Miami-Dade County (south-east Florida) is among the most vulnerable regions to sea level rise in the United States, due to a variety of natural and human factors. The co-occurrence of multiple, often statistically dependent flooding drivers – termed compound events – typically exacerbates impacts compared with their isolated occurrence. Ignoring dependencies between the drivers will potentially lead to underestimation of flood risk and under-design of flood defence structures. In Miami-Dade County water control structures were designed assuming full dependence between rainfall and Ocean-side Water Level (O-sWL), a conservative assumption inducing large safety factors. Here, an analysis of the dependence between the principal flooding drivers over a range of lags at three locations across the county is carried out. A two-dimensional analysis of rainfall and O-sWL showed that the magnitude of the conservative assumption in the original design is highly sensitive to the regional sea level rise projection considered. Finally, the vine copula and Heffernan and Tawn (2004) models are shown to outperform five standard higher-dimensional copulas in capturing the dependence between the principal drivers of compound flooding: rainfall, O-sWL, and groundwater level. The work represents a first step towards the development of a new framework capable of capturing dependencies between different flood drivers that could potentially be incorporated into future Flood Protection Level of Service (FPLOS) assessments for coastal water control structures.

Florida is more vulnerable to sea level rise (SLR) in terms of housing and population relative to local mean high-tide levels than any other state in the country (Strauss et al., 2012). Miami-Dade County, located in the south-east of Florida, is particularly vulnerable due to its gently sloped low-lying topography, densely populated coastal areas, and economic importance (Zhang, 2011). Miami, the county's principal metropolitan area, is consistently ranked among the world's most exposed and vulnerable cities to coastal flooding (e.g. Hallegatte et al., 2013; Kulp and Strauss, 2017). While debate surrounds the region's vertical land motion (Parkinson and Donoghue, 2010), the contribution of SLR to nuisance or tidal flooding (Wdowinski et al., 2016) as well as its role in escalating socio-economic impacts such as climate gentrification is becoming increasingly apparent (Keenan et al., 2018). The future rates of SLR in the region are expected to be greater than the global average due to variations in the Florida Current and Gulf Stream (Southeast Florida Regional Climate Change Compact, 2015). Higher baseline ocean levels allow storm surges to propagate further inland whilst also reducing pressure gradients in rivers hampering efficient drainage; hence, SLR also increases the fluvial flood potential (Schedel et al., 2018).

In low-lying coastal areas flooding arises because of the interplay between metrological, hydrological, and oceanographic drivers including rainfall,
river discharge, groundwater table, storm surge, and waves. In Miami Beach, for instance, Wdowinski et al. (2016) found that most flooding events
between 1998 and 2013 occurred after heavy rain (

Miami-Dade County is underlined by the highly transmissive and porous (predominantly limestone) Biscayne aquifer, which is also the region's main source of potable freshwater (Randazzo and Jones, 1997). The lateral intrusion of saltwater into the unconfined aquifer as a recirculating saltwater wedge is widely acknowledged (Provost et al., 2018). SLR along with an increased likelihood of recurring drought during the winter–spring season, associated with changes in the climate system, enhances the risk of contamination of the water supply (Bloetscher et al., 2011). Furthermore, the county's population is expected to increase by nearly 20 % in the next 20 years (Florida Office of Economic and Demographic Research, 2015), increasing flood exposure and demand on water resources. The South Florida Water Management District (SFWMD) is responsible for managing and protecting the water resources of south Florida. The SFWMD must balance demand for potable water and agricultural and landscape irrigation with flood mitigation, whilst ensuring the water table remains sufficiently high to prevent saltwater intrusion and achieve other ecological objectives (SFWMD, 2016). Their aim is to meet these objectives through the continuous operation of an extensive network of drainage canals, storage areas, pumps, and other control structures. The Biscayne aquifer has a direct hydraulic connection to the natural and man-made surface water bodies, a consequence of its shallow depth and high porosity, and is therefore considered a part of this integrated hydrologic system (Randazzo and Jones, 1997).

In heavily managed urbanized catchments, antecedent groundwater conditions are an essential initial condition for hydraulic–hydrological models for robust flood risk analysis (Hettiarachchi et al., 2019). Rainfall is often employed as a surrogate for river discharge (e.g. Zheng et al., 2013; Wahl et al., 2015; Bevacqua et al., 2020). Physical properties such as the size, gradient, and permeability of a catchment influence the river response to a given rainfall event (Svensson and Jones, 2002; Zheng et al., 2013; Hendry et al., 2019). Verhoest et al. (2010) demonstrated that the return period of a rainfall event may differ significantly from that of the corresponding discharge, depending on the antecedent wetness of a catchment. In south-east Florida, approximately half of the average annual rainfall is lost to evapotranspiration (Bloetscher et al., 2011); hence rainfall is unlikely to constitute a suitable proxy for discharge.

Due to the unusually high connectivity of ground and surface water hydrology, south-east Florida has a high propensity for pluvial flooding. The
concurrence of heavy precipitation and high antecedent soil moisture is the dominant flood-generating mechanism for most catchments without
significant snowmelt (Berghuijs et al., 2016, 2019). Many recent studies (Moftakahri et al., 2017, 2019; Bevaqua et al., 2017; Couasnon et al., 2018,
2019; Paprotny et al., 2018; Ward et al., 2018; Serafin et al., 2019) statistically model river discharge and surge (or coastal water level in the
case of Ganguli and Merz, 2019), or their relevant proxies (Kew et al., 2013), as opposed to rainfall and surge, implicitly accounting for catchment
properties and pre-existing groundwater level (Lamb et al., 2010). Not accounting for groundwater level explicitly, especially in areas like
Miami-Dade County where groundwater levels are highly responsive (and potentially correlated) to rainfall and O-sWL, precludes a robust assessment of
the risk of pluvial flooding. Therefore, in this work, statistical models will be tested for their ability to capture the joint probability
distribution of rainfall, O-sWL (tide

Traditional multivariate probability distributions are often restrictive in terms of the choice of marginal distributions; i.e. all the margins are required to be the same type of distribution. For example, fitting a bivariate Gaussian distribution to extreme tides and corresponding freshwater flows required Loganathan et al. (1987) to assume Gaussian marginal distributions. Copulas allow the dependence and marginal modelling to be carried out independently, providing more flexibility in the choice of marginal distributions than traditional multivariate models (Patton, 2006). Consequently, bivariate copulas have been used extensively in the modelling of compound flooding induced by rainfall and surge (e.g. Wahl et al., 2015) and from discharge in multiple rivers at their confluences (Wang et al., 2009; NCHRP, 2010; Chen et al., 2012; Bender et al., 2016; Peng et al., 2017, 2018; Gilja et al., 2018). Higher-dimensional multivariate parametric copulas are limited in the sense that they assume homogeneity in the type of dependence between each pair of variables (Aas et al., 2009). Pair-copula constructions (PCCs) in contrast take advantage of the rich array of bivariate copulas and overcome this limitation by decomposing higher-dimensional probability density functions (pdf's) into a cascade of bivariate copulas (Bedford and Cooke, 2002). Bevacqua et al. (2017) implemented PCC to model the conditional joint pdf of river discharge and sea levels (given meteorological predictors) to assess compound flood risk in Ravenna, Italy. The method proposed by Heffernan and Tawn (2004; referred to hereafter as HT04) is an alternative to higher-dimensional multivariate parametric copulas requiring no assumptions regarding the type of dependence between variable pairs.

Water control facilities for the Central and South Florida Project (CSFP) authorized by the Flood Control Act (1948) (Pub. L. 80-858, 46 Stat. 925, 1948). were designed by the US Army
Corps of Engineers in the 1950s and 1960s. The project included hydrologic and hydraulic design for canals, many of which terminate in flood–salinity
control structures. The control structures are operated by the SFWMD to maintain the water level to prevent saltwater intrusion and release canal
water to the sea (typically via tidally modulated channels), alleviating potential flooding. The design of the canal saw a design O-sWL, typically
obtained from tide tables, paired with a design storm under the assumption of full dependence; i.e. the bivariate design event associated with a return
period is obtained by pairing the O-sWL and peak rainfall with the corresponding univariate return periods. Groundwater level conditions were
accounted for through the rainfall input. For instance, in the Greater Miami area, it was assumed that the first 0.1 m of rainfall of the design
storm would be used to replenish the groundwater storage. The SFWMD's Flood Protection Level of Service (FPLOS) project is beginning to examine the flood protection existing coastal water control structures afford to urban areas, adopting a more holistic approach compared with their original design. FPLOS uses design storms, which are run through hydrologic models
with initial conditions given by groundwater stages. For coastal structures, the O-sWL represents an additional downstream boundary condition
described by a stage hydrograph. Peak stages in the boundary condition hydrographs are derived using frequency analysis, and hence in FPLOS
assessments rainfall, O-sWL, and groundwater level are assumed to be fully dependent. Consequently, any correlations

The overall aim of the paper is to assess the different drivers of compound flooding in coastal areas of Miami-Dade County. This will be achieved by meeting three objectives. The first objective is to determine whether there is any statistically significant correlation between extreme rainfall, O-sWL, and groundwater level, while accounting for relevant time lags. The second objective is to assess the conservative nature of the original design approach. This includes a bivariate statistical analysis, akin to those in previous studies but also including regional SLR scenarios to assess how long it will take for any safety margin (that is implicitly included by assuming full dependence between drivers) to be exhausted. The third and final objective is to incorporate antecedent catchment conditions into the statistical model and to provide robust estimates of the joint probabilities (using a variety of approaches) of extreme rainfall, O-sWL, and the groundwater table that can potentially be incorporated in future FPLOS assessments.

Miami-Dade is situated in south-east Florida (Fig. 1a). The Everglades Water Conservation Areas comprise the western portion of the county whilst heavy engineered water infrastructure and flood control systems have facilitated agricultural and urban development farther east. Three case study sites, differentiated by the colours in Fig. 1b (named after the structures where O-sWL is measured), were selected to allow an assessment of the variation of the hydrological behaviour with latitude. The study is undertaken using in situ observations with each site defined by a rainfall gauge, stage gauge (to measure the O-sWL), and groundwater well.

Study site location and data completeness.

Rainfall data consist of daily precipitation totals obtained from the National Oceanic and Atmospheric Administration's (NOAA's) National Climatic Data Center's archive of global historical weather and climate data. The rainfall record at Miami International Airport is complete, while the records at Perrine and Miami Beach contain a substantial number of missing values, constituting 22.85 % and 4.80 % of the total time series, respectively. The highly localized nature of individual rainfall events in the region along with the spatial and temporal resolution of rainfall measurements renders the estimation of missing daily rainfall values using neighbouring gauges impractical (Pathak, 2001).

Stage gauges are attached to flood–salinity control structures operated by SFWMD. The stage time series downstream of the relevant structures (here termed O-sWL) were extracted from DBHYDRO (SFWMD's corporate environmental database) and converted to daily maxima. O-sWL refers to the still water level (i.e. the water level discounting waves/wave set-up) that comprises mean sea level, the astronomical tidal component, and non-tidal residual (Pugh, 1987). O-sWL values are given in metres above the National Geodetic Vertical Datum of 1929 (NGVD 29).

Groundwater wells (maintained by the United States Geological Survey) closest to each stage gauge and with record lengths similar to the O-sWL time series were identified and daily maximum water level records extracted from DBHYDRO. An analysis of the distribution of the missing O-sWL and groundwater observations indicated the presence of long gaps in some of the records, prohibiting linear interpolation of the record to infill missing values. However, both the O-sWL and groundwater records showed a high degree of linear correlation with corresponding records at nearby sites. Missing values were therefore imputed through a linear regression of the observations at the location of interest on those at nearby sites (Fig. S1 in the Supplement), starting with the closest site to the location of interest. Any remaining non-consecutive missing values were imputed through linear interpolation (see Figs. S2–S6).

A fundamental assumption of the standard extreme-value theory statistical models is that the analysed datasets consist of independent and identically distributed (IID) random variables. The models thus require stationarity; i.e. the statistical parameters such as mean and variance should remain constant over time and be free of “trends, shifts, or periodicity” (Salas, 1993). It is standard practice to transform the data to achieve stationarity through detrending (e.g. Wyncoll et al., 2016). The long-term mean sea level signal is superimposed onto inter-annual to multi-decadal sea level variability caused by tidal modulations associated with the nodal (18.61 year) and perigean (8.5 year) cycles, as well as other oceanic–atmospheric processes (e.g. Valle-Levinson et al., 2017). Here, a moving window approach is applied to the O-sWL series to remove long-term sea level rise and seasonality effects (Arns et al., 2013). In the procedure, the estimate of the trend is subtracted from the original time series value yielding a residual, which is then added to the mean sea level derived from the last 5 years of data to represent the most recent mean sea level conditions. The groundwater level was detrended in an identical manner. The detrended series are shown alongside the imputed observational records in Figs. S7 to S12.

Nonstationarity in the dependence between rainfall and O-sWL can occur as a consequence of a range of anthropogenically and climatically induced stressors. In
this study, the dependence is assumed to be stationary, i.e. that the copula parameters remain unchanged over time. The overlapping records at the three
sites are of insufficient length to robustly test the stationarity assumption. However, Wahl et al. (2015) did not detect any significant change in
Kendall's

Section 3.1 introduces the measures for assessing the strength of the dependence between the drivers and identifying the type of dependence in their joint tail regions. Section 3.2 describes methods employed for the bivariate analysis of rainfall and O-sWL, before the choice of hazard scenario is scrutinized. Finally, Sect. 3.3 provides a description and justification for the statistical models adopted for the trivariate analysis including groundwater level.

Kendall's rank correlation coefficient

Extremal dependence falls into one of two classes: asymptotic dependence or asymptotic independence (Ledford and Tawn, 1997). If

Svensson and Jones (2002) proposed a bootstrap procedure to test for asymptotic dependence. The two records are independently sampled with replacement
using a sample size the same length as the original concurrent record. The samples are subsequently paired to create a dataset identical in size to
the original but with the dependence removed. The process is repeated to create a large number

Here, a two-sided sampling approach similar to that in Wahl et al. (2015), which involves deriving two conditional samples where each variable is conditioned on in turn, is implemented to identify bivariate extreme events. Due to the relatively short length of the overlapping records and wastefulness of the block maxima approach, the threshold exceedance method is first used to identify univariate extremes. In practice, the method of Smith and Weissman (1994) is applied to the rainfall time series to identify cluster maxima which are paired with simultaneous O-sWL values and vice versa to create two two-dimensional time series. For more details on the choice of thresholds see Sect. 4.2.

A copula is a multivariate probability distribution with uniform marginal distributions. If

For a range of thresholds, the best fitting of 40 competing copulas plus the independence copula is determined via the Akaike information criterion
(AIC), using the

Schematic illustrating the approach by Bender et al. (2016) for combining two isolines of level

As opposed to the univariate case where the region containing dangerous events is uniquely defined, in the bivariate and higher-dimensional
settings hazard scenarios are required to specify this region. For a

There are several definitions of hazard scenarios, including OR, AND, Kendall (Salvadori et al., 2004), and survival Kendall (Salvadori et al., 2013), each offering different perceived strengths and limitations (e.g. bounded vs. unbounded subcritical layer, mathematical vs. physical valid interpretation) (e.g. Salvadori et al., 2011; Gräler et al., 2016). Due to the absence of any physical interpretation, Salvadori et al. (2016) suggest the procedures à la Kendall be reserved for preliminary assessments to gauge the expected probabilities of multivariate occurrences. The OR scenario has been extensively applied in the context of compound flooding at river confluences (e.g. Wang et al., 2009; Bender et al., 2016). Recently, Moftakhari et al. (2019) proposed incorporating the AND scenario to estimate the joint return period of river discharge and ocean levels in the FEMA (2015) procedure for assessing compound flood hazard in tidal channels and estuaries. In line with this recommendation (and many other previous applications where ocean levels and pluvial/fluvial flood drivers were analysed) the AND hazard scenario is adopted in this study.

A methodology for deriving design events when adopting a conditional sampling method with two joint probability distribution functions, as proposed in
this paper, is put forward and implemented in Bender et al. (2016). The approach exploits the strict monotonicity of the joint distribution functions,
by defining the (quantile-)isoline functions, for level

The choice of hazard scenario should reflect the type of dangerous event, e.g. a mechanism of failure, but is often an arbitrary and subjective choice (Serenaldi, 2015; Gouldby et al., 2017). Volpi and Fiori (2014) noted the typical disparity in the return period of structural failure compared with that of the loading variables and consequently proposed the so-called structure-based return period. The structure-based return period is derived by propagating the joint distribution of the basic variables through a structure or response function, describing the physical dynamics of a system. Hence, the return period of a response variable is calculated directly, typically empirically from a (large) sample of the basic variables after fitting a multivariate statistical model (Gouldby et al., 2017). The approach thus negates the need for a practitioner to define a hazard scenario (Salvadori et al., 2016). Serenaldi (2015) argues the concept of the return period in univariate frequency analysis is prone to misconceptions, only exacerbated in the multidimensional domain, and that the risk of failure offers a more transparent and suitable measure of risk. A full risk analysis is beyond the scope of this study but recommended as future work.

This section provides a description of three types of multivariate statistical models – standard higher-dimensional copulas (Sect. 3.3.1), pair-copula constructions (Sect. 3.3.2), and the HT04 model (Sect. 3.3.3) – applied here to capture the dependence between extreme rainfall, O-sWL, and groundwater levels.

Copulas were first introduced to the field of hydrology in De Michele and Salvadori (2003), where an Archimedean copula was used to describe the
dependence between storm duration and average rainfall intensity. The Archimedean copula family comprises a rich array of radially asymmetric and
symmetric copulas covering a diverse range of upper and lower tail dependence. The strengths of all pairwise dependencies are captured by a single parameter; thus standard Archimedean copulas are symmetric for any permutation of indexes (exchangeability). The exchangeability of Archimedean
copulas is often considered strongly restrictive in higher-dimensional applications, as it implies all pairwise dependencies are identical (Di
Bernardino and Rullière, 2016). Elliptical copulas, as the name suggests, are simply copulas of elliptical distributions and consequently possess
many of the useful traceable properties of these multivariate distributions (Fang et al., 1990). Elliptical copulas are radially symmetric with a
correlation matrix of parameters describing the strength of the pairwise dependencies. Consequently, they are non-exchangeable, only assuming the type
of dependence within each tail is identical. The trivariate Gaussian copula

Whilst bivariate applications are extensive in hydrology, trivariate applications of standard copulas are scarce. From analysing the dependence
between drought duration, intensity, and severity in New South Wales (Australia), Wong et al. (2008) found the Gumbel copula outperformed the Gaussian
copula. In a similar application in the Weihe River basin (China), Ma et al. (2013)
reported that the trivariate Gaussian copula gave a better fit than the Student

Approaches to increase the flexibility of standard higher-dimensional copulas include techniques to remove the exchangeability property of Archimedean copulas (e.g. Di Bernardino and Rullière, 2016) as well as the development of (meta-elliptical) copulas for various meta-elliptical distributions (Fang et al., 2002). Pair-copula construction (PCC) provides greater flexibility and a more intuitive way of extending bivariate copulas to higher dimensions than these approaches (Aas et al., 2009).

PCC, originally proposed by Joe (1996), decomposes a

The class of regular vines is considered relatively broad and encompasses a range of possible pair-copula decompositions. The canonical (or C) vine and D vine are special cases of regular vines, defining specific ways of decomposing a multivariate probability density. Each of the three possible decompositions of a three-dimensional copula density are simultaneously both a C and a D vine (see Fig. 3 for one example).

General structure of three-dimensional C/D-vine copula.

Gräler et al. (2013) applied a bivariate copula to annual maximum peak discharge and its volume, as well as a trivariate vine copula, by also
including duration, to investigate the effect of different modelling choices on design events. They found evidence of design quantiles shrinking as
the number of variables considered grows (bivariate vs. trivariate), referred to as the

An alternative pair-copula decomposition of a higher-dimensional joint probability distribution is the nested Archimedean construction (NAC) (e.g.
Embrechts et al., 2003). In NAC, only

The HT04 approach models the conditional distribution of the remaining variables given a specified variable exceeds a suitably high threshold. By repeating the procedure for each variable in turn, the model captures the dependence structure between a set of variables when at least one takes on an extreme value. The HT04 approach thus requires no assumptions regarding the nature of the dependence in the joint tail regions between a set of variables.

As opposed to the standard copula methodology, the HT04 model is generally implemented using Gumbel marginal distributions given by

An outline of the steps involved in the well-established Monte Carlo procedure for generating a realization

sample

independently sample a joint residual

calculate

reject sample

The extremes observed during such temporal dependent and spatially varying events may not occur concurrently. Keef et al. (2009a) addressed this
limitation by fitting the HT04 model to the distribution of the variable at location

In this section, the results of the correlation analysis (Sect. 4.1), bivariate analysis (Sect. 4.2), and trivariate analysis (Sect. 4.3) are discussed in turn. In Sect. 4.2 and 4.3 results pertain to site S22; analogous results for the other two sites are provided in the Supplement.

Assessment of correlation between the flooding drivers at site S20

Rainfall, O-sWL, and groundwater level exhibit small (

The empirical estimates of

Comparison of the design events (diamonds) obtained using the two-sided conditional sampling approach and the approach used in the original design (triangles) for return periods of

To capture the dependence between rainfall and O-sWL, the approach outlined in Sect. 3.2 was applied for a range of thresholds. The choice of copula family is relatively insensitive to the selected threshold (see Figs. S13 to S15). The threshold is selected as a trade-off between the bias and variance in the copula parameter estimates. For each of the conditioned samples a threshold of the 0.98 quantile of the conditioning variable was deemed appropriate at each of the sites. Attention from hereon in focuses on site S22 (detailed results for the other sites are included in the Supplement), where the 0.98 quantile threshold gives an average of 6.3 and 5.2 events per year when conditioning on O-sWL and rainfall, respectively. The conditioning variable was fitted to a GPD while relevant non-extreme parametric distributions were fitted to the non-conditioning variable. The Birnbaum–Saunders(logistic) distribution was selected to model the rainfall(O-sWL) data in the sample where O-sWL(rainfall) is conditioned to exceed its 0.98 quantile, as it was consistently among the best fitting of the candidate distributions at the three sites (see Figs. S16 to S18).

The quantile isolines for several return periods are shown alongside the observations in Fig. 5. The coloured contours on the isolines represent the
relative likelihood of events. The most-likely strategy is used as a simple way to derive possible design events associated with a given return
period

Figure 5 illustrates two types of design events, indicating that the system experiences a change in behaviour between 20-
and 50-year return periods. To further investigate the return period at which the change in design event type occurs, design events were calculated
for return periods from 1 to 100 years at a yearly interval. The processes of simulating samples from the fitted copulas, estimating the relative
likelihood along the isolines and extracting the most-likely event, was then repeated to give 100 design events associated with the 1- to 100-year
return periods. The results showed that the change occurs for return periods between approximately 20 and 40 years. For small return periods
(

Most-likely design events that are surge only (for smaller bivariate return periods) will potentially produce very different water levels at a structure (response variable) compared to compound events (for higher bivariate return periods), ultimately resulting in substantially different design conditions. For several flood defences in England, Gouldby et al. (2017) illustrated the sensitivity of the return period of a response variable – overtopping discharge – to the choice of return period definition. To account for the variability in design event selection, approaches have been developed to replace single design events with ensembles of possible design realizations (Gräler et al., 2013). Testing an ensemble of design events or adopting a structurally based return period, where extremes are defined in terms of response variables directly, will produce a more robust analysis. Implementation of these approaches would be particularly beneficial at sites S20 and S28, where, although all design events can be classified as surge only, probability density is non-zero along other parts of the isolines (see Figs. S19 and S20). In many cases implementing these approaches requires running complex and computationally expensive process-based models and is therefore beyond the scope of our analysis.

The conservative nature of the original design approach is further explored by assessing how long it will take under a given SLR for the 100-year design events selected with the two different methods (i.e. full dependence assumption vs. bivariate dependence modelling) to intersect. In other words, the amount of SLR and how long it will take under different emission scenarios, for the diamonds (i.e. bivariate design events) in Fig. 5 to move vertically and close the gap to the triangles (i.e. design events under full dependence assumption, used in the original design), is assessed. The low, intermediate, and high scenarios from Sweet et al. (2017) are considered (see Fig. 6, top).

The top shows regional SLR projections for Miami Beach given in Sweet et al. (2017). The bottom shows the number of years before the O-sWL in the 50-year design event derived using the bivariate approach reaches the corresponding value obtained using the original design approach according to the three SLR scenarios.

The results are highly sensitive to the SLR scenario considered. For instance, the time before the O-sWL in the 50-year bivariate approach reaches
that of the corresponding event derived using original design approach ranges from 16 years to greater than 80 years (Fig. 6, bottom). The times before
the O-sWL in the design events given by the original design and bivariate approaches with return periods from 1 to 100 years become equal according to
the three scenarios shown in Fig. 7. The change in the characteristic of the design events (i.e. the shift from O-sWL dominated to compound
driven) between return periods of around 20 to 40 years is apparent. For events with return periods

The disparity of the rainfall totals composing the design events given by the bivariate and original design approaches is greatest for low return
periods (

Time before the O-sWL in the bivariate design event derived from the two-sided sampling approach reaches the corresponding value obtained from the original design approach (i.e. full dependence assumption) under the low (green), intermediate (blue), and high (red) SLR scenarios given in Sweet et al. (2017). Shaded regions denote 95 % (basic) bootstrap confidence intervals.

In this section, the bivariate analysis is extended by also incorporating groundwater level into the analysis. First, the marginal extremes are analysed separately for each flooding driver. The method of Smith and Weissman (1994) was applied to each time series to identify cluster maxima. For each variable, cluster maxima and excesses above a sufficiently high threshold were fitted to a GPD. The GPD was combined with the empirical distribution below the threshold. The threshold choice was guided by appropriate criteria, predominantly mean residual life plots (Coles, 2001). Diagnostic goodness of fit demonstrated the adequacy of the fit plots of the GPD for the rainfall and groundwater level series, whilst the fit to the O-sWL series was less robust (see Figs. S21 to S29). The study area is exposed to several flood-generating mechanisms including storms associated with tropical cyclones, mesoscale convective systems, and extratropical systems. Hence, a single distribution is fitted to events that are likely coming from several different populations. The fit of the GPD was particularly poor for the three largest O-sWL events. The five highest recorded O-sWL values are associated with tropical cyclones, consistent with an analysis by Villarini and Smith (2010). Nevertheless, observational records of the length available for this study contain relatively few tropical cyclone events. Consequently, risk assessments in areas exposed to tropical cyclone storm surges commonly utilize synthetic records of such events, generated based on historical observations (e.g. Nott, 2016). To generate synthetic records, wind and pressure fields simulated from statistical models of tropical cyclone behaviours are used to drive hydrodynamic storm surge models (Haigh et al., 2014). Replacing the observational record with a longer synthetic record could thus be an avenue to improve the marginal fit of the O-sWL distribution and ultimately the robustness of the proposed approach. This is beyond the scope of the present study, where the focus is on developing appropriate frameworks for capturing and modelling dependence between the different flood drivers.

(First row) Observed events at site S22 (black dots) superimposed with the

The multivariate model fitting also requires sets of independent events. Gouldby et al. (2014) used a notional flooding level, a function of the primary variables of interest, to de-cluster the offshore loading time series data before fitting the HT04 model. In other applications of the HT04 approach, marginal de-clustered excesses of the conditioning variable are paired with concurrent values of the remaining variables. The nonlinear regression model (Eq. 7) is then fitted to the set of events and the process is repeated conditioning on each variable in turn. In the absence of a suitable response function that can be evaluated without employing hydraulic–hydrologic models, this is also the approach adopted here in the application of the HT04 model. Standard higher-dimensional copulas and vine copula models are often applied conditioning on a single variable to derive a set of independent events. However, only conditioning on a single variable may result in the removal of the most extreme values of the other variables. Therefore, in this work the models are applied to the entire dataset, as implemented before for higher-dimensional copulas in Wong et al. (2008) and for vine copulas in Bevacqua et al. (2017), among others.

At all three sites, the Gaussian and Student

The return periods conditional on a range of antecedent groundwater levels for the four bivariate (most-likely) design events, accounting for the
dependence between rainfall and O-sWL (diamonds in Fig. 5), according to the three types of trivariate models are shown in Fig. 9. The trivariate
return periods are calculated empirically from the samples in Fig. 8. The bivariate events with return periods of 50 and 100 years were assigned
return periods of

Sensitivity of the return period of the four bivariate design events, derived using the approach described in Sect. 3.1 and displayed in Fig. 4, to the antecedent catchment condition. The trivariate return periods are calculated using the Gaussian copula (green), vine copula (blue), and HT04 (red) approach.

This paper puts forward a framework for assessing the different drivers of compound flooding in coastal areas of south Florida in Miami-Dade
County. The framework was derived through a gradual transition from the original structural design approach (based on the assumption of full
dependence between rainfall and O-sWL and ignoring groundwater levels) by meeting three objectives. The first objective was to determine whether there
is any statistically significant correlation between extreme rainfall, O-sWL, and groundwater level in the area. At all three study sites, rainfall,
O-sWL, and groundwater level exhibit small but statistically significant pairwise correlations over a range of relevant time lags. The second
objective was to assess the conservative nature of the original structural design approach that assumes full dependence

The output of the bivariate and particularly trivariate applications can also act as boundary conditions for coupled hydrologic–hydraulic models for assessing flood risk and designing flood defence structures, among other purposes (e.g. Serafin et al., 2019). Rigorous implementation of the bivariate and trivariate methodologies, e.g. by adopting a structure-based return period approach or using an ensemble of events, will potentially facilitate more effective flood risk management in low-lying coastal catchments. A natural next step would be to explore the influence of the more robust boundary conditions on the design specifications of the water control structures at the three sites. Meanwhile, the accuracy of the GPD fit to O-sWL at the study sites (especially in the trivariate analysis; see Figs. S22, S25 and S28) could also be improved by utilizing synthetic tropical cyclone events and associated storm surges. The methodologies introduced here are readily transferable and applicable to other locations, assuming sufficiently long overlapping records of the different variables are available.

Code and data used to complete this study are available in the

See Code availability section. The data used in this paper are also freely available through NOAA's National Climatic Data Center's (NCDC) archive of
global historical weather and climate data at

The supplement related to this article is available online at:

The study was conceived by JO and TW. RJ developed the methodology, undertook the analyses, and wrote the paper under the guidance of TW. LC and JO contributed by generating ideas, providing valuable insights during technical discussions, and editing the manuscript.

Thomas Wahl is on the Editorial Board of this special issue.

This article is part of the special issue “Advances in extreme value analysis and application to natural hazards”. It is a result of the Advances in Extreme Value Analysis and application to Natural Hazard (EVAN), Paris, France, 17–19 September 2019.

Robert Jane was supported by funding from the South Florida Water Management District. Thomas Wahl acknowledges financial support from the USACE Climate Preparedness and Resilience Community of Practice. This material is based in part on work supported by the National Science Foundation under grant AGS-1929382 (Thomas Wahl).

This paper was edited by Sylvie Parey and reviewed by Hamed Moftakhari and one anonymous referee.