# Flood Risk Analysis

## Summary and Keywords

Floods affect more people worldwide than any other natural hazard. Flood risk results from the interplay of a range of processes. For river floods, these are the flood-triggering processes in the atmosphere, runoff generation in the catchment, flood waves traveling through the river network, possibly flood defense failure, and finally, inundation and damage processes in the flooded areas. In addition, ripple effects, such as regional or even global supply chain disruptions, may occur.

Effective and efficient flood risk management requires understanding and quantifying the flood risk and its possible future developments. Hence, risk analysis is a key element of flood risk management. Risk assessments can be structured according to three questions: What can go wrong? How likely is it that it will happen? If it goes wrong, what are the consequences? Before answering these questions, the system boundaries, the processes to be included, and the detail of the analysis need to be carefully selected.

One of the greatest challenges in flood risk analyses is the identification of the set of failure or damage scenarios. Often, extreme events beyond the experience of the analyst are missing, which may bias the risk estimate. Another challenge is the estimation of probabilities. There are at most a few observed events where data on the flood situation, such as inundation extent, depth, and loss are available. That means that even in the most optimistic situation there are only a few data points to validate the risk estimates. The situation is even more delicate when the risk has to be quantified for important infrastructure objects, such as breaching of a large dam or flooding of a nuclear power plant. Such events are practically unrepeatable. Hence, estimating of probabilities needs to be based on all available evidence, using observations whenever possible, but also including theoretical knowledge, modeling, specific investigations, experience, or expert judgment. As a result, flood risk assessments are often associated with large uncertainties. Examples abound where authorities, people at risk, and disaster management have been taken by surprise due to unexpected failure scenarios. This is not only a consequence of the complexity of flood risk systems, but may also be attributed to cognitive biases, such as being overconfident in the risk assessment. Hence, it is essential to ask: How wrong can the risk analysis be and still guarantee that the outcome is acceptable?

Keywords: floods, risk, probability, extremes, failure, damage

Introduction

Flooding, the situation in which water temporarily inundates areas where it normally doesn’t, can occur in different forms. River or fluvial floods inundate floodplains along rivers, whereas coastal floods are caused by storm surges from the sea. Flash floods and pluvial floods occur when the intensity and/or volume of heavy precipitation is too large to be accommodated by the soil or the drainage system, leading to local inundation. There are other, less common, types such as dam break floods or glacial lake outburst floods (GLOF), such as downstream inundation when the dam containing a reservoir or a glacial lake fails. Different flood types have different characteristics. For example, fluvial floods along major rivers develop comparatively slowly, whereas flash floods in steep areas are characterized by sudden onset and high flow velocities, posing a challenge for forecasting, warning, and emergency measures. The approaches to quantify flood risk vary between different flood types. This article focuses on risk analysis for river floods, where the state of the art is most mature. However, the underlying concepts are the same for the different flood types and can be transferred from river floods to other types.

Floods affect more people worldwide than any other natural hazard, and their global expected average annual loss in the built environment is estimated at US$104 billion (GAR, 2015). Figure 1, based on the NatCatSERVICE of Munich Re (2016), shows that all continents are affected by floods, although there are hotspots, particularly in Asia and Europe. The list of the ten costliest floods compiled by Munich Re (2016) contains the Thailand floods in 2011, with US$43 billion losses and more than 800 fatalities, the “Great Mississippi and Missouri Rivers Flood” of 1993, with US$21 billion losses, the floods along the Yangtze River in China in 1998, with US$16 billion losses and 3,600 fatalities, and the summer floods in Central Europe in 2002, with US$16.5 billion losses. Population growth, accumulation of assets in flood-prone areas, climate change, and other drivers of global change are expected to increase flood risk in the future.

To reduce flood impacts with appropriate measures, quantification of flood risks is crucial. Flood risks results from the interaction of three components: hazard, exposure, and vulnerability. This definition is very popular and is used, for instance, by the insurance industry (Kron, 2005), the United Nations Office of Disaster Reduction (GAR, 2015), and the Intergovernmental Panel on Climate Change (IPCC, 2012). Hazard is the potential to cause harm and is typically quantified by the probability of a damaging flood situation. It depends on the physical processes of flood generation. Exposure denotes the elements at risk, e.g., people, infrastructure, or ecosystems that may be affected by flooding. Vulnerability describes the susceptibility of the elements at risk that might be adversely affected. Hence, a more vulnerable area experiences higher losses when it is affected by a flood.

## Flood Risk Analysis and Risk Reduction

Over the past one or two decades, there has been a shift in flood management from flood protection to flood risk management (Sayers et al., 2015). Traditionally, flood management strove to reduce the magnitude or probability of flooding. Assessments of flooding were limited to the hazard component and were focused on the physical processes of flood generation, on the quantification of return periods of high river flow, or on the derivation of failure probabilities for flood defense structures. The shift from flood protection towards flood risk management has broadened the interest from flood hazards to flood impacts and flood risks.

This shift also involves a much broader range of events that need to be included in flood risk assessments (Merz, Hall, Disse, & Schumann, 2010a). Traditionally, flood protection concentrated on the specification of a design flood event, typically a flood of a given return period, and on the flood defense systems that were expected to prevent flooding should this event occur. In the early 21st century, flood risk management aims at considering all flood events. This includes events that exceed the design standard. Although these events may lead to failure of the flood defense, catastrophic consequences may be avoided by minimizing failure effects. It also includes events that lead to adverse consequences although the events are below the design standard. An example is intrusion of groundwater into buildings behind dikes as consequence of prolonged high groundwater levels related to river floods, as occurred during the 2002 flood in Dresden, Germany (Kreibich, Thieken, Grunenberg, Ullrich, & Sommer, 2009).

The change towards risk management is further accompanied by a more integrated view on risk management measures. The emphasis on structural flood defense is being replaced by more integrative strategies, taking into account also non-structural measures such as spatial planning regulation, insurance solutions, warning systems or flood-proofing of buildings, and infrastructure in flood-prone areas.

Finally, there has been an increased interest in risk-informed decision making. A risk-based approach to flood risk management attempts to balance the costs and benefits in an explicit manner. Here, costs and benefits have to be seen as broad terms that may encompass not only monetary outcomes but also any other relevant, for instance social or ecological, aspects (e.g., Lind, 2002). The efforts invested in risk reduction are proportionate to the magnitude of the risk and the effectiveness with which that risk may be reduced. In an ideal situation, the process of estimating risk is transparent; the results are accessible and are used to inform multiple decision makers, including the general public (Merz et al., 2010a). This shift towards risk-based deliberations can be seen, for instance, in the recommendation of the U.S. Corps of Engineers (2006), that flood damage reduction studies should use a risk-based approach, or the Flood Directive of the European Union (European Commission, 2007), which requires member states to map the flood hazard and risk, including the potential adverse consequences, of different flood scenarios.

This shift from ensuring safety through flood protection to managing risk requires increasingly sophisticated risk analyses, encompassing a much wider range of failure scenarios and management options, and including the exposure and vulnerability components of flood risk. Notwithstanding the specific strategies and criteria that are applied in flood risk management, rational risk management requires a clear understanding of the existing flood risk, its possible future development, and the risk reduction options. Hence, flood risk analyses are the basis for rational and transparent disaster risk reduction.

This article starts with discussion of flood processes, then introduces the concepts to quantify flood risk. Special consideration is given to the estimation of probabilities for damage scenarios and the problem of validation of flood risk analyses. Finally, new developments that go beyond the state of the art are outlined.

Flood Risk Processes

A whole range of processes is involved in the generation of river flooding and flood damage. Flood risk is thus the result of a chain or network of processes, from the flood-triggering processes in the atmosphere and catchment, to the river system processes, and the damage processes that may occur, for instance, even by supply chain interruption, outside the directly flood affected areas.

In the following, a short overview about these three types of processes is given. The interplay between atmosphere and catchment processes shapes the flood regime, understood as the collection of flood discharges of a catchment, their exceedance probabilities, and their seasonal timing (Hall et al., 2014). The specific flood-causing climatic situation has a decisive role in the generation of a flood event. Extreme floods can be generated by convective storms, typically of very high intensity but with a small spatial scale of a few kilometers and short duration of a few hours or less. On the other hand, extreme flooding can also be produced by synoptic rainfall events covering much larger spatial scales, longer durations, but lower intensities. River flooding can further be caused by snowmelt or by combinations, such as rain-on-snow floods, where both rainfall and melting snow contribute to flood runoff. Recently, the concept of “flood hydroclimatology,” proposed by Hirschboeck (1988) has gained increasing attention (Merz et al., 2014). This concept extends the typical catchment-focused perspective on floods and includes also the large-scale or even global climatological context. Based on the hypothesis that unusually large floods may be related to large-scale atmospheric circulation anomalies, the impact of large-scale climatological processes and their interaction with catchment meteorology and hydrology on the flood regime has been studied. Examples are correlation studies, such as the analysis of Ward, Eisner, Flörke, Dettinger, and Kummu (2014) on the relation between the Southern Oscillation index and flood peaks, or classifications studies relating the occurrence of floods to atmospheric circulation patterns (e.g., Nied, Pardowitz, Nissen, Ulbrich, Hundecha, & Merz, 2014; Wilby & Quinn, 2013). More process-based studies on the climate-flood link like, for instance, the investigation of Nakamura, Lall, Kushnir, Robertson, and Seager (2013), show that 20 major flood events in the Ohio River basin have nearly identical storm tracks, moisture source, and delivery patterns, or like studies on the role of atmospheric rivers in flood generation (e.g., Lavers, Villarini, Allan, Wood, & Wade, 2012).

Important roles in flood generation are also played by the runoff generation and runoff routing processes. Precipitation or snowmelt is transformed to runoff through a variety of processes, including infiltration excess, saturation excess, and subsurface stormflow. This locally generated runoff is further routed to the streams via overland or subsurface flowpaths. The interplay of the catchment—the topography, soils, and land cover—and of the climatic forcing—duration and intensity of event precipitation and catchment wetness—shapes the connectivity of flowpaths, influences water storage, and can produce non-linearities in the catchment response (Rosbjerg et al., 2013). For instance, with increasing rainfall amount or intensity, runoff generation processes may switch from subsurface stormflow to saturation excess, leading to a step in the flood frequency curve (Kusumastuti, Struthers, Sivapalan, & Reynolds, 2007). Merz and Blöschl (2003) develop a flood typology where flood events are classified according to their dominant causing mechanisms, and they point to the role of different flood types in shaping the flood frequency curve, whereas the upper tail may be dominated by a particular flood type. For risk analyses that need to extrapolate to very small probabilities, it is of highest relevance to understand how these processes affect the flood frequency curve and, in particular, its upper tail.

Besides the atmospheric and catchment processes, the processes in the river system determine the flood hazard. The flood runoff generated in the catchment travels through the river network. The characteristics of the river network—for instance, topology, longitudinal river slope, and cross sections—have a substantial influence on flood hydrographs and inundation. The superposition of flood waves can aggravate flooding when floods from different parts of the catchment coincide. This temporal coincidence is influenced by a number of processes, such as the space-time pattern of precipitation, the spatial distribution of catchment wetness, the location of runoff source areas, and the typology of the river network (e.g., Blöschl, Nester, Komma, Parajka, & Perdigão, 2013; Pattison, Lane, Hardy, & Reaney, 2014; Seo & Schmidt, 2013). It can also be influenced by river training measures that may accelerate or decelerate flood waves and change the probability of temporal coincidence (Vorogushyn & Merz, 2013).

A further influence is exerted by the interactions between river channel and floodplains. Flood waves are attenuated due to increasing wetted perimeter and typically high roughness, when the flood water exceeds bankful water levels and spreads out across the floodplain. The retention effect depends on the capacity of the floodplain, and thus, on the existence of dikes, mobile flood protection, and detention basins along the river. Dike failures can have a large role in changing the spatial inundation patterns during a flood event (De Bruijn, Diermanse, & Beckers, 2014). Once a dike fails, its hinterland is flooded; however, the downstream flood peak may be significantly reduced. Apel, Merz, and Thieken (2009) show for the Lower Rhine that dike breaches lead to significant retention effects, which in turn modify the upper tail of the flood frequency curve downstream of diked river stretches.

Finally, the processes leading to damage need to be accounted for. Different types of adverse consequences of flooding are differentiated, typically classified according to two criteria: direct vs. indirect and tangible vs. intangible. Direct consequences are directly related to the location and timing of the flood, whereas indirect damages are ripple effects and occur outside the flooded area or after the flood event. Tangible damages are those that can be easily monetized, whereas intangible consequences are difficult to monetize. Examples for direct, tangible damages are damages at flooded buildings and infrastructure, evacuation, and cleanup costs. Direct, intangible consequences are, for instance, injuries or fatalities, damages to cultural heritage, or environmental losses. Examples of indirect, tangible damages are business interruption of companies outside the flooded area or relocation of companies. Among the indirect, intangible consequences are psychological effects on flood victims or loss of trust in public authorities. Most often, flood risk assessments are limited to direct, economic damages. Other consequences, such as fatalities, or indirect economic effects are less often considered. Some consequences, such as psychological distress and trauma, cannot be quantified and are omitted in risk assessments. Further, it should be noted that floods can have substantial positive effects, such as the provision of water and nutrients for agriculture (e.g., Nguyen, Nguyen, Hung, Kummu, Merz, & Apel, 2015).

When quantifying direct, economic damage, the variety of damaging processes is represented by simple approaches. (For a review on the assessment of economic flood damage see Merz, Kreibich, Schwarze, & Thieken, 2010b). The elements at risk are pooled into classes of similar objects. For each class a simplified “disturbance-response function” is assumed. For damage to buildings and infrastructure, this function typically relates the damage to the inundation depth. This simplification ignores other influences that can be of high importance. Based on investigations of flood loss data from actual events, a range of potential influences have been identified, such as the duration and velocity of the inundation, whether the flood waters are contaminated or not, the state of precaution of flood-affected households, or the lead time of warnings (e.g., Thieken, Müller, Kreibich, & Merz, 2005). However, due to the variety of possible damage-influencing variables, their large randomness, and the scarcity of flood damage data, it has been difficult to derive generic models or relationships that can be transferred to other situations. A few models consider, in addition to inundation depth, further variables—for instance, the “Multi-Coloured Manual” in the United Kingdom (Penning-Rowsell et al., 2013), HAZUS in the United States (Scawthorn et al., 2006), or FLEMO in Germany (Kreibich, Seifert, Merz, & Thieken, 2010; Schröter, Kreibich, Vogel, Riggelsen, Scherbaum, & Merz, 2014).

It should be noted that inundation depth plays a secondary role for certain damage types. Flood damage to agricultural crops depends decisively on the timing of the flood—date of occurrence in the year, and duration of the inundation (Klaus, Kreibich, Merz, Kuhlmann, & Schröter, 2016). Flood fatalities depend also on flow velocity, rate of rise of the water level, flood warning, possibilities for fleeing or sheltering, or health state of the affected people (e.g., Di Mauro, De Bruijn, & Meloni, 2012).

Floods can have disastrous effects outside the inundated area. A recent example is the impact of the Thailand floods in 2011 through disruptions of global supply chains (Haraguchi & Lall, 2015). The widespread flooding started in summer 2011 and persisted in some areas until January 2012. Seven industrial parks with more than 800 companies, in particular from the automotive and electronics industries, were flooded. Thailand is one of the centers of global automobile production. Honda required 174 days to resume its production cycle (Haraguchi & Lall, 2015). The massive disruption of supply chains in the manufacturing sector had a large influence on Thailand’s economy, and it has been estimated that this flood reduced the global industrial production by 2.5% (Haraguchi & Lall, 2015). Estimating such ripple effects is difficult, since data on the interconnectedness of the economy are hardly available. Economic models, in particular input-output models, have been used to estimate these effects (e.g., Hallegatte, 2008; Li, Crawford-Brown, Syddall, & Guan, 2013).

Quantification of Flood Risk

Quantitative risk analysis has been developed over the last decades. A number of fields, such as structural engineering, nuclear reactor safety, or airplane safety have been particularly relevant for the development of quantitative risk analysis approaches. Notwithstanding the specific risk that is scrutinized, quantitative risk analysis can be described by answering a set of three questions (Kaplan & Garrick, 1981): (a) What can go wrong? (b) How likely is it that it will happen? (c) If it goes wrong, what are the consequences? Hence, risk RI can be expressed as the set of triplets:

where S_{i} is a scenario identification or description, P_{i} is the associated probability, D_{i} is the associated damage, and N is the number of scenarios. If the risk analysis delivers a complete set of scenarios, as in all scenarios that are relevant for the risk under study, then the set of triplets is a comprehensive quantification of the risk.

Figure 2 illustrates flood risk and its components for fluvial flooding. The hazard is quantified by the flood frequency curve (Figure 2c), which is a signature of the range of flood generation processes in the atmosphere, catchment, and river network (Figure 2a). The flood frequency curve denotes, for the given river section, the exceedance probability of high discharge values. Frequently, the inverse of this probability is given, that is, the return period or return interval, describing the average period between the occurrences of floods exceeding this magnitude. For a selection of scenarios that are typically related to different probability levels, such as the 50- or 100-year flood, the associated river discharge values are transformed into inundation areas. The superposition of the inundated area with the affected elements at risk delivers the exposure for the given hazard scenarios (e.g., the 50-year inundation area, see Figure 2d). Exposure data typically relates to population and assets, such as buildings, infrastructure, or agricultural crops. The vulnerability of the affected elements is often represented by depth-damage curves (Figure 2e), showing the relative loss as a function of inundation depth. Different depth-damage curves are applied for different types of affected objects. Finally, hazard, exposure, and vulnerability are combined in the risk curve, which relates the damage of the scenarios to their probability (Figure 2g). In this example, there are N scenarios S_{i}, represented by their respective discharge value, probability P_{i}, and damage D_{i}.

Flood risk is frequently expressed as the expectation of the damage, or the Expected Annual Damage (EAD):

where $D\left(q\right)$ is the damage depending on the annual maximum discharge $q$, ${f}_{q}\left(q\right)$ is the distribution function of $q$, and ${q}_{D}$ is the threshold discharge above which flood damage occurs (Figure 2f). Since the continuous distribution function ${f}_{q}\left(q\right)$ is approximated by a limited number of scenarios with given probabilities and associated damage, equation 2 has to be replaced by:

where ${D}_{j}$ and $\Delta {P}_{j}$ are the average flood damage and probability increment for the j-th interval, respectively, and M is the number of probability increments:

Risk has the same unit as the damage indicator related to the time interval $\Delta t$, for which the probabilities are given. Since $\Delta t$ is one year, typical risk units are “economic loss per year” or “fatalities per year.”

The expected annual damage is the most widespread flood risk indicator, and it is convenient to use in cost-benefit assessments where different risk reduction options are compared in terms of their economic efficiency. However, the EAD should be used carefully. One of its shortcomings is that it does not distinguish between “high probability/low damage” scenarios and “low probability/high damage” scenarios. The underlying assumption of EAD is that decision makers and people at risk are risk-neutral, whereas people tend to be risk-averse (e.g., Bohnenblust & Slovic, 1998). Risk aversion refers to the fact that people are more concerned about events with catastrophic consequences, even if their probability of occurrence is very small, and consequently, their damage expectation is very small, too. Merz, Elmer, and Thieken (2009) show that, for river flooding in Germany, the EAD is dominated by high probability/low damage events. Low probability/high damage events have a minor contribution to the EAD, although they are typically perceived as the important events by society. Hence, the EAD may not reflect societal preferences.

The EAD condenses the risk curve (Figure 2g) into a single number, but already Kaplan and Garrick (1981, p. 14) pointed out that “A single number is not a big enough concept to communicate the idea of risk.” Recently, the concepts of resilience and robustness have found increasing attention in flood risk management (e.g., de Bruijn, 2004; Klijn, Knoop, Ligtvoet, & Mens, 2012; Mens, Klijn, De Bruijn, & Van Beek, 2011; Merz et al., 2010a). Robust safety systems perform well under a large range of flood magnitudes and situations. Resilient systems are able to recover from floods. Robust and resilient safety measures typically differ from optimal strategies, which provide the maximum benefit for the most likely development. However, highly optimized, cost-efficient safety systems may be particularly vulnerable to unexpected eventualities. Mens et al. (2011) propose quantifying robustness by three criteria of system response (e.g., economic losses) to a disturbance (e.g., flood peak): the threshold beyond which damage occurs, the proportionality with which the response increases with increasing disturbance, and the manageability that describes the ability to keep the response level below a point beyond which recovery becomes difficult. Hence, the characteristics of the complete risk curve are important to understand the performance of the system under study under a range of disturbances. Hence, deriving and communicating the full risk curve is recommend, and not only the EAD.

The example outlined in Figure 2 is a simple, maybe the simplest, version of a quantitative risk analysis for river floods. More elaborated approaches and the pitfalls of flood risk analyses are discussed in the following sections.

## System Definition

When performing a quantitative flood risk analysis, the system boundaries, the processes to be included, and the detail of the analysis need to be carefully selected. These aspects are determined by the purpose and scale of the analysis. To understand the probability and consequences of a dam break, and to reduce the associated risk to the people and their assets downstream of the dam, a detailed analysis of all kinds of initiating events and how they could evolve into a dam failure would be performed. The analysis of the consequences would include state-of-the-art hydrodynamic simulations to provide dam break inundation scenarios including spatial patterns of flow velocity, water depth, and available warning time to understand the risk of injuries and fatalities and the possibilities for early warning and evacuation. This could include considering the temporal dynamics of exposure—for example, the time-varying number of people staying in touristic places in the river valley downstream of the dam. A very different risk analysis would be performed by a re-insurer providing coverage for globally acting companies. The insurer would be interested in the probability and consequences of disruption of their supply chains, considering factors such as spatial distribution of facilities, diversification of sources of procurement, assistance from other partner companies in the same supply chain, or degree of recovery of their customers (Haraguchi & Lall, 2015).

In each case, it has to be carefully determined which processes and aspects are important and need to be represented in a certain detail, and which processes can possibly be neglected or parameterized in a simple way. A large-scale or national risk analysis may assume that dikes along rivers impede inundation as long as the events are smaller than a specified return period, that is, the standard of protection; whereas the assessment of the risk to an industrial park may include detailed investigations on dike failure mechanisms and probabilities. For flood risk maps at the national scale, or the river basin scale, for understanding the risk for single objects, at-site assessments can be pieced together. To understand the joint flood risk for a whole area, the spatial dependence needs to be considered. Flood scenarios need to represent the joint probability of flooding at different locations.

Even for a given task, such as quantifying the flood risk for a city, very different approaches can be taken. On one end of the spectrum, a comprehensive chain of models could be established, consisting of a stochastic weather generator, possibly parameterized on climate indicators to account for temporal variations in flood-causing rainfall, a hydrological model describing runoff generation and runoff routing in the catchment, a hydraulic river network model for the simulation of the routing and superposition of flood waves, a flood defense module estimating the failure of dikes and other defense structures, another hydraulic model for simulating inundation in the city, and a damage model. On the other end, the risk analysis could be based on a flood frequency analysis using the observed time series of discharge close to the city, transforming a selection of flood discharges at the river gauge by simple GIS (Geographic Information System) operations into inundation areas and estimating the damage based on inundation extent and depth. De Moel, Jongman, Kreibich, Merz, Penning-Rowsell, and Ward (2015) further discuss how the detail and choice of methods depends on the spatial scale of the risk analysis.

## What Can Go Wrong?

Identification and definition of the scenario set have not received much attention in all areas of flood risk analysis. One challenge is to identify all relevant scenarios but still constrain the scenario set so that the scenarios can be handled in the quantification. Often, risk assessments restrict the scenario set to a too small number. Extreme events, which are beyond the experience and perception of the analysts and users of the assessment, may be missing. Another challenge is to identify failure scenarios that come as a surprise. A case in point are secondary effects, such as flood-induced contamination, failure of hydraulic structures due to driftwood blocking, or inundated basements due to higher groundwater as a consequence of persisting high river flow. There are systematic methods for scenario identification, such as “Failure Mode and Effect Analysis” (Rausand & Hoylan, 2004), where all components of the system under study are screened to identify failure modes and their effects. Unfortunately, these methods have not found widespread application in flood risk analysis, with some exceptions such as dam safety studies.

Estimation of Probabilities

In quantitative risk analysis, we attempt to associate a probability to each failure or damage scenario. Ideally, this probability statement should express the probability that the specified damage occurs. However, flood risk analyses associate the derivation of probabilities with the hydro-meteorological processes in the atmosphere, catchment, and river system. The scenario probability is typically considered the same as the chance for the storm or the flood discharge to occur. The damage is then attached to this probability, that is, the probability of the storm or of the runoff is used as proxy for the probability of damage. This simplification is a consequence of the past focus on the hazard component, but it may also be a result of the lack of flood loss data. Because damaging events are rare, flood loss data for a given location are scarce or not existing at all, and probabilities are derived using available observations, either meteorological or hydrological data.

However, along the chain from the hydro-meteorological processes to the damage, mechanisms may occur that modulate the probabilities. Falter et al. (2015) have shown that the same river discharge value might lead to significantly different damage, depending on the specific characteristics of the flood hydrograph and pathways between river gauge and damage location. For instance, a longer duration flood may lead to dike failure, and consequently, much larger damage, compared to a short-duration flood with the same peak flow. Recently, methods have been developed for a more comprehensive probability assessment (Falter et al., 2015; Gouldby, Sayers, Panzeri, & Lanyon, 2010; Hall, Dawson, Sayers, Rosu, Chatterton, & Deakin, 2003; Vorogushyn, Merz, Lindenschmidt, & Apel, 2010; Vorogushyn, Lindenschmidt, Kreibich, Apel, & Merz, 2012). They attempt to include the pathway of the flood water in the probability calculation by estimating, for example, the probability that flood waters would reach a specific location where the damage occurs.

Risk is inherently connected to chance and probability. Garrick (2008) notes that a common perception of probability is that it has to be derived from data and represents the frequency with which a certain event occurs. This requires that you have data about the risk before you can quantify it. For some risks, such as automobile accidents, this data-based, frequency-centered approach is viable. For flood risk, this approach is only partially helpful. We need tools that are applicable to all risks, including those for which there is meager data or no data at all. For instance, the breaching of a large dam is a unique event. In the rare case that it breaks, it will not be reconstructed in the same way. Transferring the probability of dam failure from a large sample of actual dam breaks is not possible due to the large heterogeneity of dams, loading conditions, and safety measures. We follow Garrick (2008) in taking a broader approach: Estimating probabilities is based on all available evidence. Observations of the system under study are used whenever possible, but evidence also includes theoretical knowledge, modeling, specific investigations, experience, or expert judgment.

## Probabilities Derived From Observational Data

In case there are sufficient data available for the variables of interest, probabilities can be directly derived from the observations. We use flood frequency analysis as example for this direct approach.

Historically, flood frequency analysis, using observed discharge data, plays an essential role in flood risk assessments. Following the theory of Extreme Value statistics, flood time series are derived from the continuous streamflow measurements, using either the Block Maxima or the Peak Over Threshold approach (e.g., Coles, 2001). A distribution function, such as the Generalized Extreme Value distribution, is fitted to the sample, and the fitted distribution is used to estimate the flood discharge associated with large return periods, such as the 100-year flood peak. This approach is standard in hydrological practice, and there are guidelines in different countries on its sound application; see, for instance, Bulletin 17B for the United States (Interagency Advisory Committee on Water Data, 1982), and the Flood Estimation Handbook for United Kingdom (Centre for Ecology and Hydrology, 1999).

Flood frequency analysis has been criticized (e.g., Klemeš, 2000) because its development and application have often focused on the statistical estimation problem. In recent years, a broader perspective has emerged. Merz and Blöschl (2008) have proposed to substitute flood frequency analysis by flood frequency hydrology—an approach that assembles a range of hydrological information including evidence from similar events in the region; information about historical floods using chronicles, flood marks, or geoarchives; and causative information about the underlying processes. This spatial, temporal or causal transfer of information may help in explaining characteristics of the flood frequency curve and in constraining the uncertainty of the upper tail estimation (e.g., Rosbjerg et al., 2013).

## Probabilities Derived From Models

In many cases, there are no sufficiently large data sets or no data at all to derive probabilities directly from observations, and probabilities must be derived using models. A case in point is the failure probability of river dikes. It can be represented by fragility curves, which indicate the probability of dike failure as a function of load variables, such as water level or duration of the flood wave (Apel, Thieken, Merz, & Blöschl, 2004; Hall et al., 2003; Vorogushyn, Merz, & Apel, 2009). For each relevant failure mechanism, such as overtopping, piping, or slope instability, the failure pathways leading to dike collapse are identified. A failure condition, or limit state condition, along these pathways is defined as the exceedance of a load factor over a resistance factor. For instance, in the case of dike collapse due to overtopping and subsequent erosion of the dike crest, the actual overflow (the load factor) is compared to the critical overflow (the resistance factor) at which erosion of the dike surface occurs (Apel et al., 2004). Since essential variables, such as the turf quality, are variable in space and time and/or are uncertain, several of these variables are typically represented as random variables. Hence, the limit state equation compares the probability density function of the load with the probability density function of the resistance, from which the failure probability is derived (Figure 3). In most cases, these models cannot be integrated analytically, so probability statements are derived from Monte Carlo simulations.

This example illustrates the conceptual approach for estimating probabilities for the numerous cases where the variables or states of interest cannot be observed in sufficient quantity. The failure of a certain dike reach is a rare or even unrepeatable event, and its probability of occurrence needs to be based on causal models representing the available knowledge about the underlying cause-effect relationships. For some of the underlying variables, like geotechnical characteristics, sufficient data might be available to obtain probability density functions, for some of them expert judgment might be required. The credibility of the resulting probability statement depends on the causal model, its input parameters, and its integration.

## End-to-End Flood Risk Modeling

We define end-to-end flood risk assessments as approaches that encompass the complete flood risk processes, from the initial flood-triggering event to the resulting damage. Such approaches use coupled models, consisting of a number of modules to simulate the different processes, and they run in a Monte Carlo framework.

These approaches often follow an event-based sampling approach. From the probability distribution function of the initiating events, a large number of flood events are randomly drawn. A simple solution is to randomly pick one annual maximum flood per year from the flood frequency curve, but models that start with storm generation have been proposed as well (Rodda, 2001). These initiating events propagate through the model chain, and the resulting sample of the damage values is used to construct the risk curve. Another scenario is given when the risk shall be derived over the lifetime of a project, say, 50 years for the design of a flood defense system. In this case, the randomly drawn flood events are chronologically ordered, and the sequence of 50 floods is one possible realization of the next 50 years. Since it cannot be expected that the system remains unchanged over 50 years, temporal variations can be introduced, such as effects of climate change, land use, and asset changes due to population and economic growth, degrading defense systems, or any other important time-varying variable. Changes that interact with flood occurrence could be considered, for instance, with improved dikes or increased safety standards after an extreme event (Harris, Dunn, & Deering, 2010). Generating many realizations of this 50-year time sequence allows the derivation of a risk curve for each year. Examples of event-based end-to-end risk assessments are the preliminary national risk assessment for England of Hall et al. (2003), the analysis for the Lower Rhine by Apel et al. (2004), or the assessment for three catchments in England by Gouldby et al. (2010).

Falter et al. (2015) has presented, to the author’s knowledge, the first end-to-end flood risk model in continuous-simulation mode. Similar models have been implemented earlier (Grimaldi, Petrosellid, Arcangelettid, & Nardie, 2013; McMillan & Brasington, 2008), but without considering damages. A stochastic weather generator provides spatial fields of weather, in particular precipitation and temperature, at the daily resolution for the river basin under study. The weather generator includes the spatial dependence between locations and the dependence between climate variables. These meteorological fields are used as input into a distributed hydrological model, yielding time series of hydrological states and fluxes throughout the catchment. The subsequent river network model routes the output of the catchment model through the river network, and calculates inundation behind the dikes in case they are overtopped. Finally, a damage model estimates spatial pattern of damages along the rivers. This approach has a number of advantages. It simulates the complete flood hydrograph, which is required for a number of applications, such as design of flood retention basins or estimation of agricultural damages. It delivers spatially consistent flood events throughout the catchment, including river/floodplain and damage processes. This is a particularly important aspect, as many applications require considering the spatial dependence of flood processes. Finally, it allows deriving the probability of damage directly from the generated damage values. Hence, it does not rely on the simplified assumption that the damage probability is equal to the probability of the storm or the discharge values.

Uncertainty and Validation

## The Problem of Validation

Validation of predictions and model results is good practice in natural sciences and engineering. The so-called scientific method comprises the testing of hypotheses by systematic observation, measurement, and experiment. Some components of the flood risk process chain have undergone validation procedures. For example, inundation simulation models have been benchmarked against actual event data (e.g., Bates, Horritt, Aronica, & Beven, 2004; Werner, Blazkova, & Petr, 2005). But how to validate the flood risk curve that has been estimated, using a collection of models, assumptions, and data, for a certain city? This risk curve encompasses a whole range of scenarios, from frequent floods, to events with very small probability, and ideally, to the worst-case flood. There are at most a few observed events where data on the flood situation, such as the inundation extent and the inundation depth, and on the loss are available. That means that even in the most optimistic situation, there are only a few data points to validate the risk curve, which typically spans several magnitudes along the probability and the damage axes. The situation is even more delicate when the risk has to be quantified for important infrastructure, such as breaching of a large dam or flooding of a nuclear power plant. Such events are practically unrepeatable—in case such an event occurs, the system under study would certainly be heavily modified, either by decommissioning the installation or by significantly increasing the safety measures. The rarity or even uniqueness of events precludes the typical validation procedure, where the modeled variables of interest are compared against measured data. The risk estimate can only be evaluated through reasoning about the various elements in the process of constructing the prediction (Hall & Anderson, 2002).

Merz, Kreibich, Thieken, and Schmidtke (2004) attempt to validate their risk assessment, based on state-of-the-art models, by comparing the observed economic damages of an actual flood with the simulated damages for a synthetic flood that has the same return period. Figure 4 shows this comparison, where the damages are aggregated to the municipality level. It is obvious that the flood damage estimation is plagued by large uncertainty and errors. In two out of seven municipalities, the reported damage lies even outside the 99% uncertainty interval of the estimated damage. Since validations of flood risk estimates are scarce, it is open how representative this modest skill in estimating flood risk is. It should serve as a warning that flood risk estimates might be grossly wrong.

Every possible effort should be made to collect observations and data that can be used to validate models and their components. The analysis of actual flood disasters, but also of near misses where substantial failures and/or damages have been avoided, can play an important role in understanding the limitations of current models and assumptions. The Integrated Research on Disaster Risk has launched FORIN, the forensic investigation of disasters, to understand why natural hazards do, or do not, develop into disasters by comprehensive in-depth analysis (ISDR, 2013).

## Uncertainty Analysis

Risk estimates are inherently uncertain. They describe the occurrence of possible future events, some of which have very small probabilities. There seems to be a consensus that risk assessments should not only estimate the risk, but should also describe the uncertainty associated with the risk quantification (e.g., Hall & Solomatine, 2008; Kaplan & Garrick, 1981; Merz et al., 2010a; Sayers et al., 2015). Information about the uncertainty of a risk estimate can be highly relevant for decision making. A very uncertain risk estimate tends to trigger higher safety measures compared to a situation where the estimated risk is the same but can be estimated with high reliability. Palmer (2000) demonstrates that probabilistic weather and climate forecasts have greater economic benefits than single deterministic forecasts. Stakeholders and decision makers in different contexts, with different attitudes towards risk aversion or different cost-benefit ratios of risk reduction measures, may come to different decisions given information about the uncertainty of a risk estimate (Downton, Morss, Wilhelmi, Gruntfest, & Higgings, 2005). Unfortunately, uncertainty analyses in flood risk studies are not widespread (Hall, 2014).

Kaplan and Garrick (1981) propose to express the uncertainty by embedding the risk curve in uncertainty intervals. Merz and Thieken (2009) explore this concept in their flood analysis for the city of Cologne. For each module of the risk analysis, the important sources of uncertainty are identified and represented by alternative models. Figure 5 shows the range of plausible models and the best estimate risk curve. It contains 36 plausible models, whereas each model provides the damage for floods with return periods from 10 to 1,000 years. The uncertainty range is rather large. For the 100-year flood, it extends from €40 million to €200 million. Figure 5 also shows the available damage estimates for two floods in 1993 and 1995. Although the second flood was slightly higher, by 6 cm at the gauge Cologne, than the flood in 1993, the reported damage was less than halved. This effect has been attributed mainly to the learning effect triggered by the first flood. It exemplifies the dynamic nature of flood risk systems; temporal changes introduce additional challenges for risk analyses.

In the course of an uncertainty analysis, a number of decisions must be made. It must be decided which uncertainty sources should be included, and how these uncertainty sources should be represented; for instance, by determining the range of plausible model structures, and by selecting those parameters that are considered as uncertain. For these parameters, probability distributions are used where these can be credibly derived; in other cases, intervals or sets of probability distributions might be used (Hall & Solomatine, 2008). These uncertainties are propagated through the range of plausible models to obtain uncertainty bounds of the risk curve and other key outputs. In many cases, subjective decisions have to be taken, and we have to expect that the estimate of the uncertainty bounds is even more uncertain than the estimate of the risk curve. However, a formal uncertainty analysis forces the analyst to investigate more deeply the system under study. By identifying the most dominant sources of uncertainty, not only a better understanding of possible errors in the analysis will be obtained, but also an understanding of how the current uncertainty level can be reduced efficiently, by additional measurements.

## Limits of Predictability

The history of flood disasters is rich with surprises where the authorities, people at risk, and disaster management encountered unexpected events. Although formal uncertainty analysis methods are highly valuable, they are not designed to capture surprise. They rely on assumptions and mathematical models constrained by past observations and experience. However, the decisive events in flood risk management may be outliers or unrepeatable processes that impede learning through experience. There may be situations afflicted by indeterminacy or deep uncertainty where phenomena cannot be known, measured, or predicted, and where trustworthy models do not exist (Macgill & Siu, 2004).

Merz, Vorogushyn, Lall, Viglione, and Blöschl (2015) discuss the role of surprise in flood risk assessments. Besides the complexity of flood risk systems, such as threshold behavior or interactions between physical and societal processes, biases in human perception and thinking are identified as another source of surprise. For instance, overconfidence is a widespread phenomenon that results in a tendency to overrate one’s knowledge. Another cognitive bias is illusion of certainty, as in the psychological need for certainty even when none exists. This nurtures the belief that future developments can be forecast even in cases where the evidence suggests otherwise (Kahneman, 2011; Makridakis, 1988,). Cognitive biases may be pronounced in flood risk assessments where we deal with probabilistic events, where uncertainty is high, and where there is a lack of direct response to the outcome of decisions (Merz et al., 2015).

To better deal with surprise in flood risk analysis, Merz et al. (2015) suggest to invest more efforts on reflecting about the potential sources of surprise. This could include generating multiple hypotheses and plausible models, assembling evidence as inclusive as possible, and exploring the possible (and not only the probable). For instance, when answering the “What can go wrong” question, we should let imagination play. Kumar (2011) argues that our mindset typically focuses on constraining our models to the probable trajectory to obtain the least uncertainty. However, exploring the space of all possible trajectories is more adequate for reducing surprise. An interesting example in this respect is the study of Wardekker, de Jong, Knoop, and van der Sluijs (2010) about climate adaptation measures for the city of Rotterdam, Netherlands. It comprises a large scenario space including the collapse of the thermohaline circulation, port freezing events, port malaria incidents, a modified German water safety policy, enduring heat waves and droughts, extreme storms, and failure of the storm surge barrier during an extreme storm.

Another issue related to limits of predictability in flood risk analysis is the possible consequences of surprises. Merz et al. (2015) suggest assessing the mental and mathematical models not only according to their plausibility but also according to the harm they cause when they are wrong. When performing a flood risk analysis for a nuclear power plant, missing failure scenarios, wrong assumptions, or flawed models may lead to disastrous consequences. A risk analysis for a less critical area with the same shortcomings would lead to much less severe consequences. Hence, the robustness question (Ben-Haim, 2012) should be considered: How wrong can the risk analysis be and still guarantee that the outcome is acceptable?

Beyond the State of the Art

Increased attention has been paid to a number of developments in flood risk analyses, three of which are described here: flood risk dynamics, interactions and systems approach, and global assessments. Flood risk and its components change in time. Whereas many studies have included temporal changes in the hazard component (e.g., impact of climate change or land use change on flood peaks), and in the exposure components (e.g., impact of population growth on flood risk), attempts to consider time-variations in vulnerability are rare. Changes in risk components may act on very different time scales, from the sub-daily to the centennial time scale, and may be of different types, such as step changes, gradual trends, or periodic fluctuations. For instance, climate and land use change may gradually alter the probability of flooding over decades to centuries. River training may act on the time scale of a few years, and may impose a step change in the flood probability. Exposure has typically been considered as gradual change in population and economic assets at the decadal scale. Other variations, such as the seasonal fluctuation of people in touristic regions, or the sub-daily fluctuation of people working in industrial parks and living in residential areas, have rarely been considered. There is widespread evidence that vulnerability can change substantially with significant influence on flood risk. An example is the 1995 Rhine flood in Cologne, Germany, with less than half the damage compared to the 1993 flood, although both floods were very similar in hydrological terms. The survey of more than 750 households along the Rhine, by Bubeck, Botzen, Kreibich, and Aerts (2012), shows that the 1993 flood triggered a remarkable increase in private precautionary measures, which contributed, in turn, to the substantial damage reduction in the latter event.

Although the importance of temporal changes in all three flood risk components has frequently been acknowledged, the consideration of such changes in flood risk assessments is still very limited. More insight is required about such changes, in particular in relation to vulnerability, and how changes in different components interact with changes in other components (Di Baldassarre, Viglione, Carr, Kuil, Salinas, & Blöschl, 2013). Further, the incorporation of temporal dynamics in risk assessments is far from trivial, today often being implemented in a scenario approach. Tools for flood risk assessment are needed that are flexible enough to easily incorporate changes in all three components. Such tools could also be used to re-visit and update risk assessments on a routine basis, or when new data about changes become available. These tools would be valuable in the frame of adaptive management, the continuous and systematic process of improving policies and practices by learning from outcomes of implemented strategies (Allen, Xia, & Pahl-Wostl, 2013; Merz et al., 2010a).

Interactions and feedbacks are topics of recent attention. Flood risk systems are increasingly seen as complex, dynamic systems, shaped by interactions in space, in time, or between their elements. For example, high-resolution simulations of Vorogushyn et al. (2012) for the Elbe River, including several dike failure mechanisms, illustrate how flood retention upstream reduces the risk downstream, due to smaller downstream flood peaks and smaller dike overtopping probabilities. They also demonstrate that retention may increase the probability of other dike failure mechanisms because of more persistent high-water levels downstream of the retention area. Such counterintuitive effects may lead to surprises and unexpected risk patterns and can only be understood if the spatial interactions are explicitly considered. To date, many risk analyses are performed for smaller units and then pieced together, not representing properly the spatial interactions. Another example of the systems approach are attempts to model interactions between physical and societal processes. Di Baldassarre et al. (2013) and Viglione et al. (2014) analyze, using strongly simplified models, how the feedbacks between hydrology, economy, technology, societal risk awareness, and spatial planning shape the human settlement—floodplain system over the scales of decades and centuries. Although such “socio-hydrology” models are far from being used in flood risk analyses, they may be a valuable approach to explore the space of future developments of a given flood risk system.

The increasing availability of global data sets and high-performance computing have boosted the efforts to develop methods for global flood risk assessments. Whereas global hydrological models have been used for some decades, global inundation and flood damage models have been developed only recently. For example, Hirabayashi et al. (2013) developed a global river routing model for estimating inundation areas and flood-affected population, and Ward et al. (2013) presented a model chain to provide global flood risk maps including protection standards and damage assessment. Although there is an apparent mismatch between global flood risk models and the detail required to inform local decisions (Ward et al., 2015), such developments could inform strategic planning at the national or river basin scale. They are also of high interest, for instance, for the (re-insurance industry, to understand their risk portfolio, or for international companies to reduce their risk related to supply chain disruption.

Conclusions

Recent years have seen large efforts in developing and improving the concepts and approaches for quantifying flood risk. This is particularly valid for the vulnerability and exposure components, which had not received much attention earlier. Assessing flood risk is a challenge in many cases, and it is easy to criticize risk analyses, given the numerous and often bold assumptions that need to be made, and the large uncertainties that are typically associated with risk estimates. However, risk assessments are systematic and rigorous procedures to better understand the system under study, and to provide the best available knowledge for flood risk management decisions. A number of exciting research questions remain. These are, among others, how to quantify the temporal changes in flood risk; how to identify and take into account interactions in space, in time, and between elements of the flood risk system under study; and how to provide informative risk estimates at the regional to global scale.

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