Tools for estimating probabilities of flooding hazards caused by the simultaneous effect of sea level and waves are needed for the secure planning of densely populated coastal areas that are strongly vulnerable to climate change. In this paper we present a method for combining location-specific probability distributions of three different components: (1) long-term mean sea level change, (2) short-term sea level variations and (3) wind-generated waves. We apply the method at two locations in the Helsinki archipelago to obtain total water level estimates representing the joint effect of the still water level and the wave run-up for the present, 2050 and 2100. The variability of the wave conditions between the study sites leads to a difference in the safe building levels of up to 1 m. The rising mean sea level in the Gulf of Finland and the uncertainty related to the associated scenarios contribute notably to the total water levels for the year 2100. A test with theoretical wave run-up distributions illustrates the effect of the relative magnitude of the sea level variations and wave conditions on the total water level. We also discuss our method's applicability to other coastal regions.

Predicting coastal flooding and extreme sea level events has a focal role in the designing of rapidly evolving coastal areas, which are continuously more populated and convoluted. Such flooding events are influenced by long-term changes in mean sea level, together with short-term sea level variations and the wind-generated wave fields. These processes are further influenced by a variety of other processes and conditions like vertical crustal movements, islands, the shape of the shoreline and the topography of the seabed. Because of a rising mean sea level, the effect of sea level variations accompanied by waves might cause more damage in the future than in the present conditions. In this study, we analyse the joint effect of the still water level and wind waves on the Finnish coast.

Globally, several studies have addressed the topic of combining sea
level changes and variations with wind waves in different circumstances and
at different locations, using different methods and assumptions.

In a study conducted by

Although the changes in water level have been deemed to have the highest
impact on flooding risks by several authors,

The Baltic Sea is a shallow semi-enclosed marginal sea, connected to
the Atlantic Ocean only through the narrow and shallow Danish Straits. This
gives the sea level variations in the Baltic Sea a unique nature, which
differs from that on the ocean coasts. The components of local sea level
variations on a short timescale include wind waves, wind- and
air-pressure-induced sea level variations, currents, tides, internal oscillation (seiche)
and meteotsunamis. Long-term changes are related to the climate-change-driven
mean sea level variations; postglacial land uplift; and the limited exchange
of water through the Danish Straits, which causes variations up to 1.3 m in
the average level of the Baltic Sea on a weekly timescale

Both sea level and wind waves have been thoroughly studied separately in the
Baltic Sea area, but research into their joint effect is sparse compared to
coastal regions outside the Baltic Sea.

In Finland, there is a clear demand for flooding risk evaluation.
The irregular coastline is characterised by coastal archipelagos consisting
of tens of thousands of islands. Especially the southern part of the coast will
likely become exposed to increasing flooding risks, as the land uplift rate
no longer compensates for the accelerating sea level rise

During the record-breaking storm Gudrun in 2005, three different components
acted simultaneously in the Gulf of Finland: a high total water amount in the
Baltic Sea, a high phase of the standing waves (seiches), and severe winds
piling up the water and waves towards the shore. Gudrun caused major damage
to coastal infrastructure on both the north and south sides of the Gulf of
Finland

The earlier flooding risk estimates in Finland

In this study, we utilise location-specific probability distributions of
water level and wave run-up to obtain a single probability distribution for
the maximum absolute elevation of the continuous water mass (Fig.

This paper is structured in the following manner. In Sect.

The total water level, i.e. the maximum absolute elevation of the continuous water mass (solid blue), is a result of the (1) long-term mean sea level change, (2) short-term sea level variations and (3) wind-generated waves. On a steep shore the waves can also be fully or partially reflected (dotted blue).

The instantaneous sea surface height at any coastal site in the Baltic Sea is
affected by several physical processes on different timescales. In this
study, we use the term still water level to represent the maximum
elevation of the water level (including short- and long-term sea level
variations). Moreover, we use the term total water level

The long-term mean sea level on the Finnish coast, on decadal timescale,
is affected by the global mean sea level, the postglacial land uplift
and the Baltic Sea water balance

Short-term sea level variations on a sub-decadal timescale on the
Finnish coast range from

The wave conditions in the Baltic Sea are influenced by the limited
fetch, the topography of the seabed and the seasonal ice cover

The coastal area off Helsinki and the measurement sites used in the
study. The red box in the Baltic Sea map

We focused our calculations on three different years: 2017, 2050 and 2100.

The future scenarios of

The Finnish Meteorological Institute (FMI) operates 14 tide gauges along the
Finnish coast, most of which have been operating since the 1920s. We used
46 years (1971–2016) of instantaneous hourly sea level observations from the
Helsinki tide gauge. The Finnish sea level data are measured in relation to a
tide-gauge-specific fixed reference level, which is regularly levelled to the
height system N2000. The height system N2000 is a Finnish realisation of the
common European height system. The N2000 datum is derived from the NAP

The sea level variations are location-specific, but, as our study area is
limited to sites less than 5 km away from the Helsinki tide gauge, we
considered the sea level variability measured at the tide gauge sufficiently
representative for both study sites at Jätkäsaari and Länsikari
(Fig.

FMI conducts operational wind wave measurements at four locations in the
Baltic Sea. In the Gulf of Finland, the observations are carried out using a
Datawell Directional Waverider moored in the centre of the gulf (see Fig.

We used the open-sea measurements from the operational Gulf of Finland wave
buoy in 2000–2014 in combination with shorter time series at chosen
locations inside the Helsinki coastal archipelago. The measurements in the
archipelago were conducted at Jätkäsaari (31 days in October 2012)
and Länsikari (11 days in November 2013) (see Fig.

We chose the measurement sites at Jätkäsaari and Länsikari so that they would represent two different kinds of wave conditions: Jätkäsaari is close to the shore, in a place well sheltered from the open sea by islands. Länsikari, on the other hand, is located in the outer archipelago, relatively unsheltered from the open-sea conditions.

Most wave parameters can be defined using spectral moments:

As a first step in estimating the combined effect of the long-term mean sea
level, the short-term sea level variability and the wind waves on the
frequencies of exceedance of coastal floods, we constructed probability
distributions for each of them separately (Sect.

In this paper, we use three types of probability distributions. The
probability density function (PDF)

Since our data are based on hourly values, we converted the frequencies of exceedance from the CCDF to events per year by multiplying them with the average number of hours per year (8766). By using hourly sea level values, we practically assume a constant sea level for the entire hour. When summing a 1 h constant sea level value with a 1 h maximum wave run-up with respect to the mean water level, the result is the maximum absolute elevation within 1 h. This maximum absolute elevation during 1 h is defined as one event.

The probability distributions for the long-term mean sea level scenarios on
the Finnish coast were calculated by

Probability density functions of future mean sea level at the
Helsinki tide gauge for the years 2050 and 2100 and the long-term mean sea level
estimate of 0.19 m for the year 2017. The 5th, 50th and 95th percentiles are shown for 2050 and 2100. The data in the figure are from the
results of

We constructed the probability distribution of short-term sea level
variability from the observed sea levels in 1971–2016. The observed sea
levels practically represent the sum of the first two terms of Eq. (

We then calculated the CCDF for the short-term sea level variations and
extrapolated it with an exponential function (Fig.

CCDF of the short-term sea level variations at the Helsinki tide gauge: observed hourly values in 1971–2016, from which a time-dependent estimate for the long-term mean sea level has been subtracted.

The short time series measured at Jätkäsaari and Länsikari
(Sect.

The wave height values obtained by attenuating the open-sea data were
combined with the local measurements, and CCDFs were estimated by fitting
piecewise exponential functions to the data. For the large values of the CCDF
the exponential function was fitted to the observational data, while for the
smaller values (rarer events) a fit was made to the modelled values. These
two pieces were connected to form one continuous distribution (see Fig.

The final step was to estimate the wave run-up, i.e. the maximum vertical elevation of the water in relation to the still water level. We defined the wave run-up using the highest single wave during an hour, since this will produce one well-defined event when combined statistically with the water level data.

The highest wave during an hour was determined by assuming that the height of
the single waves are Rayleigh-distributed, following

The run-up depends on a number of parameters, but on a steep, sufficiently
deep shoreline the maximum vertical elevation is determined by the highest
single individual wave, which is further magnified by reflection. Spectral
wave measurements have been conducted at the Jätkäsaari study site

Our results should be valid also for the part of the shoreline that is not
equipped with wave-damping chambers. Based on the results of

The shoreline near the Jätkäsaari wave buoy. A part of the
shoreline is equipped with wave-damping chambers. Reprinted from

One traditional distribution used to describe the significant wave height at
a certain location is the Weibull distribution

These distributions have different properties: shape, expected value and
typical magnitude relative to the sea level variations, with probability
functions (PDFs and CDFs)

Wave run-up distributions for the two locations in the Helsinki archipelago: Jätkäsaari and Länsikari.

The different theoretical wave run-up distributions and observation-based
still-water-level distribution used for the theoretical test. The
Weibull scale parameter (

PDFs

The theory for determining the probability distribution of the sum of two
random variables can be found in textbooks

Let

The goal is to define the cumulative distribution function

For practical purposes

We applied the method for calculating the probability of the sum of two
random variables (Sect.

For the present conditions (year 2017), we calculated

Finally, we calculated a still-water-level distribution to be used in the
theoretical test by simply taking the distribution of the short-term sea
level variability (

As a second step, we calculated the CDF of the full three-component sum (Eq.

This calculation of the three-component sum was performed for the still-water-level distributions for 2017, 2050 and 2100 combined with the observation-based wave run-up distributions at Jätkäsaari and Länsikari, as well as for the zero-mean still-water-level distribution combined with the six theoretical wave run-up distributions.

We applied the presented method in the Helsinki archipelago, located at the
northern coast of the Gulf of Finland in the Baltic Sea. The calculations were done
for two locations, where Jätkäsaari is situated deep inside the
archipelago near the shoreline, while Länsikari is more exposed to the
open-sea wave conditions (Fig.

We calculated

The total water levels for a location closer to the open sea (Länsikari)
are up to 1.2 m higher compared to the values for the sheltered shore
location (Jätkäsaari). This clear difference follows from the
difference in the wave run-up distributions (see Fig.

The impact of the future mean sea level change is evident in the

As we used the same mean sea level scenario for both Jätkäsaari and
Länsikari, the effect of the mean sea level change is similar for them
even in the

Still water levels (in metres relative to N2000) corresponding to certain frequencies of exceedance for three years (2017, 2050 and 2100) based on the observed sea level variability and mean sea level scenarios for the Helsinki tide gauge.

Total water levels (metres relative to N2000), as the sum of still water level and wave run-up, for three different years (2017, 2050 and 2100) for Jätkäsaari and Länsikari.

CCDFs for the still water level alone (

CCDFs for the still water level and the total water level, obtained by applying six theoretical wave run-up distributions.

Results of the theoretical test, i.e. values for different
frequencies of exceedance for the still-water-level distribution SL, the six
theoretical wave run-up distributions W1a–W3b and the total water level
distributions SL,W1a (the convolution

The total water level distributions SL,W1a, SL,W1b etc. obtained by combining the
distribution of the short-term still water level with the theoretical wave
run-up distributions, are shown in Fig.

For the first pair of wave run-up distributions (W1a and W1b; see Table

In the second pair (W2a and W2b), neither the wind waves nor the sea level
variations are clearly dominant. The contribution of the waves is now larger
compared to the first pair. Even the total water level with a frequency of

In the case of the third pair (W3a and W3b), the contribution of the larger
waves becomes evident. The total water levels are up to 3.5 m (

With the first pair, the total water levels (SL,W1a and SL,W1b) differed by at
most 0.1 m from the sum of the still water levels and expected values of the
wave run-up distributions, namely SL

However, as soon as the contribution of the waves increases, the situation
changes. In the situation where the wave height and sea level variations are of the same order
(SL,W2a and SL,W2b vs. SL

Finally, in the case where the waves dominate (SL,W3a and SL,W3b vs.
SL

The observed wave run-up distribution at Jätkäsaari (Fig.

The same applies for the distributions of the total water level in 2050: the effect of waves adds 0.93–0.96 m to the still-water-level distribution at Jätkäsaari and 2.05–2.11 m at Länsikari. The distributions for 2100, however, behave differently. For them, the contribution of waves increases with decreasing frequency of exceedance: from 0.74 to 0.89 m at Jätkäsaari and from 1.85 to 2.02 m at Länsikari. It is also noteworthy that the contribution of waves is smaller in 2100 than in 2017 or 2050.

The effect of waves on the distributions of the total water level at Jätkäsaari and Länsikari in 2017 and 2050 can thus be quantified with a fixed wave action height, but – similar to the theoretical distributions SL,W2a and SL,W2b – this value clearly exceeds the expected value of the wave run-up distribution.

In general case, the relationships between the wave height, wave run-up and sea level variations are complex. In this study, we made several assumptions and simplifications. The aim of this section is to discuss the validity of our results, as well as to help the reader to estimate whether this method could be used at a certain location or with specific data available.

The essential prerequisites for applying the method presented above are as follows:

An estimate for the long-term mean sea level is needed. In its simplest form,
this can be a single mean sea level height value. If the mean sea level is changing,
however, an estimate for this change is needed. Again, a simple estimate could be a
time-dependent mean sea level value – a linear trend, for instance. Using an ensemble
of estimates for the future scenarios (like was done by

An estimate for the range of the short-term sea level variability is
needed – technically, in the form of a good-quality probability density function. In the case
of the Finnish coast, we have found that several decades of observations with hourly
time resolution are needed to get a reliable estimate for the extent of the local sea
level variations. Additionally, to estimate total water levels with low frequencies of
exceedance, such as

An estimate for the wave run-up distribution is needed to account for the effect
of waves on the coast. In this paper we have used the simplest formula for a steep shore
using the highest single wave, which was estimated from the significant wave height

We based our analysis on a simplifying assumption that the sea level variations
and wave run-up are independent. This makes it possible to calculate the
distribution of the sum from the marginal distributions without additional
assumptions. In practice, the independence of the variables can be, at least
partly, achieved for locations with a constant beach profile, such as deep
and steep shores. Strong wind-independent components in the sea level also
decrease the dependence of the sea level and the wave run-up. In the Baltic
Sea, such a component is the total Baltic Sea water volume, which, although
expressing a strong correlation with the wind conditions

As long as the above conditions are met, we consider the method presented here applicable also for other places than the Finnish coast. Naturally, as the most important factors causing sea level variations are different in different places, this needs to be taken into account. For instance, in places where the tidal variations dominate over storm surges, a different analysis of the short-term sea level variability might be appropriate.

In our approach, we treated the still-water-level variations and the wave run-up as independent variables as a first approximation. The limited amount of wave data available for this study imposed challenges in the construction of the full joint distribution, which would have taken into account the possible dependencies between these variables. The dependency might be affected by the location-specific circumstances, and further studies are needed to determine the conditions under which the use of the full two-dimensional distributions is preferable to assuming independence.

Using block maxima of sea level variations – such as the monthly maxima used
by

We calculated the future scenarios for the flooding risks by simply combining the mean sea level scenarios with the present-day short-term sea level variability and wave conditions. Thus, we implicitly assumed that those will not change in the future. A potential improvement, to get deeper insight into the changes of flooding risks in the future, would be to include scenarios of short-term sea level variability or wave conditions. As these both mainly depend on short-term weather (wind and air pressure) conditions, this would require scenarios for the short-term weather variability.

Safe coastal building elevations are usually estimated for structures with a
designed lifetime of at least several decades, but the relevant safety
margins differ between commercial buildings, residential buildings and e.g.
nuclear power plant sites. We therefore need to consider scenarios up to 2100
and frequencies of exceedance as rare as

In this study, a location-specific statistical method was used for the first time on the Finnish coast to evaluate flooding risks based on the joint effect of three components: (1) long-term mean sea level change, (2) short-term sea level variability and (3) wind-generated waves. We conducted an observation-based case study for two locations with steep shorelines and performed a test with theoretical wave run-up distributions.

The case study at the Helsinki archipelago (Sect.

We found the coastal flooding risks at our case study location to increase
towards the end of the century. This behaviour in our results is due to the
projected mean sea level rise as well as increasing uncertainties in these
projections

Our test with the theoretical wave run-up distributions showed that, in a situation where the sea level variations dominate over waves, simply adding the expected value of the wave run-up on top of the still-water-level distribution produces results close to the distribution of the sum. However, when the contribution of the waves increases, such addition leads to an underestimation of the effect of waves on the total water levels. Finally, when the waves are clearly dominant, their effect starts to depend on the frequency of exceedance and cannot be quantified as a constant value to be added on top of the still water levels anymore.

Please see the Supplement for our data.

The supplement related to this article is available online at:

The research question was proposed by KK. Sea level data were analysed by UL and MJ, and wind wave data by JVB and KK. The mean sea level scenarios were prepared by HP. KK, MJ and UL combined the probability distributions of sea level variations and wave run-up. The compiling of the manuscript was initiated by UL and LL, and it was mainly written by UL, JVB and MJ with contributions from all co-authors.

The authors declare that they have no conflict of interest.

This research was partly funded by the Finnish State Nuclear Waste Management Fund (VYR) through SAFIR2018 (the Finnish Research Programme on Nuclear Power Plant Safety 2015–2018), the City of Helsinki, and Arvid och Greta Olins fond (Svenska kulturfonden, 15/0334-1505). This study has utilised research infrastructure facilities provided by FINMARI (Finnish Marine Research Infrastructure network). We would like to thank the three referees for constructive comments, which helped us to improve the manuscript. Edited by: Piero Lionello Reviewed by: Jose A. Jiménez and two anonymous referees