GLOFRIS global fluvial flood hazard maps of Winsemius et al. (2013) are used as flood hazard data in this work, which are developed using the GLOFRIS modelling cascade provided in Ward et al. (2013) and Winsemius et al. (2013). The GLOFRIS modelling cascade first simulates daily discharge using the PCRaster GlobalWater Balance (PCR-GLOBWB) global hydrological model (Beek et al., 2008, 2011). Based on daily discharge, daily flood volumes are simulated using the PCR-GLOBWB extension for dynamic routing, DynRout (PCR-GLOBWB-DynRout) (Ward et al., 2013; Winsemius et al., 2013). In the next step, flood volumes, for different return periods, 2, 5, 10, 25, 50, 100, 250, 500, and 1000 years, are obtained using the annual time series for maximum flood volumes by fitting a Gumbel distribution. These flood volumes are then converted into inundation maps (30 arcsec, ca. 1 km) using the inundation downscaling model of GLOFRIS (Winsemius et al., 2013). In the appendix materials, we provide flood maps for the 50- and 500-year return periods (Fig. A1). The maps show that the inundation depth highly varies in China. Railway lines in eastern coastal China and south China are faced with the most severe floods.

The main river basin used in this work is shown in Fig. 1, which includes nine river basins: Continental Basin, Haihe River basin, Huaihe River basin, Pearl River basin, Songhua and Liaohe river basins, Southeast Basin, Southwest Basin, Yellow River basin, and Yangtze River basin. The data are from the Data Center for Resources and Environmental Sciences, Chinese Academy of Sciences, accessible from the Resource and Environment Data Cloud Platform (http://www.resdc.cn/, last access: 19 May 2020).

The geographic information, time table data, and passenger capacity data of Chinese railways are collected. The geographic information of the railway system is from OpenStreetMap (OSM), which provides the spatial distribution of the Chinese railway system (Fig. 1). The timetable data, which include the daily number of trains and associated routes from the Railway Service Website, and the passenger capacity data are obtained from https://www.china-emu.cn/ (last access: 19 May 2020) for China high-speed rail (G Train, D Train, and C Train) and https://zh.wikipedia.org/wiki (last access: 19 May 2020) for others (Z Train, T Train, K Train, etc.).

To ensure the estimation is as accurate as possible for an event-based flood risk assessment based on the Monte Carlo sampling, a large number of independent flood events are required (Speight et al., 2017; Wu, 2019; Zhu et al., 2020). In the following subsections, we will describe the procedures to generate flood events, including input flood hazard maps, the function fitting procedure, and the Monte Carlo analysis, in more detail.

In this study, we assume that a flood event within one basin will produce a flood with the same intensity (return period) within that entire basin, whilst we assume that floods between different basins are independent of each other (Fraiture, 2007; Rojas et al., 2013). In this work, the flood hazard data are extracted from the GLOFRIS global fluvial flood hazard maps of Winsemius et al. (2013). To get the basin-scale flood hazard data, we divide China into nine major river basins according to the main river system.

For each grid cell, the GLOFRIS maps estimate the flood depth for the nine aforementioned return periods (2, 5, 10, 25, 50, 100, 250, 500, and 1000 years). To estimate the flood depth for any return period between 2 and 1000 years, we fit a quadratic spline function to develop an inundation depth-exceedance probability function (P) for each return period interval for each grid cell (Marsden, 1974; Vandebogert, 2017; Meshram et al., 2018). The quadratic spline is a method that uses a piecewise quadratic function to obtain the best-fitting curves. This interpolation method allows us to obtain a smooth, continuous curve through the provided flood depths for the different return periods.

The method is applied as follows, and examples of the inundation depth-exceedance probability function of grid cells are shown in Fig. 3a.

For each grid cell gx,y (x, y are respectively the horizontal and vertical coordinates of grid cell centre), the annual exceedance probability flood depth DT is calculated by Eq. (1): 1P(DT)=1T, where DT is the magnitude of a flood depth with a return period of T years, and P(DT) is the exceedance probability of DT. DT is between [D1, D1000], with D1=D2≤D5 … ≤D1000. We assume that D1 is equal to zero (i.e. 1-year event

Considering that the inundation depth-exceedance probability is from 0 to 1, when T=2-year event, the P=1/2, and when T=5-year event, the P=1/5. When we impose the 1-year event, the P=1, which can make the inundation depth-exceedance probability from 0 to 1.

To produce a time series of flood events based on the created inundation depth-exceedance probability functions, we use a Monte Carlo sampling method. The basic idea of the Monte Carlo sampling method is that when the number of simulations is sufficiently large, the frequency of an event approximates the probability of the occurrence of the event (Baker, 2008; Speight et al., 2017). The flood event generation procedure is presented in Fig. 3 and Appendix Fig. A2 and can be summarized in two steps. First, we generate independent events at each basin and combined them into a national event. For an event Eji (“i” is the sequence number of simulated flood event; “j” is the sequence number of basin number, which belongs to (1, 9)) and for each basin Bj, a random number Pji between 0 and 1 is generated from a uniform distribution. The flood depth of the cells in basin Bj for event Eji can be calculated using Pji and the inundation depth-exceedance probability function based on the assumption that a flood event in one basin will produce a flood with the same intensity. For a national-scale flood event, basin-specific floods of nine basins can be randomly combined into a national-scale flood by assuming independence between the flood events among different basins; this concept is presented in Fig. 3b. Second, we repeat this process 10 000 times to generate a set of national-scale independent flood events as presented in Fig. 3c.

For each basin to obtain 10 000-year events (we assume that 10 000 years of events are sufficient to cover almost all probable scenarios), we therefore apply a Monte Carlo method to sample 10 000 exceedance probabilities. For each of these exceedance probabilities, we estimate the inundation depth for each cell within that basin. We repeat this procedure for each basin, which results in a 10 000-year set of flood events for each basin. We then combine these sets into a national-scale flood event set by assuming independence between the flood events in the different river basins (Fig. 3c). Hence, for each of the 10 000 years, we simply take the estimated flood depths for each basin. For example, in year 1, basin 1 may have an exceedance probability of 0.5, whilst basin 2 may have an exceedance probability of 0.98. For year 1, the resulting national-scale flood map would therefore have values for a flood event with an exceedance probability of 0.5 in basin 1, a flood event with an exceedance probability of 0.98 in basin 2, and so forth. This procedure results in a 10 000-year national-scale flood event set.

We also assess the system vulnerability by calculating the impacts that could occur throughout China if a flood with a given return period were to occur within an individual basin. To do this, for each basin and each return period we draw 10 000 events for all other basins assuming independence. In total, this leads to a set of 810 000 events (10 000 events × 9 return periods × 9 basins).

Railway systems are commonly represented through spatially explicit networks as an analogy for their structure and flows (Rodrigue, 2016). This network representation can be used to calculate system performance metrics based on network theory. In this work, the Chinese railway system was modelled as a directed weighted network, which consists of a group of nodes (stations) connected by edges (railway lines) with daily train trips, where the edges have a travel direction associated with them. Based on the geographic information of the railway system and the timetable data (see Sect. 2.1.3), we build the Chinese railway network. As our method is primarily concerned with flood risk along rail segments between cities and not within cities, for simplicity, we combine multi-stations into one node using the location of the highest-capacity station in each city. In total, 2240 nodes are combined into 1790 nodes. The final extracted railway network has a total length of 90 600 km for (merged parallel) lines connecting two stations, consisting of 1973 edges and 1790 nodes (Fig. 1). Figure 1 shows the spatial distribution of the railway network and average daily number of trains. Topology and traffic flows vary greatly in space. The network density reduces greatly moving from eastern China to western China. For the traffic flow, the railways connecting large cities, like the railways from Beijing to Guangzhou, Harbin, and Shanghai and railway from Shanghai to Changsha have higher flows.

We assume that a railway is impassable when the water level on the railway line is higher than the failure threshold Wd of the railway service after drainage (CRPH, 2012; Espinet et al., 2018). The water level after drainage WLx,y of grid cell gx,y is calculated by Eq. (4): 4WLx,y=DTx,y-Wldx,y⋅Dc, where DTx,y is the flood depth of a flood event, Wldx,y is the water level of the design standard of grid cell gx,y, and Dc is the drainage capacity rate.

The rail segment lij between two stations' failure condition is defined by Eqs. (5) and (6): 5Fcij=∏xyijZxy,6Zxy=0,WLx,y≥Wd1,WLx,y<Wd, where Fcij is the failure condition of component lij, which has two states, namely normal (denoted by 1) and disrupted (denoted by 0). Zxy is the failure condition of grid cell gx,y; when the water level after drainage is larger than Wd, Zxy=0; otherwise, Zxy=1.

In this study, we consider a failure threshold of 0.2 m after drainage, according to the railway transportation emergency plan (CRPH, 2012; Espinet et al., 2018). The flood design standard of the culverts, bridges, and embankments of the Chinese national railway system is designed for 100-year water depth, according to the standard for flood control (CRPH, 2016). Furthermore, we assume that the drainage capacity rate is 0.8

The value and the concept of the drainage capacity rate are from Espinet et al. (2018), which is defined as the drainage capacity of embankment, bridge, and culvert. In this work, the value is 0.7 for bridges and culverts in Mozambique. Considering China is more developed than Mozambique, we assume the infrastructure in China has a higher drainage capacity, and a value of 0.8 is assigned.

Failure hotspots of railway segments lij can be calculated by Eq. (7): 7AFij=∑eEFcijeN, where AFij is the failure probability to the railway segments, E is the N-year flood event catalogue, and Fcije is the failure condition of railway segment lij under flood event e.

Once a flood occurs, trains may be affected in two ways: (i) increased travel time or (ii) cancellation. The number of daily affected trains Netol is calculated by Eq. (8): 8Netol=Nec+Ned, where Nec is the number of daily cancelled trains and Ned is the number of daily detoured trains after a flood event.

We assume that the average number of passengers is 80 % of the train's capacity (Rezvani et al., 2015; Wei et al., 2017). Therefore, the number of affected passengers Petol can be defined by Eq. (9): 9Petol=∑iNetolCAi⋅0.8, where CAi is the capacity of the ith train.

Once a flood occurs, some trains will detour to complete their journeys. The daily detoured trains Ned can be calculated based on four assumptions as follows (in order of descending priority), which is also presented in Appendix Fig. A3:

Stations are not repeated along the routes.

The total increased time Tetol for detoured trains is calculated by Eq. (11): 11Tetol=∑iNdTie-∑iNdTi, where Tie is the running time of the ith train under flood event e, and Ti is the original travelling time of the ith train.

The average increased time is calculated by Eq. (12): 12Teave=TetolNed, where Teave is the average increased time under flood event e.

Once a flood occurs, some trains may be cancelled if there is no alternative route possible or when the re-routing time is too long (greater than 24 h). The daily cancelled trains Nec are calculated by Eq. (13): 13Nec=NS-Nes, where Nes is the number of running trains in the system after a flood event, and NS is the original number of trains in the system.

The daily passengers influenced by cancelled trains Pec can be defined by Eq. (14): 14Pec=∑iNecCAi⋅0.8.

Each performance loss metric is calculated for each flood event. System vulnerability curves are generated to present the relationship between performance loss and flood intensity (return period). We use the expected daily affected trains, cancelled trains, detoured trains, affected passengers, and increased time for detoured trains to present the flood risk to the railway system according to Eq. (15): 15ARs=∑eEVeN, where ARs is the expected daily flood risk level to the railway system, E is the N-event flood catalogue, and Ve is the performance loss metric, i.e. Ned, Nec, Netol, Ped, Pec, Petol, Tetol, and Teave under flood event e, which are defined in Eqs. (8)–(14).

By applying an uncertainty analysis (UA), we identified the range of model output for imprecisely known input parameters (De Moel et al., 2012). A sensitivity analysis (SA) aims to determine the parameter effect on the model output (Koks and Haer, 2020). Parameters with greater effect should attract more additional attention to deal with the uncertainty they bring (Koks and Haer, 2020; De Moel et al., 2012). Detailed methods of uncertainty and sensitivity analysis can be found in previous studies by De Moel (2011) and Koks and Haer (2020).

In this study, we make assumptions on the train disruption threshold using three parameters (the water level failure threshold, drainage capacity rate, and design standard) based on emergency code and design code standards (CRPH, 2012). However, it should be noted that these standards are not known exactly for each asset and will change over time, such as dynamically changing protection standards and ageing infrastructure. Within a railway system, a lot of different asset types exist, with varying design standards. This implies that the capacity to cope with the hazard does vary from location to location. Therefore, it is worthwhile to perform a sensitivity analysis on these key parameters (De Moel and Aerts, 2011; Horacio et al., 2019). Hence, we perform an uncertainty and global sensitivity analysis in which we assess the performance loss metrics for a range of different values for these parameters. For water level failure, we use a range between 0.1 and 0.5 m. For the drainage capacity rate, we use a range between 0.7 and 0.9, and for the design standards, we use a range between 50 and 100 years. The list of all assumptions taken in this study and their range in the sensitivity analysis can be found in Appendix Table A1. In total, we create a set of 1000 different random parameter value combinations in the sample space.