Modelling multiple hazard interrelations remains a challenge for practitioners. This article primarily focuses on the interrelations between pairs of hazards. The efficacy of six distinct bivariate extreme models is evaluated through their fitting capabilities to 60 synthetic datasets. The properties of the synthetic datasets (marginal distributions, tail dependence structure) are chosen to match bivariate time series of environmental variables. The six models are copulas (one non-parametric, one semi-parametric, four parametric). We build 60 distinct synthetic datasets based on different parameters of log-normal margins and two different copulas. The systematic framework developed contrasts the model strengths (model flexibility) and weaknesses (poorer fits to the data). We find that no one model fits our synthetic data for all parameters but rather a range of models depending on the characteristics of the data. To highlight the benefits of the systematic modelling framework developed, we consider the following environmental data: (i) daily precipitation and maximum wind gusts for 1971 to 2018 in London, UK, and (ii) daily mean temperature and wildfire numbers for 1980 to 2005 in Porto District, Portugal. In both cases there is good agreement in the estimation of bivariate return periods between models selected from the systematic framework developed in this study. Within this framework, we have explored a way to model multi-hazard events and identify the most efficient models for a given set of synthetic data and hazard sets.

A multi-hazard approach considers more than one hazard in a given place and
the interrelations between these hazards (Gill and Malamud, 2014).
Multi-hazard events have the potential to cause damage to infrastructure and
people that may differ greatly from the associated risks posed by a single
hazard (Terzi et al., 2019). Here, natural hazards (which we will also refer
to as “hazards”) will be defined as a natural process or phenomenon that may have negative impacts on society (UNISDR, 2009). For modelling
purposes, we consider two main mechanisms in natural hazard interrelations
(Tilloy et al., 2019): (i)

Meteorological phenomena such as extratropical cyclones or convective storms
often lead to the combination of multiple drivers and/or hazards and can
therefore be related to compound events as defined by Zscheischler et al. (2018). This research concentrates on cascading and compound interrelations
between natural hazards (e.g. a storm can include rain, lightning and hail,
with rain and hail both potentially triggering landslides). Case examples of
meteorological phenomena influencing natural hazard interrelations include
the following:

In 2010, storm Xynthia hit the west coast of France. The storm itself was not particularly extreme for the season, but the compound effect of extreme wind, high tides, storm surges, extreme rainfall and the fact that the soils were already saturated led to huge damage due to wind and flooding (CCR, 2019).

In summer 2010, Russia experienced a heatwave. Low precipitation in spring 2010 led to a summer drought that contributed to the heatwave having a large magnitude (Barriopedro et al., 2011; Hauser et al., 2015; Zscheischler et al., 2018). The co-occurrence of extremely dry and hot conditions resulted in widespread wildfires, which damaged crops and caused human mortality (Barriopedro et al., 2011).

Extreme thunderstorms occurred in the Paris region in 2001, involving lightning and extreme rainfall, with the rainfall triggering flooding, mudslides and ground collapse, with subsequent damage to railway networks (CCR, 2019).

In this context, the quantification of interrelations between natural hazards can play an important role in risk mitigation and disaster risk reduction. Some of the natural hazards presented in the above examples are extreme occurrences of environmental variables (e.g. extreme temperature) which have different characteristics and statistical distributions (e.g. wind and landslides). Natural hazards can be interrelated with different mechanisms (i.e. compound, cascade). For a given mechanism, interrelations also vary in strength and intensity. Additionally, as highlighted in Tilloy et al. (2019), different modelling approaches have been developed to quantify interrelations between variables. Here we focus on stochastic models that include copulas (Nelsen, 2006; Genest and Favre, 2007; Salvadori et al., 2016) and multivariate extreme models (Heffernan and Tawn, 2004), limiting our analysis to the bivariate case. The potential for misinterpretation of the dependence structure of two variables clearly presents a problem when end users try to account for hazard interrelations.

We choose six distinct bivariate models able to handle different types of
tail (extreme) dependence: one non-parametric (JT-KDE), one semi-parametric
(Cond-Ex) and four different parametric copulas (Galambos, Gumbel, FGM,
normal; see Sect. 2 and Table 2). The fitting capacities of each model are
compared with the estimation of

Examples of joint and conditional probabilities are given in Fig. 1. A joint probability is the probability of two events occurring together where both variables are extreme (also called AND probability; Fig. 1a), and a conditional probability is the probability of an event given that another has already occurred (Fig. 1b). Figure 1 illustrates the concepts of joint probability and conditional probability, with daily rainfall data from a high-resolution gridded dataset of daily meteorological observations over Europe (termed “E-OBS”; Cornes et al., 2018) and daily maximum wind gust data at Heathrow Airport provided by the Met Office (2019). A wind gust here is defined as the maximum value, over the observing cycle, of the 3 s running-average wind speed (WMO, 2019). These datasets and the interrelation between extreme rainfall and extreme wind are discussed in Sect. 4.1.

Illustration of joint and conditional extremes with daily rainfall

Joint and conditional probabilities are relevant metrics for practitioners and have been studied and used in several studies in the environmental sciences (e.g. Hao et al., 2017; Zscheischler and Seneviratne, 2017). However, as the most widely used level curve is the joint probability curve, we initially focus on it. To analyse our results and compare the performances of the models, we designed diagnostic tools that are presented in Sect. 3.2.

This paper is organized as follows. We first (Sect. 2) provide a theoretical background on key concepts used in this study and present the models and methodology used. We then (Sect. 3) discuss the characteristics of our synthetic dataset and present the results of the simulation study. The diagnostic tools used to compare models are also discussed (i.e. joint return level curves and dependence measure). As a result, a map exhibiting the strength and weaknesses of our six models is presented. It aims to provide objective criteria to justify the use of one model rather than another for a given set of hazards. Two applications to pairs of natural hazards that can impact energy infrastructure are presented in Sect. 4.

The main purpose of these data applications is to illustrate our methodology, but the natural hazard interrelations studied have the potential to negatively impact energy infrastructure. The first application looks at compound daily rainfall and wind in the United Kingdom. The combination of these two hazards can result in different and greater impacts than the addition of impacts due to extreme wind and extreme rainfall (e.g. wind destroys roof leading to greater damage, power plants flooded and rescuers slowed down by strong winds; Martius et al., 2016). The second application studies extreme hot temperatures and wildfires in Portugal. Extreme temperatures can lead to damage to infrastructure (e.g. rail track deformation) and put pressure on the energy infrastructure by increasing the demand (Hatvani-Kovacs et al., 2016; Vogel et al., 2020); they also increase the probability of wildfires (Witte et al., 2011; Perkins, 2015) which have the potential to cause fatalities and destroy infrastructure (Tedim et al., 2018). We finish (Sect. 5) with a discussion and conclusions.

We are interested in modelling interrelations between hazards in the extreme
domain. This implies the use of methods and concepts coming from the broad
area of extreme value theory (EVT). Amongst the six models compared in this
study, four are directly linked to EVT (JT-KDE, Cond-Ex, Galambos, Gumbel).
Extreme value theory has its roots in univariate studies (Coles, 2001) and
has been extended to the multivariate framework (Pickands, 1981; Davison and
Huser, 2015). A theoretical background on extreme value theory is given in Sect. S1.1 in the Supplement. In this study, we focus on modelling the
dependence between two variables. Bivariate extreme value models developed
within the statistical community (Resnick, 1987; Heffernan and Tawn, 2004;
Cooley et al., 2019) have recently been used for environmental applications
and therefore natural hazard interrelations (De Haan and De Ronde, 1998;
Zheng et al., 2014; Sadegh et al., 2017). In order to reproduce the
complexity and variety of natural hazard interrelations, we use 60 synthetic
datasets to compare the fitting performances of the models. In these
synthetics datasets we vary two main attributes of the bivariate datasets:
the dependence structure and the marginal (individual) distributions. Of
these 60 different synthetic datasets, 36 datasets have

In this section, we first present the two types of asymptotic behaviour in bivariate extreme value statistics – asymptotic dependence and asymptotic independence – and discuss different dependence measures for the estimation of the relationship between two variables (Sect. 2.1). The six bivariate models are then described (Sect. 2.2). Finally, we discuss the concept of the return level in the bivariate framework (Sect. 2.3).

Let

The variables

Using models that take the assumption of asymptotic dependence (independence) in the case of asymptotically independent (dependent) variables can lead to a large overestimation (underestimation) of the probability of joint extreme events (Ledford and Tawn, 1996; Mazas and Hamm, 2017; Cooley et al., 2019). The multivariate extreme value and regular variation theories presented in Sect. S1.2 provide a rich theory for asymptotic dependence (De Haan and Resnick, 1977; Pickands, 1981) but are not able to distinguish between asymptotic independence and full independence.

A popular method to analyse hazard interrelationships is to compute dependence measures (Zheng et al., 2013; Petroliagkis, 2018). Dependence measures aim to describe how two (or more) variables are correlated.

When focusing on the dependence in the tails or extreme part of distributions, linear or rank dependence measures might not be accurate and other coefficients appear more relevant (Hao and Singh, 2016). Dependence between variables in the joint tail domain has been widely studied in the statistics community (Coles and Tawn, 1991; Ledford and Tawn, 1997; Coles et al., 1999; Heffernan and Tawn, 2004; Zheng et al., 2014). As explained in Sect. 2.1.1, in the tails, two variables can be either asymptotically independent or asymptotically dependent; different diagnostics and coefficients previously developed are summarized in Heffernan (2000).

In this study, we use the following tail dependence measures:

the extremal dependence measures

the coefficient of tail dependence

The three coefficients used in this study to assess the dependence
between two variables at an extreme level. In the upper part of the plot
(blue), the coefficient

Dependence measures are empirical measures which estimate the strength of the correlation or interdependence between two (or more) variables. Despite the fact that these measures provide crucial information, they do not allow for the modelling of joint (or conditional) exceedance probabilities. To model joint exceedance probabilities which represent the joint occurrence of hazards (here represented by extremes of environmental variables) in time and space, the use of stochastic models is required. In this section we present the three stochastic approaches for multivariate modelling that are used in the simulation study: parametric copulas, the semi-parametric conditional extremes model, and a non-parametric approach based on multivariate extreme value theory (see Sect. S1.2) and kernel density estimation.

In the bivariate case, a copula is a joint distribution function which
defines the dependence between two variables independently from the marginal
distributions of these variables (Heffernan, 2000; Nelsen, 2006; Genest and
Favre, 2007; Hao and Singh, 2016). Let the random variables (

However, extreme-value copulas are by definition asymptotically dependent as they follow the rules of multivariate extreme value theory (see Sect. S1.2). The two types of extremal dependence were presented in Sect. 2.1 and show that it is important to also consider asymptotic independence. Many copulas are asymptotically independent, including the normal copula and the Farlie–Gumbel–Morgenstern (FGM) copula (Heffernan, 2000). These two copulas will be used in the simulation analysis as asymptotically independent models (Sect. 3).

In the present study, the application of a copula model can be summarized in
four main steps:

fitting marginal distributions to the two variables and then an empirical cumulative distribution function below a threshold and generalized Pareto distribution (GPD) above this threshold,

transforming the variables to uniform margins – the transformed datasets no longer have information on the marginal distributions but keep the information about the dependence structure (Nelsen, 2006),

fitting the copula function to the pseudo-observations by estimating the copula parameter(s) with an estimator (Genest and Favre, 2007),

estimating the probability of joint events with the copula function previously fitted.

The conditional extremes model (Heffernan and Tawn, 2004; Keef et al., 2013) is a semi-parametric model designed to overcome several limitations of copulas and other approaches such as the joint tail methods in which all variables must become large at the same rate. The aforementioned methods can typically handle only one form of extremal dependence, either asymptotic dependence or asymptotic independence. The conditional extremes model has the ability to be more flexible with asymptotic dependence classes; it can account for asymptotic independence and asymptotic dependence (Heffernan and Tawn, 2004; Keef et al., 2013). It can also be used to analyse more than two i.i.d. variables more easily than copula-based methods (Winter and Tawn, 2016); we restrict the theory provided here to the bivariate case. The conditional model has been used for different purposes: spatial or temporal dependence between extremes (Winter and Tawn, 2016; Winter et al., 2016), dependence between extreme hazards (Zheng et al., 2014) and even financial purposes (Hilal et al., 2011).

The conditional extremes model assesses the dependence structure between
several variables conditional on one being extreme and aims to model the
conditional distribution. As in joint tail models, the first step is to
transform the marginal distributions; here the preferred marginal choice is Laplace (or Gumbel) margins (Heffernan and Tawn, 2004; Keef
et al., 2013). Let the random variables (

Formally, the application of the conditional extremes model can be summarized
in four main steps:

fitting marginal distributions to the two variables, an empirical cumulative distribution function below a threshold and generalized Pareto distribution (GPD) above this threshold;

transforming those distributions onto Laplace (or Gumbel) margins;

estimating the dependence parameters using non-linear regression;

estimating the probability of joint events by simulating new extreme data through the conditional model.

The non-parametric approach used in this paper is an adaptation of the
non-parametric approach presented by Cooley et al. (2019). Moreover, the
dependence measures

The kernel density estimation (KDE) method has the advantage of being a
non-parametric way to estimate the joint distribution of

The kernel density estimator is used here to estimate an empirical density
distribution

After estimating the joint survival distribution of the two variables with a
kernel density estimator, the cumulative distributions

Therefore,

Extrapolation in a regularly varying tail for a distribution in
the max domain of attraction of some multivariate extreme value
distributions. Black circles represent an asymptotically dependent bivariate
dataset. In order to estimate the extreme joint probability

The methodology presented above is only valid when the two variables

The specificity of this approach (presented below) is that it combines a non-parametric estimation of the joint density and the framework of multivariate extreme values presented in Sect. S1.2. It can deal with both asymptotic dependence and independence. The coefficient of tail dependence estimation has an influence on the extrapolation process in the asymptotic independence case. Here we used the estimator presented in Winter (2016) which is derived from the joint tail model of Ledford and Tawn (1997).

Formally, the application of the joint tail KDE model can be summarized in
five main steps:

estimating the joint cumulative distribution of the variables with a kernel density estimator,

fitting marginal distributions to the two variables – empirical distribution below a threshold and generalized Pareto distribution (GPD) above this threshold,

transforming those distributions into Fréchet margins,

determining whether variables are asymptotically dependent or asymptotically
independent by estimating the coefficients of tail dependence

Estimating the probability of joint events and extrapolating the base isoline to an objective isoline.

Studying natural hazards as multivariate – and particularly bivariate – events is a growing practice in multiple disciplines, including the following: coastal engineering (Hawkes et al., 2002; Mazas and Hamm, 2017), climatology (Hao et al., 2017, 2018; Zscheischler and Seneviratne, 2017) and hydrology (Zheng et al., 2014; Hao and Singh, 2016). There has been debate among scientists trying to define a “multivariate return period” (Serinaldi, 2015; Gouldby et al., 2017). Serinaldi (2015) defined seven different types of probabilities that can be considered as bivariate probabilities of exceedance. These can be expressed through copula notation.

Let the random variables (

Types of probabilities for bivariate (

The function

Graphical representation of two bivariate (

In 2D space, probabilities of exceedance (or quantiles) are not represented
by a single value but by a curve with an infinite number of points with the
same probability of exceedance. However, as shown in Fig. 4, these
probabilities are defined by (i) the domain where these are computed and
(ii) the critical region corresponding to the probability type. For the AND
probability, the computation domain remains similar when moving along the
curve while the critical region evolves constantly. For the COND1
probability, both computation domain and critical region evolve when moving
along the curve (see Fig. 4). Bivariate probabilities of exceedance
are curves. These curves have been given various names in different research
papers including the following:

Here we are interested in comparing the abilities of six different models
presented in Sect. 2.3 to reproduce a given dependence structure.
We create 60 different synthetic dataset types with varying marginal
distributions and dependence structures. By doing this, we aim to produce
bivariate synthetic datasets comparable to the ones studied in bivariate
hazard analysis (Zheng et al., 2014; Hendry et al., 2019). This will allow
us to confront the six models against the synthetic datasets, as a reference
for bivariate hazard interrelation analysis (see Sect. 4). The
six models compared in this simulation study are

the conditional extremes model (Cond-Ex; Sect. 2.3.2),

the non-parametric joint tail model (JT-KDE; Sect. 2.3.3),

the Gumbel copula (Gumcop; Sect. 2.3.1),

the normal copula (Normalcop; Sect. 2.3.1),

the Farlie–Gumbel–Morgenstern (FGMcop) copula (Sect. 2.3.1),

the Galambos copula (Galamboscop; Sect. 2.3.1).

Among the four copulas used here, two are asymptotically dependent (Gumbel and Galambos) and two are asymptotically independent (normal and FGM). A description of the six models is given in Table 2. Table 2 synthesizes a range of information about all the six models used in this simulation study including their type (non-parametric, semi-parametric, parametric), equation, parameter range (if there is a parameter) and asymptotic modelling domain. This latter information is important to interpret the result of the simulation study in Sect. 3.3.

Description of the six statistical models compared in this article. The description includes the model name and abbreviation (used throughout the article), type of model (parametric, semi-parametric, non-parametric), the mathematical description, the parameter range (where relevant) and the asymptotic modelling domain (AI for asymptotic independence and AD for asymptotic dependence).

In this section, we first describe and display the synthetic data that have been generated to conduct this study. We shall then present the measures used in this study to compare the level curves and the dependence measures estimated from the six models presented in Table 2. Finally, results of the simulation will be displayed and analysed.

Synthetic datasets are often used to compare different statistical models (Chebana and Ouarda, 2011; Zheng et al., 2014; Cooley et al., 2019). Here we generated 60 bivariate synthetic datasets representative of environmental data such as daily rainfall, daily wind gust and daily wildfire occurrences (see Sect. 4). The number of synthetic data points we use here has been fixed to 5000 for each dataset. For the asymptotic dependence case, 36 distinct datasets are generated from a Gumbel copula (see Sect. S1.3.1); for the asymptotic independence case, 24 datasets are generated from a normal copula (see Sect. S1.3.2). Each synthetic dataset set of parameters has been used to generate 100 realizations to produce confidence intervals.

The synthetic datasets are generated from two marginal distributions and a dependence model (i.e. copula). Both marginal distributions are log-normal; the log-normal distribution has been used (among others) for the modelling of a wide range of natural hazards, including wind, flood and rainfall (Malamud and Turcotte, 2006; Clare et al., 2016; Loukatou et al., 2018; Nguyen Sinh et al., 2019).

Random variables

We can characterize log-normal distributions with the coefficient of
variation

The standard deviation

Marginal distributions and copula used for the synthetic datasets.

where

We use three different coefficients of variations

The dependence structure is represented by a Gumbel copula in the case of
asymptotic dependence (AD) and a normal copula in the case of asymptotic
independence (AI) as no copula can be both asymptotically independent and
asymptotically dependent (Heffernan, 2000; Coles, 2001). The Gumbel copula
is an extreme-value copula that is asymptotically dependent (see Eq. S19). The Gumbel copula function only has one parameter

The 60 different synthetic bivariate datasets used in our
simulation study. On the

To compare the fitting capabilities of the different models presented in
Sect. 2.3, we vary several characteristics of the synthetic
dataset:

There are many diagnostic tools to assess the goodness of fit of parametric
bivariate models (Arnold and Emerson, 2011; Couasnon et al., 2018; Genest et
al., 2009, 2011; Genest and Nešlehová, 2013; Sadegh et al., 2017).
Amongst these, some of the most popular are the following:

Cramér–von Mises statistic (Arnold and Emerson, 2011)

Kolmogorov–Smirnov test (Arnold and Emerson, 2011)

Akaike information criterion (AIC; Akaike, 1974)

Bayesian information criterion (BIC; Schwarz, 1978).

These criteria are designed to fit on the dependence structure of the whole dataset and not on the extreme dependence structure which can be different.

In our study we aim to compare parametric and non-parametric models.

However, we are interested in fitting capabilities in the extremes. The
models will then be compared on the estimation of two attributes of the
synthetic data detailed below:

the

the tail dependence measures

We present here the diagnostic tools related to the level curve. The tools
used to compare tail dependence measures can be found in Appendix A. Here we chose to compare our six models with respect to their ability to
reproduce a reference level curve from the underlying bivariate (

Each modelled and reference level curve is normalized by dividing its coordinates by their maximum values. With that process, the curves are bounded in the [0, 1] by [0, 1] space. The different indicators are then computed in this normalized space.

Cartesian coordinates (

Each modelled and reference level curve is discretized via linear interpolation into points. Each point corresponds to an angle value (triangles and dots on the curves in Fig. 6).

Points from both the modelled and reference level curves with the same angle are coupled. Indicators are computed at each of the 80 couples of points (see Fig. 6).

Procedure for computation of the goodness-of-fit
indicators. Two variables are given,

We used a weighted Euclidean distance (

Two analyses are conducted in parallel, one for asymptotic dependence
(AD) and one for asymptotic independence (AI). In the case
of asymptotic dependence, the Gumbel copula is used with 5000 data points.
The

The marginal distributions do not have any impact on the dependence structure (Nelsen, 2006; Genest and Favre, 2007). We show in Appendix A that marginal distributions also have a very small impact on the estimation of dependence measures. All the methods used in this study include a transformation of marginal distributions and the fitting of a GPD above an extreme threshold (Sect. 2.3). By varying the marginal distribution of the variables of our synthetic dataset, we aim to capture uncertainties and errors arising from both the fitting of the marginal distributions and the dependence structure.

For both asymptotic dependence AD and asymptotic independence
AI, the objective level curve

In an analogous way, for each of the diagnostic tools presented in
Sect. 3.2, three values are computed: (i) the 2.5 % quantile,
(ii) the median and (iii) the 97.5 % quantile. To assess more accurately
whether the models manage to represent the synthetic data in the large value
extremes, we compared their fitting capabilities to a naïve approach.
Here, the naïve approach is an empirical level curve. For each of the
60 synthetic datasets, we compute the

Weighted normalized Euclidean distance (

It is important here to note that we tested more AD (36 %–60 %) than AI (24 %–40 %) cases. To assess the flexibility of
models, additionally to comparison to the naïve approach, we also
consider the proportion of cases where models have a

The Gumbel and normal copulas, which have been used to generate the synthetic datasets with AD and AI, generally outperform all the other models in AD and AI cases respectively.

The conditional extremes model and the joint tail KDE model are the most
flexible models tested here as they can handle (Cond-Ex) 98 %
[72 %, 100 %] and (JT KDE) 97 % [65 %, 100 %] of the situation with a

The normal copula, even if asymptotically independent, is the most flexible
copula model with

Gumbel and Galambos copulas have representative fits to only 57 % of the
AD datasets. Among the 36 AD cases, they fail to represent only two
with

The FGM copula can only handle one type of extremal dependence, which is
asymptotic independence (AI) with

Higher shape parameters of the margins are associated with poorer
goodness of fit for all models. It is particularly striking with the
conditional extremes approach which exhibits high uncertainty and high

The Cond-Ex and JT-KDE models provide close results according to Fig. 7
despite adopting very different approaches. Thus, their flexibility arises
from their semi-parametric nature. Figure 7 also displays the
uncertainty in the estimate of

However, the copulas are penalized by the weighting function as they usually
reproduce quite well the naïve part of the curve. By considering again
the percentage of situations with a criterion below 0.1, the normal copula
has its performances reduced by the weighting function (

Results from the simulation study presented in the previous section (Sect. 3) can provide useful insights when modelling the interrelations between two natural hazards. In this section, we will show how results previously presented can be useful to identify the most relevant models for a given dataset according to its visual characteristics. The concordance (or discordance) of the relevant models can also increase (or decrease) confidence around the results.

The methodology for model selection presented here is composed of five steps
to select the most relevant model estimate joint exceedance probability
level curves:

The two-tail dependence measures are estimated empirically with a 95 % confidence interval. The dataset with a tail dependence measure falling into that confidence interval is suggested as analogue to the studied bivariate dataset. To select relevant combinations of marginal distribution, a scatterplot is compared visually to density plots for the 60 different datasets simulated in Sect. 3 and displayed in Fig. 5.

From the aforementioned 60 datasets, a set of one to six analogous datasets (i.e. with similar bivariate distribution) is taken.

A confidence score is used to compare the abilities of each model for the
datasets selected in step (ii). For each model, the confidence score is

Models are fit to the bivariate hazard dataset, and level curves from the most relevant models are kept.

Tail dependence measures are estimated using the most relevant model with a possible new iteration of the four previous steps according to the value of the dependence measures.

To produce a confidence interval as was carried out in the simulation study (Sect. 3) and to visually measure the uncertainty associated with each level curve as in Sect. 3, we use a non-parametric bootstrap procedure. The function tsboot from the R package “boot” (Davison and Hinkley, 1997; Canty and Ripley, 2019) is used to generate 100 bootstrapped replicate datasets with the same number of observations as the original (but some are repeated). Our six models are then fitted to the original dataset and on the 100 bootstrapped replicates.

Here, we study the interrelation between daily extreme wind gusts (

The bivariate dataset used to study the interrelation between wind gusts and
rainfall at Heathrow Airport is composed of the following data:

From 1 January 1971 to 31 May 2018 there are a total of 17 318 d
(including leap years). Our bivariate wind gust–rainfall dataset is composed
of those values where there are both non-zero rainfall

Violin plots of daily wind gust

From Fig. 8 we observe a seasonality in daily wind gust speed.
January is the month with the highest median (diamond symbol) and range of
most values in the violin plot, while July is the month with the lowest
median and range of most values in the violin plot. The daily non-zero
rainfall median per month varies between 2.5 mm in February and 3.5 mm in
June, with the highest individual daily values occurring in October (53.3 mm d

Days where there are recorded both daily wind gust (m s

Extreme rainfall and extreme wind have a compound interrelation according to
Tilloy et al. (2019). We then estimate the joint exceedance probability
curve, corresponding to a

We now go through the four steps presented for rainfall and wind gusts in Heathrow.

From Figs. 5 and 9, along with empirical estimates of

This then gives us four analogous datasets, and it is then possible to
visually infer from Fig. 6 which models are the most suitable for
these conditions. The four analogous datasets are the following:

The

Euclidian weighted distance (

Estimates of dependence parameters

According to these three first steps, the conditional extremes model appears
to be the most suitable. However, we selected the four most relevant models
for the bivariate dataset of daily rainfall and daily wind gust at London
Heathrow Airport. The conditional extremes model, the JT-KDE model, the
normal copula and the FGM copula all have low

For illustration and/or confronting our models with the data, the six models
are fit to the dataset and joint exceedance level curves are produced with a
joint exceedance probability set at

In Fig. 10 are displayed the level curves produced from the four models that were selected after steps (i) to (iii) above (Cond-Ex, JT-KDE, Normalcop and FGMcop), and their corresponding values are presented in bold numbers in Table 4.

Level curves for a

From Fig. 10, we can observe that the conditional extremes model,
the FGM copula and the normal copula all produce very similar joint exceedance
curves and that their confidence intervals overlap. Table 5
displays the estimates (with bounds of the 95 % confidence interval) of
the two dependence parameters

Here we present a second example of applying our models to natural hazards data, using as a case study daily temperature and daily number of wildfires in Portugal. Wildfire variables such as daily number and burned area depend on many influences such as wind speed, direction and gustiness; topography; and type of fuel and soil moisture (Hincks et al., 2013). The aim of our study is not to decipher the processes leading to a wildfire but rather to provide an exemplar study examining the relationship between the two variables, daily temperature and daily number of wildfires, in a given case study area.

It has been shown that dry and warm conditions increase the risk of wildfire (Littell et al., 2009; AghaKouchak et al., 2018). Witte et al. (2011), establishing a direct link between a persistent heatwave and wildfire outbreaks in Russia and eastern Europe in 2010. The northern Mediterranean countries (Portugal, Spain, France, Italy and Greece) are particularly affected by summer fires (Vitolo et al., 2019).

Among these, Portugal holds the highest number of wildfires per land area (Pereira et al., 2011). There are many environmental and anthropogenic factors influencing the rural fire regime in Portugal and making its territory a fire-prone area. However, the majority of rural fires is recorded during hot and dry conditions in the summer (Pereira et al., 2011).

Here, we used the mainland Continental Portuguese Rural Fire Database, which includes 450 000 fires, covers the
period 1980–2005 (Pereira et al., 2011) and includes data for all 18
districts in Portugal. This database is the largest such database in Europe
in terms of total number of recorded fires in the 1980–2005 period (Pereira
et al., 2011) and includes fires recorded down to a size of 0.001 ha. From
the Continental Portuguese Rural Fire Database, we chose to focus on Porto District,
which was the worst affected in the period (out of the 18 Portuguese
districts) in terms of number of wildfires with 21.6 % of the total fire
recorded in the dataset between 1980 and 2005. Porto District is
situated in the northern part of Portugal (see Fig. 11), has an
area of 2395 km

Portugal study area for the interrelation between extreme hot temperature and wildfire-burned areas. The red area represents Porto District in Portugal, containing the studied wildfire-burned areas. The blue tiles represent cells from the high-resolution gridded dataset of daily climates over Europe (E-OBS; Cornes et al., 2018) containing mean daily temperature data. Satellite image retrieved with ggmap (Kahle and Wickham, 2013). © Google Maps (2020).

The bivariate dataset used to study the interrelation between extreme
temperature and wildfire-burned areas in Porto District is composed of
the following data:

The 26 years from 1980 to 2005 have a total of 9496 d. Of these, a total of 3442 d (36 % of the days) have both non-zero days for number of wildfires and a mean temperature value, which are used in our final bivariate dataset. An overview of both daily mean temperature and daily number of wildfires is displayed in Fig. 12 in the form of monthly violin plots.

Violin plot of those days with both daily mean temperature (red,
upper violin plots)

Scatterplot of temperature dependence on wildfire
occurrence in Porto District, Portugal, for the period 1980–2005, for those
days where there are recorded both mean daily temperature (

From Fig. 12 we observe the seasonality in daily mean temperature
with January the coldest month (median

As discussed in the beginning of this section, extreme (hot) temperature and wildfire are interrelated. Indeed, extreme (hot) temperature may promote the development of wildfires (Witte et al., 2011; Sutanto et al., 2020) According to Tilloy et al. (2019), this is a change condition interrelation (i.e. one hazard changes an environmental parameter that causes a move toward a change in the likelihood of another hazard). We then estimate the conditional exceedance probability curve (Sect. 2.3).

We now go through the four steps introduced at the beginning of Sect. 4.

From Figs. 5 and 13, along with empirical estimates of

This then gives us four analogous datasets, and it is then possible to know
from Fig. 8 which models are the most adapted to these conditions.
The four datasets are the following:

The confidence score for each model is the average of the

Weighted Euclidian distance (

According to these three first steps, we can identify the most relevant model for the bivariate dataset of daily maximum temperature and daily wildfire occurrence in Porto District: the Gumbel copula, Galambos copula, JT-KDE model and conditional extremes model are the most relevant models for our dataset.

For illustration and/or the confronting of the models with the data, the six
models are fit to the dataset and the joint exceedance level curves are
produced with a joint exceedance probability set at

In Fig. 14 are displayed the conditional level curves produced from the four models that were selected after steps (i) to (iii) and shown as bold values in Table 6 (Cond-Ex, JT-KDE, Gumcop and Galamboscop).

Level curves for a

From Fig. 14, we can observe that JT-KDE and the Gumbel copula produce very similar conditional exceedance curves and that their confidence intervals strongly overlap. However, the conditional extremes model provides a lower estimate than the other approaches, the number of wildfires being conditional on the temperature being above a given threshold.

In Table 7, we present the estimates (with bounds of the
95 % confidence interval) of the two dependence parameters

Estimates of dependence parameters

Quantifying and measuring the interrelations between different natural hazards is a key element when adopting a multi-hazard approach (Gill and Malamud, 2014; Leonard et al., 2014). In this study, we focused on statistical approaches that are often used to characterize and model interrelations between hazards. Another focus has been on modelling relationships between hazards at an extreme level. In total six statistical models with different characteristics (nature of asymptotic dependence, parametric and semi-parametric) were compared. Some of these models have already been used to study compound extremes in hydrology and climatology (Hao et al., 2018; Liu et al., 2018; Sadegh et al., 2018; Cooley et al., 2019). However, these have not been compared over a broad range of bivariate datasets and applied to the same natural hazards in the same location.

This section will discuss the following three themes before a short conclusion: (a) choices influencing the results of the simulation study, (b) uncertainties at the interface between asymptotic dependence and asymptotic independence, and (c) possible extensions of this approach to more than two hazards.

There are many copulas other than the four selected in this study (Nelsen, 2006; Sadegh et al., 2017) that have been developed. Nevertheless, we believe the four copulas used in this study are suitable for bivariate extreme value analysis and are amongst the most widely used in the literature (Genest and Favre, 2007; Genest and Nešlehová, 2013). Another influential choice in this study has been the number of synthetic data points generated in each realization of the dataset.

The number of data points and dataset size is an important influence on uncertainty in natural
hazard modelling and probabilistic approaches (Frau et al., 2017; Liu et
al., 2018). For each simulation, we simulated

Recent research conducted suggest pair-copula construction (Bedford and Cooke, 2002; Hashemi et al., 2016; Bevacqua et al., 2017, Liu et al., 2018) and non-parametric Bayesian networks (NPBNs; Hanea et al., 2015; Couasnon et al., 2018) can be used to model multi-hazard events with more than two hazards. The vine copula framework allows one to select different bivariate copulas for each pair of variables, providing a great flexibility in dependence modelling (Brechmann and Schepsmeier, 2013; Hao and Singh, 2016). Non-parametric Bayesian networks, which are associated with the structure of Bayesian networks and copulas (Hanea, 2010; Hanea et al., 2010, 2015), have been used to study multiple dependences between river discharge and storm surges in the USA during a hurricane (Couasnon et al., 2018).

In conclusion, we have compared and examined the strengths and weaknesses of six distinct bivariate extreme models in the context of hazard interrelations. These six models are grounded in multivariate extreme value theory and represent the diversity of approaches (e.g. non-parametric vs. parametric) currently applied to hazard interrelation analysis. With this study we aimed to contribute to a better understanding of the applicability of bivariate extreme models to a wide range of natural hazard interrelations. The methodology developed in this article is aimed to be widely applicable to a variety of different hazards and different interrelations, here represented by the 60 synthetic datasets created.

Abilities of each model have been assessed with two metrics: (i) dependence measure; (ii) bivariate return level (level curves). These two metrics and the different diagnostic tools developed in this study offer new intuitive ways to decipher the dependence between two variables. We recommend selecting a range of models rather than one when studying interrelations between two hazards. To highlight the benefits of the systematic framework developed, we studied the dependence between extremes (natural hazards) of the following environmental data: (i) daily precipitation accumulation and daily maximum wind gust (maximum over a period of 3 s) at Heathrow Airport (UK) over the period 1971–2018 and (ii) daily mean temperature and daily number of wildfires in Porto District, Portugal, over the period 1980–2005. The two datasets represent different hazard interrelations: (i) compound interrelation between extreme wind and extreme rainfall and (ii) change condition interrelation where higher air temperatures change conditions for wildfire occurrence. In both cases, a sample of the most relevant model among the six used in this study has been selected and fitted to the bivariate datasets. The good agreement in the estimation of the bivariate return period between models corroborates the relevance of the comparison metrics we developed.

Tail dependence measures

For the non-parametric joint tail approach, the

To compare the estimated dependence measure to the reference value, the root-mean-square error (RMSE), a measure of efficiency that accounts for both the bias and variance in the estimates, is used, similarly to Zheng et al. (2014). Similarly to the metrics used in Sect. 3, the RMSE is calculated from 100 realizations of the 60 datasets.

The estimation of the dependence measure is an important step in bivariate
analysis (Coles et al., 1999; Heffernan, 2000; Zheng et al., 2013, 2014;
Dutfoy et al., 2014). Models have also been compared on their ability to
estimate the dependence measures

From Fig. A1, we observe the following:

Marginal distributions do not have a significant impact on the accuracy of the estimation of these measures for the copulas.

Marginal distributions have a small impact on the estimation of the dependence measures for the conditional extremes model and the joint tail model; however this impact is not as important as for the level curve estimation.

All copulas estimate very accurately the dependence measure within their
operating range (AI for normal copula, near independence for FGM copula and
AD for Gumbel and Galambos copulas). However, only the conditional extremes
model and the joint tail model can estimate both

The dependence measure estimator used in the joint tail KDE approach offers
slightly more accurate estimation, particularly for

Estimation performance of both joint tail KDE and condition extreme models
decreases when approaching the interface between asymptotic dependence and
asymptotic independence. The RMSE at

RMSE (root-mean-square error) in the estimated dependence measures compared to the reference for all 60 different datasets. Fitting capacities of each model are represented. Values in cells and colours represent the median RMSE from low (dark green) to high (red). Thickness of cell borders represent the 95 % uncertainty around the median value.

The codes used to produce synthetic data and the six models used in this study
are publicly available on GitLab:

We acknowledge the E-OBS dataset from the EU FP6 project UERRA
(

The supplement related to this article is available online at:

All authors discussed the whole plan of this article. AT designed and implemented all the experiments, prepared all the data, and finished the draft, including all figures in the article. BDM, HW and AJL revised the article. HW supported the methods and techniques.

Bruce D. Malamud is on the editorial board of this journal (the article has been overseen independently by another editor of this journal).

This article is part of the special issue “Advances in extreme value analysis and application to natural hazards”. It is not associated with a conference.

The first author was supported by an EDF R&D PhD studentship.

Aloïs Tilloy was supported by an EDF Energy R&D PhD studentship.

This paper was edited by Thomas Wahl and reviewed by two anonymous referees.