Determination of Hydrodynamic Design Parameters for Coastal Protection Structures on the German Baltic Sea Using Copula Models ^†

Abstract

The design of coastal protection structures requires design parameters that accurately represent the hydrodynamic conditions along the coast. Currently, these input variables are based on univariate probability models, which do not consider the joint probability of water level and statistical wave parameters such as significant wave height. Bivariate probability modeling using copula models offers an alternative. Copulas can be used to describe the dependencies between water level and significant wave height and to compute joint probabilities of occurrence. First, various copulas are fitted to samples of physically consistent combinations of water level and significant wave height extracted from storm surge events along the German Baltic Sea coast of Mecklenburg-Western Pomerania. Next, the most appropriate copula model is used to compute design combinations of water level and significant wave height for selected return periods. The bivariate design parameters are compared with the univariate ones in a simplified design example for wave run-up on a dike. The validation of various models shows that the Frank copula best describes the dependence structure. The bivariate design parameters obtained for the same return periods are lower than those determined using the univariate method. The available data only allow a limited application of the copulas for engineering design in the study area. Nevertheless, copulas have the potential to replace univariate methods for determining design parameters and thus contribute to more reliable and cost-efficient coastal protection structure design.

1. Introduction

Like many countries around the world, the German federal state Mecklenburg-Western Pomerania (MP) uses a range of methodologies to protect the region against storm surges. Such methodologies include the construction of embankments, dunes, breakwaters, groins, and various longitudinal structures such as seawalls and revetments [1]. The functional and structural design of these structures is critical in ensuring their resilience to the extreme loads imposed by high water levels in combination with high waves. The hydrodynamic design parameters are determined with the aim of ensuring that the structures can withstand these extreme loads. Important design parameters are the design water level (DWL) and the design wave conditions (DWC). These parameters are instrumental in the design, sizing, and safety evaluation of coastal protection structures [2]. It is important to note that both parameters are location-dependent and represent the safety level considered necessary to protect the coastal area [3].

The sea state is described by characteristic parameters such as the significant wave height, the peak period, and the mean wave direction. The significant wave height H_1/3 is derived from measurements or numerical simulations of a wave field and represents the mean of the highest one-third of waves within a given time interval, such as an hour, providing a statistical measure of the actual sea state at a given location [4]. S_W denotes the peak value of the water levels observed during a storm surge event. H_1/3 and S_W form the basis for the derivation of the DWL and DWC.

In current practice in MP the DWL and DWC are determined using the methods of univariate extreme value statistics. The methodology involves extracting extreme values from the time series of parameters using an appropriate procedure. The parameters of suitable extreme value distributions are estimated using fitting methods. The design event is then extrapolated using the previously defined probability of occurrence. In MP, the return period (R) for the DWL and the significant wave height of the DWC is set at 200 years, which corresponds to a non-exceedance probability of p = 0.995 [3].

A disadvantage of the univariate method for determining DWL and DWC is that both parameters are determined independently of each other. However, various publications have demonstrated a moderate dependence between S_W and H_1/3 in several sea areas during storm surges (e.g., [5,6,7]). Recent studies for the Baltic Sea region confirm this behavior and show that the dependence is moderate and event-dependent rather than strictly simultaneous [8,9]. The univariately determined parameters DWL and DWC do not consider this dependence and may not reflect the real conditions during storm surges with sufficient accuracy [10]. This can result in suboptimal dimensioning of coastal protection structures. The failure of such structures can be attributed to either under-dimensioning, leading to functional and structural failure, or over-dimensioning, which can result in increased construction costs and reduced public acceptance.

The statistical description of the dependence between two or more variables and the computation of the joint probabilities of occurrence is facilitated by copula models. The theoretical foundation of these models is the theorem formulated by Abe Sklar [11]. This theorem enables the decomposition of each multivariate distribution into its marginal distributions and a dependence structure, expressed by the copula function C:

$${\mathrm{F}}_{{\mathrm{X}}_1,\dots ,{\mathrm{X}}_{\mathrm{n}}\left({\mathrm{X}}_1,\dots ,{\mathrm{X}}_{\mathrm{n}}\right)}=\mathrm{C}\left({\mathrm{F}}_{{\mathrm{X}}_1}\right({\mathrm{x}}_1),\dots ,{\mathrm{F}}_{{\mathrm{X}}_{\mathrm{n}}}({\mathrm{x}}_{\mathrm{n}}\left)\right)$$

(1)

In the bivariate case, the left-hand side of Equation (1) describes the cumulative distribution function, while the right-hand side describes the two continuous marginal distributions of the random variables. Copulas thus offer a high degree of flexibility in modeling complex dependencies with different forms of distribution. This represents a considerable advantage over multivariate distributions (e.g., [5]).

Copula models were originally widely used in risk management, especially in financial applications (e.g., Frees & Valdez [12], Cherubini et al. [13], Kole et al. [14], Bouyé et al. [15], Fan & Patton [16]). The models are now also being used to address coastal engineering issues (e.g., Ferreira & Guedes Soares [17], de Waal & van Gelder [18], Salecker et al. [19] and Corbella et al. [20]). A framework and a practical guideline for performing a multivariate analysis of hydrodynamic parameters are presented by Salvadori et al. [21,22]. More recent studies further emphasize the importance of multivariate approaches for representing compound events and improving the reliability of design conditions (e.g., Latif & Simonovic [23], Gomze-Rave et al. [24]), particularly by demonstrating the need to account for interacting hydrodynamic drivers in compound coastal flood events.

While modeling the joint probability of S_W and H_1/3 is not a new concept in coastal engineering—having been applied, for example, to the Ravenna coast (Italy) by Masina et al. [25] and to Noordwijk (Netherlands) by Li et al. [26]—the German Baltic coast of MP represents a different sea area with distinct hydrodynamic conditions. The Baltic Sea is a semi-enclosed, micro-tidal basin where water levels are primarily driven by wind forcing, which may lead to different dependencies between water levels and wave heights and limits the direct transferability of existing approaches.

In this study, storm surge characteristics and significant wave heights are analyzed at six locations along this coast. This comparative analysis is essential to determine whether the DWL and DWC can be represented by a single probability model or require multiple models. A new approach for sampling is developed that extracts extreme combinations of S_W and H_1/3 while maintaining a stable dependence between the parameters. Based on the samples, the temporal dependence between the parameters during storm surges is investigated. Furthermore, DWL and DWC are derived using copula-based bivariate probability models and compared with univariate approaches in a conceptual design example.

By introducing a data-driven sampling strategy based on the stabilization of Kendall’s τ, this study provides a systematic and reproducible approach for representing the dependence between S_W and H_1/3. The results demonstrate that accounting for this dependence leads to systematically lower design values compared to conventional univariate approaches and thus has direct implications for the design of coastal protection structures.

2. Characterization of the Study Area

The Baltic Sea is an inland sea of the Atlantic Ocean, located in north-eastern Europe (see Figure 1). It is comparatively young, with an estimated age of about 8000 years. At the end of the last glacial period (Weichselian Glacial), the Scandinavian glaciers melted and filled the Baltic Sea basin with meltwater. The sea covers an area of 420,000 km² and has a volume of 22,000 km³ [27]. Key characteristic values are summarized in

Table 1. Characteristic values of the Baltic Sea.

Characteristic Value	Size
north-south extent	1300 km
west-east extent	1000 km
maximum width	300 km
mean depth	52 m
maximum depth	460 m

.

The study area comprises the coast of MP in the south-western Baltic Sea (see Figure 1). The coastline can be divided into three main morphological types: large bay coast, equalizing coast, and Bodden-type equalizing coast. Its highly indented geomorphology results in locally varying hydrodynamic conditions, which directly influence water level and sea state. To further characterize the coastal sections,

Table 2. Bathymetric characteristics of the coastal sections [28].

Coastal Section	Distance to 15 m Isobath	Seabed Slope
Boltenhagen	71.6 km	0.02%
Warnemünde	10.0 km	0.15%
Zingst	15.0 km	0.10%
Varnkevitz	1.6 km	0.94%
Göhren	2.4 km	0.63%
Koserow	60.0 km	0.03%

summarizes key bathymetric parameters, including the nearshore seabed slope and the distance to the 15 m isobath, based on Dimke [28]. During storm surges, the local hydrodynamic conditions are primarily controlled by geomorphology, local water depth, and the fetch length determined by the geographical exposure of the coastline. Two main factors trigger storm surges in the Baltic Sea: elevated reference water levels along a given coastal section and strong onshore winds. In the south-western Baltic Sea, additional contributing factors include the basin filling level, seiche oscillations, wind setup, and, depending on the location, bay setup. Among these, wind setup typically has the greatest influence. In MP, large parts of the coastline are protected by dikes (218 km) and dunes (106 km) [1].

3. Data and Methodology

3.1. Data Basis

The determination of hydrodynamic design parameters is based on time series of water level and significant wave height, representing the observed local sea state climate over the measurement period. These series constitute a single realization of the local sea state climate rather than all theoretically possible extreme events.

For extreme value statistical evaluations, hydrodynamic data must fulfill various criteria. According to the federal and state guidelines on hydrometry [29], the data must be free of gaps (continuity), free of anthropogenic influences (homogeneity), free of outliers (plausibility) and collected and processed in a standardized and systematic manner (consistency). Wind data are also included, as they will be used to model H_1/3 via the wind-wave correlation method (WWC method) [30]. The WWC method is applied in this study primarily to extend the available wave data records and support statistical analysis, rather than to provide exact deterministic predictions of individual wave conditions.

Before the copula-based analysis, the available datasets are evaluated with respect to completeness and consistency to determine their suitability for the subsequent analyses. Both numerically generated and measured data are available for the study area. However, the recent comprehensive water level and sea state data from the CoastDat datasets [31], generated using different input data and model setups, are not consistent and therefore unsuitable for describing dependencies between water levels and significant wave heights or for computing joint probabilities of occurrence. In addition to numerical data, measurement data are also available, generally collected near the coast.

Sea state data have been collected near the coast through various measurement campaigns by the Chair of Geotechnics and Coastal Engineering at the University of Rostock (UHRO) and the State Offices for Agriculture and the Environment (StALU). Most campaigns span only a few months to a few years and are therefore not suitable for extreme value analyses. An exception is the Internal Measuring Coast Network (IMK) operated by StALU, which has continuously recorded wind, sea state, and water level data for 24 years (as of 2022) at six homogeneously distributed coastal stations with a temporal resolution of Δt = 20 min [32]. The wave parameters used in this study are provided as time-domain estimates derived from measured time series using a zero-downcrossing approach, rather than from spectral analysis. However, it is not documented whether the IMK data undergo manual or automated plausibility checks.

Additional water levels are measured by the Waterways and Shipping Offices (WSA) at 19 heterogeneously distributed locations along the Baltic coast of MP. Depending on the gauge location, these high-resolution coastal water level records (Δt = 60 min) extend back up to 60 years. While no explicit information is available on data verification, it can be assumed that plausibility checks are performed as part of the WSA’s operational duties [33].

Due to the micro-tidal conditions of the Baltic Sea, the influence of astronomical tides is very small, typically only a few centimeters (e.g., [34]), and is therefore negligible compared to meteorologically induced water level variations. Consequently, no explicit tidal correction was applied.

Wind data in Germany are officially recorded by the German Weather Service (DWD), which operates 292 stations nationwide and provides open access to data [35,36]. All stations are operated in accordance with World Meteorological Organization (WMO) regulations, and data undergo multiple manual and automated quality control procedures [37,38]. Along the Baltic Sea coast of MP, DWD records wind data at nine stations.

Within the study area, only the 24-year IMK sea state time series have sufficient temporal coverage for extreme-value analysis and for deriving the DWC. A plausibility comparison of the IMK water level and wind records against corresponding WSA and DWD data revealed long periods with significant deviations. Therefore, the WSA water level and DWD wind time series closest to the IMK locations are used instead. To ensure temporal consistency, the time series of H_1/3 are extended using the WWC method based on the DWD wind data. Using this approach, the significant wave height data at all six IMK locations are extended to 38 years, resulting in longer time series suitable for extreme value statistical analysis. Figure 2 shows the coastal sections of the IMK stations used in the following analysis.

Table 3. Assignment of the WSA gauges and DWD wind stations to the IMK coastal sections.

IMK Location	WSA Gauge (Water Level)	DWD Station (Wind)
Boltenhagen	Wismar	Boltenhagen
Warnemünde	Warnemünde	Warnemünde
Zingst	Warnemünde	Warnemünde
Varnkevitz	Sassnitz	Arkona
Göhren	Sassnitz	Putbus
Koserow	Koserow	Greifswald

lists the corresponding WSA and DWD stations assigned to each IMK section.

3.2. Methodology for the Analysis of the Hydrodynamic Parameters

Once the data base has been compiled, the available hydrodynamic parameters along the coast of MP are analyzed and compared. This is necessary to decide whether a single probability model can describe the parameters at the various locations or whether multiple models are required due to the high spatial variability of S_W and H_1/3 along the coast. For this purpose, the spread and central tendency of water levels and significant wave heights are compared across the available coastal sections using boxplot diagrams. In addition, peak values, frequencies, and durations of storm surge events are evaluated in relation to storm surge categories to assess spatial and temporal variability.

3.3. Analysis of the Temporal Dependence Between Peak Water Level and Maximum Significant Wave Height

Many coastal protection structures are subjected to high hydrodynamic loads when the S_W and the maximum significant wave height (H_1/3max) occur simultaneously. Therefore, understanding their temporal relationship is essential. Three possible temporal sequences can be distinguished: S_W occurs before H_1/3max, both maxima occur simultaneously, and S_W occurs after H_1/3max. The three states are shown in Figure 3.

To ensure comparability of the temporal dependence of both parameters across the coastal sections, the times of S_W and H_1/3max are expressed relative to the length of a storm surge event:

$${\mathrm{t}}_{\mathrm{r}\mathrm{e}\mathrm{l}}\left(\mathrm{X}\right)={\displaystyle \frac{\mathrm{t}\left(\mathrm{X}\right)}{{\mathrm{t}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}}},\hspace{1em}\mathrm{X}∊\left\{{\mathrm{S}}_{\mathrm{W}},{\mathrm{H}}_{1/3\mathrm{m}\mathrm{a}\mathrm{x}}\right\}$$

(2)

Here, t_rel is the relative time (dimensionless), t(X) denotes the time of occurrence of parameter X, and t_total is the total duration of the storm surge event. The variable X represents either S_W or H_1/3max. If, for example, a storm surge with a total length of 10 h reaches its peak after 6 h, t_rel(S_W) = 0.6. In this way, it is possible to determine the time sequence and the time interval between the maximum S_W and H_1/3max in relation to the total duration of an event. To quantify the temporal difference between the occurrence of both maxima, the parameter

$$\mathrm{d}={\mathrm{t}}_{\mathrm{r}\mathrm{e}\mathrm{l}}\left({\mathrm{H}}_{1/3\mathrm{m}\mathrm{a}\mathrm{x}}\right)-{\mathrm{t}}_{\mathrm{r}\mathrm{e}\mathrm{l}}\left({\mathrm{S}}_{\mathrm{W}}\right)$$

(3)

is introduced. The value of d ranges from −1 to 1. Positive values indicate that S_W occurs before H_1/3max, negative values indicate that it occurs afterwards, and d = 0 denotes simultaneous occurrence (see

Table 4. Possible states of the factor to describe the temporal sequence and the temporal distance of S_W and H_1/3max.

Factor d	Temporal Sequence	Description
d > 0	t(SW) < t(H1/3max)	SW occurs earlier than H1/3max
d = 0	t(SW) = t(H1/3max)	SW and H1/3max occur simultaneously
d < 0	t(SW) > t(H1/3max)	SW occurs later than H1/3max

). The further d is from 0, the greater the relative time interval between S_W and H_1/3max. Based on this parameter, the temporal relationship between S_W and H_1/3max within the storm surge events along the Baltic Sea coast of MP is investigated.

3.4. Development of an Approach to Determine the Joint Probabilities of Occurrence of S_W and H_1/3

In the design, dimensioning, and safety assessment of coastal protection structures, it is essential to determine the magnitude of extreme events corresponding to specific return periods (e.g., 200 years). These aspects can be analyzed using both univariate and bivariate probability models. While univariate models treat each variable independently, bivariate models based on copulas account for the dependence between S_W and H_1/3. The univariate marginal distributions form the basis for the copula models, which link these distributions to represent the joint probability structure. Therefore, as a first step, the parameters of several univariate distribution functions are fitted to the samples of S_W and H_1/3 along the coast of MP.

3.4.1. Bivariate Probability Analysis

The steps involved in univariate extreme value analysis are well established and extensively documented in the literature (e.g., Coles [39], Dutfoy [40], Gomes [41]). Parameters of the selected extreme value distributions are estimated using the maximum likelihood method (ML method). The goodness of fit can be assessed through the Kolmogorov–Smirnov (KS test), the root mean square error (RMSE) between empirical and fitted distributions, and the Akaike Information Criterion (AIC).

Copulas are commonly categorized according to their mathematical construction, which determines the types of dependence they can represent. Well-known classes are the elliptical copulas, the extreme value copulas, the Archimedean copulas and vine copulas. In multivariate probability analysis of hydrodynamic variables, Archimedean copulas, first proposed by Kimberling [42], are of particular interest. Archimedean copulas are specifically designed for bivariate dependence structures and offer a wide range of models that can map different types of dependencies, including asymmetry and tail dependence. The parameterization, interpretation, and estimation of these models are straightforward, making them particularly suitable for modeling two parameters [43]. Vine copulas, on the other hand, have been designed for high-dimensional dependence structures by combining a large number of bivariate copulas in a tree structure. However, when reduced to only two parameters, a vine copula is reduced to a single bivariate copula, so there is no advantage over a directly selected Archimedean copula. The additional complexity and flexibility of vine copulas only become apparent when there are more than two variables [44].

The adaptability results from the construction of the copulas via the so-called generator function φ. The class of suitable generator functions is large, which means that many dependence structures can be generated. In this study, the Archimedean copulas are considered for modeling the dependence between S_W and H_1/3.

An important step for the adaptation of copula models is the extraction of the bivariate sample. Extreme combinations of the parameters S_W and H_1/3 are particularly relevant for the design of coastal protection structures. A generally valid definition of extreme events is difficult to establish in coastal engineering. As in the case of breakwaters, for example, extreme events do not always lead to the greatest loads [2]. The extraction of the sample therefore depends on the underlying question and must represent the local hydrodynamic conditions in the case of the design or safety review of coastal protection structures.

Bender [45] has summarized common methods in the literature shown in Figure 4, for bivariate sampling: the componentized block maxima model according to Tawn [46], the threshold model according to Resnick [47], the point-process model [48] and the conditional extreme events according to Heffernan and Tawn [49]. The sampling methods mentioned are not suitable for this study, as they do not necessarily account for the physical and temporal coupling between S_W and H_1/3 and all require the specification of thresholds, which must be set subjectively. Since different coastal sections with varying hydrodynamic conditions are analyzed, a common threshold is not meaningful. Therefore, a sampling approach was developed based on the extended block maxima method of Smith [50].

In this approach, storm surge events are first detected, and their peak value together with the corresponding significant wave height are extracted. For each coastal section, the r largest S_W–H_1/3 combinations from each temporal block (1 year) are then added to the bivariate sample. To ensure a stable dependence between S_W and H_1/3, parameter combinations are successively added to the bivariate sample for each coastal section. Starting from the highest annual maxima of S_W together with their corresponding H_1/3 values, additional S_W–H_1/3 combinations are included step by step until the dependence measure, expressed by Kendall’s τ, stabilizes. The corresponding number of combinations added per year is denoted as r. The parameter r thus represents the number of annual extreme S_W–H_1/3 combinations contributing to the final sample. Kendall’s τ is used as a diagnostic tool to assess the statistical stability of the dependence structure. By allowing more than one parameter combination per block, this approach provides flexibility across coastal sections and reduces bias associated with relying on single extreme values, thereby ensuring a stable dependence structure in the resulting sample.

As with univariate distributions, various adjustment tests can be performed to identify suitable copulas. These tests can be divided into formal and graphical tests. A detailed overview of the tests can be found in Fermanian [51], Berg [52], Genest [53], and Fermanian [54], among others. The Cramér-von-Mises test (CvM test), the RMSE and the AIC are used to validate different copula models against the samples. In addition, an empirical tail dependence analysis following the approach of Frahm et al. [55] is performed to examine the strength and stability of the upper-tail dependence between S_W and H_1/3. This analysis quantifies whether extreme storm surge peaks are systematically associated with extreme significant wave heights.

3.4.2. Comparison of the Univariate and Bivariate Design Parameters Determined Using a Design Example

Since dikes are a common coastal protection measure in MP (Section 2), the DWL and DWC determined in the coastal sections are applied in a dike design example to assess wave run-up and to compare the results of the univariate and bivariate probability analyses. The wave run-up height (R_u2%) on dikes is defined as the distance between the DWL and the highest point of the wave run-up and is computed using the methods of the EurOtop Manual [56] as

$${\displaystyle \frac{{\mathrm{R}}_{\mathrm{u}2\mathrm{\%}}}{{\mathrm{H}}_{1/3}}}=1.65·{\gamma }_{\mathrm{b}}·{\gamma }_{\mathrm{f}}·{\gamma }_{\beta }·{\xi }_{\mathrm{m}-1,0}$$

(4)

where ξ_m−1,0 is the surf similarity parameter (breaker parameter or Iribarren number), γ_b is the influence factor for a berm, γ_f is the influence factor for roughness elements on a slope and γ_β is the influence factor for oblique wave attack. Figure 5 shows the input parameters of the method. The aim of the design is to dimension the dike so that the wave run-up height on the seaward slope does not exceed the crest elevation.

The following assumptions are made: The dike toe and the foreshore are at mean water level. Both are covered with low-roughness material (e.g., concrete or short grass). Thus, the foreshore and seaward slope have no significant wave-damping effect. The dike has a 1:5 slope without berms. The design waves approach the dike perpendicularly.

The input values for determining the wave run-up heights must be available at the foot of the dike. The wave period used in the run-up calculation is derived from the data used in this study, based on a linear regression with significant wave height, applied separately for each coastal section. Since DWC refers to deep-water conditions, it is transformed to the location of the dike toe using the SWAN model (v40.72AB) [57]. S_W does not appear explicitly in Equation (4) but is indirectly accounted for as a boundary condition in the wave transformation performed with the SWAN model. Thus, S_W influences the wave characteristics at the dike toe, which serve as input for the run-up calculation. The resulting wave run-up heights are then computed for both univariate and bivariate design parameters and compared at locations along the coast of MP.

4. Results

4.1. Analysis of the Hydrodynamic Parameters

4.1.1. Water Levels

For the systematic analysis of the water levels, the statistical key figures of water levels in the coastal sections are first determined and visualized in box plot diagrams (see Figure 6). All analyses presented in this section are based on a consistent 38-year period, as described in Section 3.1. The water levels of the sections Warnemünde and Zingst as well as Varnkevitz and Göhren are evaluated together, as they are assigned the same gauge (see Table 3).

The statistical key figures of the water levels along the Baltic Sea coast of MP are presented in Figure 6 in order from west to east. Boltenhagen is the westernmost and Koserow the easternmost section of the coast (see Figure 2). The median water levels rise from west to east. In Boltenhagen, the median is 0.03 m above NHN and increases eastward. NHN (Normalhöhennull) is the official German vertical reference system for elevations above mean sea level. The highest median, of 0.1 m above NHN, occurs in Koserow. This gradient is primarily influenced by dominant westerly winds, which push water eastward (see Section 3), but is also affected by geomorphological conditions and coastal configuration.

A contrary trend can be identified for the extreme water levels. During the measurement period, the highest water levels were measured in Boltenhagen at up to 2 m above NHN. In comparison, lower peak values were measured in the Varnkevitz/Göhren and Koserow sections, which are around 1.5 m above NHN. The variability of the extreme water levels can be explained by the interaction of geomorphological factors [28], in addition to the influences on storm surge development mentioned in Section 2. The most important geomorphological factors include the slope of the seabed, the shape and exposure of the coast and the fetch length.

StALU and other authorities classify storm surges according to their peak water level.

Table 5. Classification of storm surges [1].

Storm Surge Category	Peak Value (m Above NHN)
elevated water levels	0.50 m–0.99 m
light storm surge	1.00 m–1.24 m
medium storm surge	1.25 m–1.49 m
heavy storm surge	1.50 m–1.99 m

shows the StALU classification of storm surges. Figure 7 shows the frequency analysis of elevated water levels and storm surges in the coastal sections investigated. The number of storm surge events is lowest in the Varnkevitz/Göhren section due to its island location and the absence of a significant wind setup effect in bays. Elevated water levels occur with similar frequency in the other sections. Light storm surges occur most frequently in the Boltenhagen and Warnemünde/Zingst sections, while medium storm surges are mainly observed in Boltenhagen and Koserow. The variability can be attributed to the interaction of geomorphological factors and meteorological conditions.

The mean storm surge duration shown in Figure 8 indicates that surges in the western sections are generally shorter than those in the eastern sections. In bay-like sections, such as Boltenhagen, wind setup is the dominant factor controlling extreme water levels, and the same may apply to Warnemünde during prolonged storms. In contrast, wind setup is less dominant in the eastern sections, where basin filling and seiche oscillations, which have greater inertia, are likely to contribute to the longer observed surge durations.

4.1.2. Significant Wave Heights

After analyzing the water levels in the coastal sections, H_1/3 values are now examined, which were modeled using the WWC method (see Section 3.1). Figure 9 shows that the lowest median of H_1/3 was modeled in the Boltenhagen section. This section has a low inclination of the seabed, which means that comparatively less energy is transported into the coastal area by waves [28]. The median H_1/3 values in the Warnemünde, Zingst, Göhren and Koserow sections are comparable—they range between 0.1 m and 0.17 m. In Varnkevitz, the median is higher at 0.27 m, which suggests that the waves here are comparatively higher on average. The extreme H_1/3 values vary between the sections. The highest H_1/3 is modeled in Boltenhagen at approx. 2 m. In Warnemünde, Varnkevitz and Koserow, the highest H_1/3 values reach a height of 3 m to 4 m. In the Göhren section, the highest H_1/3 is modeled at approx. 5 m.

Figure 10 summarizes and compares the statistical metrics of investigated hydrodynamic parameters in the coastal sections: mean S_W, mean H_1/3 at S_W, mean duration, and frequency of storm surge events. The coordinates of the points correspond to the mean S_W above the 0.9 quantile (x-axis) and the corresponding mean H_1/3 (y-axis). The frequency of storm surge events is represented by the diameter of the inner points, while the diameter of the outer circles represents the mean duration. The results clearly show that the hydrodynamic parameters show considerable spatial variation along the coast, indicating that DWL and DWC cannot be represented by a single model, but require separate models for each coastal section.

4.2. Analysis of the Temporal Dependence of Peak Value and Maximum Significant Wave Height

Figure 11 shows the relationship between S_W and the factor d (see Table 4). Blue dots indicate a positive factor d, yellow dots a factor d equal to zero and orange dots a negative factor d. The sections Boltenhagen, Warnemünde, Zingst and Varnkevitz show similar characteristics. Only in 18% to 21% of cases does S_W reach its maximum before H_1/3max. S_W and H_1/3max rarely occur simultaneously, only in 4% to 8% of cases.

In most cases (>70%), H_1/3max occurs before S_W. In the sections with an east-facing coastline (Göhren and Koserow), the relative proportion of events in which H_1/3max occurs after S_W is higher than in the other coastal sections, at 38% and 35%, respectively. Simultaneous occurrence of the maxima is also rare in these sections.

Figure 11 also shows that the positive and negative factors approach 0 as the peak value increases. This indicates that with increasing S_W, the relative time intervals between the two maxima become smaller, while H_1/3max still tends to occur before S_W in most cases. Events with d = 0 occur nearly uniformly across all peak values. At low S_W, the time dependence between the maxima is comparatively low. This can be explained by the different factors influencing the development of storm surges. While the waves are mainly generated by the wind, the seiche oscillations and the base filling level, among other things, influence the formation of storm surges (see Section 2).

4.3. Determination of the Univariate and Bivariate Parameters

4.3.1. Sampling Method

To identify a suitable sampling range that ensures stable dependencies between S_W and the corresponding H_1/3, the samples were generated using the sampling approach described in Section 3.4.1, applying different values of r annual maxima per year. Figure 12 shows the variation of Kendall’s τ as a function of r for the six coastal sections. At low r-values (1–6), Kendall’s τ fluctuates considerably, indicating unstable dependencies due to limited sample sizes and inconsistent event representation. With increasing r, the correlations gradually converge. To reduce subjectivity in the selection of r, the stability of the dependence structure was assessed based on the change in Kendall’s τ between consecutive values, defined as

$$\Delta \tau \left(\mathrm{r}\right)=\left|\tau \left(\mathrm{r}\right)-\tau \left(\mathrm{r}-1\right)\right|$$

(5)

Stability was assumed when the maximum change across all coastal sections remained consistently below a threshold of ε = 0.04 for at least two consecutive values of r.

It should be noted that the choice of the threshold ε is to some extent subjective. However, the selected value is consistent with the observed magnitude of Δτ (≈ 0.02–0.03 in the stable range) and ensures that the stabilization of the dependence structure is identified without being overly sensitive to minor fluctuations. The results show a reduction in Δτ up to r ≈ 7, after which the changes remain small (Δτ ≈ 0.02–0.03), indicating that the dependence structure has reached an approximately stable plateau. Therefore, r = 7 was selected as the smallest value at which the dependence can be considered stable. This interpretation is also supported by the visual assessment of Figure 12, where the curves clearly flatten from r ≈ 7 onward.

Based on this selection, for each year, the seven highest S_W values together with their corresponding H_1/3 were included in the bivariate sample, ensuring that the resulting dataset provides a balance between statistical robustness and the preservation of physically meaningful S_W–H_1/3 combinations. The samples for the coastal sections are shown in Figure 13. Events with S_W < 0.5 m above NHN were excluded from the dataset, as they do not represent extreme events with high load potential for coastal protection structures. The analysis is based on absolute water levels rather than isolated storm surge components. Consequently, negative anomalies (e.g., related to inverse barometer effects) are not represented in the selected extreme event sample. Additionally, physically implausible parameter combinations were removed.

Although Figure 13 indicates an overall increasing trend between water level and significant wave height, the relationship is not strictly linear and shows considerable scatter. The dependence is only moderate (τ ≈ 0.23–0.34), and the variability of the significant wave height increases with higher water levels. For design applications, the joint probability of extreme events is more relevant than the average trend, which motivates the use of copula-based approaches.

4.3.2. Determination of the Univariate Marginal Distributions

To identify suitable marginal distributions for the stochastic modeling of S_W and H_1/3, a univariate goodness-of-fit analysis was performed. Several candidate distributions were assessed—including Normal, Weibull, Generalized Extreme Value (GEV), Logistic, and Lognormal distributions—based on the KS test, the RMSE, and the AIC. Figure 14 presents the results for S_W. Both Lognormal and GEV distributions demonstrated the best overall performance. The KS test indicates that the Lognormal distribution yields p-values above the significance level α = 0.05 for most coastal sections, suggesting no significant deviation from the empirical data.

This finding is supported by low RMSE and AIC values, which confirm that the Lognormal model provides a consistent representation of the upper tail and extreme water levels. The GEV distribution performs comparably but shows slightly higher residuals in some cases, while the Logistic distribution tends to underestimate high water levels. Consequently, the Lognormal distribution was selected as the marginal model for S_W. The estimated parameters of the Lognormal distribution for all coastal sections are summarized in

Table 6. Estimated parameters for adjusting the LogN distribution to the water levels in the coastal sections.

Section	μ	σ
Boltenhagen	−0.1750	0.3060
Warnemünde	−0.2469	0.2873
Zingst	−0.2469	0.2873
Varnkevitz	−0.3714	0.2869
Göhren	−0.3671	0.2887
Koserow	−0.1792	0.3184

.

Figure 15 shows the results for H_1/3. Among the tested models, the GEV distribution exhibits the most consistent behavior across all coastal sections, with the highest KS p-values and the lowest RMSE and AIC values. These results indicate that the GEV distribution adequately represents the statistical characteristics of H_1/3, particularly in the upper range of the data. The Normal and Logistic distributions, in contrast, underestimate the tail behavior, while the Weibull distribution shows higher deviations for extreme events. Therefore, the GEV distribution was selected as the marginal model for H_1/3. The corresponding parameter estimates for all coastal sections are provided in

Table 7. Estimated parameters for adjusting the GEV distribution to the wave heights in the coastal sections.

Section	γ	μ	σ
Boltenhagen	−0.0714	0.3157	0.5000
Warnemünde	−0.0133	0.3497	0.6526
Zingst	−0.0315	0.4228	0.6868
Varnkevitz	−0.1355	0.4911	1.0287
Göhren	0.04448	0.5314	0.6907
Koserow	0.05527	0.3464	0.5600

.

It should be noted that Figure 14 and Figure 15 only display the probability distributions that successfully passed the KS test (p > 0.05) for each coastal section. Distributions that did not meet this criterion were excluded from graphical representation to ensure that only statistically valid models are compared. Overall, the combination of a Lognormal distribution for S_W and a GEV distribution for H_1/3 provides a statistically consistent and physically meaningful description of the univariate extremes. These marginals form the basis for the subsequent bivariate dependence modeling described in Section 4.3.3.

4.3.3. Determination of the Bivariate Joint Distributions

To describe the dependence structure between S_W and H_1/3, several Archimedean copulas were tested, including Clayton, Frank, Gumbel, and Normal copulas. The copula parameters are estimated using the ML method. Figure 16 shows the resulting goodness-of-fit evaluation for all coastal sections based on the CvM test, the RMSE, and the AIC.

Among the tested families, the Frank copula provides the most consistent and statistically robust fit across all coastal sections. Across individual coastal sections, differences in copula performance are relatively small, indicating similar dependence structures between S_W and H_1/3 along the coast. It shows the lowest RMSE values and the highest CvM test p-values in most cases, indicating that the Frank copula adequately captures the central dependence structure between S_W and H_1/3 without overemphasizing either tail. The Gumbel copula performs comparably but tends to overrepresent upper-tail dependence, while the Clayton copula overestimates the lower-tail behavior. The Normal copula generally shows weaker performance, particularly for extreme events. The Göhren section shows reduced fit performance, with higher RMSE values and lower p-values, likely caused by increased local variability or data-related uncertainties. Overall, the Frank copula was selected as the most suitable dependence model for the subsequent bivariate extreme value analysis, as it provides a statistically consistent and physically plausible representation of the dependence between S_W and H_1/3. The estimated copula parameters for all sections are summarized in

Table 8. Estimated parameters for the Clayton, Frank, Gumbel, and Normal copulas in the coastal sections.

IMK Location	Frank	Clayton	Gumbel	Normal
Boltenhagen	1.018	3.354	1.509	0.505
Warnemünde	0.797	2.749	1.399	0.433
Zingst	0.941	3.146	1.471	0.482
Varnkevitz	0.611	2.206	1.306	0.359
Göhren	0.618	2.225	1.309	0.362
Koserow	0.971	3.227	1.485	0.491

.

To evaluate the co-occurrence of extreme S_W and H_1/3, the empirical upper-tail dependence λ^U(u) was estimated following the approach of Frahm et al. [55]. This method quantifies the conditional probability that one variable exceeds a high quantile given that the other exceeds the same quantile, thereby measuring the strength of dependence between simultaneous extremes. The analysis was applied to the observed bivariate samples of S_W and H_1/3 from all coastal sections (see Figure 13). Figure 17 presents the variation of λ^U as a function of the quantile threshold u, illustrating how the strength of dependence evolves toward the upper tail of the joint distribution.

The empirical analysis of the upper-tail dependence λ^U(u) shows moderate positive dependence between S_W and H_1/3 across all coastal sections, with values mostly between 0.2 and 0.5. The Boltenhagen section exhibits the strongest and most stable dependence, whereas the eastern sections (Göhren, Koserow) show slightly weaker and more variable dependence. For u > 0.98, λ^U becomes unstable due to limited sample sizes.

An infinite number of parameter combinations can be generated using the parameterized copula model, for which either the non-exceedance probability or the corresponding return period can be determined. This allows the identification of all combinations S_W and H_1/3 associated with a given probability of occurrence. A graphical representation of these combinations is provided by isolines, which connect all pairs of S_W and H_1/3 values corresponding to the same return period. Figure 18 illustrates the isolines for R = 5 a, R = 25 a, and R = 100 a for all coastal sections. Each isoline theoretically contains an infinite number of possible parameter combinations, and there is no inherent criterion for selecting a single design event along the curve. Therefore, a direct comparison of events between different coastal sections is challenging. To support a consistent interpretation, the black points (P_maxD) indicate the locations of maximum probability density along each isoline, following the approach of Salvadori et al. [58].

The shape of the isolines in Figure 18 further supports the findings of the dependence analysis. In all coastal sections, the isolines exhibit a pronounced curvature toward the upper right corner, indicating a positive dependence between S_W and H_1/3. The curvature is strongest in Boltenhagen, reflecting a stable coupling between high S_W and H_1/3. In the central sections (Warnemünde and Zingst), the dependence is moderately expressed, while in the eastern sections (Göhren and Koserow) the isolines flatten slightly, suggesting weaker joint extremes and a higher degree of variability. These spatial patterns correspond well with the tail-dependence results shown in Figure 17.

With the adapted univariate and bivariate probability models, it is now possible to compute and compare S_W and H_1/3 for defined R in each coastal section. The red lines in Figure 19 represent the univariate results, while the black lines correspond to the bivariate combinations derived from the copula models. Water levels are indicated by dots and wave heights by squares. The curves of the univariate and bivariate determined S_W show only minor differences in magnitude, and from R = 50 a onward, both curves progress almost linearly. In all sections, the univariate determined S_W values are higher than the corresponding bivariate ones. In contrast, the univariate determined H_1/3 show both linear and exponential tendencies across the coastal sections. In Boltenhagen, the increase in H_1/3 remains almost linear, while in Göhren a pronounced exponential rise is observed. At the points P_maxD, the H_1/3 values exhibit no significant increase beyond R = 50 a, indicating an approximately linear progression. The deviations between univariate and bivariate determined H_1/3 are considerably larger than those of the water levels.

Accounting for the dependencies between S_W and H_1/3 using the copula models leads to systematically lower parameter magnitudes for the same return periods across all coastal sections. This suggests that univariate analyses tend to overestimate design values, while the bivariate approach provides more realistic estimates by incorporating the physical dependence between S_W and H_1/3.

Section 4 presented the results of the data analysis, univariate and bivariate probabilistic modeling of S_W and H_1/3 along the Baltic Sea coast of MP. The findings revealed distinct spatial variations in the hydrodynamic parameters and a consistent positive dependence between S_W and H_1/3 across all coastal sections. Univariate analyses identified the Lognormal distribution as the most suitable model for S_W and the GEV distribution for H_1/3. The bivariate dependence was successfully represented by the Frank copula, which captured the moderate upper-tail dependence and provided statistically robust fits. The comparison of univariate and bivariate design values demonstrated that neglecting the dependence between S_W and H_1/3 leads to an overestimation of design parameters, particularly for the wave height.

Overall, the combined use of univariate and copula-based bivariate models provided a reliable and physically consistent framework for determining joint design parameters. These results form the basis for the subsequent design example in Section 5, where the derived design parameters are applied to calculate the wave run-up on coastal dikes.

5. Design Example for Determining the Wave Run-Up on Dikes

For simplification and comparability, the influencing factors γ_b, γ_f and γ_β are set to 1 in all sections when determining the wave run-up heights. As a result, berms, surface roughness, and wave direction have no influence on the computed run-up height. This simplification is deliberately chosen to isolate and demonstrate the effect of univariate and bivariate design parameters on wave run-up and crest elevation. It should be noted that wave run-up depends not only on hydrodynamic conditions (water level, wave height, and wave direction) but also on dike geometry, vegetation, and foreshore characteristics.

First, the univariate design parameters determined for R = 10 a, 25 a, 50 a, 100 a, 150 a and 200 a are transformed to the location of the dike toe using the numerical sea-state model SWAN [57], and the corresponding wave run-up heights are computed. When applying the bivariate design parameters, the procedure must be modified, as it is initially unknown which combination of water level and wave height for a given R produces the maximum run-up. Therefore, the relevant parameters are first modeled at the dike toe for a sufficiently large number of combinations of R within each coastal section. The wave run-up height is then computed for each combination, and in the bivariate case, the combination producing the highest run-up for the respective R is selected for comparison. Figure 20 compares the resulting univariate and bivariate wave run-up heights.

The comparison shows that the run-up heights are generally higher in the univariate case than in the bivariate case. This is attributed to the DWLs, which are higher when derived from univariate probability analysis. Higher water levels result in increased wave energy reaching the dike, leading to greater run-up. In both cases, DWCs computed in deep water show no significant influence on run-up, as the waves are transformed across the dike foreland. The required crest elevation of the dike is then computed as the sum of S_W and the wave run-up height. Figure 21 presents the results. The light-blue and yellow bars represent the DWL for each return period, while the dark-blue and orange bars correspond to the required crest elevation in the univariate and bivariate cases, respectively. A freeboard of 0.5 m, typically applied in MP [1], was not considered in this computation. The bivariate analysis results in lower crest elevations for all coastal sections, although the magnitude of the difference varies. While Boltenhagen shows nearly identical crest levels for both approaches, substantial differences appear in Koserow.

6. Discussion

6.1. Data Basis and Limitations

Measured and modeled water level and wave data are available for the study area. However, the modeled datasets were not used, as differences in atmospheric forcing and model configurations result in a lack of physical consistency between water levels and sea state conditions. Instead, observational data from the IMK network, DWD and WSA were employed, providing consistent measurements of water levels, wave parameters, and wind data at six coastal locations.

A key limitation of the IMK dataset is the relatively short record length of 24 years, which includes temporal gaps and does not cover major historical storm surge events prior to 1998. This restricts the robustness of extreme value analyses and the derivation of hydrodynamic design parameters. To partially address this limitation, the H_1/3 time series were extended to 38 years using the WWC method based on DWD wind data.

While the WWC method provides a pragmatic approach to extend the dataset, it introduces additional sources of uncertainty. It does not fully capture the variability of wave generation under identical wind conditions, especially during extreme events [59]. Moreover, the transferability of wind data to realistic sea state conditions is limited in coastal areas, where the correlation between wind and waves decreases due to local processes such as land–sea interactions, atmospheric turbulence, and bathymetric influences. These effects can introduce additional scatter and pseudo-correlations in the data. Furthermore, swell components originating outside the local domain may only be weakly correlated with local wind conditions, increasing the uncertainty of reconstructed wave heights [60,61].

These uncertainties directly affect the copula-based dependence modeling. Inconsistent or noisy data can distort the estimated dependence structure between S_W and H_1/3, potentially leading to both over- and underestimation of joint extremes. Local processes and unresolved variability may artificially strengthen or weaken apparent dependence, thereby reducing the stability and interpretability of the copula fit. In addition, the presence of data uncertainties complicates the identification of a unique best-fitting copula model, as different models may yield similar goodness-of-fit metrics (see Section 4.3.3), limiting the diagnostic value of conventional performance measures.

Despite these limitations, the overall conclusions of the study are considered robust. The WWC method is primarily used to improve the statistical basis of the analysis rather than to reproduce exact wave conditions. Furthermore, the focus is on statistical dependencies and relative patterns between S_W and H_1/3, which are generally less sensitive to moderate uncertainties in individual wave estimates.

For a more reliable application of copula-based approaches, physically consistent hydrodynamic model datasets are required that jointly represent water levels and sea states. Such datasets, driven by long-term atmospheric reanalysis data (e.g., COSMO or ERA5), would enable the inclusion of major historical storm surge events and provide a more robust basis for the estimation of joint extremes [62,63]. In general, internally consistent time series with a sufficient number of extreme events are essential to ensure a reliable characterization of the upper tail of the joint distribution. While a quantitative uncertainty analysis was not performed, the impact of the WWC-based extension on the resulting design parameters can be qualitatively assessed. The main effect is expected in the magnitude and variability of H_1/3, which directly influences the derived DWC and, consequently, the wave run-up and crest elevations.

In contrast, the dependence structure between S_W and H_1/3, which forms the basis of the copula modeling, is considered less sensitive to moderate uncertainties in individual wave estimates, as it is governed by relative patterns rather than absolute values. As a result, uncertainties in the WWC-derived wave data may lead to deviations in the absolute design parameters, particularly for extreme return periods, whereas the overall conclusions regarding the dependence structure and the relative differences between univariate and bivariate approaches are expected to remain robust. A quantitative assessment of these effects, for example through sensitivity analyses or ensemble-based approaches, is identified as an important topic for future research.

6.2. Analysis of the Hydrodynamic Parameters and the Temporal Dependence of S_W and H_1/3max

The hydrodynamic conditions along the Baltic Sea coast of MP show pronounced spatial variability. These differences are primarily related to variations in storm surge generation mechanisms, coastal morphology, and exposure to prevailing wind directions. As a result, the application of a single probabilistic model for the entire coastline—whether univariate or bivariate—is not appropriate, as it would introduce significant uncertainty in the derivation of DWL and DWC. Instead, location-specific modeling approaches are required.

The temporal relationship between S_W and H_1/3max depends on event intensity. For low S_W, both parameters exhibit weak coupling, with H_1/3max occurring before and after S_W. With increasing S_W, the relative time lag decreases, although H_1/3max still tends to precede S_W in most cases, while simultaneous occurrence remains rare. This behavior is physically plausible, as storm surge water levels respond more slowly than waves due to the inertia of the water body.

Previous studies confirm that the temporal relationship between S_W and H_1/3max is highly variable and site dependent. Kudryavtseva [8] reported weak correlations for the southern Baltic Sea, whereas stronger dependencies and more frequent simultaneous occurrences were observed in northern regions such as the Gulf of Bothnia. Johansson et al. [9] showed for the Gulf of Finland that S_W and H_1/3max do not necessarily occur simultaneously, although a moderate positive correlation (τ ≈ 0.2) was identified. Comparable findings have also been reported outside the Baltic Sea, for example by Bricheno et al. [64], who observed event-dependent time lags between wave and storm surge maxima in the Irish Sea. These results indicate that the temporal coupling between S_W and H_1/3max cannot be generalized and is strongly influenced by regional hydrodynamic conditions.

The interpretation of temporal dependencies is further constrained by the data base. Both S_W and H_1/3 are available at hourly resolution, and the statistical extension of the H_1/3 time series introduces a loss of temporal detail. As a result, short-term dynamics and exact time lags may not be fully resolved, which may affect the accuracy of the relationships derived. Higher resolution and physically consistent datasets would be required for a more reliable assessment.

Overall, the results emphasize the need to account for spatial variability and temporal mismatch between S_W and H_1/3max. Neglecting these effects may lead to biased estimates of hydrodynamic loads and thus affect the reliability of coastal protection design.

6.3. Univariate and Bivariate Probability Analyses

In literature and coastal engineering practice, predefined threshold values are commonly used to identify extreme events (see Section 4.1.1, Table 5). These thresholds are typically defined for practical reasons and applied uniformly along the coast of MP. However, they do not reflect the spatial variability of hydrodynamic conditions observed in the study area. Approaches such as those presented by Bender [45] rely on the selection of one or more threshold values, which—although often guided by experience—inevitably involve a degree of subjectivity. Given the pronounced spatial variability, the use of a single threshold may introduce systematic biases or excessive variance in the extracted samples.

The sampling approach developed in this study addresses this limitation by replacing subjective threshold selection with a data-driven criterion. Instead of defining a fixed threshold, the sampling density r was determined based on the stability of the dependence between S_W and H_1/3, expressed through Kendall’s τ. This ensures statistically stable samples and an appropriate representation of the dependence structure for copula-based modeling of joint extremes. Nevertheless, the uncertainties in the data base discussed in Section 6.1 also affect this approach and should be considered when interpreting the results. The selection of the stability threshold ε introduces a degree of subjectivity, as different values may lead to slightly different sampling densities. However, the chosen value is consistent with the observed stabilization behavior of Kendall’s τ and results in a robust and physically plausible sample size.

The univariate analysis identified the Lognormal distribution as the most suitable model for S_W and the GEV distribution for H_1/3. Both distributions provided statistically consistent fits across all coastal sections. The GEV distribution also performed satisfactorily for S_W, indicating that both distributions are generally appropriate for describing extreme water levels in the study area. For H_1/3, the GEV distribution showed the most consistent performance, although local deviations were observed, particularly in the Göhren section, likely due to limited sample size and the influence of WWC-based data.

Despite the overall good statistical performance, the reliability of the fitted distributions is constrained by the limited length of the available time series. With a total duration of 38 years, rare extreme events are only sparsely represented. The absence of major historical events, such as the 1872 [65] storm surge, limits the characterization of the upper tails and affects extrapolation to long return periods (e.g., R > 100 a), which may lead to over- or underestimation of design parameters.

The bivariate dependence analysis confirmed a consistent positive correlation between S_W and H_1/3 across all coastal sections. Among the tested Archimedean copulas, the Frank copula provided the most stable and statistically robust fit, characterized by low RMSE values and high CvM test p-values. Its symmetric structure enables a balanced representation of dependence without emphasizing tail asymmetry, which is consistent with the observed data behavior. However, differences between the tested copulas were generally small (see Figure 16, Section 4.3.3), which can be attributed to the limited number of extreme S_W–H_1/3 combinations.

This interpretation is supported by the empirical upper-tail dependence analysis. The estimated λ^U(u) values (Figure 17) indicate moderate positive dependence across all coastal sections, typically ranging between 0.2 and 0.5. Spatial differences are evident: Boltenhagen exhibits the strongest and most stable coupling, which is also reflected in the lower variability of extreme S_W–H_1/3 combinations (Figure 13). A plausible explanation is the geographical setting within Lübeck Bay, where bay-induced water level amplification enhances the coupling between water level and wave height. In contrast, eastern coastal sections are more influenced by basin-scale processes such as basin filling and seiche oscillations (see Section 2), which reduce the direct correlation between S_W and H_1/3.

Future research should evaluate the robustness of the selected marginal distributions and copula models using longer and physically consistent datasets that better represent extreme events. The proposed sampling approach should be validated for other coastal regions and hydrodynamic conditions. In addition, the sensitivity of copula selection to data uncertainty and sample size should be systematically assessed. The development of physically consistent hydrodynamic modeling frameworks, as outlined in Section 6.1, is essential to provide coherent long-term datasets, improve the estimation of joint extremes, and reduce uncertainties associated with the WWC method.

6.4. Design Example

For the application of the univariate and bivariate design parameters, wave run-up on a conceptual dike was calculated. Deep-water wave conditions were transformed to the dike toe using the SWAN model, and wave run-up was computed with the EurOtop approach. Since the governing combination of S_W and H_1/3 for a given return period is not known a priori, multiple parameter combinations were evaluated for each return period. The results show that bivariate design parameters consistently lead to lower wave run-up heights and crest elevations compared to the univariate approach across all coastal sections. This reflects the consideration of the dependence between S_W and H_1/3. In line with the temporal analysis presented in Section 6.2, the maxima of S_W and H_1/3 rarely occur simultaneously. Consequently, univariate approaches tend to overestimate hydrodynamic loads and may result in conservative or overdesigned coastal protection structures. The magnitude of these differences varies between locations, indicating the influence of spatially variable dependence structures.

A key limitation of the copula-based approach is that, for a given return period, an infinite number of S_W–H_1/3 combinations exist along the same isoline of joint probability. The selection criterion proposed by Salvadori et al. [58], based on maximum probability density, is mathematically consistent but not necessarily representative of the design-relevant condition. The governing parameter combination depends on the structure type, dominant load mechanism, and design objective. Therefore, engineering judgment remains essential when applying bivariate methods in practice. In addition, evaluating a large number of parameter combinations may become computationally demanding in practical applications. Although wave heights typically peak before maximum water levels, simultaneous extremes may still occur. Given the data-related uncertainties, such compound events should be considered in design to ensure structural safety. Overall, the use of bivariate design parameters enables a more physically consistent and potentially more efficient design of coastal protection structures.

The applicability of the results is limited by the available data, particularly the absence of major historical storm surge events and uncertainties associated with the WWC-based estimation of H_1/3. With improved, physically consistent datasets—such as those derived from integrated hydrodynamic modeling frameworks (Section 6.1)—copula-based approaches have the potential to replace univariate methods for determining hydrodynamic design parameters in coastal engineering.

Future research should focus on validating the approach using extended datasets and assessing the sensitivity of design results to the selection of dependence models and sampling strategies. Furthermore, multivariate approaches should be extended beyond two variables, and alternative methods such as the Metastatistical Extreme Value Distribution [66] should be explored for improved representation of rare events.

7. Conclusions

This study investigated the determination of hydrodynamic design parameters, namely S_W and H_1/3, for coastal protection structures along the Baltic Sea coast of MP using copula-based bivariate probability models. The motivation was to overcome limitations of the currently applied univariate approach, which neglects the dependence between peak water levels and significant wave heights during storm surge events.

Hydrodynamic conditions were analyzed at six coastal sections, including the temporal relationship between both variables. A dependence-guided, physically motivated sampling approach was developed to extract extreme and statistically consistent S_W–H_1/3 combinations for copula modeling.

The results demonstrate pronounced spatial variability along the coast, indicating that a single probabilistic model is not sufficient to represent all locations. The temporal analysis further showed that H_1/3 typically reaches its maximum before S_W, while simultaneous occurrence is rare.

Univariate analysis identified the Lognormal distribution for S_W and the GEV distribution for H_1/3 as suitable marginal models. The dependence structure was best represented by the Frank copula, indicating a moderate but consistent positive dependence across all coastal sections.

The comparison of univariate and bivariate design values revealed that the univariate approach tends to overestimate hydrodynamic design parameters, particularly for H_1/3, due to neglecting parameter dependence. This was confirmed in the design example, where bivariate design parameters resulted in lower wave run-up heights and crest elevations. Accounting for dependence structures therefore improves the physical realism of design conditions and can contribute to more efficient coastal protection design.

However, the applicability of the results is limited by the available data, including the short record length, the absence of major historical storm surge events, and uncertainties associated with the WWC-based estimation of H_1/3. Therefore, the application of copula-based methods in practice should currently be treated with caution.

Nevertheless, the study demonstrates the potential of multivariate approaches for coastal engineering and introduces a transferable framework for deriving joint design parameters. However, its applicability to other coastal regions may be limited by the specific hydrodynamic and morphological conditions of the southwestern Baltic Sea. Future work should focus on longer and physically consistent datasets to validate the methodology and improve the reliability of multivariate design approaches.

This article is a revised and expanded version of a paper entitled “Determination of the hydrodynamic design parameters water level and wave height using copula models for the design of coastal protection structures on the Baltic Sea of Germany”, which was presented at the International Probabilistic Workshop (10–12 September 2025, University of Rostock, Germany) [67].