Open question stated by Fedeski and Gwilliam (2007) in their GIS-based vulnerability and risk assessment methodology for urban areas subject to flood and geological hazards: given that hazard-specific damages are estimated separately and combined by summation, when do temporal and physical dependence between hazard events make this simple summation invalid, and when are interaction models required? Later literature has produced taxonomies of hazard interrelationships (Kappes et al. 2012; Gill and Malamud 2014; Liu et al. 2016; Tilloy et al. 2019), statistical machinery for dependent drivers and compound events (Zscheischler and Seneviratne 2017; Zscheischler et al. 2018; Tilloy et al. 2020; Pflug et al. 2026), local-scale coupled-damage models with state-dependent vulnerability (Korswagen et al. 2019), and one fully quantitative compound-flooding framework (Ming et al. 2022). No published work states general conditions under which additive aggregation of hazard-specific damages is valid, nor supplies validated urban-scale interaction models covering flood combined with geological hazards. The open core — a quantitative validity boundary for additive damage aggregation under hazard dependence, and transferable interaction models where it fails — remains open.

Background

In disaster-risk research, risk is assessed by combining hazard (the probability and severity of a dangerous event), exposure (the assets in harm's way), and vulnerability (the susceptibility of those assets to damage). Fedeski and Gwilliam (2007) developed, within the UK ASCCUE project (Adaptation Strategies for Climate Change in the Urban Environment), a GIS-based methodology — GIS, geographic information system, meaning that spatial hazard layers are overlaid on a building dataset — for assessing risk to buildings from fluvial (river) and coastal flooding and from geological hazards such as landslides, shrink–swell clays, ground dissolution, and compressible or collapsible deposits. For each building they predicted damage as damage=RR×DV\text{damage}=\text{RR}\times\text{DV}, where RR (replacement ratio) is the building's construction value expressed in units of an average house and DV (damage value) is the expected fraction of that value lost, itself a function DV=f(SI,VI)\text{DV}=f(\text{SI},\text{VI}) of a hazard severity index SI and a building vulnerability index VI, the latter built from externally observable building factors weighted by the Analytical Hierarchy Process (a structured pairwise-comparison weighting technique). Flood severity was taken from Environment Agency probability and depth layers, and geological hazard potential from the British Geological Survey GeoSure datasets, which give qualitative susceptibility classes rather than quantitative severity or probability. Building-level damages are aggregated by arithmetic summation to neighbourhood and city scales, and damages for different hazards are combined by summing the hazard-specific estimates. In a Lewes (UK) case study of 1503 buildings under a 1-in-100-year flood (an event with roughly 1% annual exceedance probability), the method's two provisional damage functions produced aggregated totals of 500 and 890 housing-equivalents, correlating strongly at neighbourhood scale with three established loss-estimation approaches. Among six gaps the authors listed for future work, the fifth is the question posed here: the temporal and physical dependence between different hazard events was not investigated, so it remained unknown when this simple summation of hazard-specific damages is valid and when interaction models are required.

Later work has organized the problem without closing it. Kappes et al. (2012) reviewed the challenges of multi-hazard risk analysis, including the comparability and combination of single-hazard results. Gill and Malamud (2014) catalogued 90 interaction relationships among 21 natural hazards, in which one hazard triggers another or increases its probability of occurrence. Liu et al. (2016) classified relationships between hazards as independent, mutex (mutually exclusive), parallel, or series based on shared triggering factors, and stated the summation problem in its modern form: existing multi-hazard risk assessments compute each hazard's loss separately, assuming unchanged vulnerability between events, and then sum the losses, which misrepresents real losses whenever an earlier event has changed the vulnerability of exposed assets. Tilloy et al. (2019) reviewed methodologies for quantifying hazard interrelationships — triggering, change-condition, compound, independence, and mutually exclusive — identifying 19 modelling approaches in stochastic, empirical, and mechanistic families with sparse quantitative coverage across hazard pairs. On the statistical side, Zscheischler and Seneviratne (2017) showed for compound hot–dry events that dependence between drivers can change the frequency of joint extremes by a large factor relative to the independence assumption; Zscheischler et al. (2018) established compound events — impact-relevant combinations of multiple climate drivers and/or hazards — as a research field, noting that traditional risk assessment considers one driver at a time and can underestimate risk, and Zscheischler et al. (2020) proposed a typology distinguishing, among others, consecutive events in which an earlier event preconditions impacts of a later one. The standard statistical tools are copulas (functions coupling marginal distributions into a joint distribution, so that dependence can be modelled separately from the marginals) and bivariate extreme-value models (statistical models of the joint tail behaviour of two variables), for which Tilloy et al. (2020) provide model-selection guidance for pairs of hazards, and Pflug et al. (2026) introduced Poisson-type dependence models between event processes of different hazard types. On the physical side, Korswagen et al. (2019) demonstrated a probabilistic framework for cumulative structural damage from coupled hazards (an earthquake followed by an earthquake-triggered flood, for an unreinforced masonry house behind a levee) in which the first hazard structurally weakens the building so that subsequent damage exceeds what the unconditioned damage function predicts; such state-dependent vulnerability is precisely what additive summation ignores. Ming et al. (2022) built a fully quantitative framework for compound flooding — joint statistics of heavy rainfall, extreme river flow, and storm surge via copulas, coupled to a 2D hydrodynamic inundation model and damage functions — applied to Greater London and the Thames estuary. Ward et al. (2020), reviewing natural-hazard risk assessments at the global scale, concluded that damage from interacting multi-hazard chains has been assessed only at local scales and that truly comprehensive multi-hazard risk assessment remains unachieved. What no published work supplies is the piece Fedeski and Gwilliam asked for: general conditions under which the summation is valid, and validated interaction models for the settings where it fails, at the spatial scales and hazard combinations of urban planning practice.

Problem Statement

Fedeski and Gwilliam's methodology, like most planning-scale multi-hazard assessments, estimates the damage from each hazard separately — per building, via damage functions DV=f(SI,VI)\text{DV}=f(\text{SI},\text{VI}) (DV, the expected fraction of construction value lost; SI, a hazard severity index; VI, a building vulnerability index), aggregated by summation — and combines hazards by summing the hazard-specific damage estimates. The source poses, as an open problem, the investigation of dependence between different hazard events, covering temporal dependence (statistical dependence between the occurrence, timing, or severity of events of different hazard types) and physical dependence (interaction of the events themselves, including changed vulnerability of exposed assets after an earlier event and joint forcing beyond the superposition implicitly assumed by separate damage functions), in order to determine when this simple summation of hazard-specific damages is valid and when interaction models are required.

Determine, in quantitative terms, the conditions under which summed hazard-specific damage estimates reproduce the aggregate multi-hazard loss, as a function of: (a) the risk metric adopted, e.g., expected periodic damage as in the source methodology versus exceedance probabilities or quantiles of aggregate loss; (b) the statistical dependence structure between hazard events, including temporal clustering, common drivers, and joint extremes; and (c) the state-independence of vulnerability and damage functions across events, e.g., recovery and structural weakening after an earlier hazard. For settings where summation fails, interaction models are required that quantify the discrepancy at the spatial scales used in urban risk assessment, from individual buildings to whole cities: they must cover at least the flood and geological hazard combination of the source setting, transfer to other hazard pairs, and admit independent validation. The question is posed at the generality of the hazards combined in such assessments; for the validity conditions, no restriction to particular hazard pairs, metrics, or scales is implied.

References

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