The quantification of low-probability–high-consequences events: part I. A generic multi-risk approach
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Abstract
Dynamic risk processes, which involve interactions at the hazard and risk levels, have yet to be clearly understood and properly integrated into probabilistic risk assessment. While much attention has been given to this aspect lately, most studies remain limited to a small number of site-specific multi-risk scenarios. We present a generic probabilistic framework based on the sequential Monte Carlo Method to implement coinciding events and triggered chains of events (using a variant of a Markov chain), as well as time-variant vulnerability and exposure. We consider generic perils based on analogies with real ones, natural and man-made. Each simulated time series corresponds to one risk scenario, and the analysis of multiple time series allows for the probabilistic assessment of losses and for the recognition of more or less probable risk paths, including extremes or low-probability–high-consequences chains of events. We find that extreme events can be captured by adding more knowledge on potential interaction processes using in a brick-by-brick approach. We introduce the concept of risk migration matrix to evaluate how multi-risk participates to the emergence of extremes, and we show that risk migration (i.e., clustering of losses) and risk amplification (i.e., loss amplification at higher losses) are the two main causes for their occurrence.
Keywords
Multi-hazard Multi-risk Extreme event Monte Carlo Markov chain1 Introduction
Multi-risk assessment is still in its infancy and often only refers to the analysis of multiple single hazards in a same framework (Grünthal et al. 2006; Carpignano et al. 2009; Schmidt et al. 2011). Major catastrophes however remind us that multi-risk is not simply the sum of individual risks but that correlations between natural hazards, technological hazards and our complex socioeconomic networks lead to greater risks (e.g., 2005 hurricane Katrina, USA; 2010 eruption of Eyjafjallajökull, Iceland; 2011 Tohoku earthquake, Japan). Innovative methods have been proposed in recent years to tackle the problem of hazard interactions (e.g., Marzocchi et al. 2012) and of other dynamic aspects of risk, such as time-dependent vulnerability and exposure (e.g., Selva 2013) or network failures (e.g., Adachi and Ellingwood 2008). However, so far, only a limited number of scenario-based and/or site-specific multi-risk studies have been proposed due to the difficulty and novelty of the task.
Development of a comprehensive multi-risk framework is hampered by the following requirements: (1) large amount of input data (2) cross-disciplinary expertise and (3) innovative risk assessment methods. The first two points are generally solved in dedicated multi-risk projects at the national (e.g., HAZUS-MH, http://www.fema.gov/hazus), international (e.g., CAPRA, http://www.ecapra.org/) or private sector levels (e.g., Grossi and Kunreuther 2005; Schmidt et al. 2011). The third point remains to be solved. As indicated by Kappes et al. (2012), “despite growing awareness of relations between hazards, still neither a uniform conceptual approach nor a generally used terminology is applied”. Similarly, but based on feedback from civil protection stakeholders, Komendantova et al. (2014) noted that: “two areas are most problematic. These are (1) the absence of clear definitions and (2) the lack of information on the added value of multi-risk assessment”.
In the present study, we present a novel, generic, multi-risk framework based on the sequential Monte Carlo Method (MCM) to allow for a straightforward and flexible implementation of hazard interactions, which may occur in a complex system. Real-world examples of hazard interactions include: earthquake clustering, storm clustering, tsunamis following earthquakes or landslides, landslides or fire following earthquakes, storm surges associated to hurricanes, technological accidents triggered by natural events (i.e., NaTech events). Time-variant vulnerability and exposure related to hazard clustering are also considered, although not the primary focus of this study. More generally, time-variant vulnerability may refer to different processes, such as structure ageing, not-repaired pre-damage due to past events or damage conditioned on the co-occurrence of several events. Time-variant exposure supposes the evolution of assets value with time, which may be due to socioeconomic factors or to previous losses.
Our goal is specifically to capture and quantify extreme (i.e., low-probability–high-consequences) events using inductive generalization (e.g., Bier et al. 1999) and by following the recommendation of Kameda (2012), which is to “mobilize “scientific imagination” in the process of decision”—by incorporating extreme events in risk modelling if no observations but sound scientific bases are available (Note that the term “reasoned imagination” is used by Paté-Cornell 2012). Our approach differs from site-specific and scenario-based studies (e.g., Adachi and Ellingwood 2008; Marzocchi et al. 2012; Selva 2013) in that we do not define any specific hazard or risk interaction but a framework to implement any type of interaction. With such an objective, real interaction processes have to be abstracted to more basic concepts and engineering methods by-passed. The proposed framework is described in Sect. 2, in which the concept of hazard correlation matrix is introduced.
Validation of our framework, which should be considered as a proof-of-concept, is made using generic data and processes defined heuristically. This strategy, that is the use of intuitive judgment and simple rules, allows for the solving of problems that are otherwise difficult to consider. Based on an extensive literature survey, we generalize the concepts of peril, of peril characterization and of hazard interaction. Our aim, by abstracting these concepts into basic categories, is to provide some general guidelines for extreme event quantification. The proof-of-concept is presented in Sect. 3, and a discussion on the applicability of the proposed framework to real-world conditions is given in Sect. 4.
List of the symbols used in the present study
Symbols | Description |
---|---|
A, B, C, … | Peril identifier |
A _{ i }, B _{ i }, C _{ i }, … | Stochastic event identifier |
i, j, k | Increment |
n | Number of events |
N _{sim} | Number of simulations |
t | Time in the interval Δt = [t _{0}; t _{max}] |
ε | Time lapse between two correlated events (ε ≪ Δt) |
λ | Long-term occurrence rate |
λ _{mem} | Time-variant occurrence rate |
β | Event frequency–intensity ratio |
ι | Hazard intensity |
δ | Mean damage ratio (i.e., system damage) |
μ | Mean of lognormal vulnerability curve |
σ | Standard deviation of lognormal vulnerability curve |
γ | Calibration factor for conditional vulnerability curve |
Λ | System loss |
Ε | System exposure |
e | Reconstruction function |
Pr(j|i) | Probability of occurrence of event j conditional on event i |
α | Shape parameter of the lognormal distribution for event repeat |
ΔT _{ ij } | Time shift in occurrence rate of event j due to event i |
f | Coupling factor |
ϕ | Index of dispersion |
2 Generic multi-risk framework
2.1 Sequential Monte Carlo Method
The proposed multi-risk framework is formed of a core simulation algorithm based on the MCM. We adopt the MCM for its flexibility when dealing with complex systems. We generate N _{sim} time series, sampling events from a Poisson distribution (homogeneous or non-homogeneous process). Each time series represents one risk scenario, and the analysis of N scenarios allows for the probabilistic assessment of losses and for the recognition of more or less probable risk paths. These risk paths emerge naturally from the system implemented in the MCM.
A set of stochastic events is defined as input for the MCM, with each event characterized by an identifier, a long-term occurrence rate λ and a loss Λ. The loss is defined as Λ = δ Ε with δ the mean damage ratio and Ε the system exposure. δ is derived from the hazard intensity ι with each event being represented by one unique hazard footprint. Each event is therefore implicitly related to one specific source, e.g., an earthquake related to a given fault segment, a storm related to a given track (see Sect. 3.1.1 for the definition of a stochastic event set).
- Multi-hazard assessment: define the simulation set with simulation identifier, event identifier and event occurrence time t.
- 1.
Generate N _{sim} random time series: Sample N _{sim} sets of events over the time interval Δt = [t _{0}; t _{max}] drawn from the Poisson distribution with each stochastic event i characterized by the long-term rate parameters λ _{ i }. Affix an occurrence time t to each event following the random uniform distribution. Record the time series in the simulation set S _{0}, which represents the null hypothesis H _{0} of having no interaction in the system. Fix increment j = 1, which indicates the occurrence of the first event in the time series.
- 2.
For each of the N _{sim} simulations, record the characteristics of the jth event, which occurs at t _{ j }, in simulation set S _{1}. Resample events k occurring in the interval [t _{ j }; t _{max}] if the conditional probability Pr(k|j) exists. This conditional probability is defined in the hazard correlation matrix, described in Sect. 2.2. Affix t _{ k } = t _{ j } + ε with ε ≪ Δt. Fix j = j + 1.
- 3.
Repeat step 2 while t _{ j } ≤ t _{max}.
- 4.
Fix j = 1.
- 1.
- Multi-risk assessment: update the simulation sets S _{0} and S _{1} with event loss Λ.
- 5.
For each of the N _{sim} simulations, calculate the mean damage ratio δ _{ j } due to the jth event, which is potentially conditional on the occurrence of previous events. The implementation of time-variant vulnerability is described in Sect. 2.3.
- 6.
For each of the N _{sim} simulations, calculate the loss Λ _{ j } due to the jth event, which is potentially conditional on the occurrence of previous events. The implementation of time-variant exposure is also described in Sect. 2.3. Record Λ _{ j }.
- 7.
Repeat steps 5 and 6 while t _{ j } ≤ t _{max}.
- 5.
2.2 Hazard correlation matrix
We introduce the concept of hazard correlation matrix to quantify hazard interactions. It should be noted that we use a loose definition of the word interaction, as we also refer to one-way causal effects by this term (noted→). The hazard correlation matrix is illustrated in Fig. 1b where trigger events are represented in rows i and target/triggered events in columns j. A given peril P consists of n events P _{ i } with 1 ≤ i ≤ n. Each cell of the square matrix indicates the 1-to-1 conditional probability of occurrence Pr(j|i) = Pr(P _{ j }|P _{ i }) over Δt, which is used as input in the MCM. We also consider the n-to-1 conditional probability by incorporating a memory element to the correlation matrix. Various interaction processes may be implemented, based on empirical, statistical or physical laws. Those are run in the background with only the conditional probability represented in the hazard correlation matrix. However, the memory element is defined such that it can alter the process in the background by informing it of the sequence of previous events. This is illustrated by several examples in Sect. 3.1.2. The approach is different from a strict Markov chain in the fact that it is not memoryless and because the matrix does not require \(\mathop \sum \limits_{j} \Pr \left( {j|i} \right) = 1\) (i.e., finite chains of events).
We define various terms (noted in italics) to categorize different types of interactions based on the concept of hazard correlation matrix: An event repeat is described by P _{ i } → P _{ i }, an intra-hazard interaction by P _{ i } → P _{ j } and an inter-hazard interaction by A _{ i } → B _{ j } with A and B two different perils. Moreover, perils can be separated into primary perils A when λ _{ A } > 0 and secondary perils B when λ _{ B } = 0 and Pr(B|A) > 0, with λ the long-term occurrence rate. Invisible events i, which do not yield direct losses, should be included in the system if they trigger events j that do (case Pr(j|i) > 0, Λ _{ i } = 0 and Λ _{ j } > 0).
2.3 Time-variant exposure and time-variant vulnerability
The clustering of events in time may also influence the vulnerability, which can be described by the conditional mean damage ratio δ _{ j|i }. The dependence on the trigger event i may take different forms, independently of the framework developed here. An example of vulnerability dependence on hazard intensity is given in Sect. 3.1.3. Time-dependent vulnerability δ _{ i }(t), such as ageing, is not considered.
3 Proof-of-concept
3.1 Generic data and processes
We generate generic data and processes by following the heuristic method and by abstracting the concepts of peril, of peril characterization and of hazard interaction into basic categories. Our approach provides some general guidelines for extreme event quantification and a dataset for testing the generic multi-risk framework described in Sect. 2. It follows the existing recommendations on extreme event assessment (Bier et al. 1999; Kameda 2012; Paté-Cornell 2012) by combining inductive generalization and “scientific imagination” to include known examples of extremes as well as potential “surprise” events in a same framework. We intentionally do not consider the case of networks (Adachi and Ellingwood 2008). Data and processes are then implemented in the MCM, and the results analysed in Sects. 3.2 and 3.3. In the present study, we use N _{sim} = 10^{5}, Δt = [t _{0} = 0; t _{max} = 1] and ε = 0.01. It should be noted that the numerous assumptions made below are not a requirement of the proposed multi-risk framework but are working hypotheses for basic testing purposes.
3.1.1 Building a stochastic event set
Using risk as a common language, we resolve the problem of hazard comparability. Event loss Λ _{ i } is then defined by Λ _{ i } = δ _{ i } Ε where Ε = 1 the system exposure. Since a unique vulnerability curve is used for the full exposure, all losses given in the present study are mean loss values.
Stochastic event set defined as input for the MCM
Event id. | λ | Λ | Event id. | λ | Λ |
---|---|---|---|---|---|
A _{1} | 0.100000 | 0.00003 | B _{6} | 0.031623 | 0 |
A _{2} | 0.079433 | 0.00008 | B _{7} | 0.025119 | 0 |
A _{3} | 0.063096 | 0.00018 | B _{8} | 0.019953 | 0 |
A _{4} | 0.050119 | 0.00038 | B _{9} | 0.015849 | 0 |
A _{5} | 0.039811 | 0.00073 | B _{10} | 0.012589 | 0 |
A _{6} | 0.031623 | 0.00131 | B _{11} | 0.010000 | 0.00003 |
A _{7} | 0.025119 | 0.00220 | B _{12} | 0.007943 | 0.00018 |
A _{8} | 0.019953 | 0.00350 | B _{13} | 0.006310 | 0.00096 |
A _{9} | 0.015849 | 0.00532 | B _{14} | 0.005012 | 0.00394 |
A _{10} | 0.012589 | 0.00778 | B _{15} | 0.003981 | 0.01283 |
A _{11} | 0.010000 | 0.01099 | B _{16} | 0.003162 | 0.03415 |
A _{12} | 0.007943 | 0.01505 | B _{17} | 0.002512 | 0.07620 |
A _{13} | 0.006310 | 0.02006 | B _{18} | 0.001995 | 0.14608 |
A _{14} | 0.005012 | 0.02611 | B _{19} | 0.001585 | 0.24509 |
A _{15} | 0.003981 | 0.03326 | B _{20} | 0.001259 | 0.36727 |
A _{16} | 0.003162 | 0.04156 | B _{21} | 0.001000 | 0.50000 |
A _{17} | 0.002512 | 0.05104 | B _{22} | 0.000794 | 0.62872 |
A _{18} | 0.001995 | 0.06173 | B _{23} | 0.000631 | 0.74063 |
A _{19} | 0.001585 | 0.07359 | B _{24} | 0.000501 | 0.82986 |
A _{20} | 0.001259 | 0.08662 | B _{25} | 0.000398 | 0.89471 |
A _{21} | 0.001000 | 0.10079 | B _{26} | 0.000316 | 0.93851 |
A _{22} | 0.000794 | 0.11605 | B _{27} | 0.000251 | 0.96593 |
A _{23} | 0.000631 | 0.13227 | B _{28} | 0.000200 | 0.98189 |
A _{24} | 0.000501 | 0.14948 | B _{29} | 0.000158 | 0.99104 |
A _{25} | 0.000398 | 0.16751 | B _{30} | 0.000126 | 0.99564 |
A _{26} | 0.000316 | 0.18634 | B _{31} | 0.000100 | 0.99799 |
A _{27} | 0.000251 | 0.20581 | C _{1} | 0 | 0.01 |
A _{28} | 0.000200 | 0.22559 | C _{2} | 0 | 0.1 |
A _{29} | 0.000158 | 0.24661 | C _{3} | 0 | 1 |
A _{30} | 0.000126 | 0.26717 | D _{1} | 0 | 0.01 |
A _{31} | 0.000100 | 0.28847 | D _{2} | 0 | 0.1 |
B _{1} | 0.100000 | 0 | D _{3} | 0 | 1 |
B _{2} | 0.079433 | 0 | E _{1} | 0 | 0.01 |
B _{3} | 0.063096 | 0 | E _{2} | 0 | 0.1 |
B _{4} | 0.050119 | 0 | E _{3} | 0 | 1 |
B _{5} | 0.039811 | 0 |
3.1.2 Populating the hazard correlation matrix
Characteristics of generic perils and processes and analogy with real ones
Peril | Type | Parameters | Analogy | |
---|---|---|---|---|
A | Primary | β _{ A } = ln(10), μ _{ A } = ln(5), σ _{ A } = 0.4 | E.g., earthquake | |
B | β _{ B } = 0.5ln(10), μ _{ B } = ln(6), σ _{ B } = 0.1 | E.g., volcanic eruption | ||
C | Secondary | δ _{1} = 0.01, δ _{2} = 0.1, δ _{3} = 1 | E.g., tsunami | |
D | E.g., Natech (i.e., technological accident with natural hazard trigger) | |||
E | E.g., technological accident |
Category | Effect | Memory | Parameters | Analogy |
---|---|---|---|---|
Repeat | Pr ↓ | Yes | α = 1 | E.g., earthquake on same fault |
Repeat | Pr = 0 | Yes | – | E.g., technological accident on same non-repaired infrastructure impossible |
Repeat | Pr = 0 | No | – | E.g., second tsunami after same earthquake impossible |
Intra-hazard | Pr ↑↓ | Yes | f = 0.1 | E.g., earthquakes on different faults |
Inter-hazard | Pr ↑↓ | Yes | E.g., volcanic eruption → earthquake | |
Inter-hazard | Pr ↑ | Yes | – | E.g., → technological accident only if infrastructure still functional |
Inter-hazard | Pr ↑ | No | – | E.g., earthquake → tsunami; hurricane → storm surge |
3.1.3 Considering time-variant vulnerability
3.2 Definition of extremes
Extreme events may be defined as events, which are “rare, severe and outside the normal range of experience of the system in question” (Bier et al. 1999). This definition however assumes that extreme events are somewhat anomalous. In this line of reasoning, quantification of extremes would make them normal, i.e., non-extreme. In the present study, extreme events are simply defined as low-probability–high-consequences events, whether they seem normal or abnormal. Here, extremes do not only refer to individual events, but also to groups of events of which only the overall impact is considered. Therefore, the definition of an event remains blurry and depends on the level considered in the system (i.e., an event may be composed of sub-events and a meta-event of events). Then, extremes are categorized into individual events (Sect. 3.2.1) and multiple events, i.e., coinciding events or triggered chains of events (Sect. 3.2.2).
In recent years, anomalous events, or outliers, have been repackaged into fancier animals, some of which made their way into popular culture. Taleb (2007) coined the term “black swan” to describe rare events, which in principle cannot be anticipated. Another popular term is “perfect storm”, which refers to an event resulting of a rare combination of circumstances (Paté-Cornell, 2012). Finally, the concept of “dragon king” introduced by Sornette and Ouillon (2012) explains that outliers are due to a physical mechanism not represented in the distribution tail considered. These different terms are not used in the present study.
3.2.1 Individual extreme events: concept of heavy tail
Rare unknown perils could also contribute to overall risk, showing the potential instability of the risk process, as any added information on possible extreme events could significantly, if not dramatically, alter the risk measure (Fig. 7). The level of knowledge on extreme individual events is directly linked to the length of the available records (Smolka 2006). New approaches, such as the study of myths (Piccardi and Masse 2007) or of odd geomorphological structures (e.g., Scheffers et al. 2012) within the scope of scientific imagination, should help improving these records and reassess the overall risk. This is however out of the scope of the present study.
3.2.2 Coinciding events and triggered chains of events
List of tested hypotheses H
H | Primary peril interactions | Secondary peril interactions | Time-variant vulnerability | Time-variant exposure |
---|---|---|---|---|
H _{0} | × | × | × | × |
H _{1} | ✓ | × | × | × |
H _{2} | ✓ | × | ✓ | × |
H _{3} | ✓ | ✓ | ✓ | × |
H _{4} | ✓ | ✓ | ✓ | ✓ |
3.3 Emergence of extremes
3.3.1 Risk migration
In hypothesis H _{1}, events from perils A and B interact following the rules described previously (Fig. 3; Eqs. 6–7). The number of events per simulation is compared to the one expected in the null hypothesis H _{0} in Fig. 8. Using the akaike information criterion (AIC), we find that the number of event occurrences k per simulation for simulation set S _{1} (in dark grey) is better described by the negative binomial distribution than by the Poisson distribution, which indicates over-dispersion (i.e., clustering of events). Here, we use a coupling factor f = 0.1 (see Sect. 3.1.2 for the definition of f), which gives the index of dispersion (variance/mean ratio) ϕ = 4.7. 0 < ϕ < 1 represents under-dispersion and ϕ > 1 over-dispersion. A higher f yields a higher ϕ, more coupling yielding more clustering. Let’s note that one could directly simulate risk scenarios by sampling from the negative binomial distribution instead of from the Poisson distribution in step 1 of the MCM and by not applying steps 2–3 (see Sect. 2.1). This approach applies when clustering is not due to event interactions but to a higher-level process. The negative binomial distribution is frequently used in storm modelling for instance (Mailier et al. 2006; Vitolo et al. 2009). This is however not tested in the present study.
In Fig. 8, the probability of having k = 5 events is Pr(k = 5) ~ 0.04 in set S _{1} in contrast with Pr(k = 5) ~ 0.003 in set S _{0}. For k = 7, Pr(k = 7) ~ 0.02 in set S _{1} while Pr(k = 7) ~ 0.0004 in set S _{0}. With N _{sim} = 10^{5}, we find a maximum number of events per simulation k _{max} = 7 in set S _{0} and k _{max} = 41 in set S _{1}. For hypothesis H _{3}, in which the domino effect A → C → D → E is added (Fig. 3), the index of dispersion increases to ϕ = 7.0, and we get Pr(k = 5) ~ 0.05, Pr(k = 7) ~ 0.03 and k _{max} = 49 in set S _{3}. This demonstrates that risk migrates to lower-probability–higher-consequences events when hazard interactions are considered. Here, we talk about meta-events, the term consequence being defined as the aggregated loss over the k events of the cluster (assuming in a first time a homogeneous distribution of losses Λ over k).
3.3.2 Risk amplification
Let’s finally remark that aggregated losses saturate to ∑Λ = Ε _{0} = 1 in hypothesis H _{4} if the exposure is not reconstructed after a loss. Moreover, the renewal process of event repeats limits the number of occurrences of any given event (Figs. 3, 4; hypotheses H _{1} and H _{3}). Such processes counteract risk amplification and avoid the emergence of exploding chain reactions. Since the data and processes defined in Sect. 3.1 were carefully selected to be representative of existing perils and interactions based on an extensive survey of the literature, our results (risk migration and amplification) are believed to represent some common characteristics of multi-risk. It is evident that both aspects strongly depend on site conditions and that only the analysis of real sites will permit to determine the real impact of multi-risk. We have demonstrated that the proposed framework could be used for such a task.
3.3.3 A multi-risk metric: the risk migration matrix
To better evaluate the role of multi-risk in the emergence of extremes, we introduce the notion of risk migration matrix, which is shown in Fig. 11. It is defined as the difference in the density of risk scenarios observed between two hypotheses. To avoid a pixellated result, the densities are first calculated using a Gaussian kernel, here with a standard deviation large enough to focus on the first-order migration patterns. The right column shows the case H _{ i } − H _{0}, and the left column shows the case H _{ i } − H _{i−1} with i the hypothesis number (Table 4). Risk scenarios of the first and second hypotheses are represented by white and black points, respectively. An increase in risk is represented in red and a decrease in blue. The proposed approach allows us to visualize how the risk migrates as a function of frequency and aggregated losses when new information is added to the system. From H _{0} to H _{3}, we see a progressive shift of the risk to higher frequencies and higher losses (from yellow to red), indicating that the risk is underestimated when interactions at the hazard and risk levels are not considered. The shift is particularly pronounced in the case of domino effects with A → C → D → E. Finally, the risk migration matrix H _{4} − H _{3} shows how time-dependent exposure yields a saturation of losses. Feedback from civil protection stakeholders also showed that a risk matrix view might be preferable to the use of loss curves for communicating multi-risk results. This is in the context of this feedback that the concept of risk migration matrix was developed (Komendantova et al. 2014).
4 Applicability to real-world conditions
Testing of the proposed multi-risk framework has been made possible thanks to the heuristic method and the definition of generic perils and processes, which is based on the idea of “scientific imagination” (Kameda 2012; Paté-Cornell 2012). We have shown that the hazard correlation matrix, by adding knowledge of potential interaction processes, allows for the capture of low-probability–high-consequences events. In particular, we have shown the role of risk migration and of risk amplification for their occurrence. The present work should be seen as a proof-of-concept as we did not attend to fully resolve the difficult problem of extremes. We only considered a selected number of possible interactions, but while the chains of events that emerge in the system may seem obvious, adding more perils and more interactions will yield more complex risk patterns. We thus recommend a brick-by-brick approach to the modelling of multi-risk, to progressively reduce epistemic uncertainties. A more realistic modelling of low-probability–high-consequences events will also require the consideration of additional aspects, such as uncertainties (e.g., Barker and Haimes 2009), domino effects in socioeconomic networks (e.g., Adachi and Ellingwood 2008; Buldyrev et al. 2010) and long-term processes (Grossi and Kunreuther 2005; Smolka 2006), such as climate change (Garcin et al. 2008), infrastructure ageing (Rao et al. 2010) and exposure changes (Bilham 2009). While the concepts developed in the present study can suggest the theoretical benefits of multi-risk assessment, identifying their real-world practicality will require the application of the proposed framework to real test sites.
Notes
Acknowledgments
We thank two anonymous reviewers for their comments. The research leading to these results has been supported by the New Multi-HAzard and MulTi-RIsK Assessment MethodS for Europe (MATRIX) project, funded by the European Community’s Seventh Framework Programme [FP7/2007-2013] under Grant Agreement No. 265138.
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