## 1 Introduction  
For favorable adaptability of dam foundations, full utilization of local materials and building-excavated materials, low construction cost, and low cement consumption, earth-rockfill dams are one of the most promising dam types in dam construction [1; 2]. In order to obtain hydropower, many high earth-rockfill dams have been built in China, such as the Nuo Za Du (261.5 m high), Chang He Ba (240 m high), Pu Bu Gou (186 m high), Xiao Lang Di (154 m high) and Shi Zi Ping (136 m high). Catastrophic consequences can result from dam breaks, thus the safety of these high earth-rockfill dams has been a matter of increasing concern. According to the previous statistical works reported in the literature [3; 4; 5] crest cracking is one of the most common damage types for earth dams and embankments. Crest cracking had also been observed in some of the high earth-rockfill dams mentioned above (i.e., the Nuo Za Du, Pu Bu Gou, Xiao Lang Di and Shi Zi Ping dams) during construction or operation period.  
It is well known that crest cracking is closely related to the deformation behavior of earth dams and embankments, specifically the differential settlements [3; 6; 7; 8; 9; 10; 11; 12; 13; 14]. The deformation of high earth-rockfill dams can be accurately predicted using the Finite Element Method (FEM) with calibrated model parameters based on the in-situ deformation monitoring data [15; 16; 17; 18]. The crest cracking hazard of high earth-rockfill dams can be evaluated with the settlement inclination computed from FEM [3; 10]. If the computed settlement inclination is greater than a certain threshold, the dam crest is prone to crack, but it does not mean that the dam crest will crack immediately. However, if the abnormal state (i.e., the settlement inclination of the dam crest exceeds the threshold value) lasts for a long time, the potential of crest cracking becomes large and increases with the duration of abnormal state. Due to complex physical mechanisms and stochastic factors, the duration of abnormal state before crest cracking is uncertain and can be regarded as a random variable. Consequently, the probability of crest cracking during the period of abnormal state can be calculated.  
Since the relevant historical data is scarce, especially for a specific earth-rockfill dam, the Probability Distribution Parameters (PDPs) for the duration of abnormal state before crest cracking cannot be obtained by classical (frequentist) statistical analysis. Fortunately, Bayesian theory can be applied to estimate the probability distribution of the PDPs given the observed time intervals between the starting of abnormal state and the occurrence of crest cracking [19]. Moreover, the probability distribution of the PDPs can be dynamically updated along with the accumulation process of the observed evidence. Therefore, in the Bayesian framework, crest cracking risk of earth-rockfill dams can be dynamically evaluated according to the progressively updated probability distribution of the PDPs. In addition, early warning of the crest cracking can be issued in terms of the probabilistically estimated cracking time range.  
The objectives of our works are mainly to address (1) how to evaluate crest cracking risk of earth-rockfill dams with duration of abnormal state of crest settlement, (2) how to estimate the cracking times corresponding to a given probability interval for early warning of crest cracking and (3) how to update the probability distribution for parameter \(\lambda\) in the formulation of crest cracking risk. The intellectual merit of our works lies in the idea of using duration of abnormal state of crest settlement to evaluate crest cracking risk of earth-rockfill dams. The remainder of this paper is organized as follows. In Section 2, brief descriptions of an actual earth-rockfill dam and its crest cracking are presented. In Section 3, the methodology is detailed. In Section 4, the results and discussion for the case study are given. In Section 5, some conclusions are drawn.