# Probabilistic Seismic Safety Assessment of Railway Embarnments
## 1 Introduction  
Transportation systems play a significant role in the world's economy. They may affect economic growth and thus, they can be considered key facilitators of sustainable development [1]. Railways are known as fundamental components of transportation networks [2]. They can provide a safe platform for large-scale transportation of goods and passengers over long distances in reasonable durations [3]. Accordingly, damage to railway infrastructures due to natural hazards (e.g., earthquakes, floods, etc.) may lead to significant consequences [4; 5]. Strong ground motions can be considered potential threats to railways. Improper performances of railways during earthquakes revealed that they are seismically vulnerable. For instance, The Izmit earthquake in Turkey in 1999 interrupted rail transportation because of track failure [6]. Following the Niigata Ken Chuetsu earthquake in Japan in 2004, several local train lines were temporarily closed because of track failure [7]. Most Northern Line railway services in New Zealand have been suspended for more than 100 days following the Kaikoura earthquake in 2016 [8]. Earthquake-induced damage to railways can occur in several forms, such as failure of cuts and slopes, damage to infrastructures (such as bridges and tunnels) and track deformation due to fault surface rupture [6; 9]. In addition to the aforementioned failure modes, embankment failure is one of the most frequently observed forms of damage following seismic events. The importance of proper seismic performance for railway systems has highlighted the necessity of assessing the seismic safety of railway structures. Quantifying the potential for damage as a function of earthquake intensity is a substantial requirement for the seismic safety evaluation of a particular structure. Fragility curves relate the probability of reachingor exceeding a certain damage state to the level of seismic hazard. Hence, they can provide an appropriate framework for railway seismic safety assessment [10; 11; 12]. Several methodologies have been developed to generate fragility curves, i.e., empirical, analytical, expert judgment and hybrid methods [13; 14; 15]. Several researchers performed empirical fragility curves with respect to post-earthquake damage data and observations. The lack of observed damage surveys on the studied structures for different intensities of seismic loading is the main drawback of this method. Moreover, a rapid fragility-curve estimation can be developed considering expert judgment. In addition to the above two methods, analytical techniques have become more popular because of the recent developments in computer and software capabilities. In other words, they make the analysis of a problem at different boundaries and seismic loading conditions possible. Moreover, when one of the above methods cannot sufficiently cover the entire range of the fragility curve, a hybrid method is used. This method combines two or more of the above methods of fragility analysis. The seismic behavior of an embankment can be established using several test methods, including the shaking table [16] and dynamic centrifuge test [17]. Although laboratory tests expose the seismic behavior of the embankment, they are costly. Moreover, in some cases, there are some limitations to utilizing full-scale test models because of the capacity and size of the testing apparatuses. Accordingly, employing numerical techniques instead of test methods can be an appropriate way to simulate the seismic performance of embankments. However, numerical methods have shortcomings themselves. Numerical instability and the divergence of results are some of the common problems encountered during numerical analysis. During the past decades, several researchers have focused on numerical analysis methods in geotechnical engineering [18; 19; 20; 21; 22; 23; 24; 25]; however, to the knowledge of the authors, reported literature on the seismic vulnerability of embankments has been very limited so far. [PERSON] et al.