## 3 Methodology for Dynamic Risk Evaluation and Early Warning of Crest Cracking
### Risk Estimation and Early Warning of the Dam Crest Cracking  
In order to implement dynamic risk assessment of the dam crest cracking, after each crest cracking, the distribution of \(\lambda\) is updated according to the observed value of \(T_{AS}\) first, and then the cracking probabilities, for an arbitrary duration of abnormal state, are calculated by Equation (3) using Monte-Carlo simulation with the updated distribution of \(\lambda\)[24]. Furthermore, the mean value, 5% and 95% quantiles of the simulated cracking probabilities can be derived. Figure 13 shows the estimated mean value and confidence interval of cracking probability based on the distribution of \(\lambda\) updated with the information on last crest cracking event, as well as the observed next cracking time and the corresponding computed mean value of cracking probability, where the origin of the time axis (abscissa) corresponds to the beginning of abnormal state for next cracking event. It can be seen that the cracking probability of dam crest increases with the duration of abnormal state and is gradually approaching 100%. In addition, the increment of cracking probability is larger in the initial stage than that in the later stage. It is also noticed in Figure 13 that as the uncertainty in \(\lambda\) diminishes with the accumulation of the observed data on the duration of abnormal state before crest cracking, the width of confidence interval of the estimated cracking probability progressively contracts. Thus, it is indicated that the increase of observed relevant information is helpful to improve the accuracy of the risk evaluation model and reduce the uncertainty in cracking risk assessment. It is worth noting that the computed mean value of cracking probability corresponding to the observed cracking time is not a fixed value and ranges from 39.96% to 58.07%. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **After the 1 st** & **After the 2 nd** & **After the 3 rd** & **After the 4 th** & **After the 5 th** \\  & **Cracking** & **Cracking** & **Cracking** & **Cracking** \\ \hline Maximum likelihood & 0.436\% & 0.509\% & 0.568\% & 0.627\% & 0.724\% \\ probability & 133 & 113 & 117 & 118 & 109 \\ Maximum likelihood value (d) & 170.3 & 162.4 & 162.0 & 159.7 & 148.6 \\ Standard deviation (d) & 72.0 & 68.9 & 65.1 & 62.0 & 59.0 \\ \hline  & **After the 6 th** & **After the 7 th** & **After the 8 th** & **After the 9 th** & **After the 10 th** \\  & **cracking** & **cracking** & **cracking** & **cracking** \\ Maximum likelihood & 0.821\% & 0.928\% & 1.008\% & 1.110\% & 1.198\% \\ Maximum likelihood value (d) & 104 & 99 & 99 & 96 & 95 \\ Mean value (d) & 140.0 & 131.3 & 127.2 & 121.0 & 116.9 \\ Standard deviation (d) & 55.4 & 51.4 & 48.1 & 44.4 & 41.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Statistical characteristics of parameter \(\lambda\). There is a requirement for early warning of crest cracking (i.e., prediction of the cracking time) in safety management of the dam. The cracking time can be estimated using Equation (4) with a given cracking probability. Therefore, the prediction accuracy of cracking time depends on the specific value of probability adopted for early warning. As mentioned above, the computed probability related to the actual cracking time is random, hence reasonable values of probability for early warning could be selected by statistical analysis of the calculated probabilities at the observed cracking times with the updated distributions of \(\lambda\). Figure 14 shows the mean values and confidence intervals of the calculated probabilities at the cracking times for all the past cracking events. It should be noted that the mean value and confidence interval are statistically derived from the calculated probabilities at all the previously observed cracking times. It can be seen that the mean values and confidence intervals of the calculated probabilities essentially remain constant for the studied dam.