Incorporating geographic interdependencies into the resilience assessment of multimodal public transport networks

.


Introduction
The effective functioning of the public transport system is placed at risk by naturally-occurring, spatially-defined events such as earthquakes, rainfall, flooding and landslides.These events have the capacity to disrupt transport infrastructure and, in areas affected by such events, discrete transport networks carrying separate public transport modes (e. g. railway and bus) are at risk of concurrent disruption.For example, heavy snowfall in March 2018 caused widespread railway and road closures in Scotland affecting train and bus services (Network Rail, 2020).More recently, in July 2021, heavy and prolonged rainfall caused extensive flooding in central Germany and Belgium, leading to damage of railway lines, roads and bridges (Fathom Global, 2021).Humancaused climate change is expected to intensify and alter patterns of extreme weather around the world in the coming decades, including increases in heavy precipitation, flooding and heat, thereby increasing the risk of transport disruption (Lee et al., 2023).
Infrastructure networks which are located in close proximity to each other are said to be geographically (or spatially) interdependent if a hazardous event can disrupt both networks at the same time (Rinaldi et al., 2001;Zimmerman, 2004;Dudenhoeffer et al., 2006).As such, discrete public transport networks located in corridors defined by the natural landscape or built environment are particularly susceptible to concurrent disruption.From the public transport users' perspective, geographical interdependency reduces the added benefit of flexibility which exists when more than one transport mode is available.
The resilience of a transport system represents its ability to resist, absorb, adapt and quickly recover from the impact of hazardous events (Jenelius and Mattsson, 2020).Redundancy is a key component of transport system resilience (Bruneau et al., 2003) and in its simplest form is equal to the total number of (feasible) options, such as paths and transport modes, which exist between locations (Berdica, 2002).By weighting each option by a function of travel deterrence, a generalised measure of redundancy is obtained which is equivalent to established measures of accessibility (Ben-Akiva and Lerman, 1985;Anas, 1983;Xi et al., 2018).High levels of redundancy help maintain connectivity between locations in the event of disruption.However, it is also evident that spatially-defined events may reduce redundancy in geographically interdependent systems.Although a number of studies have considered the redundancy of single and multimodal networks with shared infrastructure (e.g.Frappier et al., 2018;Liao and van Wee, 2017), and whilst there is a growing body of research addressing geographical interdependencies between civil infrastructure systems (Patterson and Apostolakis, 2007;Johansson and Hassel, 2010;Pant et al., 2016;Thacker et al., 2017;Kays et al., 2023), to the authors' knowledge there has been no attention paid to the effect of geographical interdependencies on the redundancy of discrete transport networks carrying separate public transport modes.
Recently the concept of substitutability was introduced into the literature (Van Wee et al., 2019).Substitutability is defined as the level of accessibility which is preserved in the event of the unavailability of a preferred option and reflects the capacity of the transport system to absorb the impact of disruptive events.As with redundancy, the level of substitutability will be reduced as a result of geographical interdependencies between public transport modes.A comparison of the concepts of redundancy and substitutability, with and without consideration of geographical interdependency, is given in Fig. 1.Whilst the attractiveness of preferred option (A) relative to the attractiveness of other options (B, C and D) does not affect the level of redundancy (or accessibility) offered by the set of options, it does affect the level of substitutability.Consequently, it is argued that both redundancy and substitutability can give important and complementary insights into public transport system resilience when considered within an accessibility framework.
Redundancy and substitutability are closely related to but distinct from the robustness of a transport system which reflects its capacity to withstand an adverse event without disproportionate consequences (Bruneau et al., 2003).Thus, they contribute to robustness (Agarwal, 2015;Van Wee et al., 2019) and are, by extension, negatively correlated to vulnerability.Previous studies have assessed the vulnerability of single and multimodal transport systems in terms of loss in network performance (from a topological or system-based perspective) that would result from the failure of network infrastructure (Mattsson and Jenelius, 2015).This is achieved by failing one network component at a time to identify those components which are most critical to network performance (Rodríguez-Núñez and García-Palomares, 2014; Cats and Jenelius, 2014).A variant of this approach, known as the cell-space method, has been designed to capture the vulnerability of geographically interdependent networks to spatially-defined events, which involves overlaying the networks with a regular grid of cells and then computing, for each cell in turn, the impact of the failure of all infrastructure within cells (Johansson and Hassel, 2010;Ouyang, 2014).The vulnerability of multimodal transport networks with geographic interdependencies has been also evaluated by linking road and railway networks at points where a motorway bridge crosses the railway network and vice versa (Ferrari and Santagata, 2023).
Whilst transport vulnerability has been comprehensively researched in recent years, these studies often reveal issues stemming from a lack of system redundancy.Therefore, gaining insight into the redundancy (and substitutability) of a system has the potential to shed light on the root cause of these issues and to identify key areas for improvement.Methodologically, vulnerability studies require the development of disruption scenarios with which to test the network whereas redundancy analyses do not require any prior assumption of disruption.Moreover, these scenarios are predominately based on the failure of single links (or adjacent links within defined cells) whereas redundancy focusses on the quality of route options between Origin-Destination (OD) pairs.This route-based approach is particularly pertinent when considering geographical interdependency since higher levels of interdependency reduces the level of redundancy between OD pairs which is not something that would be evident from vulnerability analysis.This study aims to assess the impact of geographic interdependencies on the redundancy and substitutability of discrete public transport infrastructure networks.This aim is achieved by developing two resilience measures which consider the contributions of each network to accessibility whilst accounting for the spatial proximity between network components.The measures are then applied to the Scottish public transport network consisting of long-distance bus and railway services.
The rest of the paper is organised as follows.Section 2 provides a review of relevant literature and Section 3 includes the research methods.In Section 4 the case study is presented, followed by the results in Section 5. Finally, Section 6 includes conclusions and discussion.

Key literature on the redundancy and substitutability of transport networks
In the past decade, the redundancy of transport networks has received growing attention.Xu et al. (2018) proposed two measures capturing the number of connections with realistic travel times between locations and applied these to an urban road network.Similar approaches were developed by Jing et al. (2019) who measured redundancy between each pair of stations in a metro network, and by Yang et al. (2016) who proposed a network-level redundancy index reflecting the average number of routes between all stations of an urban rail network.The metrics discussed above assess redundancy from a topological perspective.Taking a different approach, Liao and van Wee (2017) developed an indicator for the diversity of travel options considering each option's travel time and the extent of overlap between options, and subsequently applied it to a regional multimodal network.Mamun et al. (2013) measured the redundancy of bus travel options between census zones by developing and applying an index that considered the number of available routes and their associated travel times.Li et al. (2024) proposed route diversity measures reflecting the number of routes between origin-destination pairs, their corresponding travel costs and the level of overlap between them, and subsequently used these to evaluate the redundancy offered by urban multimodal bus and metro networks.
In contrast, limited work has been undertaken on substitutability within the context of resilience.Van Wee et al. (2019) defined substitutability as the reduction in accessibility occurring when the preferred option becomes unavailable.Building on this idea, Bondemark et al. (2021) measured substitutability between transport modes as the reduction in accessibility of locations, when a mode becomes unavailable.Chan et al. (2023) developed a metric that captures available options based on their travel time and monetary cost, and further examined how unappealing alternatives could be adjusted to improve spatial equity of options, thus providing insights into how substitutability of options can be enhanced.Another body of research has explored substitutability between travel modes from an impact-based perspective.For example, Ouyang et al. (2015) measured the extent to which rail can substitute the airline network of China when the latter experiences disruptions, by comparing the accessibilities of locations offered by the airline network with and without considering the substitute rail services.Taking a similar approach, Hong et al. (2017) examined how well the urban bus and subway networks can substitute for each other, when one of them is disrupted.Although still in its infancy, current research shows that, by focusing on the attractiveness of options, substitutability provides further insights into the characteristics of resilience of networks, and how well alternative options can replace a preferred one.
These substitutability measures pertain to events affecting only one option at a time and cannot be applied directly to the situation when area-wide events concurrently disrupt more than one modal option.While geographic interdependencies have been explored between transport and other networks (e.g., Dong et al., 2020;Li et al. (2019); Pant et al., 2016;Patterson and Apostolakis, 2007;Zorn et al., 2020), no study has assessed this interdependency between transport modes which use discrete infrastructure networks.

Network representation
Each public transport network is represented by a graph G consisting of a set of nodes N = {n 1 , n 2 , …, n K } and a set of segments E = {e 1 , e 2 , …, e M }, where K is the number of nodes and M is the number of segments.Each node corresponds to a public transport stop and each segment represents the transport infrastructure connecting two nodes (e. g., railway tracks, roads) consecutively serviced by a public transport trip.The segment is characterised by a shape, which is dictated by the travel paths that public transport vehicles take along the infrastructure.A set of trips, denoted by L = {l 1 , l 2 , …, l S } operates on each segment, where S is the total number of trips.Each trip is associated with a schedule defined by a sequence of nodes.Thus a trip l is expressed as l = {n lO , n l1 , n l2 , …, n lT }, where n lO is the origin station and n lT is the terminal station of the trip.
In the following sections indicators of redundancy and substitutability are proposed which reflect the potential for concurrent disruption to alternative routes from spatially-defined events.These indicators are extensions of previous work which adjusted redundancy or substitutability on the basis of the shared use of infrastructure as discussed above.Here, the adjustment reflects the extent to which alternative routes fall within a specified distance of each other and thus would be exposed to the impact of the same event.
Redundancy can be viewed as the total accessibility offered by alternative routes minus the contribution to accessibility from spatially proximate infrastructure.Similarly, substitutability can be viewed as the total accessibility minus the accessibility offered by the preferred route and the contribution to the remaining accessibility from spatially proximate infrastructure.

Redundancy Indicator
The following redundancy indicator is based on an indicator of "robustness"1 proposed by Liao and van Wee (2017) which is adapted in this paper to consider the degree of geographic interdependency between two networks by introducing a term based on spatial proximity.
When considering a single mode m 1 , let p m1 denote the least-cost route connecting an origin-destination pair, OD.The accessibility offered by m 1 is expressed in Eq. ( 1) (Liao and van Wee, 2017).The negative exponential form is derived from the widely used gravity-based measure, based on which higher costs of options result in lower accessibility values (Geurs and Van Wee, 2004).
Where C ( p m1 ) represents the cost of travel along p m1 and β m1 indicates the maximum travel cost acceptable to travellers.If the indicator refers to the same region and the same types of destinations, then β m1 can be set arbitrarily as all indicator values are corrected equally (Liao and van Wee, 2017).Now consider that mode m 2 is an alternative to m 1 .Following Liao and van Wee (2017), the measure in Eq. ( 1) can be extended by adding the accessibility offered by m 2 , acc m2 OD .When routes p m1 and p m2 are not spatially proximate to each other, and, thus, not subject to geographic independencies, the contribution of m 2 to the overall level of accessibility is acc m2 OD and hence the redundancy between OD is the sum of acc m1 OD and acc m2 OD , as shown in Eq. ( 2) below.This extended measure reflects the degree of redundancy offered by the two modes.
However, in case that p m2 is in the vicinity of p m1 , a correction factor is introduced to account for the geographic interdependency between m 1 and m 2 .This factor is referred to as the neighbourhood coefficient and denotes the share of length of p m2 which lies within a specified distance of p m1 .This formulation is presented in Eq. ( 3).The reduced contribution of m 2 to the accessibility of OD via m 1 , due to the proximity between the routes is included in Eq. ( 4). where ) is the length of p m2 , and l ( p m2 , p m1 ) is the length of p m2 in the neighbourhood of p m1 .The neighbourhood of routes was represented by buffer zones constructed around them.Fig. 2 illustrates an example of p m2 lying partially in the neighbourhood of p m1 .
Eq. ( 4) includes the positive effects of the alternative mode m 2 to the primary mode m 1 in connecting a pair of locations, whilst considering geographic interdependencies.
Similarly, acc m2←m1 OD reflects the redundancy offered by m 2 and m 1 , when taking into account the spatial proximity of the two routes.These two indicators are not necessarily equal, because the share of length of p m2 in the neighbourhood of p m1 is not equal to the share of length of p m1 in the neighbourhood of p m2 , and therefore both indicators were computed.For example, for a pair of locations where accessibilities acc m1 and acc m2 are approximately equal, if the share of p m2 in the neighbourhood of p m1 is lower than the share of p m1 in the neighbourhood of p m2 , i.e., R c ( , then the redundancy acc m1←m2 will be higher than acc m2←m1 .This in turn indicates that using m 1 as the primary mode of travel and m 2 as alternative provides more resilient connectivity between the pair of locations than when m 2 is the primary mode. If p m2 is entirely within the neighbourhood of p m1 , then R c ( p m2 , p m1 ) takes the value of 1, and therefore acc m1←m2,y=1 OD takes its minimum value given by Eq. (1), while if p m2 is entirely outside of the neighbourhood of ) is zero and, as such, acc m1←m2,y=1 OD takes its maximum value (Eq.( 2)).
The redundancy indicator can be aggregated by origin i or destination j, as shown in Eqs. ( 5) and ( 6) respectively, to assess the redundancy of travel options from each origin to all zones (Eq.( 5)) or from all zones to each destination (Eq.( 6)).
where j = {1, 2, …N} is the set of destinations and i = {1, 2, …M} is the set of origins.For this study, the redundancy indicator was used to determine the loss in redundancy because of area-wide events concurrently affecting two modes m 1 and m 2 .
The opportunities available at the destination zones were not considered, as the scope of this work focuses on the impacts of geographic interdependency on the connectivity offered by two discrete networks, rather than the wider implications on the activity system.
In the absence of empirical data, it was assumed that both β m1 and β m2 were equal to 12 h which was considered as the maximum travel time that users of both m 1 and m 2 are willing to travel.In reality these coefficients may not be equal because users may place different limits on the maximum time they would be willing to spend on different modes, for example due to variations in the levels of comfort provided.
The redundancy indicator was computed as follows: Case I.The positive effects of m 2 were added to the level of accessibility, without accounting for geographic interdependencies between m 1 and m 2 .
For each OD pair, the neighbourhood coefficient was zero, and therefore acc (m1←m2)I ij was estimated from Eq. ( 2).The redundancy indicator was computed for each origin zone, Acc (m1←m2)I i using Eq. ( 5).
Case II.The positive effects of m 2 were added to the level of accessibility, accounting for geographic interdependencies between m 1 and m 2 .
For each OD pair, the redundancy acc (m1←m2)II ij was quantified using Eq. ( 4).The value of neighbourhood coefficient was computed using Eq.

Substitutability indicator
The model of substitutability developed by Van Wee et al. ( 2019) was adapted to incorporate geographic interdependencies between two modes.Van Wee et al. ( 2019) defined substitutability as the change in accessibility when the least-cost option is unavailable (Eq.( 9)).The normalised substitutability, which ranges between 0 and 1, is shown in Eq. ( 10).When the normalised measure is close to 0, the substitutability between the OD pair is very poor, whilst when the value is 1, the preferred option can be fully substituted by alternatives without any accessibility loss.
where S OD is the degree of substitutability for an OD pair and ŜOD is the normalised substitutability measure.LS OD is the total accessibility (i.e. the maximum expected utility) of all options between OD under normal conditions and LS Y=i OD is the accessibility of remaining options when the preferred choice i is unavailable.
Although Eqs. ( 9) and (10) remain unchanged, the remaining utility was adapted to incorporate the geographic interdependency between m 1 and m 2 .When both p m1 and p m2 are available, where the former is the preferred option and the latter is the alternative, their maximum utility, LS m1←m2 OD ,is as shown in Eq. ( 11).When p m1 becomes unavailable and the two routes are distant from each other, and thus there are no geographic interdependencies, the remaining utility LS m1←m2,y=1 OD is given in Eq. ( 12).
where acc m1 OD and acc m2 OD can be computed using Eq.(1).However, when p m2 is in the vicinity of p m1 , the neighbourhood coefficient of Eq. ( 3) is introduced to reflect the potential for that part of p m2 which lies in the neighbourhood of p m1 , to be concurrently disrupted.The remaining utility in this case is shown in Eq. ( 13).
In case that p m2 is entirely within the neighbourhood of p m1 , and thus R c ( p m2 , p m1 ) is one, the remaining utility of Eq. ( 13) approaches negative infinity and, therefore, substitutability is zero.
Similarly with redundancy, the normalised substitutability indicator was obtained for origins, without accounting for geographic interdependencies between the two modes (Case I) and when accounting for them (Case II).In the former case, the neighbourhood coefficient of substitute routes was set to zero, while in the latter case it was computed using Eq.(3).Consequently, the absolute and relative losses in substitutability for the origins were computed in a similar way to the indicator of redundancy.

Application to rail and long-distance bus networks in mainland Scotland
The services provided by the public transport system range from local, operating mainly in urban areas (e.g., local bus, subway), to regional and long-distance services, which provide connectivity between cities and regions (e.g., certain railway, long-distance bus/coach services).Because the focus of this work is on services that can act as alternatives to each other, modes operating on the same functional scale were selected.Thus, discrete public transport networks for long-distance travel consisting of railway and coach/bus services in mainland Scotland were used to illustrate the application of measures described in Section 3.
The main data sources for the networks were the publicly-available General Transit Feed Specification (GTFS) data for railway (Association of Train Operating Companies, 2020) and bus (Traveline, 2019).For each mode, the data contains information on the stops, routes, trips and timetable of services of the relevant operating companies.The GTFS datasets were initially filtered based on the rail stations and bus stops located within the mainland of Scotland.Contrary to the rail dataset that Fig. 3. Travel paths of the long-distance public transport network in mainland Scotland, along with selected Scottish cities and towns and their respective population.relates to regional and long-distance routes exclusively, the available dataset for bus made no distinction between urban and long-distance services (i.e.all bus routes were recorded as belonging to the same route type) and, therefore, it was not possible to directly extract data associated exclusively with long-distance bus travel.While most longdistance services in Scotland are operated by certain companies, such as National Express, Stagecoach and Megabus, there are multiple operators for urban travel that also provide longer-distance services.Therefore, with a view to retaining only long-distance bus routes, the length of bus routes was used to characterise long-distance travel.Because there is no consensus on how long-distance travel is defined (Aultman-Hall, 2018), with thresholds ranging between 24 km in the UK (van de Velde, 2013) and over 80 km in the United States (Outwater et al., 2015), a 30 km threshold was selected in this study which removed bus routes within urban areas and also between adjacent urban areas from the final dataset.OpenTripPlanner (OTP) (OpenTripPlanner, 2022) was used to obtain travel distances between the termini, and only those routes longer than 30 km long were retained.
The GTFS data was then related to spatially accurate models of the railway and bus infrastructure networks.For bus, the OS MasterMap Highways Network (Ordnance Survey (GB), 2021) was used and, for railway, a model was created using spatial data on railway lines and junctions provided by Network Rail (Network Rail, personal communication, 7 June 2021).In each case, routes were constructed by finding the shortest path between consecutive stops/stations for each unique service contained in the GTFS data.The final representation of these two discrete public transport networks is shown in Fig. 3.
To measure accessibility between locations, a grid was formed dividing the study area into cells representing travel zones.A hexagonal grid was selected because it is preferred when exploring connectivity (Birch et al., 2007).Using the same reasoning as that for the identification of long-distance bus routes, the cell size was selected to have a 30 km long diagonal to exclude short trips.Proximity analysis between localities (cities and towns in Scotland with population of more than 500) revealed that only 8% of locality pairs were related to separation distances of less than 30 km.These are adjacent localities (e.g., Prestwick to Ayr) forming parts of the same continuous urban area and thus travel between them was not considered long-distance.Therefore, it was concluded that the selected cell size was appropriate for this work.For zones partly located outside of Scotland, only the part lying within the country was considered in the analysis.Furthermore, zones separated by un-spanned stretches of coastal water were subdivided to avoid problems in connectivity with the rest of the zoning system.As the focus is on locations serviced by both modes, only zones containing at least one stop of each mode were considered.
OTP was used to estimate the least-cost routes between zones.Two joint public transport-walk networks were constructed, one for each mode, using the street network provided by OpenStreetMap (Open-StreetMap, 2022) and relevant GTFS dataset.Owing to the nature of the GTFS data, the models were schedule-based, rather than frequencybased, as they use the actual timetable of services.Subsequently, route analysis was performed between all OD pairs.The day of journeys was set to Monday and potential times of departure were set for a time window between 7:30 am and 9:30 am, the former time being the earliest possible time of departure and the latter being the latest.This time window was selected to be relatively long to avoid excluding from the analysis infrequent rail and bus services that connect rural and remote areas.
Since access to long-distance services can be achieved via various modes, e.g.walking, bus and taxi, the start and end point of travel were taken to be the stops closest to the centroid of origin and destination zones.The maximum walking distance when transferring between services was restricted to 5 km to prohibit excessive walking, and the maximum number of transfers was set to 2 on the assumption that travellers making longer-distance journeys have a higher willingness to transfer between routes than those on shorter journeys.
Then, route analysis was performed between all combinations of candidate stops in the origin and destination zones, and the least-cost route identified across all possible departure times within the twohour time window was selected on the basis of travel time.
The output itineraries derived from OTP included the start and end journey times, number of transfers, duration of journey, service (or services) used, boarding and alighting stops, and boarding and alighting times for each service.Owing to the travellers' flexibility to start their journey within the defined time window, there was no waiting time at the start of each journey, however the waiting time arising from transfers between services was included.A spatial representation of these itineraries was then constructed by identifying the routes of services used by the traveller and linking them to the spatially accurate travel paths of networks (Fig. 3), to enable geographical proximity between alternative bus and railway routes to be estimated.

Results
Based on the travel times of least-cost routes between OD pairs, the accessibility offered by bus and rail routes were calculated using Eq. ( 1).For simplicity, routes connecting OD pairs served by both modes will be henceforth referred to as alternative routes.Fig. 4 shows the distribution of accessibility values characterising the alternative routes.
Fig. 4 shows that high accessibility values, especially those higher than 0.8, are more prevalent to each mode than low values.Furthermore, the distributions show that travel by rail is associated with slightly higher accessibility than bus which was expected as travel times by train are generally lower than travel times by bus.
To assess the geographic distribution of accessibility offered by rail and bus jointly, accessibility values of routes were aggregated for origins and destinations (Eqs.( 5) and ( 6)), as shown in Fig. 5.
Fig. 5 reveals significant similarities in the accessibility for travel from origins and to destinations.In both cases, a cluster of high accessibility is observed in the central part of Scotland ("Central Belt"), where the largest and most populated cities are located, such as Glasgow and Edinburgh.Zones containing less populated localities in North Scotland (e.g., Oban, Mallaig, Inverness) and South Scotland (e.g., Ayr, Stranraer, Dumfries), but also in the area above the Central Belt are associated with lower accessibility values.Finally, other zones where smaller settlements are located (e.g., Elgin, Thurso, Wick) are related to the lowest accessibility values.

Redundancy of options accounting for geographic interdependencies between modes
To assess how the positive effects of redundancy diminish due to potential geographic interdependencies, the redundancy indicator (Eq.( 4)) was computed for various buffer widths and the losses in redundancy due to geographic interdependencies were obtained (Fig. 6).
Fig. 6 reveals that, as expected, for hazards with a 100 m footprint, the redundancy of OD pairs exhibits slight losses, and as the buffer widths increase, the losses continue to increase gradually.For the most localised hazard, the relative losses for most OD pairs are less than 15%, which indicates that only a small part of the alternative route is within the 100 m-wide buffer of the primary route and, thus, the contribution of former to the redundancy is reduced only by a small percentage.On the other hand, for the most large-scale hazard, losses can be as much as 60%, revealing that a significant part of the substitute route is in the 10 km-wide buffer of the primary, consequently reducing markedly the contribution of substitute mode to the redundancy.Post-hoc comparisons of redundancy values were performed between the case where proximity is ignored and all other cases using Dunn's test (Table S1) which revealed that the losses are statistically significant in all cases, indicating that, regardless of their spatial extent, hazards markedly influence redundancy of travel options.
For small buffer widths, the outliers of Fig. 6 represent location pairs related to particularly high redundancy losses and thus particularly susceptible to geographic interdependencies.The OD pairs related to outliers for 100 m-wide buffers, along with the sections of routes that contribute to this susceptibility, are shown in Fig. 7.
The results reveal that the most susceptible OD pairs are similar for rail and bus (Fig. 7(a) and Fig. 7(b)).It is observed that these pairs of locations share the same region, particularly the northern part of the country, while a few OD pairs are also scattered in the rest of the country.Sections of routes that contribute to these high losses are identified in Fig. 7(c) and Fig. 7(d).Most of these sections are located in North Scotland and are located in very close proximity to each other, thus revealing that even the most localised events may concurrently disrupt these alternative rail and bus routes and result in high accessibility losses.
Finally, the redundancy indicator was aggregated by origin to ascertain the geographic distribution of potential redundancy loss due to geographic interdependencies, shown in Fig. 8. Based on the mean relative losses in redundancy between OD pairs due to geographic interdependencies (Table S1), the buffer widths of 100 m, 1.5 km and 10 km were selected for the zone-level aggregation.
Fig. 8 reveals that, for the 100 m hazard footprint, all zones experience low absolute losses in redundancy (Fig. 8(a)), and only two origins suffer slightly higher relative losses (Fig. 8(d)).For the 1.5 km footprint, absolute losses in redundancy increase for densely-populated zones in Central Belt (Fig. 8(b)).For the rest of Scotland, while absolute losses remain low, they are high in relative terms (Fig. 8(e)), ranging for most zones between 13% and 26%, but also 26% to 39% in some cases.This indicates that as the hazard footprint increases, zones may lose a relatively large percentage of their initial redundancy due to alternative routes connecting them being concurrently disrupted.Finally, for the 10 km footprint, absolute losses are very high in Central Belt, and considerably lower in the rest of the country (Fig. 8(c)).However, when  considering these in relative terms (Fig. 8(f)), less populated zones outside of Central Belt are the most susceptible to losses arising from geographic interdependencies.Generally, it is observed that relative losses due to large-scale hazards result in significantly high losses across the entire country and especially in less populated zones, revealing that in many cases those zones may lose larger percentage of their initial accessibility than urban zones, when the correlated risk of alternative routes being concurrently disrupted is considered.
Fig. S1 shows the loss in redundancy for origins due to geographic interdependencies when bus is the primary mode and rail is substitute, and the results are very similar to those of Fig. 8.

Substitutability of options accounting for geographic interdependencies between modes
Fig. 9 shows the distributions of normalised substitutability losses for alternative routes for various buffer widths.
As with redundancy, the box plots of Fig. 9 reveal that losses in substitutability of OD pairs exhibit an upward trend as the buffer widths around primary routes increase.Post-hoc comparisons using Dunn's test (Table S2) showed that geographic interdependencies result in significantly different substitutability values, even for the smallest-scale hazards considered.
As with the redundancy indicator, the routes associated with the high outliers of substitutability losses in the case of 100 m-wide buffers were identified, as shown in Fig. 11, along with the section of public transport routes that contribute to these losses.The results reveal that the OD pairs related to these outliers are in similar locations to those of redundancy (Fig. 7), however they appear to be significantly more, when considering either mode as primary.This shows that while significant discrepancies in the redundancy and substitutability losses do not exist, it further reinforces the observation that it is possible for an OD pair to experience very high losses in substitutability but not in redundancy.Thus, for an OD pair where the primary route is characterised as high accessibility and the alternative route is entirely within the neighbourhood of the former, the redundancy indicator will still be high, while substitutability will be zero.
Regarding the sections of the rail and bus networks that contribute to these high losses in substitutability, Fig. 10(c) and Fig. 10(d) show that these are the same as those for redundancy (Fig. 7), thus revealing that regardless an OD pair is characterised by extremely high losses in redundancy, substitutability (or both), its corresponding routes use specific parts of the network that contribute to its susceptibility to geographic interdependencies.
The normalised substitutability of routes was aggregated for origins for the 100 m, 1.5 km and 10 km footprints, as shown in Fig. 11.
The results for localised hazards (Fig. 11(a) and Fig. 11(d)) suggest that all zones exhibit very low losses in substitutability, which is consistent with those of redundancy (Fig. 8(a) and Fig. 8(d)).However, as the scale of hazard increases, differences between the two indicator values are revealed.While the geographic distribution of absolute losses in redundancy for the 1.5 km-wide buffers (Fig. 8(b)) shares significant similarities with that of normalised substitutability (Fig. 11(b)), the distributions of relative losses exhibit differences as Fig. 11(e) indicates that fewer zones are related to high substitutability losses due to geographic interdependencies than Fig. 8(e).These differences are even  This indicates that while a zone may experience very high losses in terms of redundancy, its losses in terms of substitutability may be lower.It is further worth noting that zones that rank high in substitutability losses but low in redundancy are not observed, indicating that using the redundancy indicator to assess losses in accessibility of locations provides more conservative results than the substitutability indicator.
The origin-level losses in substitutability were also computed (Fig. S2) when considering bus as primary mode and reveal very similar results to those of Fig. 11.

Conclusions and discussion
In this paper, an approach is presented to assess the role of geographic interdependencies between two discrete public transport networks for two components of resilience, namely redundancy and  Losses in normalised substitutability values of alternative routes of each mode for buffers of varying widths.The box plots for bus (orange colour) show substitutability losses when bus is the primary mode and railway is substitute and express the reduction in the extent to which railway routes replace the corresponding bus routes when the latter become unavailable.Likewise, box plots for railway (green colour) reflect the drop in extent to which bus routes replace the accessibility provided by railway routes.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)substitutability of travel options.Measures were developed to represent each of these components using an accessibility-based approach.The degree of geographic interdependencies was introduced by reducing the contributions of total accessibility of alternative routes based on the proximity between them.The results reveal that while an alternative mode provides significant resilience benefits, its contributions are potentially reduced when geographic interdependency is considered.The extent of this reduction depends on the spatial footprint of hazards and the degree of proximity of alternative routes, highlighting the importance of careful selection of buffer size.A very small value will result in narrow buffers that underestimate the risk of routes being concurrently disrupted by large-scale events, while wide buffers could overestimate the risk of concurrent disruptions caused by localised events.In the example presented, the redundancy and substitutability of most routes between OD pairs were not significantly affected by smallscale events, however losses became significantly more noticeable for larger-scale hazards.
Furthermore, the results of the example show that, although urban, densely-populated areas are associated with the highest redundancy and substitutability losses in absolute terms, rural areas that are less densely populated lose a higher percentage of their initial values as a result of area-wide events.This is because, although absolute losses in zones located in less populated areas due to geographic interdependency were low, their initial values in redundancy and substitutability (i.e., when ignoring geographic interdependency) were also low.In contrast, absolute losses due to geographic interdependency in urban zones were high, but their initial values in redundancy and substitutability were also high; therefore, these high absolute losses were only a small proportion of their initial indicator values.An important observation of the results highlighted differences in the ranking of OD pairs and locations in terms of redundancy and substitutability, which is attributed to the fact that the former metric places more emphasis on the contribution of the primary option to the accessibility, while the latter focuses more on the remaining accessibility when the primary option is unavailable.These indicators therefore complement each other when assessing the accessibility of locations, with or without accounting for geographic interdependency.The work presented comes with several limitations.Firstly, the redundancy and substitutability measures were assessed in terms of travel time, however other elements of travel deterrence could be considered, such as distance and economic cost of travel.Secondly, for the proposed indicators, rail and bus operating on the same functional level were identified based on the travel distance of their services.However, although the modes selected were largely interchangeable, each has different capacities and flexibilities.For example, typically a bus service may more easily detour in the event of a road closure, while rail can put on replacement buses when trains are disrupted.Thirdly, due to the fact that the available GTFS data for bus services in Scotland made no distinction between local and long-distance routes, the latter were selected based solely on the length of each bus route.This selection could be further refined by retaining bus services with termini in different localities which exceed a given route distance threshold.Another limitation is that the impact of geographic interdependencies on components of resilience was assessed for a certain time-window on a weekday.Choosing a different day or time-of-day may have resulted in different itineraries and, as such, different redundancy and substitutability values.Repeating the analysis for various time-windows (e.g., peak and off-peak times) would allow ascertaining hours-of-day and days-of-week, where resilience of networks is mostly affected by geographic interdependencies.
Furthermore, the geographic interdependencies were estimated using a buffer-based approach where only those parts of alternative routes which lie within the buffer of preferred route were considered, while those parts lying outside of that buffer were ignored.The method could be extended to avoid this "cliff-edge" effect by allowing the degree of geographic interdependency to decay with separation distance between routes.However, capturing the distance-decay effect would be challenging and computationally intensive, as proximity between alternative routes varies along their length.Another limitation of this work is that the potential of two travel options being concurrently disrupted arises purely from their proximity.Whilst horizontal distance is an important determinant for concurrent failures, other factors may influence this, depending on the hazard of concern, such as vertical separation of routes and slope.To better account for these factors, risk maps showing the spatial footprint and intensity of hazards of interest could be used instead of buffers.
Finally, because the focus of this work was to assess the effects of the geographic interdependency between two discrete public transport modes, only the shortest path route of each mode was considered.The method could also include other feasible routes of these modes following the approach of Liao and van Wee (2017), in which the accessibility offered by each additional route is reduced by the extent to which it falls within the buffers of routes which have already been included in the calculation.Likewise, this approach could be extended to include more than two transport modes, as well as the attractiveness of destinations.A general model of redundancy that considers both the value of destination opportunities and the connectivity provided by multiple options is presented in the Supplementary Material.
Despite these limitations, the findings provide novel indications on the impacts of geographic interdependencies related to area-wide events on components of resilience of transport networks.The approach presented enables policy makers and network managers to explore the potential severity of consequences of area-wide events on multimodal transportation networks and identify areas of the network which, if impacted, would lead to the highest losses in accessibility.

Fig. 1 .
Fig. 1.Comparison of redundancy and substitutability, with and without geographical interdependency.All figures show a choice set containing a total of four options (A-D) which is equal to the unweighted redundancy of options.The size of options indicates the attractiveness of each option; thus, Option A is the preferred option.The intersection of options A ⋂ B plus C ⋂ D is the degree of geographical interdependency in each choice set.In Fig. 1(a) and (c), the sum of weighted options (A ⋃ B ⋃ C ⋃ D) is the weighted redundancy which is equivalent to the accessibility presented by the options.In Fig. 1(b) and (d), Option A is unavailable, thus the remaining accessibility is equal to (((A ʹ ∩ B) ⋃ (C ⋃ D)) which is the substitutability of each choice set.

Fig. 2 .
Fig. 2. Example of a route of the alternative transport mode (purple line) located in the buffer-based neighbourhood (grey area) of the primary route (red line).(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4 .
Fig. 4. Histograms of accessibility for the (A) railway and (B) bus routes for OD pairs served by both modes on Monday between 07:30 am and 09:30 am.Accessibility values for a particular mode range from approximately 0 to 1. Low values of accessibility indicate long travel times close to 12 h, while high accessibility values indicate short travel times.

Fig. 5 .
Fig. 5. Maps showing accessibility-based redundancy values of zones connected by both modes for (a) travel from origins (b) travel to destinations, when ignoring geographic interdependencies between the two networks.Non-shaded zones are those not served by both modes.Zones in lighter colours are characterised by lower accessibility than those in darker colours.

Fig. 6 .
Fig. 6.Relative losses in redundancy of OD Pairs when considering geographic interdependencies related to various neighbourhood sizes.Larger values indicate higher susceptibility of travel options between an OD pair to accessibility loss due to geographic interdependencies, while lower values indicate lower susceptibility.

Fig. 7 .
Fig. 7. OD Pairs associated with particularly high redundancy losses where primary mode is (a) rail and (b) bus in the case of 100-m wide buffers, along with sections of (c) rail and (d) bus (road) networks (in red) used by routes connecting these OD pairs.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 8 .
Fig. 8. Losses in redundancy for origins (a) in absolute terms due to hazards of 100 m footprint (b) in absolute terms due to hazards of 1.5 km footprint (c) in absolute terms due to hazards of 10 km footprint (d) in relative terms due to hazards of 100 m footprint (e) in relative terms due to hazards of 1.5 km footprint, and (f) in relative terms due to hazards of 10 km footprint, when rail is considered as primary travel mode and bus as alternative.Non-shaded zones are those origins not served by both modes.Zones in lighter colours are less susceptible to losses due to geographic interdependencies.

Fig
Fig. 9.Losses in normalised substitutability values of alternative routes of each mode for buffers of varying widths.The box plots for bus (orange colour) show substitutability losses when bus is the primary mode and railway is substitute and express the reduction in the extent to which railway routes replace the corresponding bus routes when the latter become unavailable.Likewise, box plots for railway (green colour) reflect the drop in extent to which bus routes replace the accessibility provided by railway routes.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 10 .
Fig. 10.OD Pairs associated with high outliers of substitutability losses for travel by (a) rail and (b) bus, in the case of 100 m-wide buffers, along with sections of (c) rail and (d) bus (road) networks that the routes connecting these OD Pairs use.

Fig. 11 .
Fig. 11.Losses in normalised substitutability for origins (a) in absolute terms due to hazards of 100 m footprint (b) in absolute terms due to hazards of 1.5 km footprint (c) in absolute terms due to hazards of 10 km footprint (d) in relative terms due to hazards of 100 m footprint (e) in relative terms due to hazards of 1.5 km footprint, and (f) in relative terms due to hazards of 10 km footprint, when rail is considered as primary travel mode and bus as substitute.Non-shaded zones are those origins not served by both modes.Zones in lighter colours are less susceptible to losses due to geographic interdependencies.