In contrast to recent advances in projecting sea levels, estimations
about the economic impact of sea level rise are vague.
Nonetheless, they are of great importance for policy making with
regard to adaptation and greenhouse-gas mitigation. Since the
damage is mainly caused by extreme events, we propose a stochastic
framework to estimate the monetary losses from coastal floods in
a confined region. For this purpose, we follow
a Peak-over-Threshold approach employing a Poisson point process and
the Generalised Pareto Distribution. By considering the effect of
sea level rise as well as potential adaptation scenarios on the
involved parameters, we are able to study the development of the
annual damage. An application to the city of Copenhagen shows that
a doubling of losses can be expected from a mean sea level increase
of only 11

From extreme sea levels to damage.

Considering current

Adverse effects from sea level rise are particularly expected from
storm surges, presupposing the coincidence of extreme tidal and storm
conditions

For estimating
the annual flood damage at a specific site (that
is the sum of all damages caused within a year), information on the
occurrence of flood events, their magnitude as well as the
corresponding damage is required. Due to the stochastic nature of
extreme events, the annual damage cannot be predicted for a specific
year and is characterised by its average value over a longer time
period. In reality, the actual damage fluctuates around this
expected annual damage with a certain variability. For
instance, there are years without any damage and others where a very
unlikely flood event (e.g. a

Overview of employed symbols, their meanings, and section in which they are introduced.

Considering sea level rise, we find analytic relations describing the damage for asymptotic parameter values (i.e. for very large changes) and show that they represent good approximations for the behaviour of damage under current conditions. Furthermore, studying the mitigation effects due to coastal protection measures in an analogous way, we provide three potential decays of residual damage, depending on the shape of the sea level distribution. In general, our analytical relations are capable of describing the development of damage for all parameter variations.

The paper is organised as follows. Section

Our proposed methodology (illustrated by Fig.

Extreme events are commonly characterised by employing extreme value
theory

In our context,

Section

Illustrative time series of sea levels and their probability
density function for several scenarios.

We want to study the impact of sea level rise as well as potential
protection measures on the flood damage. As illustrated in
Fig.

We assume that a rise in mean sea levels results in a shift of today's
sea level distribution towards higher water levels without deformation
of the distribution

The implementation of a coastal protection measure will be considered
in such a way that any damage from flood levels up to a specific
protection height

Given the GPD parameters with respect to a threshold

After having information about the occurrence of flood events, the
resulting damage is obtained by means of a (stage-)damage function

Most commonly, damage functions are applied on the building scale

The combination of the methodologies above provides the probability
distribution of the annual flood damage in a specific region
(Fig.

Given the extreme value parameters

We start in a general setting where the GPD parameters

All other parameters are kept constant in the following.
As described in Sect.

Average magnitudes of excesses over the threshold

We find an increase of the annual damage by means of two separate
effects (as described in Sect.

In the case

In contrast, we find a less steep relation if the water
levels are bounded tailed (i.e.

Here, the two effects are superposed: the average damage of an event
increases with exponent

For the heavy-tailed case (

which holds for

The expected annual damage only represents average values, and the
actually occurring losses fluctuate considerably. Therefore, we also
examine the uncertainty of our estimations by means of the standard
deviation of the damage,

This uncertainty measure represents just a lower bound since it
includes only the aleatory uncertainty from the fact that one does not
know when the extremes occur and does not take into account
additional epistemic uncertainties due to a lack of knowledge, e.g.
stemming from the stage–damage relation

Expected annual damage (dark blue) and standard deviations
(light blue) in

Besides sea level rise, which is regarded as the main driver for
higher and more frequent extremes

We would like to illustrate how the respective variables behave in
real examples and compare our analytic derivations from the previous
section with numerical calculations (as described in
Sect.

For the estimation of extreme value parameters in the two case
studies, extreme sea level records from closely located gauges were
preprocessed by subtracting a linear trend of 0.45 cm (Copenhagen)
and 0.16 cm (Kalundborg) per year (derived from mean sea level
data, available at

The available damage functions support our presumption from
Sect.

Once having this information, the annual damage can be calculated
numerically (as described in Sect.

In practice, one is often interested in a temporal development of
damage.
In order to further elaborate on this issue, our approach
requires a projection of mean sea levels.
For the city of Copenhagen, such have been extracted from the
Dynamical Interactive Vulnerability Assessment (DIVA) tool

At this point, it is important to bear in mind that the severity of
a flood disaster is not only determined by environmental factors but
also to a significant extent by human decisions

Expected annual damage (dark green) and standard deviations
(light green) in

Regarding the case studies Copenhagen and Kalundborg,
Fig.

Beside the point process approach, the method of block maxima using
the GEV distribution is a common
approach in extreme value theory

Complementary to the work in hand, an analogous analysis using the
block maxima instead of the point process approach has been carried
out recently

In contrast, an increasing variability in the sea levels, reflected in
a changing scale parameter

Finally, investigating increasing protection levels, the results of the two approaches again coincide. This is not surprising, since for high protection levels, inundations are very rare and more than one flooding per year is very unlikely. Consequently, the disregard of additional floods becomes negligible and the annual flood damage is typically determined by one – the most severe – flood event.

Both approaches are based on extreme value theory but differ in the
extreme sea levels that are taken into account. Since the point
process approach presented in this work is able to consider
all relevant flood events, it can be considered as
advantageous, particularly for the investigation of sea level rise
impacts. However, the choice of the threshold

Despite the accurate analytical formulation of the work at hand, some weaknesses need to be noted.
For instance, the occurrence probability of a flood event on a specific day is assumed to be
independent from the other days. In the short-term there is a strong correlation between sea levels.
This becomes apparent when considering the fact that storm surge events typically last for several days.
In addition, it has been shown, that sea level records also comprise long-term correlations

Studying the effect of sea level rise, we find that in any case the expected damage increases super-linearly with the mean sea level, when considering typical values of the shape parameter. This means that the losses always increase at a higher rate than the sea levels – a universal result that needs to be explored when the climate change impacts of sea level rise are discussed economically.

Our work also shows that the upcoming losses from sea level rise are
mostly determined by the type of sea level extremes (i.e. the sign of
the parameter

In general, our results show how the complexity of climate change,
adaptation, and flood damage can be disentangled by surprisingly simple
and general expressions which are applicable to arbitrary regions and
case studies. These relations are the basis for understanding the
effect of sea level rise on coastal flood damage and are of great
importance for the development of broad-scale assessment models in the
context of climate change

The main text is complemented in two ways. Firstly, an additional
analysis of flood damage in Copenhagen and Kalundborg as a function of
the scale parameter

Expected annual damage (red) and standard deviations (orange)
in

In addition to varying the parameters

In this section we derive the asymptotic relations from the main text.
The section comprises three parts, each part considering the effects
of changing parameters

The provided expressions describe the damage for asymptotically large
parameter values or, in case they are bounded, for parameters
approaching their limit. This means that the numerically calculated
values divided by the analytic result obtained converge to a non-zero
constant number for increasing parameter values. In the whole
section, the Generalised Pareto probability density function
with regard to the threshold

The investigation of the occurrence rate

With these expressions, the frequency of events is fully described for
large parameter values and, in combination with the following section,
the behaviour of the annual damage is derived in
Sect.

Not only the number of flood events is affected by evolving
parameters. As described in Sect.

Let the water levels above a threshold

The relations for

As mentioned above,

Let the water levels above a threshold

The proof corresponds to the case of increasing

Finally, we derive expressions for the dependence on the protection
height. As can be found in

Let the water levels above a threshold

Let

For

For the case

As stated in the main text, the total annual damage

The asymptotic behaviour of the number of annual flood events

We would like to thank Stéphane Hallegatte, Carlo S. Sørensen, and Jacob Arpe for the provision of data, as well as Luís Costa, Boris F. Prahl, and Dominik E. Reusser for fruitful discussions and comments. The research leading to these results has received funding from the European Community's Seventh Framework Programme under Grant Agreement No. 308 497 (Project RAMSES). We also express our thanks to the Potsdam Research Cluster for Georisk Analysis, Environmental Change and Sustainability (PROGRESS) for their financial support. Edited by: P. Tarolli Reviewed by: G. Le Cozannet and one anonymous referee