Whole value at risk for flood damage estimates through spatial data analysis

Effective disaster risk reduction (DRR) for flooding requires a comprehensive estimate of the whole value at risk (WVAR) to inform appropriate and proportionate mitigation expenditure. Conventional flood risk estimation methods focus on the direct effects of inundation on community value and generally ignore collateral effects on assets and populations outside the flooded area. Consequently, conventional methods tend to underestimate the cost of flooding, leading to an underestimate of the return on DRR investment. Using spatial data analysis in an urban case study for Toronto, Canada, we identify and capture the collateral value at risk (ColVaR) to estimate the WVAR more comprehensively. In our case study, ColVaR (mean estimate) amounts to 70% of direct losses (ColVar = $344 M; direct losses = $475 M CAD), ranging from 20%–150% (ColVar $100–$740 M) when spanning the 90% confidence intervals of our Monte Carlo simulations. Thus, we demonstrate that if the collateral value at risk is ignored, WVAR can be significantly underestimated, potentially leading to reduced disaster risk reduction resource allocations and thereby adding risk exposure for communities. We present an accessible, seven-step process using existing spatial analysis tools and techniques that infrastructure stakeholders and planners can use to estimate ColVaR and better formulate DRR measures for their communities.


Introduction
According to the parliamentary budget officer for Canada flooding, and the damages associated with it, constitutes the most expensive natural hazard in Canada, accounting for 75% of all Federal disaster relief expenditure annually [1][2][3].Data suggests that flooding as a result of climate change is increasing globally in both frequency and intensity [4][5][6], and the effects on societies are increasing, as reflected in the UNOCHA global humanitarian overview [7].Urban populations are also increasing [8], thus exposing more assets, particularly critical infrastructure (CI) components and the humans that depend on them, to increasing flood risk [1].Higher intensity land occupation also drives greater concentration of habitation and consequently a greater proliferation of impermeable land cover surfaces amplifying storm water runoff effects and producing flooding where it was not previously present [9][10][11][12][13].In combination, these factors are putting communities, the humans who inhabit them, the environmental footprint they occupy, and the value they embody, at increasing risk.In this paper we propose a systematic, spatially based framework for quantifying this risk while accounting for collateral risk invoked through infrastructure dependencies to a degree that is not currently addressed in the literature.
Community value comprises the natural, built, human and virtual components that together can achieve a thriving and flourishing community, one that is both economically viable and aesthetically desirable [14].Hipel and Kilgour describe it as a vitae system of systems (VSOS) where community value lies at the intersection of the natural, built, human, and virtual domains.Value is amplified when all four domains are in a balance appropriate to their context [15].The value in modern, flourishing and increasingly urban communities depends on uninterrupted essential services such as water, electricity, transport, communications etc., and the intricate, dependent and inter-dependent system of infrastructure components, that supply them [16].Ensuring the continuity of those services is therefore increasingly vital to preserving the community value that Hipel and Kilgour describe.
Disaster risk reduction (DRR) is the collective term for all actions taken to reduce the effect of disasters, whether floods, wildfires or hurricanes, on community value as articulated in the sendai framework [17,18].DRR addresses two primary requirements.First, to systematically identify assets at risk, the consequences to communities of their exposure to hazards, and to identify achievable mitigation strategies.Secondly, DRR seeks to consistently identify the costs of various disaster types with a view to reliably estimating mitigation assistance and/or relief funding requirements.The fundamental principle applied to the calculation of disaster cost is 'direct exposure' [19][20][21], meaning that only assets and populations directly affected by a disaster are considered in damage estimations.This is also true for the insurance industry which typically only insures losses that result from direct contact with the hazard [22][23][24].We argue, however, that flood risk effects are propagated outside the directly affected flood zone and inflict measurable and predictable losses on community value in adjacent areas.We call these outcomes collateral losses and the value they represent as collateral value at risk (ColVaR).
Borrowing terminology from the financial industry, a community can be thought of as a portfolio of assets distributed across the VSOSs categories that, taken together, has both tangible and intangible value.Disaster inflicts losses that reduce the value of this portfolio, which in turn reduces community value by eroding its desirability as a place to thrive and flourish, as it did in New Orleans after Hurricane Katrina [25][26][27].Community value can therefore be portrayed as the whole value at risk (WVAR), defined as the maximum value of loss for all combined disaster outcomes, a key component of which is clearly associated with the probability of flooding.
In modern cities where the population and businesses rely on increasingly dependent and interdependent infrastructure systems, losses can be propagated through those systems via pathways of exposure to risk and cascades of consequence outside the direct hazard exposure area, and as we will show, can represent up to 130% of the direct losses incurred in a single event [28][29][30].These losses are typically uninsurable, are generally ineligible for disaster relief funding and are therefore typically borne by the community members and business owners affected.But in many cases businesses can no longer operate, therefore they are not generating revenue and they often must lay off employees as experienced in the aftermath of hurricanes Katrina and Sandy [31][32][33].Such financial burdens inhibit response and recovery by reducing the cash readily available to pay for it.Additionally, assets adjacent to the flood zone such as community centres and schools that were intended for use as emergency shelters to support response and recovery become unusable for that purpose because they lack essential services like electricity, gas, water and sewage as expected, as residents in adjacent New York boroughs experienced during Super Storm Sandy in October 2012 [26,34,35].
Finally, we show that WVAR can be significantly underestimated if ColVaR is ignored, resulting in a low return on investment for mitigation measures and sometimes the avoidance of justifiable DRR investments.We present here a method to more comprehensively estimate ColVaR, illustrated through a case study on the Lower Don River in Toronto, Ontario, Canada.We propose spatial data analysis techniques, following the general methods outlined by Dall'ebra et al in the international encyclopaedia of human geography 2009 [1,36] to identify collateral losses, estimate their costs, and determine a WVAR more representative of the community value at risk in the absence of appropriate DRR measures.

Literature review
Risk, at its most elemental level, is 'the possibility of loss' [37].The existence of risk requires that an asset of value is exposed to a hazard and that the consequences of the exposure diminishes the value of that asset [17,18,38].The extent to which the value of the asset, and the service it provides, is diminished becomes the value at risk (VaR) [39], and invokes a series of considerations and calculations concerning its protection from the hazard(s), the cost of that protection, and the relative benefit gained for the protection expenditure [34,35], or the return on mitigation investment.
Loss is typically categorized as direct or indirect.Direct losses are generally considered to be those sustained by direct physical contact with the hazard (fire, flood, blast etc.), whereas indirect losses comprise business interruption or income loss incurred as a result of the physical damage to the asset [40,41].In the case of flooding, predictive methods are employed to determine the extent and severity of future flooding events based on the extrapolation of historic flood data into the future.An inventory of potentially exposed assets is compiled, and a loss/damage prediction estimate is calculated for all the direct and indirect losses likely to be sustained by the assets exposed to the various severities of flood [30,[42][43][44].
Loss/damage predictions are used by multiple stakeholders for a variety of purposes [45].Insurance companies use them to estimate their policy risk exposure and calculate the premiums for insured properties and businesses in the flood zone [46].Public sector agencies use them in combination with flood maps to plan public infrastructure, flood hazard mitigation measures, emergency service capabilities, and to inform land use decisions [47,48].Businesses are increasingly using both products to plan asset locations, supply chains, and distribution networks [49].
Business operations and their assets do not exist in isolation, however.They depend on one another via supply and distribution networks (which depends on the transport network), they depend on their communities for workers and customers, and they all depend on the critical infrastructure networks that provide the essential 'enabling' services vital to their operations (electricity, water and wastewater, gas etc.).They are inextricably part of the 'VSOSs' described by Hiple et al [15] to exist, thrive and prosper [16].
Infrastructure networks are typically laid out for maximum efficiency and they take the likelihood of 'design flood' (The maximum flood probable for that area) into consideration at the time of their design [50].Infrastructure assets that have to be located in flood plains are generally designed to withstand the 'design flood' applicable to their location [51][52][53].The increasing frequency and intensity of extreme weather events experienced globally (e.g. the 'atmospheric river' events in British Columbia in 2021/2022 and in California in January 2023), combined with habitual land use patterns, exposes more infrastructure to the risk of failure due to flooding [1,24,54].Inevitably this exposes their dependents (Homes, businesses and other infrastructure that depend on the infrastructure component(s) at risk) to service interruption, whether they themselves are exposed to flood effects or not [55].Examples include electrical transformer stations in a flood zone that serve customers outside it, wastewater treatment plants that become inundated and release untreated effluent into surface runoff and potentially into surface or groundwater potable water sources, or solid waste disposal sites that become flooded and release toxic leachate into the surface water runoff, contaminating both surface and groundwater sources used by surrounding communities.
While flood loss damage prediction within hazard zones has been widely researched [2,[56][57][58][59][60][61][62], to date, the matter of collateral losses, or losses experienced outside the flood zone as a result of infrastructure within it failing, is not directly addressed.Koks et al capture the concept at a high level by accounting for Socio-Economic Indicators and GDP impact [45], but this approach is not specific or sufficiently localized to be actionable by local authorities when planning DRR interventions.Likewise, Jonkman et al classify 'damage for companies outside the flooded area' as 'indirect, tangible and priced losses' but do not provide a method to identify and localize them to the point they can be discretely priced or mitigated [63].Jha et al also make tangential reference to it as an indirect effect at the macro-economic level, but also do not propose a method to identify, localize and mitigate the effects [9,12].As Hammond points out, estimating the impacts of flooding on infrastructure and the subsequent losses both inside and outside the hazard area is particularly complicated, consequently it is a comparatively under researched area [64].
Recognizing the complexity of the undertaking, we propose an approach to isolating and estimating this under-researched dimension of disaster risk using spatial analysis techniques, illustrated in a case study of the Lower Don River Valley.The method presented contributes to a more comprehensive estimation of the WVAR (Direct + Indirect + Collateral) for flood scenarios, so that evidence informed DRR investments can be made with greater confidence in the return on the mitigation investment.
Hammond et al point out, flood effects terminology is not consistent and the use of the terms direct and indirect to describe losses in relation to their causes is imprecise [56].In this case, the reference study defines direct loss and damages as those incurred because of direct contact with flood water and indirect losses as those incurred as a direct consequence of that contact, e.g.lost stock, revenue and productivity by businesses and homes that experienced direct flood exposure [43,44].To avoid confusion, we characterise losses incurred outside the flood zone, but directly attributable to infrastructure failure within it, as collateral losses, and their value as collateral value at risk (ColVaR).

Core modelling approach
The core model for flood causation in the world bank guidance for flood risk assessment by Jha et al [12,65] reproduced at figure 1 presents a generic causal chain method for assessing flood effects.In this causal chain approach excess water sources (Rainfall, Coastal Surge, Snowmelt etc.) are identified, quantified, and introduced to a digital elevation model (DEM) and various types of hydraulic and hydrological modelling tools to identify flood pathways and behaviour.Receptors, or assets exposed to the flood hazard, are identified and the consequences of their exposure are assessed.As described, the authors mention tertiary consequences occurring outside the flood pathway but do not offer a method of quantifying it at the local scale.In the example that follows we illustrate the application of Voronoi techniques to estimate infrastructure service areas, spatial analysis to conduct failure means and effects analysis (FMEA), infrastructure dependency analysis, and socio-economic impact analysis using census and business index data as extensions to the World Bank model, as indicated in the red box in figure 1.
While we focus on an extreme riverine flooding event to illustrate the technique clearly, the same method can be applied to riverine flooding at any scale or return period.Acknowledging the added complexities described by Oddo et al [66] and Wong et al [67], with appropriate contextual modifications it could also be applied to coastal inundation effects estimation.
Extending the work of Jha et al, using a case study example, we identify electrical distribution system infrastructure within the flood pathway and, through failure modes and effects analysis, determine if a service interruption will occur and if so, what would be the duration.Using a Voronoi tessellation to approximate the distribution area for the electrical distribution service (in place of the real-world service distribution map) we simulate the effects on network of homes, businesses, and dependent infrastructure.The result is an emerging picture of ColVaR in comparison to conventional calculations that do not consider collateral losses.
We acknowledge five sources of uncertainty in the study pertaining to the seven-step process we define below.These include: flood extent as modelled (Steps 1 and 2); the duration of the power loss event (Step 3); the accuracy of the Voronoi derived service areas (Derived in Step 4); the validity of the business index data (Used in Step 5); and the accuracy/validity of household income data (also used in Step 5).Each uncertainty is treated both separately and in combination in our results to provide the best estimate range possible.

The WVAR assessment method
Figure 2 illustrates the steps in the estimation workflow and the data required for each step.
Data about the 'source, pathways and receptors' shown in figure 1 is required as input data to the process, while 'consequences' are calculated outputs.'source' data is typically derived from historic values and/or downscaled climate models processed into predicted pluvial volumes and introduced to the 'pathway' flood maps via hydraulic modelling methods.
Data for infrastructure assets should include accurate asset type, location, and capability (information about what it does and what its capacity is e.g.'medium voltage transformer station').Subscriber data describing service coverage for 'receptor' infrastructure components that provide essential services such as Sources for these data inputs will vary from jurisdiction to jurisdiction.Typically, hydrological data for flood mapping can be obtained from city planning offices or, as is the case in Ontario, the conservation authorities whose mandate is specifically to manage water sheds to minimize loss to life or property [68].The source of infrastructure data will vary depending on the mix of public/private control.Much of the required information can increasingly be found in municipal open data portals such as 'open data Toronto' .
In this case direct and indirect VaR figures are supplied by the regional conservation authority (TRCA) via the tangible business related damage totals (TBRDT) in the Toronto flood risk ranking [43], summarized at table 1.An alternate source for this data may be National insurance Bureaus, such as the insurance Bureau of Canada.
To illustrate the technique and the calculations involved, we consider the example of the regional design flood (Hurricane Hazel, 1954) in the Lower Don River Valley in Toronto, Ontario.Canada.The Don River Watershed comprises a 356 km 2 catchment area, north of Toronto, Ontario.The river discharges into Lake Ontario near the city core.Over time the banks of the Lower Don River and its estuary have become constrained by industrialization and hardened with overflow protection (Grey) infrastructure.Consequently, riverine flooding because of extreme pluvial events upstream has increased.As population density has also increased in line with the global trend [17,18], more assets are being exposed to the flood hazards.
In addition, as both public and private infrastructure (e.g.roads and shopping centres) proliferate to service the growing population, impermeable ground cover is increasing, reducing the absorptive capacity of the natural domain.The result is increased runoff, storm water system overcharge and increased riverine flooding as illustrated by the cloudburst event in July 2013 which caused extensive flood damage [69].Canada's senior climatologist David Phillips was quoted as saying 'this is likely the wettest moment in Toronto's history' and more than 90 mm of rain was recorded at Pearson international airport in a two-hour period [70].Prior to this, the wettest day in Toronto's history had been 15 October 1954, when Hurricane Hazel brought 121.4 millimeters of rainfall over a 24 hour period.The average expected rainfall for Toronto in July is 75 mm over 24 hours.
In 1957, following the devastation of Hurricane Hazel, 36 Regional Watershed conservation authorities were established under the conservation authorities act, with the mandate to 'ensure the conservation, restoration and responsible management of Ontario's water, land and natural habitats through programs that balance human, environmental and economic needs' [68,71].Under this legislation, Toronto and Region conservation authority (TRCA) is responsible for flood and erosion risk management, stormwater and watershed management, greenspace management and the regulation of planning and development (review and permit approvals) on lands within its jurisdiction.TRCA is also designated as a 'source protection authority' in the clean water act which governs the protection of source drinking water in Ontario [72].TRCA is therefore the 'authoritative source' for data in this case study.
As part of their mandate TRCA engage in periodic flood risk reviews.The latest entitled 'Toronto flood risk ranking' published in 2019 presents the most recent flood risk data and flood damage estimates [43].
The TRCA area of responsibility is divided into flood vulnerability areas (FVA), as illustrated in red in the left panel at figure 3 and in more detail in the right panel, circled in red.FVA 7-4 Brickworks and FVA 26-4 Lower Don, define the areas of interest (AOI) for this case study.

Step 1: establish hazard pathways
Step 1 is to establish the Hazard pathways.In this case study we identify the 'source' as an extreme weather event, equivalent to the conditions that produced the regional design flood (Hurricane Hazel, 1954) P = 1:500, identified as the Toronto regional flood extent polygon, shown in light green at figure 4. While no return period is specified for Hurricane Hazel at the reference, it is commonly assigned the return probability P = 1:500.We therefore adopt this convention in the case study, as indicated above.
These flood profiles do not reference climate change maximum pluvial predictions derived from downscaled climate data, consequently they are historically referenced and not predictive.They are however sufficient to illustrate the technique.Practitioners are encouraged to consider the predicted pluvial maximums from a downscaled climate model for multiple representative concentration pathways specific to the AOI they are analysing.It is important to note that downscaling the output of any global climate model to local scale, particularly for precipitation values, requires compensation for several biases that inevitably arise.Care is required to calibrate or adjust the data using local atmospheric factors and 'ground truth' .
Figure 4 depicts the flood hazard pathway.The colour ramp depicts flood exceedance (depth) from 1 m-9 m as shown in the legend (top left).The light green flood extent indicates the regional design flood fringe.

Step 2: identify critical infrastructure (CI) receptors
The next step is to identify CI 'receptors' within the hazard 'pathway' defined in Step 1.We plot the locations of critical infrastructure nodes such as, but not limited to, electrical transformers, water purification and distribution stations, wastewater pumping and treatment facilities, solid waste management sites, healthcare facilities and critical transport nodes as point features on discrete layers in a geospatial information system (GIS) application, in this case ESRI Arc GIS Pro ® .Private sector infrastructure elements such as food distribution terminals, critical manufacturing sites, cold chain facilities etc. are also identified and geolocated.In this case study, we focus on electrical power distribution infrastructure, consequently, all other asset classes are omitted from the scene for the sake of simplicity.
Visual inspection of figure 4 reveals that Gerrard Street transformer station (TS) and Basin TS, (identified in red) are both infrastructure 'receptors' in the direct pathway of the design flood hazard.As indicated in the legend, both assets are expected to experience between a 4 m and 6 m of flood depth.Electrical transformer stations are 'receptors' of primary interest because if they fail, they deny an essential service, directly affecting the capacity to operate and to respond and recover from the flood effects to multiple dependent entities (end-users).For example, electrical sump pumps do not operate, refrigeration fails in homes and food distribution outlets, and emergency pumps must be independently powered.For the purposes of this case, we assume that electrical transformer equipment in direct contact with flood water will not deliver electrical service either because of preventative shut down ('safe to fail' protocol) or failure due to water exposure.
TRCA considers assets within each of the FVA flood extent polygons to be at risk of both direct loss (physical damage incurred through direct contact with flood water) and indirect loss (operational or stock losses incurred by assets within the flood polygon because of the flood but not because of direct contact with flood water), calculated as a percentage of direct loss for various asset categories [43].TRCA aggregates both losses as the Value at Risk expressed as TBRDT for a range of return periods from P = 1:2 (nuisance flooding), up to the Regional Flood for all FVAs, summarized at table 1.The baseline model for these calculations is derived from the assessment of flood risk in the city of calgary following the 2013 flood [44].
While table 1 summarizes the aggregate damage loss estimates by exceedance probability for receptors within the respective flood fringes, these estimates do not identify damage to public utilities, or the collateral losses to dependent assets (end users of the service) anticipated from subsequent service interruptions due to that damage.The following steps in the process identify and estimate these damages and their effects.

Step 3: conduct CI failure modes and effects analysis 4.3.1. Failure modes
In Step 3 we test the assumption that infrastructure assets within the flood polygon will fail (cease the flow of essential service to dependents) when they are inundated.To do this we inspect the CI 'receptors' identified in the 'pathway' to determine how they might fail and what the effects are likely to be.This process can take the form of a full engineering FMEA assessment, a constructive simulation, or a spatial analysis to determine which specific components of an infrastructure 'receptor' are vulnerable and whether simple mitigations, such as the rearrangement of technical subcomponents would avoid failure [51].In the absence of a more sophisticated model, a simple set of rules may be created relating flood depth to probability of failure to each infrastructure type.For this case study we will demonstrate the method assuming a simplified binary rule whereby failure occurs (p = 1) for all assets when flood depth is greater than the design depth of the component, and there is no failure (p = 0) otherwise.
Figure 5 presents the regional flood extent polygon from figure 4 overlaid on nadir (vertical) imagery of Gerrard Street TS.North is at the top of the image, indicated by the North mark.Here we observe two operational elements circled in red; one on the East (right) bank of the river and the other on the West (left) bank, both falling within the flood fringe from figure 4 shown in translucent light blue.
Because Gerrard Street TS is within the flood 'pathway' does not automatically mean it will fail.In a 'safe to fail' design, knowledge of the impending hazard triggers an orderly shutdown protocol within the 'hazard travel time' , or the time it takes for the hazard to reach the asset from initial detection and verification.Operations are shut down, flood defences are deployed, electrical load is transferred to other stations (in a 'smart' distributed grid architecture) and the inundation is allowed to happen, predicated on the assumption of a rapid reaction, response, and recovery process after floodwaters recede.Since neither the existence of a 'smart' grid architecture nor the flood management protocols at Gerrard Street TS are known, for the purposes of this investigation we assume the P = 1 scenario (flooding equals failure), either by virtue of inundation or by the implementation of the safe to fail protocol.
To gain better perspective of the flood threat, we generate a three-dimensional view of the flood fringe from the TRCA 2015 light detection and ranging (LiDAR) contour data and the ESRI World Atlas imagery in Arc GIS Pro ® at figure 6 using the 'rubber sheet' method, the imagery is 'draped' over the DEM generated from TRCA provided LiDAR data.The flood extent data is then overlaid and presented as a flood depth colour ramp.For ease of understanding exceedance spot depths are added at key points in the scene.As we see, both the EAST and WEST Yards (circled in red) experience inundation between 5.0 and 5.5 m.
Further perspective is gained in figure 7(w)hich shows the 5.5 m flood stage projected onto both sites from ground level.The left pane is the EAST yard looking North along the Don Valley Parkway, and the right pane is the WEST yard looking south down riverside drive.From this perspective we see that the transformer infrastructure will be inundated and unlikely to sustain operations (P = 1), either through failure or deliberate 'safe to fail' shut down.Floods of this magnitude typically carry large debris fields however, that can inflict significant damage when colliding with electrical infrastructure components.The deflection wall installed at the EAST yard may provide some protection, but the WEST yard is exposed.Damage from the debris field may therefore negate immediate the benefits of 'safe to fail' protocol.
The same process is carried out for Basin Street TS, shown at figure 8.In this case there is little variance in the expected flood stage, all falling within the 3-4 meter range.Debris field damage is less likely in this case as the electrical infrastructure is protected by buildings to the North.
Figure 9 shows the ground level perspective for inundation levels at Basin Street TS.Components can be seen at the edge of the inundation extent; therefore, we assume that the 'safe to fail' protocol would be enacted to preserve operational capability for response and recovery operations.Unlike Gerrard Street TS, there is no obstacle to the dissipation of water at Basin Street, such as the identified constraints at the Don River estuary.Consequently, any inundation no matter the severity, is likely to dissipate into the lake and be of short duration.Of note is the portlands energy centre, seen to the South of Basin Street TS, across the ship channel, which supplies much of the back-up generation capacity to downtown Toronto.Similar conditions as those described are likely to apply, however the consequences cannot be assessed without access to the interior plans and layout of the system components.Such analysis, while possible using building information management data and dependency analysis techniques, is out of the scope of this study.

Failure effects
Having tested failure modes (How failure might be caused) we determine failure effects (what are the likely consequences).Effects in this context have two dimensions.First there is the effect of the hazard on the CI node and its ability to deliver essential service per its design.Secondly is the propagation of service failure to direct and indirect dependents, and the potential 'cascade of failures' to multiple 'n' order dependents.Two factors must be determined at this point.First, we must estimate the likely duration of the service interruption and secondly, we must estimate the service coverage area that will be affected.
The Toronto flood risk ranking does not provide restoration of service profiles for civil infrastructure; however, it does provide estimates of building restoration times for each building type in the study.The closest category is Warehouses and industrial buildings, giving an estimate of 100 d restoration time per vertical (z) meter of flooding [43].With an expected flood stage (depth) of 5.5 meters, this calculation produces an expected restoration time of 550 d.
Even in Hurricane Katrina or Super Storm Sandy [73] electrical power was typically restored to the vast majority (90%-95%) of subscribers that could accept it within 14 d.In some cases, power was available within 7 d.Excluding the most severely damaged locations or those that were abandoned completely, the longest restoration period appears to be 21 d [27].Consequently a 14 d service interruption period was selected as the mean, with upper and lower bounds of 21 d and 7 d respectively.
From the failure modes and effects analysis we therefore conclude that P = 1 (failure) condition prevails at both sites for the flood profile P = 1:500 and that a mean 14 d service interruption is a reasonable consequence to consider under these conditions.We determine the service area as a separate spatial analysis operation in the next step.

Step 4: determine critical infrastructure service area
The preferred approach to determining a utility service area is to obtain the subscriber coverage map from the service provider.For security reasons, utilities guard this information closely.While city engineers may have access to it, other stakeholders may not.Consequently Step 4 presents a method to determine service area coverage for essential services, such as electrical power, in the absence of either a dependency model or full systems mapping, using a Voronoi Tessellation and Tobler's first law of geography.Tobler's law states: 'everything is usually related to all else but those which are near to each other are more related when compared to those that are further away' [74].
Tobler's law suggests that the distance from any primary service node (or 'seed') to all the dependent nodes in a distribution network will tend to be the shortest possible (those that are nearer), suggesting that a Voronoi tessellation, such as the one figure 10, could be used to approximate the service coverage area for each transformer station.Du et al [75] provide a comprehensive explanation of Voronoi tessellation technique [75] and Yan et al [76] explain its application to power grid mapping.The result is a framework of electrical distribution 'cells' called Thiessen Polygons that contain all the points closest to the 'seed' transformer station and not closer to another (i.e.those that are farther away).
We begin by mapping all the Transformer Station locations in Toronto and designating them as the 'seeds' , as shown in figure 10.The subsequent 'cells' (Thiessen polygons) generated by the tessellation produces the approximate service areas of each transformer station, as shown at figure 10.The calculation of the Thiessen Polygons is undertaken by a GIS application, in this case QGIS © , and rendered as a layer, following the example of Mureddu et al [77] for 'Islanding' the German power grid.Infrastructure owners or municipalities doing this analysis and having access to accurate service area boundary definitions and can omit this step and simply use the distribution area polygon.
Figure 10 shows the estimated service area for Gerrard Street TS and Basin TS overlaid on the regional design flood pathway for the Lower Don River (the dark blue polygon derived in Step 1).Here we see the first indication of the service area affected by the interruption of electrical service at Gerrard Street TS and Basin Street TS, but that will not experience tangible building related damage losses as conventionally defined and reflected in the loss values at table 1.
All customers within the service area will experience interruption, but only those in the flood pathway are typically insured, whether directly or by government disaster relief funding.As we demonstrate, the potential financial losses to those outside the flood pathway, which we term collateral losses, are just as real and economically damaging as those in the flood pathway.They diminish community value by diverting funds to self-recovery expenditures and away from personal and community enriching activities and purchases.Measures taken to mitigate flood risk will preserve that value equally for direct and collateral loss areas.But if the benefit of mitigation is only calculated on the conventionally estimated tangible building related damage within the flood pathway, then the WVAR is undervalued and the return on DRR investments will also be undervalued.Conversely, if the collateral value at risk (ColVAR) is added to the tangible building related damage, then WVAR more accurately reflects the total effect on community value and justifies a greater mitigation expenditure in view of the greater value protection offered.The following steps demonstrate techniques to estimate the collateral value at risk (ColVaR).
The uncertainty between the real world electrical distribution network and the Thiessen polygon estimations is reported to be ±15% [78].Recalling that the purpose of using this technique here is to approximate the distribution service area for further spatial analysis activities, this uncertainty is acceptable.We deal with this dependency approximately by expanding and contracting the Thiessen polygon areas by 15% to examine the sensitivity of the model as shown in figure 11.The inner polygon represents the 85% (minimum) scenario; the middle polygon the 100% (mean) scenario; and the outer polygon the 115% scenario.For the base calculations we use the mean or 100% values and deal with the 85% and 115% scenarios as uncertainty parameters.

Step 5: determine dependents
Step 5 determines the assets at collateral risk by capturing and categorizing them within the service effects Thiessen polygon (i.e.repeated respectively for the 85%, 100% and 115% Voronoi polygons).Infrastructure dependency modelling is the preferred method of doing this where a dependency network is established as a 'digital twin' that emulates the performance of selected nodes in relation to each other under stress scenarios.The software tools to accomplish this modelling are not generally accessible however and require extensive knowledge and competence to employ.The objective of this work is to present a method accessible to most  stakeholders with commonly available tools, therefore we present an ensemble of spatial analysis techniques within the scope and capability of most GIS teams.
For this example we consider the impact of Gerrard and Basin Street TS service interruption on businesses, via their potential revenue loss, and households via income loss due to furlough or temporary layoff, as illustrated post Hurricane Katrina [26,31,32].We use Scott's Business Index ® data to locate the business entities, and derive their type of operation, revenue (high, mean and low estimate), and the number of employees they have.We also use statistics Canada (2016) census data, at the lowest aggregation level (the Census dissemination area), to derive the number of households and their median household income.
Founded in 1957, Scott's directories are a trusted source for business data in Canada.Individuals, enterprises and governments source data from Scott's for planning, sales generation and research purposes.Scott's kindly supported this work with free access to the data used in this case study.

Determine dependent businesses
Figure 12 shows the distribution of businesses in proximity to the flood zone and registered in Scott's Business Index ®.The tables at SI tables 4-6 contain all the data attributes and values used.The flood pathway is shown in darker blue, and the 100% service area Thiessen polygon is overlaid in light pink.Green point symbols indicate businesses not affected by the simulated power service interruption and light blue point symbols indicate those that are.The size of the business point symbol (blue or green) reflects the annual revenue range reported by the business in Scott's Index ® .Figure 12 provides a visualization of the dependent businesses within the service area polygon for the 100% Voronoi Scenario derived using the 'summarize within' function in ESRI ArcGIS Pro ® which summarizes all the features in the business layer that are also within the service interruption layer.The minimum, mean and maximum revenue range is the sum for all summarized businesses.

Identify households with income at risk
The economic impact of a 14 day electrical service interruption in an extreme weather event will be experienced by affected households primarily through income loss.Income data in Canada is collated at the individual, census economic family, and census household level.Using the Census dissemination area geography (a defined area containing 400-700 dwellings; the smallest area for which all data attributes are disseminated) we capture the number of households within the service area by mapping the census dissemination areas (CDA's).We then overlay the service area polygons, as shown at figure 13.Data used to perform this operation can be found at SI table 7.In the case of the 115% Voronoi scenario where the two Thiessen polygons would logically overlap, the two shapes have been merged into a single polygon.Consequently, they only 'summarize' the business entities within the polygon once.We verify this using the Scott's_Id numbers in the spatial data table to identify double instances of the same Id and find none.

Four receptor categories
Visual inspection of figures 12 and 13 reveals four distinct 'receptor' categories as shown in table 2. The table identifies the presence of flood and the absence of power in four possible combinations by the presence of an 'X' .
Category 1: Businesses and households with no flood and full electrical service.They are within affected CDAs but not in the affected service area.They will generally not experience negative effects and may indeed profit from the event as business in the neighbouring affected areas is diverted to them during response and recovery phase.Category 1 assets are not considered further here; however, the economic flows in Category 1 areas are identified as an issue for further investigation.
Category 2: Businesses, or households experiencing flood effects but have electrical service.They are inundated but do not lose electrical power.They may or may not experience property damage (direct loss), operational loss, and possible loss of stocks (indirect losses), but they can use the full range of electrical disaster response and recovery tools (electrical pumps, fans etc), allowing them to, react, respond, and recover quickly.
Category 3: Businesses and households that are inundated and lose electrical power, among other essential services.They sustain direct losses from physical contact with flood water and they experience operational interruption and stock losses (indirect losses).Electrical power interruption makes the deployment of response and recovery tools more difficult, prolonging reaction, response, and recovery time.They generally cannot self-recover.
Category 4: Businesses and households that are not inundated but lose electrical power and potentially other essential services.These assets represent a special case in that they do not experience property damage (direct loss), but they do experience operational interruption (indirect loss) at the same scale as Categories 2 and 3, but without any physical inundation damage.They typically are not insured for this type of risk and are not eligible for disaster relief funding.
Without direct physical damage as the causative factor in the operational losses to Category 4 receptors, the risk is generally not recognized, yet the businesses, home owners and employees still sustain economic loss and the value of the community is collectively diminished by that amount [2,12,79,80].Category 4 receptors experience collateral damage, or damage not directly inflicted by the flood event but directly attributable to it since without the flood, Gerrard and Basin Street TS would not have failed/shut down and there would not have been any interruption of operations.The loss of revenue, income and stocks in this category is therefore apart from the direct and indirect losses in Category 2 and 3 and can be considered collateral losses, and their value thought of as collateral value at risk (ColVaR).

Step 6: determine collateral value at risk (ColVaR)
Having identified the Category 4 businesses and households in the three service area scenarios, we now estimate ColVaR.Business revenue and household income are the primary enablers of self-recovery [81][82][83][84] and their loss directly inhibits response and recovery for the community as a whole.In extreme examples (e.g.Hurricane Katrina, New Orleans, 2005) the effect of this revenue diversion can overwhelm the self-recovery capacity of a modern city [26,85] and lead to significant depopulation.
It can be argued that while the businesses lose revenue, they retain value in that they do not pay salaries.From this perspective, accounting for Household Income loss and lost business revenue could be viewed as double accounting since the same value cannot be lost twice.But the salaries would, in this case, most likely be spent on self-recovery within the community, hence 'flowing' back to businesses if they could operate and are providing valued services.We propose therefore that if there was no interruption, salaries would be spent on the normal goods and activities that define and enrich communities.The interruption redirects that flow towards self-recovery or, if salary stops, arrests the flow altogether except where savings exist.Identifying changes in economic flows under stress is beyond the scope of this work and presents an avenue for future investigation.For the purposes of this work, we estimate the economic impact through business revenue loss and household income at risk to indicate the scale and magnitude of ColVAR, and the ultimate undervaluation of WVAR.

Annual business revenue at risk
Annual business revenue at risk for the affected businesses is extracted from the geodata using a 'summarize within' function for each Voronoi Scenario.Businesses within the Voronoi Scenario polygon, but not within the flood hazard pathway polygon are tagged as the Category 4 entities of interest.Data is then 'sorted' for Category 4 and their annual max and min revenues summed as shown at table 3. SI table 2 summarizes the output data.SI tables 4-6 show the geodata extracted for the high and low revenue range calculations and table 7 is the Scott's index reported revenue range lookup table.Figure 12 illustrates the spatial analysis for the 100% Voronoi Scenario.
Scott's Index ® provides annual business revenue in ranges (all amounts in $CAD): <$1M; $1-$5M; $5-$10M; $25-$50M; $50-$75M; $75M-$100M; and >$100M.By default, we select the midpoint of the revenue range for each business, however this is further explored in our subsequent uncertainty analysis which uses the minimum and maximum values from each range to provide bounding scenarios.Since businesses are likely to fall both below and above that midpoint value, depending on the distribution, and the total business revenue at risk results from summing hundreds or thousands of businesses, the resulting errors from using midpoint values are likely to cancel (at least partially) and thus lead to a relatively modest aggregate error.Thus, in table 3 we estimate annual business revenue value at risk for the three Voronoi Table 3. Annual business revenue value at risk for each Voronoi Scenario derived from the spatial analysis of businesses registered with Scott's index within each Voronoi Scenario Polygon and their reported revenue range.These values will subsequently be increased by the index coverage parameter (discussed below).(All amounts in $CAD).Scenarios using the minimum and maximum declared revenue figures (and aforementioned calculated mean) for each Category 4 business found within each Voronoi Scenario polygon.

Annual household income (HHI) at risk
The loss of household employment income during a prolonged emergency has a significant economic effect on community value [86], and as Feltmate and Moudrack et al report, this compounds the psychological stress of those impacted.The loss of household income in this example is a 'proxy' for the impact that the interruption of economic flow is likely to have on community value.To facilitate this, we assume that employees within the affected CDA's live in proximity to the businesses they work in, and therefore most economic effects will be felt locally, i.e. within their CDA or those immediately adjacent.This is consistent with the commuting distance and travel times reported for most cohorts in the census data for this area of the city.We also assume that the reaction of employers whose businesses cannot function in the face of this kind of disaster will be to shutter the business and lay employees off (even if only temporarily).This is consistent with the behaviours seen post Hurricane Katrina and Super Storm Sandy.As indicated, our intent is to gain insight into the magnitude of potential collateral losses rather than provide a detailed assessment of the behavioural economics governing post flood scenarios.In this context, the 'proxy' approach is satisfactory.
Household income at risk is determined by 'joining' each of the Voronoi Scenario polygons to the CDA grid polygons they subtend and 'summarizing' the percentage of each CDA within each Voronoi Scenario.Figure 14 shows all the affected CDAs, coloured to indicate the percentage of each within the 115% scenario boundary as per the legend, overlaid by the three Voronoi Scenarios.The supporting geodata can be found at SI tables 6 and 7. for ColVaR resulting from the three uncertainties (Scott's index coverage, event duration, and actual revenue within the stated range) for the 85%, 100% and 115% Voronoi scenarios, derived using a 10 000 iteration Monte Carlo Simulation in @Risk ® .For greater numerical context, the bar above the graph shows the 90% confidence interval for the middle (100%) Voronoi scenario, and percentiles for the other 2 scenarios associated with those bounds.
We multiply the number of income earners in each CDA by the median income in that CDA.The result is multiplied by the percentage of each CDA in the applicable Voronoi Scenario as shown in equation (1) Annual Household Income at Risk = (# Income Earners*Median Income*%CDA Without Power) .(1) Results for each CDA are aggregated at table 4 to show totals for each Voronoi Scenario.Data used to perform this operation may be found in SI table 3.
Although employees frequently will not live within the same CDA as the business they work in, we argue that this is a sufficient proxy to bound the approximate value at risk, to within the range of all the other existing uncertainties.
Finally, since Scott's business index officials estimate that their index covers approximately 50% of businesses in Toronto [87], the results from tables 3 and 4 must be scaled up accordingly.In the default case, this means multiplying the values by a factor of 2. In the sensitivity analysis, we consider how the results vary as the index coverage ranges from 25% to 75%.

Collateral value at risk (ColVaR)
Collateral value at risk can now be calculated as the sum of the annual business revenue at risk (table 3) plus the annual household income at risk (table 4) divided the index coverage, divided by 365 d times the estimated service interruption period (in days) as shown at equation (2).
where n = service interruption days (i.e. 14 in our mean scenario), and 'index coverage' is a scaling factor to adjust for incomplete coverage by the Scott's index (50% in our mean scenario).Using the mean value for business revenue values for all three Voronoi Scenarios from Scott's index at Using the values at table 5, the WVAR estimate for the Lower Don River P = 1:500 flood risk for all 3 Voronoi Scenarios is given at table 6.

Results and uncertainty
Results for ColVAR at table 5 indicate a non-linear relationship between the size of the Voronoi areas and ColVAR.In this case we see that ColVAR increases 24% between the 85% Voronoi Scenario and the 100%, but only 10% from the 100% to the 115% scenario.This is to be expected since businesses are not evenly distributed within cities, as illustrated in figure 12.This highlights the value of a spatial analytics approach to estimating ColVAR since a statistically inferenced approach would not expose or account for the spatial distribution of businesses on the ground.More sophisticated earth observation or ground sensor data collection techniques (such as Street scape analysis) may increase the precision of this estimation process, and thereby reduce some of the inherent uncertainty, but the estimation process itself would remain largely unchanged.
Analysing the data, we identified three uncertainties exerting influence on the results in addition to the variance in the three Voronoi Scenarios: (1) the possible variation in the electrical power interruption from 7-21 d; (2) the percentage of Toronto businesses included in Scott's Index ® (25%, 50% and 75%); and (3) the actual revenue of businesses within their reported revenue ranges per Scott's Index ® .We used a Monte Carlo simulation to estimate the influence of each uncertainty factor on ColVAR compared to the deterministically estimated results, using @Risk ® .Results are depicted as three overlaid probability density functions (PDFs) at figure 14.
To derive the PDFs, we applied triangular distributions to the three uncertainties (power interruption periods, the Scott's Index ® coverage range and the min, mean and max values for estimated revenue from table 3) and performed a 10 000 iteration Monte Carlo simulation to assess the effects of these three uncertainties on ColVAR.Figure 14 shows the results for each of the Voronoi Scenarios: 85% in blue; 100% in red; and 110% in green.The resulting distributions exhibit a distinct right skew, with 90% confidence intervals ranging from a low of CAD$100 M for the 85% Voronoi (∼20% of the direct losses from TBRDT) up to CAD$740 M for the 115% Voronoi (over 1.5 times the direct losses from TBRDT).Across this range (CAD$100-740M), the ColVar remains a substantial contributor to WVAR, amounting to ∼15%-60% of the associated WVAR totals (CAD$580-1200M).The supporting data may be found at SI table 2.
Figure 15 shows a tornado plot of the uncertainties ranked by effect on the output mean for the 100% Voronoi Scenario to illustrate the decomposition of contributing uncertainties in figure 14.The edges of the tornado bars show the output (ColVaR) mean for the simulation iterations representing lowest (respectively highest) 10% of values for each selected input; the total width of each bar is an indicator of the importance of each variable and its uncertainty on the final results.
While the annual business revenue range derived from Scott's index contributes the greatest uncertainty, all contribute similar orders of magnitude to the range in the final results.The % coverage of the index presents an inverse relationship between the contributions of the high and low inputs in the other uncertainties, suggesting that in this scenario when the business index coverage is high, ColVAR decreases.Conversely, the lower the business index coverage is, the higher ColVAR becomes.Assuming a 75% coverage rate (high input) for the business index data reduces the uncertainty about business value remaining outside the model.ColVAR is therefore calculated on the known range of values with allowances made for only a 25% uncertainty.But when only 25% (low input) coverage is assumed, a greater range of unknown value possibilities must be considered, hence a greater ColVAR will result.

Discussion
The mean ColVaR for the 100% Voronoi Scenario is CAD$344M or 72% of TBRDT, but in the 115% Voronoi Scenario and 21 d duration, ColVaR is CAD$427M or 90% of TBRDT-even before accounting for uncertainties in the business revenue range or index coverage.The 90% confidence intervals across all scenarios range from CAD$100M-CAD$740M as discussed above.
As mentioned, typically only the TBRDT (CAD$476M) is insurable or eligible for relief funding, meaning ∼CAD$100-CAD$740M of community value remains exposed.Further, when considering DRR measures to protect value in the Lower Don River valley, the benefit cost ratio would be calculated using only category 2 + 3 TBRDT losses versus the combined category 2 + 3 + 4 WVAR loss estimate of between CAD$580M and CAD$1,200M (90% confidence interval across all scenarios).This means whatever mitigation measures are considered for these flood vulnerable areas, their DRR return on investment could potentially be substantially undervalued.
Despite the large uncertainty in these estimates, they remain substantial in all scenarios.The greatest contributor in this example is the service area sizes or Voronoi Scenarios.In a municipal study this uncertainty should not exist as city staff will have access to precise subscriber data.Following this, the revenue range for each business has the next greatest influence.This uncertainty can be reduced through the correlation of municipal business tax data.Each business pays municipal taxes based on its revenue and each record has either an address or a unique geocode associated with it.The business revenue uncertainty could be greatly reduced, if not eliminated using this data.Likewise, the business index coverage uncertainty would also be reduced to near zero by using municipal tax record data.Service interruption uncertainty is likely to remain however until active resilience measures such as smart grid architecture and load rebalancing are implemented, the ROI of which can now be assessed against a comprehensive WVAR.
As the aftermath of Hurricane Katrina and Super Storm Sandy demonstrate, the economic shock of flood disaster can so diminish the embodied value of communities as to cause them to collapse and depopulate.As Katrina demonstrates particularly, the cost of the mitigations proposed by the Army Corps of Engineers for the rehabilitation of the Levees was miniscule in comparison to the TBRDT, but both were dwarfed by the ColVAR, the effects of which are still being felt in New Orleans today [31,35,85,88].
As stated, the objective of this work is to illustrate a method by which staff in any community can estimate ColVAR using existing tools and techniques to sufficient accuracy to support informed DRR resource allocation and decision making.It is not intended as a real-world analysis of this particular case.To that end, the census data used comes from the 2016 statistics Canada census of the canadian population using data collected in 2015 rather than the 2021 census data now available.While 2016 census data is sufficient to demonstrate the process, practitioners are encouraged to use current census data for real world estimation purposes.
Notwithstanding these caveats, that WVAR can potentially be so significantly underestimated indicates the value of the exercise when considering the benefit cost ratio calculation for DRR investment to protect any community's economic value in an increasingly uncertain context.

Conclusion
Where the societal value of infrastructure, and thereby its DRR priority, is determined by its economic value, it is important to consider ColVaR (Category 4 Risk) as a part of the WVAR, especially when ColVaR can be a significant portion of the TBRDT risk alone.Failure to include ColVaR can have consequences including a possible mis-prioritization of DRR resources at the local level.
Disaster response planning is based on operational assumptions that, in certain collateral damage scenarios, will be invalid.For example, the use of schools and community centres adjacent to the flood zone as emergency relief centres may no longer be feasible if electrical power and water are not present because of collateral losses.Care homes and medical facilities that are not within flood zones may be supplied by critical infrastructure components that are, and that may fail as a result, invoking emergency protocols when they were not expected, and when that facility is expected to be operating at full capacity as part of the emergency response plan.
Category 4 home and business owners subject to collateral losses from flooding are typically uninsured for that peril and usually ineligible for disaster relief funding.Consequently, they carry the financial burden for response and recovery themselves, often unknowingly.That burden typically occurs when many of the businesses that employ them cannot operate and they are temporarily laid off, constraining household income just when it is most needed.
Such effects accumulate and degrade community value.Local resources that must then be spent on response and recovery are not available for enriching family and community life.For most people and businesses, it is not a zero-sum game.People choose the place they live and work as much for the enrichment the place brings them, as they do for its economic benefit.Increasingly, people are considering the resilience and sustainability of the communities they choose to live in and avoiding those that are not inherently resilient if they can.Discovering and mitigating collateral risks must become as much an integral part of modern urban 'place making' as transport-oriented design and walkability if we are to make cities desirable and liveable for the next century and beyond.

Further work
The analysis presented here illustrates a systematic and reproducible method for quantifying collateral value at risk (ColVAR) using tools and techniques readily accessible to infrastructure stakeholders and planners, however it remains limited in scope.Although we flagged several key uncertainties in our analysis, numerous systemic uncertainties remain-requiring further analysis to bound maximum plausible flood extent (e.g. using climate and hydrological models), improve on our simplistic binary rule that failure occurs with certainty once design depth is exceeded, and incorporate a broader set of human and physical responses to flooding beyond business interruption and household income.
We also acknowledge that this study does not explore the social justice aspects of flooding through deeper investigation of the socio-economic data available in the census data.Demographic characteristics, zoning, and density of the affected areas will affect both the numerical ColVaR and WVAR, as well as the prioritization of risk reduction measures as related to vulnerable populations.Exploration of these aspects inevitably necessitates the application of data mining and spatial analysis techniques beyond both the remit and the space limitations of this article.Clearly, they are also important and represent another avenue of future investigation.
Both Mureddu et al and Wang et al [77,78] agree that planning infrastructure for the future is a problem in deep uncertainty.The conditions of operation during the service life of infrastructure components can change significantly due to climate change and other factors, yet the demand for essential services will increase as populations grow and as citizen expectations increase.The consequence of infrastructure failure on the health and wellbeing of communities therefore increases with the size of the population and the uncertainty in the evolving operational context.Consequently, infrastructure planning based on heuristic codes of historical best practice may be less useful and need to be augmented by risk-based planning methods that can model the effects of various risk scenarios and the tractability of possible mitigations.
Essential services in large cities are provided by complex, dependent and inter-dependant infrastructure 'systems of systems' .The effect of a hazard on a single component has both direct and indirect effects on others and those effects may cascade into a systems-wide failure.Risk mitigations implemented in one component may also invoke unintended and undesired consequences that can cascade across multiple systems, interrupting essential services and negatively affecting embodied community value.While spatial data analysis is a valid decision support tool to visualize the spatial effects of infrastructure systems failure, it cannot model infrastructure system level risks.This next step requires a network-based risk analysis approach.
As Pederson et al and Guidotti et al observe [89,90], the potential of directed graphs to model the risk to infrastructure systems and assess the tractability and feasibility of proposed mitigations is in its infancy.Following work will explore the application of graph theory, game theory and social justice analysis to risk-based infrastructure systems planning under the conditions of deep uncertainty with a view to providing additional accessible tools and techniques for planners and infrastructure stakeholders to use.

Figure 1 .
Figure 1.The contribution this paper makes to the pathway, receptor, consequence model of Jha et al.Adapted based on Jha et al, 2012.Reproduced with permission.Adapted from Jha et al [9].CC BY 3.0.

Figure 2 .
Figure 2. The process workflow extending the World Bank model for defining flood risk (Jha et al) to capture collateral value at risk and comprehensively estimate whole value at risk.* (CI = Critical Infrastructure).

Figure 3 .
Figure 3.The case study area of interest on the Don River Watershed showing identified flood vulnerable areas in red in the left pane, magnified for clarity on the right.Contains information made available under the Toronto and region conservation authority (TRCA)'s Open Data Licence v 1.0.Adapted with permission from TRCA (2019) Open Data Licence v 1.0.

Figure 4 .
Figure 4. Transformer station 'receptors' (Gerrard Street TS and Basin TS) in the pathway of regional design flood (P = 1:500 Hurricane Hazel, 1954).Source: Basemap data from ESRI community maps; contributors, City of Toronto, Province of Ontario, ESRI Canada, HERE, SafeGraph, Maxar technologies, © 2021.Contains information made available under the Toronto and region conservation authority (TRCA)'s Open Data Licence v 1.0.Data sourced with permission from TRCA (2019) Open Data Licence v 1.0.

Figure 5 .
Figure 5. Nadir view of the regional design flood fringe, shown in light blue overlaid on Gerrard Street TS East and west Yards.Source: ESRI ArcGIS Living Atlas of the World in ArcGIS Pro and imagery from HERE, Maxar and Sentinel.Contains information made available under the Toronto and region conservation authority (TRCA)'s Open Data Licence v 1.0.Data sourced with permission from TRCA (2019) Open Data Licence v 1.0.

Figure 6 .
Figure 6.3D flood extent map of Gerrard Street transfer station East and West Yards, looking south along the lower Don valley.Depth data for the spot heights is calculated above ground level (AGL).Z accuracy = ± 4 cm.Source: Basemap data from the ESRI ArcGIS Living Atlas of the World, HERE.Maxar and Sentinel © 2020.Scene built in ESRI Arc GIS Pro in 3D view.Contains information made available under the Toronto and region conservation authority (TRCA)'s Open Data Licence v 1.0.Data sourced with permission from TRCA (2019) Open Data Licence v 1.0.Data sourced with permission from TRCA (2019) Open Data Licence v 1.0.

Figure 7 .
Figure 7. Regional design flood depth at Gerrard Street transformer station East and West Yards showing projected inundation of the equipment.Scene produced using imagery from Google Street View © 2020.Contains information made available under the Toronto and region conservation authority (TRCA)'s Open Data Licence v 1.0.Data sourced with permission from TRCA (2019) Open Data Licence v 1.0.

Figure 8 .
Figure 8. Basin Street Transformer Station shown with the Regional Flood Fringe Overlay and spot depths.Depth data for the spot heights is calculated above ground level (AGL).Z accuracy = ± 4 cm.Scene built using imagery from ESRI ArcGIS Living Atlas of the World in ESRI Arc GIS Pro and Digital Elevation Model.Contains information made available under the Toronto and region conservation authority (TRCA)'s Open Data Licence v 1.0.Data sourced with permission from TRCA (2019) Open Data Licence v 1.0.

Figure 9 .
Figure 9. Regional Design Flood depth at Basin Street Transformer station showing projected inundation of the equipment.Scene produced using imagery from Google Street View © 2020.Contains information made available under the Toronto and region conservation authority (TRCA)'s Open Data Licence v 1.0.Data sourced with permission from TRCA (2019) Open Data Licence v 1.0.

Figure 10 .
Figure 10.Voronoi tessellation to approximate electrical transformer station service areas for the City of Toronto, with Gerrard Street TS and Basin Street TS service areas outlines in blue and overlaid on the regional flood extent Polygon.Sources: Basemap data from the ESRI ArcGIS Living Atlas of the World, HERE and Sentinel.Contains information made available under the Toronto and region conservation authority (TRCA)'s Open Data Licence v 1.0.Data sourced with permission from TRCA (2019) Open Data Licence v 1.0.

Figure 11 .
Figure 11.Nested Thiessen polygons showing the ±15% uncertainty in the estimated service interruption areas for Gerrard and Basin TS.Source: Basemap data from the ESRI ArcGIS Living Atlas of the World, HERE and Sentinel.Data sourced with permission from TRCA (2019) Open Data Licence v 1.0.

Figure 12 .
Figure 12.Spatial distribution of businesses within the estimated Gerrard Street TS and Basin Street TS service areas and the Regional Design Flood fringe for the 100% Voronoi Scenario.Sources: Basemap data from the ESRI ArcGIS Living Atlas of the World, HERE and Sentinel.Scene created using Scott's Index ® data under licence in ESRI Arc GIS Pro ® and QGIS.Contains information made available under the Toronto and region conservation authority (TRCA)'s Open Data Licence v 1.0.Data reused with permission from Scott's Business Index ® .Data sourced with permission from TRCA (2019) Open Data Licence v 1.0.

Figure 13 .
Figure 13.Census Dissemination Areas (CDAs), overlaid with the Gerrard and Basin Street TS Service Area polygons for 85%, 100% and 115% scenarios.CDAs are coloured by % of CDA without electrical power per the legend.Sources: Basemap data from the ESRI ArcGIS Living Atlas of the World, HERE and Sentinel.Census Geography data for Census dissemination Areas (CDA) from the Statistics Canada 2015 Census of the Canadian Population, reproduced under the Open Government Licence-Canada.Data sourced from Statistics Canada, Open Government Licence-Canada.

Figure 14 .
Figure14.The probability density function (PDF) for ColVaR resulting from the three uncertainties (Scott's index coverage, event duration, and actual revenue within the stated range) for the 85%, 100% and 115% Voronoi scenarios, derived using a 10 000 iteration Monte Carlo Simulation in @Risk ® .For greater numerical context, the bar above the graph shows the 90% confidence interval for the middle (100%) Voronoi scenario, and percentiles for the other 2 scenarios associated with those bounds.

Figure15.
Figure15.Tornado plot showing the mean ColVar resulting from the lower (in red) and upper (in blue) 10% realizations for each input to the Monte Carlo simulation: % Scott's index coverage, service interruption, and annual business revenue at risk for the 100% Voronoi Scenario.(All amounts in $CAD).Monte Carlo Simulation in @Risk ® .

Table 1 .
Excerpt from toronto flood risk tangible building related damage totals (TBRDT) in CAD $K.Source: Adapted from Toronto Flood Risk Ranking 2019[43].Data reproduced with permission.Adapted with permission from TRCA (2019) Open Data Licence v 1.0.

Table 2 .
Flood/Power status of four receptor categories.

Table 4 .
Annual household income at risk in the 85%, 100% and 115% Voronoi scenario.These values will subsequently be increased by the index coverage parameter (discussed below).(All amounts in $CAD).

Table 5 .
ColVAR calculated using mean revenue estimates from Scott's index.(All amounts in $CAD).

Table 6 .
Deterministically derived whole value t risk range for three Voronoi scenarios in the Lower Don Valley P = 1:500 flood hazard profile.(All amounts in $CAD).

table 3 ,
estimated ColVAR values are given at table 5. SI table 2 provides the data supporting this operation.