Most climate change impacts manifest in the form of natural hazards. Damage assessment typically relies on damage functions that translate the magnitude of extreme events to a quantifiable damage. In practice, the availability of damage functions is limited due to a lack of data sources and a lack of understanding of damage processes. The study of the characteristics of damage functions for different hazards could strengthen the theoretical foundation of damage functions and support their development and validation. Accordingly, we investigate analogies of damage functions for coastal flooding and for wind storms and identify a unified approach. This approach has general applicability for granular portfolios and may also be applied, for example, to heat-related mortality. Moreover, the unification enables the transfer of methodology between hazards and a consistent treatment of uncertainty. This is demonstrated by a sensitivity analysis on the basis of two simple case studies (for coastal flood and storm damage). The analysis reveals the relevance of the various uncertainty sources at varying hazard magnitude and on both the microscale and the macroscale level. Main findings are the dominance of uncertainty from the hazard magnitude and the persistent behaviour of intrinsic uncertainties on both scale levels. Our results shed light on the general role of uncertainties and provide useful insight for the application of the unified approach.

As climate extremes, natural hazards are an inherent part of the climate
system. There is increasing evidence that a changing climate leads to changes
in hazard characteristics and can even result in unprecedented extreme
weather and climate events

For a risk assessment of natural hazards, damage functions are employed to
translate the magnitude of extreme events to a quantifiable damage. Often the
focus is on the modelling of the hazard, while the damage assessment
receives less attention

Accordingly, the availability of damage functions is very limited. On the one
hand, empirical damage functions may not be inferable due to a lack of
observations for certain impacts or sites. On the other hand, the
correlations between loss and the explanatory variable(s) might be weak and
loss estimates could become unreliable due to the high level of uncertainty.
This results in the need for a comprehensive damage assessment in order to enable
the quantification and comparison of the impacts from different natural
hazards and their interactions

For this purpose, the work at hand provides an investigation into the common
aspects of damage functions for different hazards. It considers similarities
in damage functions and exposure for coastal flooding

Moving towards a multi-risk assessment, it is shown how this approach can be
extended to heat-related mortality. This is of particular concern since heat-related
fatalities currently comprise over 90 % of total natural hazard
fatalities in Europe and are also a major issue for developing countries

The unified damage function also provides a platform for the discussion of
potential uncertainties. Embedding the damage function in a probabilistic
framework, this study investigates the relevance of different uncertainty
sources for damage estimation. Excluding considerations about the stochastic
nature of extreme events, we consider uncertainties in the damage function
subject to a hypothetical hazard magnitude. A variance-based
sensitivity analysis (VBSA)

In Sect.

Damage functions are an important tool for an impact assessment of
climate-related hazards. For example, Fig.

Schematically,

Henceforth, we rely on the following definitions. A damage function is defined as the mathematical relation between the magnitude of a (natural) hazard and the average damage caused on a specific item (building, person, etc.) or portfolio of items. The emphasis is on direct monetary damage, but the findings can be generalized to any measurable quantity.

In this context, the microscale level relates to a single item. In contrast,
the macroscale level refers to a portfolio of independent items with similar
properties (e.g. residential buildings). With this definition, we go beyond
similar definitions that define the macro domain solely via the spatial
extent

Damage can be expressed in absolute or relative terms

In the following, we give account of a damage function that has been
frequently applied for the assessment of coastal flooding

We begin by defining a microscale damage function

The hazard magnitude may be represented by a more or less complex indicator.
Frequently the most basic indicator, maximum flood height, is chosen

The microscale damage function has a lower bound of

For a macroscale damage assessment, e.g. for a coastal city, it is assumed
that all items in the portfolio are exposed to the same hazard magnitude.
Local fluctuations (e.g. caused by obstruction or varying distance to
coast) are considered as a source of uncertainty in
Sect.

The damage ratio for the portfolio (relative damage) is given by the average
damage of the individual items:

While the above equation assumes equal monetary value for each item,
generalization is simple. Different item values can be incorporated by
weighting the sum with a normalized asset value

In order to emphasize the similarity to the storm-damage function described
in the following section, we define a discrete frequency distribution

The relationship between macroscale damage, portfolio composition, and
microscale damage function is shown schematically in Fig.

The key characteristic of this approach is the consideration of a granular portfolio of buildings, each with an observable hazard threshold. The approach is reliant on the availability of building-specific information and prior knowledge on the microscale damage function and hence represents a bottom-up approach.

In this section we give account of a storm-damage function developed by

Storm-damage functions are typically calibrated to insurance data. The data
comprise the fraction of affected buildings (claim ratio) and the damage
ratio for a defined region

It can be assumed that buildings have a specific resistance to wind (i.e. a
threshold wind speed) that depends on their characteristics

In analogy to the coastal flood example, let

For a given portfolio,

Having identified the distribution of hazard thresholds, the macroscale
damage ratio is given by the convolution of the probability density of the
hazard threshold and the microscale damage function

As in the previous case of coastal flooding, the damage function considers a granular portfolio of exposed buildings. The key difference is that in the case of wind storms a direct observation of the hazard threshold is not feasible. Instead, an implicit description of the portfolio is given by the distribution of hazard thresholds. In order to obtain this distribution, the damage function is calibrated against macroscale damage data in a top-down approach.

Simple inspection shows that Eq. (

Formally, the mathematical relationships derived in the previous sections also hold for other natural hazards such as heat-related mortality.

In general terms, the mortality rate is a measure of fatalities in a given
population over a certain period of time. While it is not always possible to
attribute fatalities to distinct causes, the effect of excess mortality due
to the impact of heat waves has been widely studied

Although it is a delicate issue to discuss human mortality in a technical
language, we believe that it allows for an intuitive and meaningful
application of the unified damage function. First, decease is expressed via
a Heaviside step function, where

Extending the regional focus,

Caution should be taken when considering the uncertainty of the hazard threshold. In contrast to the cases of coastal flood and storm damages, where building portfolios change only gradually, human heat tolerance is subject to continuous biophysical, behavioural, and environmental changes. Hence, a path dependence of the threshold exceedance is expected for ongoing heat waves.

While the stochastic occurrence of hazardous events has been subject to ample
research, the origin and propagation of uncertainty within the damage
function has received less attention. Often, a rough understanding of
sensitivity is obtained from estimating alternative scenarios

To enable a comprehensive sensitivity analysis of uncertainty from different sources, the unified damage function is cast into a probabilistic framework. We begin by defining a taxonomy of uncertainty sources that are relevant in our context.

Uncertainties arise at each step along the causal chain, from the modelling or observation of the hazard through the estimation of micro- and macroscale damage to the validation against reported losses. We focus on the propagation of uncertainties within the damage function, linking the microscale with the macroscale behaviour. For this reason, model and parametric uncertainty are excluded. Model uncertainty would arise from selecting an inadequate damage function that deviates from the actual hazard–damage relation. Parametric uncertainty relates to incomplete knowledge about the model parameters (but not the explanatory variables).

It is common to categorize uncertainties into those that are due to
statistical variability (aleatory) and those that are due to incomplete
knowledge (epistemic)

In order to maintain an intermediate level of detail, the considered
uncertainties are classified as aleatory, i.e. statistically tractable.
Having excluded model and parametric uncertainty, the remaining sources of
uncertainty can be identified from the mathematical description of the damage
function. For this purpose, Eq. (

The sources of uncertainty are summarized in Fig.

The asset values of affected items can vary significantly (e.g. different
house prices). The attribution of values to location is feasible only on
a detailed case study level, while large-scale assessment typically relies on
by-proxy estimation of average asset value

Even if structures of similar type are equally affected (i.e. at the same
threshold exceedance) their damage can differ considerably. The underlying
damaging processes are not well understood and are dependent on construction
types and employed materials. The resulting uncertainty could in principle be
reduced by modelling all physical processes involved. However, data
limitations usually permit no more than a stratification to a few predefined
asset classes

The threshold exceedance for an item is subject to uncertainty in the hazard
threshold (

On the macroscale level, the hazard magnitude is typically described by
a single indicator (e.g. the maximum flood level or gust speed). For all
practical purposes, this indicator is subject to uncertainty, stemming
either from imprecise measurement, uncertain model output, or confidence
levels estimated from extreme value statistics

For purposes of calibration and validation, model estimates are often put into comparison with reported figures of damage or economic loss. Like any observation, these figures are subject to uncertainty. For example, reported figures may be affected by gradual damage accumulation masking the effect of individual hazard occurrences, by incentives for insurance holders (e.g. deductibles), and by wealth levels that affect the construction quality and the likelihood of purchasing insurance.

A quantitative analysis of the aforementioned uncertainties requires an extension of the basic damage function. Here, we derive a comprehensive probabilistic framework for the unified damage function. The framework also forms the mathematical basis for the subsequent sensitivity analysis.

Classification of the sources of uncertainty into intrinsic and extrinsic.

We begin by defining random variables for each of the micro- and macroscale
model variables. Microscale variables are the local hazard magnitude

The Lisbon urban cluster supplied by B. Zhou

The exceedance,

The number of inundated
buildings within the Lisbon urban cluster at hypothetical flood levels between 0 and

The distribution of the relative damage caused,

Parameterization of the probabilistic damage
function for the estimation of damage from coastal flooding in Lisbon. The
variables

Parameterization of the probabilistic damage
function for the storm-damage simulation for a German building portfolio. The
variables

We define the asset-weighted damage as the product of relative damage and
normalized asset value,

The PDF of

Since the macroscale damage

Finally, uncertainty in the true hazard magnitude

Based on our taxonomy of uncertainties, we provide an exemplary parameterization of the unified damage function for two separate climate-related hazards: (i) coastal flooding in Lisbon, Portugal, and (ii) winter-storm damage for a German building portfolio comprised of 5000 individual buildings.

The Lisbon case exemplifies a bottom-up approach, where the individual hazard
thresholds are known explicitly. Since coastal flooding is not bound by
artificial administrative boundaries, we consider a cluster of
continuous urban agglomeration in the Lisbon metropolitan area
(Fig.

The portfolio of flood-prone buildings within the cluster of Lisbon is
based on statistical data provided by the Instituto Nacional de
Estatística

The number of buildings within each CORINE cell were assigned to elevation
levels obtained from the
EU-DEM

Microscale building damages in the Lisbon cluster are estimated with the
damage function employed by

A complementary top-down approach is pursued for the German building
portfolio, with an implicit description of the hazard threshold by means of
a probability density distribution. In the case of storm hazard, the
determinants of the hazard threshold are less clear-cut than for flood
damages. While they depend strongly on construction type and building age,
a strong residual uncertainty remains.

The mean microscale damage caused by severe winds is often described as
a power law with an upper bound representing complete destruction

Unlike the general features of the damage function, the nature of the
uncertainties involved is typically not well understood and their
quantification heavily relies on assumptions. Consequently, the required PDFs
of the asset value, the microscale damage, the exceedance, and the hazard
magnitude were estimated from literature, where available, and otherwise
based on own considerations. Tables

Figure

Going beyond the qualitative description of the involved uncertainties, this section focusses on their potential impact on damage estimates. From a non-linear damage function we expect potential interactions between different uncertainties that may vary with the hazard magnitude. Moreover, the analysis should take the different scales into account, as the macroscale damage is effectively an aggregation of microscale damages.

The influence of the various sources of uncertainty is assessed by performing a sensitivity analysis. Sensitivity analysis usually considers the effect of variation in one or more input variables on the dependent variable. For simple linear models, it may be sufficient to vary only one input variable at a time, since there is no interaction between different input variables. Non-linear models, in contrast, require a global sensitivity analysis, where simultaneous changes of all input variables are considered.

We employ the VBSA, which
estimates the contribution of each input variable to the total variance of
the dependent variable

The general algorithm of VBSA is summarized as follows. First, two

The total-effect index is chosen as the main metric for sensitivity. It describes the share of output variance that is due to the direct and indirect effects of an uncertain variable. The direct effect (also called first-order effect) measures the lone contribution of varying a single variable, averaged over different realizations of the remaining variables. Indirect effects (higher-order effects) are due to interactions between two or more variables, e.g. a second-order effect may arise from the interaction between the threshold exceedance and the damage level.

The total-effects index

The VBSA was applied on three distinct levels: (i) the microscale level
related to a single item, (ii) the macroscale level limited to intrinsic
uncertainty, and (iii) the macroscale level including extrinsic uncertainty.
The sample size

In order to evaluate the uncertainty of the sensitivity indices, the
bootstrap method was used to obtain uncertainty intervals. Specifically, the

Figure

The overall behaviour seen for the microscale case also holds true for the
accumulated building portfolio of Lisbon. Excluding extrinsic uncertainty,
Fig.

On the macroscale level, Fig.

This behaviour can be explained as follows. For a fixed number of affected buildings, intrinsic uncertainty outpaces the uncertainty in hazard magnitude. However, an increase in affected buildings reduces the relative magnitude of intrinsic uncertainty due to diversification. This is not the case for the uncertainty in hazard magnitude, which acts as a bias for the entire portfolio.

The sum of the total-effect indices of each variable is equal to

In the absence of interaction, the relevance of the uncertainties is
determined by their relative magnitude. In this regard, Fig.

The sensitivity results obtained for the second case study – storm damage in
a hypothetical German city – are similar to the Lisbon case.
Figure

On the macroscale level, intrinsic uncertainties show a sensitivity that is
similar to the microscale level. The curves shown in Fig.

Based on damage assessments for coastal flood and storm hazards, a unified damage function was identified and embedded into a probabilistic framework for the consideration of uncertainty.

While an exchange of information between the various hazard communities could
potentially trigger methodological improvement

Cross-hazard comparison of uncertainties within the unified approach has the
potential to provide valuable insight on the nature and relevance of
uncertainties along the causal chain. From a practitioner's point of view,
determining the most relevant sources of uncertainty is arguably more
important than quantifying each potential uncertainty source. Serving this
purpose, valuable insight could be gained from a variance-based sensitivity
analysis of the unified damage function. The analysis goes beyond similar
studies

On a general level, extrinsic and intrinsic sources of uncertainty were distinguished. Extrinsic sources manifest as a random bias for the entire portfolio (e.g. hazard magnitude), while intrinsic uncertainties arise locally and affect individual portfolio items (w.r.t. asset value, damage level, and threshold exceedance).

As demonstrated by both case studies, extrinsic uncertainty can play a crucial role as the dominant source of uncertainty. In contrast to the intrinsic uncertainties, whose aggregated effect (i.e. in terms of the standard deviation of the macroscale damage) increases sub-linearly with portfolio size due to diversification, the effect of extrinsic uncertainty is directly proportional to portfolio size. Hence, given a sufficiently large portfolio and uncertainty in the hazard magnitude, intrinsic uncertainty sources may be negligible for damage assessment. This is of particular importance in climate science, where practitioners often deal with ensemble simulations exhibiting large model spreads. It is also relevant for natural hazard research, where extreme value theory often implies broad confidence intervals for extreme events.

An example where this result allows for additional insight is the work of

Considering the relevance of intrinsic uncertainty sources, our results show that the composition of uncertainty within the microscale damage function largely determines the role of intrinsic uncertainties at the portfolio level.

Amongst the intrinsic uncertainties, the uncertainty due to local threshold
exceedance (being a combination of local hazard fluctuations and local
variations in hazard threshold) is only significant for low hazard
magnitudes. This magnitude range may not be relevant in certain cases, e.g.
focussing on high-end scenarios or including protective measures such as sea
walls. The case studies also show the extent to which variability in asset
values can dominate intrinsic uncertainty. While that uncertainty could be
reduced if spatially resolved data were available, this is typically not the
case for data-scarce regions within developing countries, which are also more
severely affected by natural disasters

Despite the different microscale damage functions used, both case studies show a similar sensitivity to uncertainties. This indicates that the validity of our conclusions on uncertainty reaches beyond the considered hazards. Moreover, the effect of different microscale damage functions (of the same one-parameter family) could be simulated by a re-scaling. For the sensitivity results, for example, a more shallow microscale damage function would result in a stretch along the hazard axis, while preserving overall behaviour.

The effect of large-scale protection measures, e.g. sea walls, was not
considered in this study for two reasons. Firstly, such measures are specific
to flood hazards and have no counterpart for other hazards, such as wind
storms. Secondly, sea walls modify the incident hazard by interrupting events
below the design protection level and are hence not an immediate component of
damage estimation. However, it is known that the probability of protection
failure, e.g. crevasses, represents a major source of uncertainty for damage
assessment

In practice, there are some limitations to the unified damage function that
arise from the simplicity of the approach. At increased cost and effort of
data acquisition, more specialized approaches could provide superior damage
estimates

Addressing the need for comprehensive approaches for risk analyses and management, we have shown that certain damage functions for coastal floods and windstorms are two facets of a unified damage function. Further, it was indicated how this unified approach could be extended to the estimation of heat-wave fatalities.

With its wide applicability to the assessment of both loss and fatalities, the unified damage function has the potential to facilitate knowledge transfer between climate-related hazards and to narrow the gap for a multi-hazard damage assessment. Moving towards this goal, the interdependence and cascading effects of climate-related hazards become of wider concern. For further research, we hence propose the extension of the unified approach to include non-stationary hazard thresholds.

For our case study region, the Portuguese coast and in particular Cascais,

Modelling flood damages, exceedance uncertainty is mostly driven by errors
related to the elevation model used. For Portugal, statistical validation of
the EU-DEM against ICESat measurements

Actuarial practice suggests that the log-normal distribution may serve as
a first approximation to the broadly skewed damage claim distributions

Regarding storm or flood damages to individual buildings, the built-up values
can be approximated by the distribution in house prices. For the case of
Tokyo,

For maximum wind gusts, which are required for the assessment of storm
damages,

Wind gusts exhibit a strong spatial variability at short ranges. This aspect
is demonstrated, inter alia, by the fact that the

In the lack of local empirical studies for the uncertainty in damage levels
or the variation in asset values, we employ an identical parameterization for
both the coastal flooding and the storm hazard case studies. The
parameterization for the damage level uncertainty and the variation in asset
values is described in Sect.

We appreciate valuable discussions with U. Ulbrich and L. Krummenauer. This work was produced using Copernicus data and information funded by the European Union EU-DEM layers. This work was supported by the European Community's Seventh Framework Programme under grant agreement no. 308 497 (Project RAMSES).Edited by: H. Kreibich Reviewed by: four anonymous referees