefore, the seepage in the rockfill can be ignored.