RESEARCH ARTICLE
10.1002/2015WR017225
A multivariate copula-based framework for dealing with hazard
scenarios and failure probabilities
G. Salvadori
1
, F. Durante
2
, C. De Michele
3
, M. Bernardi
4
, and L. Petrella
5
1
Dipartimento di Matematica e Fisica, Universit
a del Salento, Lecce, Italy,
2
Faculty of Economics and Management, Free
University of Bozen-Bolzano, Bolzano, Italy,
3
Department of Hydraulic, Environmental, Roads and Surveying Engineering,
Politecnico di Milano, Milano, Italy,
4
Department of Statistical Sciences, University of Padua, Padua, Italy,
5
Department of
MEMOTEF, Sapienza University of Rome, Rome, Italy
Abstract This paper is of methodological nature, and deals with the foundations of Risk Assessment.
Several international guidelines have recently recommended to select appropriate/relevant Hazard Scenar-
ios in order to tame the consequences of (extreme) natural phenomena. In particular, the scenarios should
be multivariate, i.e., they should take into account the fact that several variables, generally not independent,
may be of interest. In this work, it is shown how a Hazard Scenario can be identified in terms of (i) a specific
geometry and (ii) a suitable probability level. Several scenarios, as well as a Structural approach, are pre-
sented, and due comparisons are carried out. In addition, it is shown how the Hazard Scenario approach
illustrated here is well suited to cope with the notion of Failure Probability, a tool traditionally used for
design and risk assessment in engineering practice. All the results outlined throughout the work are based
on the Copula Theory, which turns out to be a fundamental theoretical apparatus for doing multivariate risk
assessment: formulas for the calculation of the probability of Hazard Scenarios in the general multidimen-
sional case (d 2) are derived, and worthy analytical relationships among the probabilities of occurrence of
Hazard Scenarios are presented. In addition, the Extreme Value and Archimedean special cases are dealt
with, relationships between dependence ordering and scenario levels are studied, and a counter-example
concerning Tail Dependence is shown. Suitable indications for the practical application of the techniques
outlined in the work are given, and two case studies illustrate the procedures discussed in the paper.
1. Introduction
Several international guidelines concerning the risk assessment in engineering practice are available in liter-
ature: see, among others, European Committee for Standardization [2002]; ISO [1998]; JCSS [2008]; ISO
[2009a, 2009b]. An interesting novel approach is outlined in the Directive 2007/60/EC of The European Par-
liament and of The Council [The European Parliament and The Council, 2007]: it concerns the assessment and
the management of flood risks, but the strategies proposed are paradigmatic, and can be adopted in all
areas of environmental engineering (e.g., those concerning droughts, rainfall storms, sea storms, etc.). In
particular, the cited Directive states that [The European Parliament and The Council, 2007, p. 30, chap. III, Arti-
cle 6.3] the flood risk management should require the implementation of suitable ‘‘flood hazard maps cov-
ering the geographical areas which could be flooded according to the following scenarios: (a) floods with a
low probability, or extreme event scenarios; (b) floods with a medium probability (likely return period 100
years); (c) floods with a high probability, where appropriate.’’ Moreover, a multivariate approach is recom-
mended [The European Parliament and The Council, 2007, p. 31, chap. III, Article 6.4], since it is suggested to
consider, for each flood scenario, the following quantities: ‘‘(a) the flood extent; (b) water depths or water
level, as appropriate; (c) where appropriate, the flow velocity or the relevant water flow.’’ In turn, the scope
of the Directive is twofold. On the one hand, the EU framework requires the specification of suitable sto-
chastic models for flood events that are per se multivariate (viz., they involve a number of nonindependent
variables for the characterization of a flood). On the other hand, relevant flood scenarios of interest are indi-
cated, each associated with prescribed probability levels.
The EU guidelines pose nontrivial questions concerning the mathematical framework used to model natural
threatening phenomena: in fact,
Key Points:
Methodological approach to
multivariate risk assessment via
copulas
Probabilistically consistent definition
of multivariate hazard scenario
Calculation of the failure probability
of different multivariate hazard
scenarios
Correspondence to:
G. Salvadori,
gianfausto.salvadori@unisalento.it
Citation:
Salvadori, G., F. Durante, C. De Michele,
M. Bernardi, and L. Petrella (2016), A
multivariate copula-based framework
for dealing with hazard scenarios and
failure probabilities, Water Resour. Res.,
52, 3701–3721, doi:10.1002/
2015WR017225.
Received 12 MAR 2015
Accepted 21 APR 2016
Accepted article online 25 APR 2016
Published online 14 MAY 2016
V
C
2016. American Geophysical Union.
All Rights Reserved.
SALVADORI ET AL. MULTIVARIATE HAZARD SCENARIOS AND RISK ASSESSMENT 3701
Water Resources Research
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1. a notion of ‘‘(extreme) event scenario’’ must be introduced;
2. scenarios should be multivariate, viz. several variables, generally nonindependent, should be considered;
3. a suitable multivariate probability law must be specified.
Note that these issues are quite general, and are related to the (extreme) events associated with quite a few
environmental phenomena (for a survey, see Chebana [2013] and Straub [2014]). In recent years, several
investigations have focused on this troublesome matter. In literature [see e.g., Reeve, 2000, chap. 5; Reeve
et al., 2004, chap. 7; Kottegoda and Rosso, 2008, chap. 9; Liu et al., 2015], the occurrence of environmental
extreme event scenarios in a multivariate framework has been addressed trying to determine the probabil-
ity corresponding to a failure region, considering failure modes with elements in series, in parallel, or mixed,
both under independent and dependent circumstances. Specifically, in hydrology, examples are the joint
occurrence of flood discharge at river confluences [Raynal and Salas, 1987; Favre et al., 2004; Wang et al.,
2009; Bender et al., 2013], the superposition of river flooding and storm surges at coasts [Kew et al., 2013], or
the important role of flood duration, besides peak discharge, for dike failure [Vorogushyn et al., 2010] and
for flood losses [Merz et al., 2013]. Analogously, storm related coastal flooding events are mainly caused by
high water levels, due to a combination of astronomical tide and storm surge, and high and long waves
incident on the coast, through the effects of wave setup and runup—see Masina et al. [2015]. Moreover,
several analyses highlighted relationships existing among the main oceanographic variables and proposed
multivariate methods for the assessment of sea defenses—see e.g., Hawkes [2008]; Hawkes et al. [2002]; and
Ferreira and Soares [2002]. Alternative methods that can deal with scenarios of arbitrary geometry can be
found in Girard and Stupfler [2015]. In addition, recent advances concerning spatial risk assessment and
quantification can be found in Gr
€
aler [2014], and references therein.
The target of this paper is to elaborate a probabilistically consistent framework that, according to various
regulation requirements, is suitable for (i) dealing with the concept of multivariate hazard scenarios, and (ii)
providing valuable tools for assessing the probability of threatening of (extreme) natural occurrences. Sev-
eral are the elements of novelty introduced:
1. the concept of Hazard Scenario is defined at a general level via the notion of Upper Set, and is identified
by (i) a specific geometry, and (ii) a suitable probability level;
2. general multidimensional formulas (for the case d 2) are derived, both concerning the probabilities of
occurrence of Hazard Scenarios and Failure Probabilities;
3. the formal connections between different Hazard Scenarios are investigated, and multivariate switching
formulas for their respective probabilities are presented;
4. the Extreme Value and Archimedean special cases are dealt with, relationships between dependence
ordering and scenario levels are studied, and a counter-example concerning Tail Dependence is shown.
Throughout the paper, several indications about the choice of the Hazard Scenario to be used in practical
applications are given, and a Structural approach is dealt with as well.
The structure of this paper is as follows. Section 2 provides a general overview of the multivariate (copula-
based) setting used in the paper. In section 3, the notion of Hazard Scenario is introduced, and several cases
are discussed and compared; in addition, suitable indications concerning the application of the techniques
outlined in the work are given. In section 4, the notion of Failure Probability is recalled, and it is shown how
the Hazard Scenario approach illustrated here is well suited to cope with it. In section 5, two practical illus-
trations are shown. Finally, some conclusions are drawn in section 6.
2. The Theoretical Background
A convenient way to deal with multivariate phenomena, where the variables at play are generally noninde-
pendent, is to use Copulas [Nelsen, 2006; Salvadori et al., 2007; Durante and Sempi, 2015]. Since the introduc-
tion of copulas in hydrology [De Michele and Salvadori, 2003], a number of papers in hydrology, as well as in
other geophysical fields, have shown the theoretical and practical advantages of using a copula approach,
and support their usage. For an overview concerning different ways of quantifying the risk of compound
events see, among others, Shiau [2003]; Salvadori [2004]; Gr
€
aler et al. [2013]; Salas and Obeysekera [2014];
and Serinaldi [2015a]. A thorough list of relevant works is also available at the STAHY website (www.stahy.org).
In particular, concerning selection/estimation/testing statistical procedures for copulas, the interested
Water Resources Research 10.1002/2015WR017225
SALVADORI ET AL. MULTIVARIATE HAZARD SCENARIOS AND RISK ASSESSMENT 3702
reader may refer to Genest and Favre [2007]; Genest et al. [2009]; Choro
setal. [2010]; Kojadinovic and Yan
[2010]; Kojadinovic et al. [2011]; De Michele et al. [2013]; Fermanian [2013]; Joe [2014]; Salvadori et al. [2014],
and references therein. In addition, freeware for working with copulas, developed for the ‘‘R’’ package [R
Core Team, 2013], is available online [Hofert et al., 2013; Gr
€
aler, 2015]. The results presented in the Case Stud-
ies (see section 5 below) have been obtained using the techniques outlined in the cited works, to which the
reader is referred.
In the following, the same notation used in Salvadori et al. [2011, 2013] is adopted. In particular, I denotes
the unit interval [0,1], and L
t
indicates the level set at t 2 I of the (joint) continuous distribution
FðxÞ5CðF
1
ðx
1
Þ; ...; F
d
ðx
d
ÞÞ. Practically, L
t
is the set of points in the d-dimensional Euclidean space R
d
such
that FðxÞ5t—L
t
will also be referred to as a ‘‘critical layer’’ of level t. Here C is the copula of the random vec-
tor X5ðX
1
; ...; X
d
Þ describing the phenomenon under investigation, with univariate margins F
i
’s (assumed
to be strictly increasing on their support), according to the representation given by Sklar’s Theorem [Sklar,
1959]. Similarly,
L
t
denotes the critical layer of the (joint) survival function FðxÞ5
PðX
1
> x
1
; ...; X
d
> x
d
Þ5
^
Cð
F
1
ðx
1
Þ; ...;
F
d
ðx
d
ÞÞ, where
^
C is the survival copula of the X
i
’s, and
F
i
512F
i
is
the survival function of X
i
, with i51; ...; d. Note that
^
C can be written in terms of the copula C by using the
Inclusion-Exclusion principle [see Joe, 2014, p. 27], since
^
CðuÞ5
Cð12uÞ for all u 2 I
d
, and C is the survival
function associated with C given by
CðuÞ512
X
d
i51
u
i
1
X
S2P
ð21Þ
#
ðSÞ
C
S
ðu
i
: i 2 SÞ; (1)
where P is the set of all subsets of f1; 2; ...; dg with at least two elements, #ðSÞ denotes the cardinality of
S, and C
S
ðu
i
: i 2 SÞ represents the marginal copula of C, with dimension equal to #ðSÞ, involving only those
indices i’s belonging to S. As will be made clear below, both L
t
and L
t
play the role as of (critical) multivari-
ate thresholds, with dimension d –1.
Furthermore, due to the assumption that the F
i
’s are strictly increasing, the Probability Integral Transform
(hereinafter, PIT) viz., the relations
U5ðU
1
; ...; U
d
Þ5ðF
1
ðX
1
Þ; ...; F
d
ðX
d
ÞÞ5T
F
ðXÞ; (2)
and
X5ðX
1
; ...; X
d
Þ5ðF
21
1
ðU
1
Þ; ...; F
21
d
ðU
d
ÞÞ5T
21
F
ðUÞ; (3)
are one-to-one. These formulas map the vector U living in I
d
onto the vector X living in the d-dimensional
Euclidean space R
d
(and vice-versa)—see Nelsen [2006]; Salvadori et al. [2007]; and Embrechts and Hofert
[2013]. Thanks to the invariance of copulas for strictly increasing transformations [Nelsen, 2006, Theorem
2.4.3], U and X share the same copula.
Note that, thanks to Sklar’s Theorem, the PIT uniquely maps the probabilities of the events in I
d
(as induced
by the copula C) onto R
d
(as induced by F5CðF
1
; ...; F
d
Þ), and vice-versa; the same holds for
^
C and F. The
role played by the univariate margins is only to geometrically remap such probabilities onto suitable
regions in the Euclidean space R
d
(and vice-versa), without affecting them. By the same token, also L
t
and
L
t
are mapped from R
d
onto I
d
(and vice-versa), thus becoming the level sets of C and
^
C, respectively.
A further notion of interest is represented by the Kendall’s function K [Genest and Rivest, 1993; Barbe et al.,
1996; Genest et al., 2011] associated with the copula C of X, which yields the following probability:
KðtÞ5PFðX
1
; ...; X
d
ÞtðÞ5PCðF
1
ðX
1
Þ; ...; F
d
ðX
d
ÞÞ tðÞ; (4)
with t 2 I. For a graphical illustration of K see, e.g., in Salvadori et al. [2011, Figure 4].
Finally, analogously to the case of distribution functions, it is possible to define an upper-orthant Kendall dis-
tribution function
^
K associated with
F, and given by
^
KðtÞ5 P
FðX
1
; ...; X
d
Þt
5P
^
Cð
F
1
ðX
1
Þ; ...;
F
d
ðX
d
ÞÞ t
; (5)
with t 2 I. The survival function associated with
^
K will be (loosely) called Survival Kendall , and will be
denoted by
K, viz.
K512
^
K—for more details, see Nappo and Spizzichino [2009]; Salvadori et al. [2013,
2014]; and Cousin and Di Bernardino [2014]. For a graphical illustration of
K see, e.g., in Salvadori et al. [2013,
Figure 2].
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SALVADORI ET AL. MULTIVARIATE HAZARD SCENARIOS AND RISK ASSESSMENT 3703
3. Hazard Scenarios
In the following, the notion of Hazard Scenario will be fundamental. In particular, the concept of Upper Set
[Davey and Priestley, 2002] will play an important role.
Definition 1 (Upper Set). SR
d
is an Upper Set if, and only if, x 2Sand y x component-wise imply
y 2S. 䊊
The notion of Upper Set well copes with the intuitive (and practical) reasoning that, if an occurrence is risky,
then also ‘‘larger’’ realizations may be threatening. In turn, a possible definition of Hazard Scenario is as
follows.
Definition 2 (Hazard Scenario). Let X be a random vector describing the phenomenon of interest. A Haz-
ard Scenario (hereinafter, HS) of level a 2ð0; 1Þ is any Upper Set S5S
a
R
d
such that the relation
PðX 2SÞ5a (6)
holds. 䊊
In order to keep the notation as simple as possible, in the following the dependence upon a will be sup-
pressed whenever no confusion may arise. As a practical interpretation, a Hazard Scenario can be conceived
as a set containing all the occurrences x’s reputed to be ‘‘dangerous’’ (i.e., possibly affecting/damaging a
given structure) according to some suitable criterion. In particular, if x 2Sand y x component-wise, then
also y could be considered as a dangerous occurrence. Last, but not least, here it is important to stress the
probabilistic foundational novelty brought by Definition 2 (viz., the a-level characterization of HS’s), which is
different from the traditional geometrical approach (as e.g., in Serinaldi [2015a, and references therein]). The
advantages of the formalization of Hazard Scenarios proposed in this work will be well appreciated later, in
the Case Studies section.
Since a Hazard Scenario is an Upper Set, it deals with situations where large values of the variables of inter-
est are associated with dangerous conditions—as is common in many environmental engineering applica-
tions. When, on the contrary, small values of some variables X
i
’s may be dangerous (for instance, think of
small discharges in case of droughts), formally it is enough to change the sign of the variables of interest, or
carrying out other suitable transformations as in AghaKouchak et al. [2014].
As will be shown below, given a realization x 2 R
d
, there exist several ways to associate x with a suitable
HS, occasionally denoted by S
x
R
d
for the sake of clarity and notational convenience. Clearly, via the PIT
(see equations (2) and (3)), there exists in I
d
a unique realization u5T
F
ðxÞ corresponding to x, as well as a
unique region S
u
I
d
corresponding to S
x
. In turn, the knowledge of the copula at play may suffice to cal-
culate the level of S
u
, and hence of S
x
.
In literature, several scenarios are usually considered. The choice of a specific HS to be used in practice may
depend upon two different, and complementary, criteria: viz., the type of events considered to be menac-
ing, and their probabilities of occurrence, as will be made clear below (see also Table 1, the examples pre-
sented later in section 3.1, and the survey in Serinaldi [2015a]).
1. ‘‘OR’’ scenario S
Ú
.Ad-dimensional OR HS is given by the region
S
Ú
x
5
[
d
i51
ðR3 3ðx
i
; 11Þ3 3RÞ; (7)
and the associated level a is
a
Ú
x
5PX2S
Ú
x
512CðF
1
ðx
1
Þ; ...; F
d
ðx
d
ÞÞ: (8)
For the realization of the event fX 2S
Ú
x
g it is sufficient that one of the variables X
i
’s, with i51; ...; d,
exceeds the corresponding threshold x
i
. The shape of a bivariate OR HS is illustrated in Figure 1a, consid-
ering the pair ðU; VÞ2I
2
(see, among others, Yue and Rasmussen [2002]; Shiau [2003], and also Salvadori
[2004]; Salvadori and De Michele [2004]; De Michele et al. [2005]).
Water Resources Research 10.1002/2015WR017225
SALVADORI ET AL. MULTIVARIATE HAZARD SCENARIOS AND RISK ASSESSMENT 3704
2. ‘‘AND’’ scenario S
Ù
.Ad-dimensional AND HS is given by the region
S
Ù
x
5
\
d
i51
ðR3 3ðx
i
; 11Þ3 3RÞ; (9)
and the associated level a is
a
Ù
x
5PX2S
Ù
x
5
^
Cð
F
1
ðx
1
Þ; ...;
F
d
ðx
d
ÞÞ: (10)
For the realization of the event fX 2S
Ù
x
g it is necessary that all the variables X
i
’s, with i51; ...; d,
exceed the corresponding thresholds x
i
’s. The shape of a bivariate AND HS S
Ù
u;v
is illustrated in Figure 1b,
considering the pair ðU; VÞ2I
2
(see, among others, Yue and Rasmussen [2002]; Shiau [2003], and also Sal-
vadori [2004]; Salvadori and De Michele [2004]; De Michele et al. [2005]).
3. ‘‘Kendall’’ scenario S
K
. Let x 2 R
d
be a given occurrence, with t5FðxÞ, and let L
t
be the level set crossing
x.Ad-dimensional Kendall HS is given by the region
S
K
t
5fy 2 R
d
: FðyÞ > tg5fy 2 R
d
: CðF
1
ðy
1
Þ; ...; F
d
ðy
d
ÞÞ > tg; (11)
and the associated level a is [Salvadori et al., 2011]
Table 1. Literature Survey (in a Chronological Order) Concerning the Usage of the Approaches OR, AND, Kendall (K.), and Survival Ken-
dall (S.K.)—See Text
a
Reference OR AND K. S.K.
Yue [2000a] *
Yue [2000b] *
Yue [2001a] *
Yue [2001b] *
Yue [2002] *
Yue and Rasmussen [2002] * *
Yue and Wang [2004] *
Shiau [2003] * *
Salvadori and De Michele [2004] * * *
De Michele et al. [2005] * * *
Shiau [2006] * *
Salvadori and De Michele [2007] * * *
Salvadori et al. [2007] * * *
Poulin et al. [2007] *
Shiau and Modarres [2009]
Karmakar and Simonovic [2009] *
Salvadori and De Michele [2010] * * *
Salvadori et al. [2011] * * *
Klein et al. [2011] * * *
Fan et al. [2012] *
Zhang and Singh [2012] *
Corbella and Stretch [2012] *
Salvadori et al. [2013] **
De Michele et al. [2013] *
Li et al. [2013b] *
Li et al. [2013a] * *
Lian et al. [2013] * *
Gr
€
aler et al. [2013] * * *
Chen et al. [2013] * * *
Zhang et al. [2013] * * *
Requena et al. [2013] * * *
Salvadori et al. [2014] *
Jeong et al. [2014] * *
Serinaldi [2015a] * * *
Mitkova and Halmova [2014] * *
AghaKouchak [2014] *
AghaKouchak et al. [2014] *
Xu et al. [2014] *
Volpi and Fiori [2014] * *
Salvadori et al. [2015] *
Ming et al. [2015] *
Liu et al. [2015] *
Serinaldi [2015b] * * * *
a
The ‘‘*’’ indicates which approach is discussed in the corresponding reference.
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SALVADORI ET AL. MULTIVARIATE HAZARD SCENARIOS AND RISK ASSESSMENT 3705
