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Bulletin of Earthquake Engineering (2019) 17:1769–1793
https://doi.org/10.1007/s1051801800531x
1 3
ORIGINAL RESEARCH
REASSESS V2.0: software forsingle‑ andmulti‑site
probabilistic seismic hazard analysis
EugenioChioccarelli1 · PasqualeCito2· IunioIervolino2· MassimilianoGiorgio3
Received: 28 March 2018 / Accepted: 29 November 2018 / Published online: 5 December 2018
© Springer Nature B.V. 2018
Abstract
Probabilistic seismic hazard analysis (PSHA) is generally recognized as the rational
method to quantify the seismic threat. Classical formulation of PSHA goes back to the sec
ond half of the twentieth century, but its implementation can still be demanding for engi
neers dealing with practical applications. Moreover, in the last years, a number of develop
ments of PSHA have been introduced; e.g., vectorvalued and advanced ground motion
intensity measure (IM) hazard, the inclusion of the eﬀect of aftershocks in singlesite haz
ard assessment, and multisite analysis requiring the characterization of random ﬁelds of
crosscorrelated IMs. Although software to carry out PSHA has been available since quite
some time, generally, it does not feature a userfriendly interface and does not embed most
of the recent methodologies relevant from the earthquake engineering perspective. These
are the main motivations behind the development of the practiceoriented software pre
sented herein, namely REgionAl, SingleSitE and Scenariobased Seismic hazard analysis
(REASSESS V2.0). In the paper, the seismic hazard assessments REASSESS enables are
discussed, along with the implemented algorithms and the models/databases embedded in
this version of the software. Illustrative applications exploit the potential of the tool, which
is available at http://wpage .unina .it/iunie rvo/doc_en/REASS ESS.htm.
Keywords Performancebased earthquake engineering· Performancebased seismic
design· Sequencebased probabilistic seismic hazard analysis· Spectralshapebased
intensity measures· Infrastructure risk analysis· Conditional spectra
* Eugenio Chioccarelli
eugenio.chioccarelli@unipegaso.it
Pasquale Cito
pasquale.cito@unina.it
Iunio Iervolino
iunio.iervolino@unina.it
Massimiliano Giorgio
massimiliano.giorgio@unicampania.it
1 Università Telematica Pegaso, piazza Trieste e Trento 48, 80132Naples, Italy
2 Dipartimento di Strutture perl’Ingegneria e l’Architettura, Università degli Studi di Napoli
Federico II, via Claudio 21, 80125Naples, Italy
3 Dipartimento di Ingegneria, Università degli Studi della Campania Luigi Vanvitelli, via Roma 29,
80131Aversa, CE, Italy
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1 Introduction
The classical (singlesite) formulation of probabilistic seismic hazard analysis (PSHA)
aims at evaluating the rate of earthquakes causing exceedance of any arbitrary ground
motion intensity measure (IM) threshold (im) at a site of interest (Cornell 1968). PSHA
lies at the basis of seismic risk assessment according to the performancebased earthquake
engineering paradigm (Cornell and Krawinkler 2000) and serves for the determination of
seismic actions for structural design in several countries.
The probabilistic assessment of the seismic threat at a site is, in principle, not straight
forward for structural engineers because it requires the employment of models and skills
they do not typically have at hand. For this reason, during the last four decades, computer
software to carry out PSHA have become available, starting from EQRISK (McGuire
1976). Other relevant codes are, for example, SEISRISK III (Bender and Perkins 1987),
OpenSHA (Field etal. 2003) and CRISIS (Ordaz etal. 2013); see Danciu etal. (2010).
Recently, the global earthquake model (GEM) foundation developed OpenQuake (Pagani
etal. 2014) that has been adopted, among others, within the EMME (Giardini etal. 2018)
and SHARE (Woessner etal. 2015)hazard assessment projects.
PSHA has been signiﬁcantly extended since its introduction in the late sixties. For
example, its classical version refers to a scalar IM, while advanced structural assessment
procedures may require hazard in terms of vectorvalued IMs (Baker and Cornell 2006b)
or, equivalently, development of conditional hazard for secondary IMs (Iervolino etal.
2010). Typically, PSHA is carried out considering spectral accelerations as the IM, while
in the last years more eﬃcient intensity measures have been introduced for more accurate
seismic structural assessment (e.g., Cordova etal. 2000; Bianchini etal. 2009; Bojórquez
and Iervolino 2011). Furthermore, PSHA, as normally implemented, only refers to main
shocks (see next section) neglecting the eﬀect of foreshocks and aftershocks on seismic
hazard for a site. In other words, PSHA only considers the exceedance of the im thresh
old of interest due to prominent magnitude earthquakes within a cluster of events; i.e.,
the typical way earthquakes occur (e.g., Boyd 2012; Marzocchi and Taroni 2014). This
is to take advantage of the ease of calibration and mathematical manageability of the
homogeneous Poisson process (HPP) (e.g., Cornell 1968; McGuire 2004). Nevertheless,
recently, a generalized hazard integral, able to account for the eﬀect of aftershocks with
out losing the advantages of HPP, was developed and named sequencebased probabilis
tic seismic hazard analysis or SPSHA (Iervolino etal. 2014). Finally, in some situations,
for example in the case of risk assessment of building portfolios or spatiallydistributed
infrastructure, hazard must account for exceedances at multiple sites jointly. In this case,
which may be referred to as multisite probabilistic seismic hazard analysis (MSPSHA),
the key issue is to account for the stochastic dependence existing among the processes
counting exceedances at each of the considered sites (e.g., Eguchi 1991; Giorgio and
Iervolino 2016).
To provide an engineeringoriented tool including a number of stateoftheart advances
in probabilistic seismic hazard analysis, a standalone software named REgionAl, Single
SitE and Scenariobased Seismic hazard analysis (REASSESS V2.0), with a graphical user
interface (GUI), has been developed and it is presented herein.1 To this aim, the remain
der of this paper is structured such that the hazard assessment methodologies considered
1 An early release of REASSESS (V1.0) was introduced in Iervolino etal. (2016a).
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are recalled ﬁrst, along with the algorithms and numerical procedures developed for their
implementation. Subsequently, REASSESS V2.0 is presented with the main input and out
put options. Finally, illustrative examples show the tools capabilities for earthquake engi
neering practice.
2 Single‑site PSHA essentials
In classical PSHA, earthquakes on a seismic source are assumed to occur according to a
HPP characterized by a rate,
𝜈
. In other words, the probability of observing, in the time
interval
ΔT
, a number of earthquakes,
N(ΔT)
, exactly equal to n is given by Eq.(1).
The objective of PSHA is to compute the rate,
𝜆im
, of seismic events exceeding the im
threshold at a site of interest. Such a rate completely deﬁnes the HPP describing the occur
rence of the events causing exceedance of im. In other words, the probability that, in the
time interval
ΔT
, the number of earthquakes causing exceedance of im at the site,
Nim(ΔT)
,
is equal to n, is given by Eq.(2).
For a site subjected to earthquakes generated at
ns
seismic sources, the rate
𝜆im
can be com
puted as illustrated in Eq.(3), known as the hazard integral.
In the equation the i subscript indicates the ith seismic source;
𝜈i
is the rate of earthquakes
above a minimum magnitude of interest
(
m
min,i)
and below the maximum magnitude
deemed possible for the source
(
m
max,i)
;
fM,X,Y,i(m,x,y)
is the joint probability density func
tion (PDF) of earthquake magnitude
M
and location
{X,Y}
;
P[
IM >im

m,x,y
]i
, typically
provided by a ground motion prediction equation (GMPE), is the exceedance probability
conditional on the magnitude and location (via a sourcetosite distance metric). GMPEs,
usually, also account for soil type, rupture mechanism and other parameters that are not
explicitly considered in the notation here for the sake of simplicity (see also Sect.2.1).
It is also for simplicity that the location is deﬁned in Eq.(3) by means of two horizontal
coordinates that can represent, for example, the epicenter. This representation is typically
used in the case of areal source zones; however, it is frequent that hazard assessments have
to account for threedimensional faults (see Sect.5.1). Moreover, it also happens that the
distance metric of the selected GMPE is not consistent with the way location is deﬁned.
In these cases, because the relationship between location and sourcetosite distance is not
necessarily deterministic, the hazard integral has to account for the probabilistic distribu
tion of the distance metric of the GMPE, conditional to the considered location parameters
(e.g., Scherbaum etal. 2004).
(1)
P
[N(ΔT)=n]=(𝜈⋅ΔT)
n
n!
⋅e−𝜈⋅Δ
T
(2)
P[
Nim(ΔT)=n
]
=
(
𝜆im ⋅ΔT
)n
n!
⋅e−𝜆im⋅ΔT
(3)
𝜆
im =
n
s
∑
i=1
𝜈i⋅∫
M
∫
X
∫
Y
P
[
IM >im

m,x,y
]
i
⋅fM,X,Y,i(m,x,y)⋅dm ⋅dx ⋅
dy
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Magnitude and location of the earthquake are often considered stochastically inde
pendent, that is
fM,X,Y,i(m,x,y)=fM,i(m)
⋅
fX,Y,i(x,y)
. The distribution
fM,i(m)
is often
modeled as an exponential distribution in the
(
m
min,i
,m
max,i)
interval; i.e., of Guten
berg–Richter (G–R) type (Gutenberg and Richter 1944); however, other choices are also
considered by literature (e.g., Convertito et al. 2006). The distribution of earthquake
location,
fX,Y,i(x,y)
, typically reﬂects the hypothesis of uniformlydistributed probability
on the source. For further details on classical PSHA, the interested reader is referred to,
for example, Reiter (1990).
Equation(3) can be numerically solved via a matrix formulation approximating the
integrals with summations. To this aim, MATHWORKSMATLAB® provides a sim
ple computing environment that can be used to evaluate this expression. The domain
of the possible realizations of the magnitude random variable (RV) is discretized via k
magnitude bins represented by the values
{
m
1
,m
2
,…,m
k}
, while the seismic source is
discretized by means of s pointlike seismic sources,
{
(x,y)
1
,(x,y)
2
,…,(x,y)
s}
. Given
these two vectors of size
1×k
and
1×s
, Eq.(3) can be approximated by Eq.(4), where
the row vector approximates
fX,Y,i(x,y)
by a mass probability function (MPF) described
by a vector in a way that each element is repeated
k
times; i.e., the ﬁrst
k
elements
are the probabilities of
(x,y)1
, the elements from
k+1
until
2k
are for
(x,y)2
and so on,
until
(x,y)s
. Thus, the row vector has size
1×(k
⋅
s)
. The ﬁrst column vector of Eq.(4)
is a
(k
⋅
s)×1
vector and accounts for the GMPE: each element represents the exceed
ance probability conditional to magnitude and location. The second column vector of
the equation collects the ﬁnite
k
probabilities of event’s magnitude, identically repeated
stimes, as shown and it is, again, a
(k
⋅
s)×1
vector. Finally, in the equation, the point
wise multiplication between matrices of the same size (i.e., the Hadamard product,
represented by the
⊗
symbol) results in a matrix of the size of those multiplied, and
in which each element is the product of the corresponding elements of the original
matrices.
Equation(4), as already discussed with respect to Eq.(3), is written in the case location
can be deﬁned by means of two coordinates and the distance metric of the GMPE is a
deterministic function of the location. Otherwise, it is necessary to account for the non
deterministic transformation of the location in sourcetosite distance, which can be done
in the same framework presented herein.
(4)
𝜆
im =
n
s
i=1
𝜈i⋅
P
(x,y)1
P
(x,y)1
…P
(x,y)1
…P
(x,y)s
P
(x,y)s
…P
(x,y)s
i
⋅
PIM >imm1,(x,y)1
PIM >imm2,(x,y)1
⋮
PIM >immk,(x,y)1
⋮
PIM >imm1,(x,y)s
PIM >imm2,(x,y)s
⋮
P
IM >im
mk,(x,y)s
i
⊗
Pm1
Pm2
⋮
Pmk
⋮
Pm1
Pm2
⋮
P
mk
i
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To compute the hazard curve, that is the function providing
𝜆im
as a function of im,
the hazard integral has to be computed for a number of values of
im
, say q in number,
discretizing the domain of IM, that is
{
im
1
,im
2
,…,im
q}
. The corresponding rates,
{
𝜆im1,𝜆im2,…,𝜆imq
}
, can be obtained via a single matrix operation conceptually equiva
lent to Eq.(4); see Iervolino etal. (2016a).
2.1 Disaggregation
Disaggregation of seismic hazard (e.g., Bazzurro and Cornell 1999) is a procedure that
allows identiﬁcation of the hazard contribution of one or more RVs involved in the haz
ard integral: e.g., magnitude and sourcetosite distance,
R
, which, as discussed, is a
function of the earthquake location. Another RV typically considered in hazard disag
gregation is
𝜀
(epsilon). It is the number of standard deviations that
log (im)
is away from
the median of the GMPE considered in hazard assessment. In fact, classical GMPEs
are of the type in Eq. (5), where
log (im)
is related to magnitude, distance and other
parameters.
In the equation,
𝜎⋅𝜀
is a zeromean Gaussian RV with standard deviation
𝜎
; often it is split
in inter and intraevent components in a way that
𝜎
=
√
𝜎2
intra
+𝜎2
inter
. The
𝜇(m,r)
term
depends on magnitude and distance, θ represents one or more coeﬃcients accounting, for
example, for the soil site class. Ultimately,
𝜇(m,r)+𝜃
is the mean, and the median, of the
logarithms of IM given
{m,r,𝜃}
. (Note that, although the majority of the GMPEs is of the
type in Eq.(5), see Douglas (2014), most of the recent models have soil factors that also
change with magnitude and distance. This representation is considered herein to discuss
some shortcuts implemented in REASSESS and that apply only in this case; see Sects.2.3,
4.1.)
The result of disaggregation is the joint PDF of
{M,R,𝜀}
conditional to the exceed
ance of an IM threshold,
fM,R,𝜀IM
, as per Eq.(6).
In the equation,
I
is an indicator function that equals one if IM is larger than im for a given
magnitude, distance and
𝜀
, while
fM,R,𝜀,i(m,r,e)
is the marginal joint PDF obtained from
the product
fM,R,i(m,r)
⋅
f𝜀(e)
.
From the engineering perspective, hazard disaggregation is useful to identify the
characteristics of the earthquake scenarios providing the largest contribution to the haz
ard being disaggregated and, consequently, for hazardconsistent seismic input selection
for structural assessment (e.g., Lin etal. 2013). Moreover, it is a required information
to compute the conditional hazard for secondary intensity measures, which is brieﬂy
recalled in the next section. Finally, note that disaggregation can also be obtained for
the occurrence of im, that is
IM =im
, and REASSESS provides also this result; i.e.,
McGuire (1995). For a discussion on whether exceedance or occurrence disaggregation
is needed in earthquake engineering, see, for example, Fox etal. (2016).
(5)
log (im)=𝜇(m,r)+𝜃+𝜎
⋅
𝜀
(6)
fM,R,𝜀
�
IM (m,r,e)=
∑n
s
i=1𝜈i⋅I[IM >im
�
m,r,e]i⋅fM,R,𝜀,i(m,r,e
)
𝜆
im
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2.2 Conditional hazard
Vectorvalued probabilistic seismic hazard analysis (VPSHA), originally introduced by
Bazzurro and Cornell (2002), provides the rate of earthquakes causing joint occurrence
(or exceedance) of the thresholds of two IMs at the site. VPSHA could improve the pre
diction of structural damage (e.g., Baker 2007). If one of the two intensity measures can
be considered of primary importance with respect to the other, conditional hazard (Ierv
olino etal. 2010) can be considered an alternative to VPSHA. Conditional hazard pro
vides the distribution of a secondary intensity measure
(
IM
2)
, conditional to the occur
rence (or exceedance) of a threshold of the primary one, that is
IM1=im1
(or
IM1>im1
). In the hypothesis of bivariate normality of the logarithms of the two IMs,
the conditional mean
(
𝜇log IM2

IM1,M,R
)
and standard deviation
(
𝜎log IM2

IM1
)
of
log (
IM
2)
,
given
IM1
, magnitude and distance, are reported in Eq.(7).
In the equation,
𝜇log IM2M,R
and
𝜎log IM2M,R
are the mean and standard deviation of
log IM2
;
𝜇
log IM
1
M,
R
and
𝜎
log IM
1
M,
R
are the mean and standard deviation of
log IM1
according to
the selected GMPE (these terms are simply indicated as
𝜇(m,r)
and
𝜎
, respectively, in
Sect.2.1);
𝜌
is the correlation coeﬃcient between residuals of
log IM1
and
log IM2
at the
site (e.g., Baker and Jayaram 2008). Thus, the conditional distribution of the logarithm
of the secondary IM is given by Eq. (8) in which
f
M,R,𝜀

IM
1
is from disaggregation and
f
log IM
2
IM
1
,M,R,
𝜀
is the distribtion with the parameters in Eq.(7).
Factually, the conditional hazard formulation of Eq.(8) allows VPSHA to be an output of
REASSESS. This is because, for example,
f
log IM
2
IM
1
multiplied by the absolute value of
the derivative of the hazard curve from Eq.(3),


d𝜆
im

, calculated in
im1
, allows to obtain
the joint annual rate of
{
IM
1
,IM
2}
for any pair of arbitrarilyselected realizations of the
two IMs,
𝜆
IM
1
=im
1,
IM
2
=im
2
, as per Eq.(9).
2.3 Logic tree andshortcuts forGMPEs withadditive soil factors
PSHA is often implemented considering a logic tree, which allows accounting for model
uncertainty (e.g., McGuire 2004; Kramer 1996); indeed, it allows the use of alternative
models, each of which is assigned a weighing factor that is interpreted as the probability
of that model being the true one. When the logic tree of
nb
branches is of concern,
𝜆im
(7)
⎧
⎪
⎨
⎪
⎩
𝜇log IM2
�
IM1,M,R=𝜇log IM2�M,R+𝜌⋅𝜎log IM2�M,R⋅
log IM1−𝜇log IM1�M,R
𝜎log IM1�M,R
𝜎log IM2
�
IM1
=𝜎log IM2
⋅
√
1−𝜌2
(8)
flog IM2IM1
log im2
im1
=∫
M
∫
R
∫
𝜀
flog IM2
IM1,M,R,𝜀log im2
im1,m,r,e⋅fM,R,𝜀
IM1m,r,e
im1⋅dm ⋅dr ⋅d
𝜀
(9)
𝜆
IM
1
=im
1
,IM
2
=im
2
=
d𝜆im
1
⋅flog IM
2
IM
1
log im2
im1
.
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is computed through Eq.(10) in which
pj
and
𝜆im,j
are the weight and the result of each
branch of the logic tree, respectively.
It should also be noted that, according to Eq.(5), and only in the case of GMPEs of this
type, it can be easily demonstrated that, if PSHA is performed without logic tree: (1) haz
ard curves for the condition represented by
𝜃
(e.g., a speciﬁc site soil class) can be obtained
shifting, in the logarithmic space, those for a reference condition when
𝜃=0
; and (2) dis
aggregation distribution does not depend on
𝜃
(i.e., disaggregation does not change with
the soil site class). Moreover, if a logic tree featuring diﬀerent GMPEs, with this same type
of structures, is adopted, the discussed translation of hazard curves can be applied to the
result of each branch, then reapplying Eq.(10) provides the hazard in the changed condi
tions (Iervolino 2016).
3 Sequence‑based probabilistic seismic hazard analysis
Classical singlesite PSHA discussed in the previous section neglects the eﬀect of after
shocks on the exceedance rate. This descends from the fact that the rates
𝜈i
,
{
1, 2, …,n
s}
are obtained removing alleged foreshocks and aftershocks from earthquake catalogs; i.e.,
they refer to the socalled declustered catalogs. This is mainly because declustering is
needed for the HPP to apply (Gardner and Knopoﬀ 1974). Recently, Boyd (2012) discussed
that mainshock–aftershock sequences occur, on each seismic source, with the same rate of
the mainshocks; i.e.,
𝜈i
of Eq.(3). Then, Iervolino etal. (2014) demonstrated the possibility
to include the eﬀect of aftershocks in PSHA still working with HPP and declustered cata
logs. On this premise, the SPSHA, was developed combining PSHA with the aftershock
probabilistic seismic hazard analysis (APSHA) of Yeo and Cornell (2009). As a result,
for any given imvalue, SPSHA provides the annual rate,
𝜆im
, of mainshock–aftershock
sequences that cause exceedance of
im
at the site, which can be computed via Eq.(11).
In the equation, the terms:
𝜈i
,
P[
IM
≤
im

m,x,y
]i
=1−P
[
IM >im

m,x,y
]i
, and
fM,X,Y,i(m,x,y)
are the same deﬁned in Eq. (3). The exponent
E[
N
A,imm(
0, ΔT
A)]
refers
to aftershocks, as indicated by the A subscript: it represents the average number of after
shocks that cause exceedance of im in a sequence triggered by the mainshock of magnitude
and location
{m,x,y}
, Eq.(12).
(10)
𝜆
im =
n
b
∑
j=1
𝜆im,j⋅p
j
(11)
𝜆
im =
ns
i=1
𝜈i⋅
1−�
M
�
X
�
Y
P
IM ≤im
m,x,y
i
⋅e−E[NA,imm(0,ΔTA)]⋅fM,X,Y,i(m,x,y)⋅dm ⋅dx ⋅dy
(12)
E[
NA,imm
(
0, ΔTA
)]
=E
[
NAm
(
0, ΔTA
)]
⋅∫
M
A
∫
X
A
∫
Y
A
P
[
IM >im

mA,xA,yA
]
i
⋅fMA,XA,YA,i

M,X,Y
(
mA,xA,yA

m,x,y
)
⋅dmA⋅dxA⋅dy
A
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The probability represented by the exponential term depends on
P[
IM >im

m
A
,x
A
,y
A]i
,
that is the probability that im is exceeded given an aftershock of magnitude and loca
tion identiﬁed by the vector
{
m
A
,x
A
,y
A}
; i.e., a GMPE for aftershocks (although
in several applications those for mainshock are considered applicable). The term
f
M
A
,X
A
,Y
A
,i

M,X,
Y
is the distribution of magnitude and location of aftershocks, which is con
ditional on the features,
{m,x,y}
, of the mainshock. This distribution can be written as
f
M
A
,X
A
,Y
A
,i

M,X,Y
=f
M
A
,i

M⋅
f
X
A
,Y
A
,i

M,X,
Y
, where
f
M
A
,i

M is the PDF of aftershock magnitude of
G–R type, and
fXA,YA,iM,X,Y
is the distribution of the location of the aftershocks and depends
on the magnitude and location of the mainshock (e.g., Utsu 1970).
E[
N
Am(
0, ΔT
A)]
is the
expected number of aftershocks to the mainshock of magnitude
m
in the
ΔTA
and, accord
ing to Yeo and Cornell (2009), can be computed via Eq.(13) in which
mA,min
is the mini
mum magnitude considered for aftershocks (often taken equal to the minimum magnitude
considered for mainshocks) and
{a,b,c,p}
are parameters of the modiﬁed Omori Law.
Finally, note that
𝜆im
is still the rate of the HPP of the kind in Eq.(2), which now regulates
the occurrence of sequences causing exceedance of im.
The matrix formulation presented in Eq.(4) for the numerical computation of PSHA,
can be extended to the SPSHA case as reported in Eq.(14). In the latter, the vectors are
arranged as discussed referring to Eq.(4), but a new column vector is introduced: it has
the same
(k
⋅
s)×1
size and each element of it accounts for the probability that none of the
aftershocks, after a mainshock of given magnitude and location, causes the exceedance of
im.
3.1 SPSHA disaggregation
Disaggregation of seismic hazard can be performed also in the case of SPSHA. Equa
tion(15) provides the PDF of mainshock magnitude and distance
(R)
, given that the ground
motion intensity of the mainshock,
IM
, or the maximum ground motion intensity of the
following aftershock sequence
(
IM
∪A)
is larger than the im threshold. In the equation,
(13)
E
NA
m
0, ΔTA
=10
a+b
⋅
(m−m
A,min
)
−10
a
p−1
⋅
c1−p−
ΔTA+c
1−p
(14)
𝜆
im =
n
s
i=1
𝜈i⋅
1−
P
(x,y)1
P
(x,y)1
…P
(x,y)1
…P
(x,y)s
P
(x,y)s
…P
(x,y)s
i
⋅
PIM ≤imm1,(x,y)1
PIM ≤imm2,(x,y)1
⋮
PIM ≤immk,(x,y)1
⋮
PIM ≤imm1,(x,y)s
PIM ≤imm2,(x,y)s
⋮
P
IM ≤im
mk,(x,y)s
i
⊗
e−ENAm1(0,ΔTA)⋅P[IMA>imm1,(x,y)1]
e−ENAm2(0,ΔTA)⋅P[IMA>imm2,(x,y)1]
⋮
e−ENAmk(0,ΔTA)⋅P[IMA>immk,(x,y)1]
⋮
e−ENAm1(0,ΔTA)⋅P[IMA>imm1,(x,y)s]
e−ENAm2(0,ΔTA)⋅P[IMA>imm2,(x,y)s]
⋮
e−E
NA
mk(0,ΔTA)
⋅P[IMA>im
mk,(x,y)s]
i
⊗
Pm1
Pm2
⋮
Pmk
⋮
Pm1
Pm2
⋮
P
mk
i
.
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similarly to what discussed in Sect.2.1,
{X,Y}
and
{
X
A
,Y
A}
vector RVs, are substituted by
R
and
RA
, respectively.
Moreover, it can be useful to quantify the probability that, given the
im
exceedance, such
exceedance is caused by an aftershock rather than by a mainshock. This probability, which
quantiﬁes the contribution of aftershocks to hazard, is recalled in Eq.(16).
In the equation,
P[
IM
∪A
>im ∩IM
≤
im


IM >im ∪IM
∪A
>im
]
it is the probability that,
given that exceedance of
im
has been observed during the mainshock–aftershock sequence,
(
IM >im ∪IM
∪A
>im
)
, it was in fact an aftershock to cause it, while the mainshock
was below the threshold; i.e.,
(
IM
∪A
>im ∩IM
≤
im
)
. All the terms of the equation have
been already deﬁned discussing Eq.(11); see Iervolino etal. (2018) for derivation of the
equation.
4 Multi‑site hazard
In the case of MSPSHA, for a set of spatiallydistributed sites, say
nsts
in number, one can
deﬁne a vector of thresholds, one for each site
{
im
1
,im
2
,.…im
nsts }
, of the IM of interest.
Given a vector of thresholds, the sought outcomes of MSPSHA can be various, for exam
ple, the probabilistic distribution of the total number of exceedances collectively observed
at the sites in the
ΔT
time interval. The main issue with MSPSHA is that, even if the pro
cess counting exceedances at each of the sites is an HPP, that is Eq.(2), these HPPs are (in
general) not independent. Then, the process that counts the total number of exceedances
observed at the ensemble of the sites over time is not a HPP. The nature and form of sto
chastic dependence, existing among the processes counting in time exceedances of ground
motion thresholds at multiple sites, is related to the probabilistic characterization of the
eﬀects of a common earthquake at the diﬀerent sites (e.g., Giorgio and Iervolino 2016).
The same reasoning discussed for one IM at multiple sites, can be applied when
MPSHA involves multiple IMs. For example, if one considers as IMs two pseudo accel
erations at two spectral periods,
IM1
=Sa
(
T
1)
and
IM2
=Sa
(
T
2)
, it is generally assumed
that, given an earthquake of m and
{x,y}
characteristics, the logarithms of IMs at the sites
form a Gaussian random ﬁeld (GRF), a realization of which is a
1
×
(
n
sts
⋅2
)
vector of
the type
{
im
1,1
,im
1,2
,…,im
1,nsts
,im
2,1
,im
2,2
,…,im
2,nsts }
. This means that the logarithms of
IMs have a multivariate normal distribution, where the components of the mean vector are
given by the
E[
log IM
1
m,r
j
,𝜃
]
and
E[
log IM
2
m,r
j
,𝜃
]
terms; two for each j, being
rj
the
distance between the site j and the location of the seismic event, and the covariance matrix,
(15)
fM,RIM>im∪IM∪A>im (m,r)=
n
s
i=1𝜈i
𝜆im
×
1−P[IM ≤im
m,r]i⋅e
−E[NAm(0,ΔTA)]⋅∫
MA
∫
RA
P[IMA>immA,rA]i
⋅fMA,RA,iM,R(mA,rAm,r)⋅dmA⋅drA
⋅fM,R,i(m,r
)
(16)
P
IM∪A>im ∩IM
≤
imIM >im ∪IM∪A>im
=
n
s
i=1
𝜈i
𝜆im
⋅�
M,R
P[IM
≤
imm,r]i
×
1−e
−E[NAm(0,ΔTA)]⋅∫
MA
∫
RA
P[IMA>im
mA,rA]i
⋅fMA,RA,iM,R(mA,rA
m,r)⋅dmA⋅drA
⋅fM,R,i(m,r)⋅dm ⋅
dr
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Σ
, is given in Eq.(17). In the equation,
𝜎inter,1
and
𝜎inter,2
are the standard deviations of the
interevent residuals of the GMPEs of the two IMs, while
𝜎intra,1
and
𝜎intra,2
are the standard
deviation of intraevent residuals of
Sa(
T
1)
and
Sa(
T
2)
, respectively;
𝜌inter(
T
1
,T
2)
is the
correlation coeﬃcient between interevent residuals at the two spectral periods in the same
earthquake, while
𝜌intra(
T
1
,T
2
,h
i,j)
is the correlation coeﬃcient between intraevent residu
als of the GMPEs of
Sa(
T
1)
and
Sa(
T
2)
for sites
i
and
j
; and
hi,j
is the intersite distance.
In this case,
Σ
is the sum of two square matrices, each of
(
n
sts
⋅2
)
×
(
n
sts
⋅2
)
size. The ﬁrst
matrix accounts for the crosscorrelation of interevent residuals which is, by deﬁnition,
independent on the intersite distance; the second matrix accounts for the intraevent resid
uals spatial crosscorrelation and is dependent on intersite distance as well as the selected
spectral periods. Assigning the mean vector and the covariance matrix completely deﬁnes
the GRF in one earthquake (e.g., Baker and Jayaram 2008; Esposito and Iervolino 2012;
Loth and Baker 2013; Markhvida etal.2018).
To compute MPSHA representing the GRF with the discussed covariance structure, in
REASSESS the Monte Carlo simulation approach has been chosen. In this framework, one
possible algorithm is the twostep procedure of Fig. 1, which is described, for simplic
ity, with reference to a single seismic source where earthquakes occur as per Eq.(1) with
assigned magnitude and location distributions.
(a) The ﬁrst step is addressed to simulate and collect realizations of the GRF conditional to
the occurrence of an earthquake of generic magnitude and location. In other words, mag
nitudes and locations of the seismic events on the source are sampled according to their
distributions and, then, the realizations of the IMs at the considered sites are simulated in
accordance with the considered GMPEs and
Σ
. This step is described in Fig.1, where
nm
,
nxy
and
n𝜀
are the indices counting the number of simulations for magnitude, event loca
tion and GRF of residuals at the sites, respectively; capital letters of the indices,
Nm
,
Nxy
and
N𝜀
are the total number of simulations for each of the three variables. Thus, the results
of the ﬁrst step are
Nm⋅Nxy ⋅N𝜀
vectors, one for each simulation, collecting the IMvalues
simulated at the sites in each event. Each vector
{
im
}
=
{
im
1
,im
2
,…,im
nsts }
represents
realizations of the random ﬁeld of IMs at the sites in one generic (i.e., considering all
possible magnitudes and locations) earthquake and, therefore, it is timeinvariant.
(b) The realizations from step (a) are the input for the second step that consists of simulating
the process of earthquakes aﬀectingthe sites, in any time interval
ΔT
of interest; i.e., the
(17)
Σ=
𝜎2
inter,1 ⋯𝜎2
inter,1 𝜌inter
T1,T2
⋅𝜎inter,1 ⋅𝜎inter,2 …𝜌inter
T1,T2
⋅𝜎inter,1 ⋅𝜎inter,2
⋱⋮ ⋮ ⋱ ⋮
𝜎2
inter,1 𝜌interT1,T2⋅𝜎inter,1 ⋅𝜎inter,2 ⋯𝜌interT1,T2⋅𝜎inter,1 ⋅𝜎inter,2
𝜎2
inter,2 …𝜎2
inter,2
sym ⋱⋮
𝜎2
inter,2
+
𝜎2
intra,1 ⋯𝜌intraT1,T1,h1,nsts ⋅𝜎2
intra,1 𝜌intraT1,T2,h1,1 ⋅𝜎intra,1 ⋅𝜎intra,2 …𝜌intra T1,T2,h1,nsts ⋅𝜎intra,1 ⋅𝜎intra,2
⋱⋮ ⋮ ⋱ ⋮
𝜎2
intra,1 𝜌intraT1,T2,hnsts ,1 ⋅𝜎intra,1 ⋅𝜎intra,2 …𝜌intra T1,T2,hnsts ,nsts ⋅𝜎intra,1 ⋅𝜎intra,2
𝜎2
intra,2 …𝜌intraT2,T2,h1,nsts ⋅𝜎2
intra,2
sym
⋱⋮
𝜎2
intra,2
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seismic history for the sites in
ΔT
. In each run of the simulation of this step, indicated
by the index z which varies from 1 to Z, the number of earthquake events on the source
is sampled from a HPP with mean equal to
𝜈⋅ΔT
. Then, a number of IM random ﬁelds,
equal to the sampled number of events, is randomly selected among those generated in
the ﬁrst step of the procedure. These random ﬁelds collectively represent one realiza
tion of the seismic history at the sites in
ΔT
. Therefore, repeating Z times this step, can
provide a sample of histories of what could occur in
ΔT
at the sites.
The simulated seismic histories can be used to compute any MSPSHA result. For example,
if one is interested in the distribution (i.e., the MPF) of the total number of exceedances
collectively observed at the sites in
ΔT
, it is suﬃcient to count in how many histories a
speciﬁc number of total exceedances of the
{
im
1
,im
2
,…,im
nsts }
vector has been observed
and divide by the total number of simulated histories. For example, the probability that
zero exceedances are observed collectively at the sites, in
ΔT
years, is equal to the number
of histories in which none of the IM thresholds set for each of the sites is exceeded, divided
by the number of simulated histories (i.e., Z).
In the case of more than one seismic source, the ﬁrst step is repeated for each of them to
simulate the random ﬁeld they individually produce. In the second step, the HPP describ
ing the event occurrence on all the sources has mean equal to
Δ
T⋅
∑i
𝜈
i
. This, similarly
to the case of a single source, is used to sample the number of earthquakes in
ΔT
and to
randomly select the random ﬁeld realizations from those of each source; the number of
realizations to be selected for each source is proportional to the probability that given that
an earthquake occurs it is from source i, that is
𝜈i�∑i
𝜈
i
. At this point the seismic history in
ΔT
for the sites is obtained in analogy to the case of a single source.
Fig. 1 Flowchart of the simulation procedure for MSPSHA in the case of single seismic source
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4.1 MSPSHA shortcuts forGMPEs withadditive factors
In this section some helpful shortcuts for MSPSHA calculations that are implemented in
REASSESS and that apply (only) in the case of GMPEs of the type in Eq.(5) are discussed.
It should be noted that the covariance of two or more RVs does not change adding constant
terms. Thus, to the aim of this section, it is required to recognize that Eq.(5) implies that the
RV representing the logarithms of IM for a site with conditions represented by
𝜃
, is obtained
adding such a coeﬃcient to the RV representing the logarithms of IM for a reference condi
tion for which
𝜃=0
; this means that the covariance matrix of the GRF is also independent
of
𝜃
(e.g., the soil class of each of the sites). As a consequence, the simulations described in
Sect.4 can be carried out considering a common site condition for all sites (e.g., rock). To
obtain GRF realizations reﬂecting the diﬀerent site conditions at the sites from those for the
reference case, it is suﬃcient to add to the logarithms of the simulated IMs the sitespeciﬁc
coeﬃcient, that is
{
𝜃
1
,𝜃
2
,…,𝜃n
sts }
, from the GMPE. Equivalent, but even simpler, is to sub
tract the
{
𝜃
1
,𝜃
2
,…,𝜃
nsts }
vector from the vectors of logarithms of the IM thresholds for the
sites. However, in closing this section, it has to be emphasized that, as mentioned, several
recent GMPEs are not of the type in Eq.(5) for what concerns the soil term, and these short
cuts do not apply (see also Staﬀord etal. 2017). Nevertheless, this same reasoning holds in
the case
𝜃
of Eq.(5) represents any other factor aﬀecting the IMs, not only soil site class.
5 REASSESS V2.0
To implement the types of hazard assessment discussed above, REASSESS V2.0 is coded
in MATLAB and proﬁts of a graphical user interface (GUI). The GUI features one input
panel and two output panels, one for PSHA/SPSHA and one for MSPSHA. In fact, the
main GUI is complemented by secondary interfaces that pop up when needed (see Fig.2).
Fig. 2 Principal and auxiliary GUIs of REASSESS V2.0
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Note that, in the case of extended analyses (e.g., several seismic sources or sites), input can
also be deﬁned via dedicated MICROSOFT®EXCEL spreadsheets, as a shortcut.
A schematic ﬂowchart of the way REASSESS V2.0 operates is given in Fig.3. First,
the user is required to deﬁne the type of analysis to be performed; i.e., PSHA, SPSHA, or
MSPSHA. Even in the case of singlesite analysis (PSHA and SPSHA) the user is allowed
to deﬁne more than one site of interest; in this case, REASSESS will run singlesite PSHA
or SPSHA separately for each of them according to Sects.2 or 3. If MSPSHA is selected,
more than one site must be deﬁned, and the analyses are performed according to what dis
cussed in Sect.4. (When SPSHA or MSPSHA is selected, the corresponding PSHA is also
performed for the considered sites, as it is considered a reference case.)
The second step refers to thedeﬁnition of the coordinates and soil condition of the
sites. It can be carried out via the GUI or via an EXCEL spreadsheet, for which a tem
plate is given. The soil conditions can be deﬁned in terms of shear wave velocity of the
top 30m of subsurface proﬁle (Vs30) expressed in meter/second or in terms of the soil
classes (A, B, C, D and E) according to the Eurocode 8 classiﬁcation of sites (CEN
2004).
The third step is dedicated to the selection of the GMPE(s). A database of alternative
GMPEs is included in the current release of REASSESS: Ambraseys etal. (1996), ﬁtted
on a European dataset, Akkar and Bommer (2010), which refers to data from southern
Europe, North Africa, and active areas of the Middle East, Bindi etal. (2011), ﬁtted on
Fig. 3 REASSESS V2.0 ﬂowchart showing singlesite and multisite modules functionalities
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Italian dataset and Cauzzi etal. (2015), based on a worldwide dataset.2 At this step, also
the discretization of the domain of the intensity measure for singlesite PSHA, which
serves to lump the hazard curves, has to be deﬁned in terms of minimum, maximum
values and number of intermediate steps (constant in logarithmic scale). In the case of
PSHA, the third step also allows the deﬁnition of a logic tree (Sect.2.3) in terms of:
(1) parameters of the magnitude distributions, (2) mean annual frequency of earthquake
occurrence on the sources and (3) GMPEs (among those available).
The choice of the IMs to be considered (e.g., spectral pseudoacceleration for diﬀer
ent natural vibration periods) for all the types of analysis (PSHA, SPSHA or MSPSHA)
is dependent on the IMs available per the selected GMPE (step 4). If a logic tree with
diﬀerent GMPE for each branch has been deﬁned, the selection is among the IMs of the
GMPE belonging to the branch with the largestweight. If diﬀerent branches have the
same (largest)weight, the selection is among the IMs of the GMPE selected for the ﬁrst
branchwith that weight.
When PSHA is of concern, REASSESS also allows to perform analysis for advanced
spectralshapebased intensity measures such as
INp
proposed by Bojórquez and Ierv
olino (2011) and reported in Eq. (18) in logarithmic terms. The
INp
is a proxy of the
pseudoacceleration response
(Sa)
spectral shape in a range of periods
(
T
1
…T
N)
and is
dependent on a reference period
(̄
T
)
belonging to the
(
T
1
…T
N)
interval and an
𝛼
param
eter. In its analytical expression
Saavg (
T
1
…T
n)
appears; it is the geometric mean of the
spectral acceleration in the
(
T
1
…T
N)
range of periods (Baker and Cornell 2006a). In
the software,
(
T
1
…T
N)
,
̄
T
and
𝛼
can be selected by the user (the periods can be chosen
among those of the selected GMPE). It is easy to see that when the
𝛼
parameter equals
one,
INp
corresponds to
Saavg (
T
1
…T
n)
.
In the case of MSPSHA, when a single spectral ordinate is selected as IM, the user is
allowed to choose the model of spatial correlation of intraevent residuals of Esposito and
Iervolino (2012) or Loth and Baker (2013). On the other hand, when the IMs at the sites
are spectral ordinates for several natural vibration periods, simulated (spatially) crosscor
related scenarios are computed adopting the models of (1) Loth and Baker (2013) for the
spatial correlation of intraevent residuals and (2) Baker and Jayaram (2008) for the spec
tral correlation of interevent residuals.
Step 5 is dedicated to the seismic source deﬁnition. In REASSESS V2.0, seismic
source zones and/or ﬁnite threedimensional faults can both be input of analysis. Faults
are discussed in Sect. 5.1; for what concerns source zones, these are deﬁned by the
coordinates of the vertices of the zone, the annual rate of occurrence of earthquakes
of Eq.(1) and the event’s magnitude distribution, which is assumed to be a truncated
exponential distribution as discussed in Sect.2; hence, the slope of the G–R relation
ship, together with minimum and maximum values of magnitude, is required. If known,
a rupture faulting style can be associated to the seismic zone. As mentioned, all the
required parameters can be alternatively given via GUI or EXCEL spreadsheet.
(18)
log
INp
=log
Sa
̄
T
+𝛼log
Saavg
T1…TN
Sa
̄
T
2 These GMPEs are of the type in Eq. (5), then the shortcuts discussed in Sects. 2.3 and 4.1 apply. Also
note that although the Ambraseys etal. (1996) GMPEs dates more than 20years ago, it has been considered
because it is the one the current oﬃcial Italian hazard model is based on (Stucchi etal. 2011).
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A number of literature databases of seismic zones are already embedded in the cur
rent version of the software. Referring to Italy, it is known that the seismic hazard study
of Stucchi etal. (2011) lies at the basis of the hazard assessment for the Italian cur
rent building code and features a logic tree made of several branches; the branch named
921 is the one producing the results claimed to be the closest to those provided by the
full logic tree. This branch considers the seismic source model of thirtysix areal zones
of Meletti etal. (2008) and the GMPE by Ambraseys etal. (1996). It is implemented
in REASSESS V2.0 and is named Meletti etal. (2008)—Magnitude rates from DPC
INGVS1—Branch 921. It is the sole database selection which implies a speciﬁc GMPE
(automatically selected). An alternative source model for Italy is named Meletti etal.
(2008)—Magnitude rates from Barani etal. (2009) in which the same source model of
Meletti etal. (2008) is considered, but the associated seismic characterization is from
Barani etal. (2009). Other databases in REASSES are the one from the SHARE project,
which covers the EuroMediterranean region, the one from the EMME project, which
covers middleeast; i.e., Afghanistan, Armenia, Azerbaijan, Cyprus, Georgia, Iran, Jor
dan, Lebanon, Pakistan, Syria and Turkey. Moreover, included databases are: ElHussain
etal. (2012), Ullah etal. (2015) and Nath and Thingbaijam (2012), referring to the Sul
tanate of Oman, Kazakhstan, Kyrgyzstan, Tajikistan, Uzbekistan and Turkmenistan, and
India, respectively. The area covered by the embedded databases is given in Fig.4. For
each of the cited databases, assuming a uniform earthquake location distribution in each
seismic source, epicentral distance is converted into the metric required by the GMPE
according to Montaldo etal. (2005). The styleoffaulting correction factors proposed
Fig. 4 Embedded databases of seismogenic sources
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by Bommer et al. (2003) are also applied to the Ambraseys et al. (1996) GMPE, in
accordance with the rupture mechanism associated to each seismic zone (if any).
When SPSHA is performed, an additional step is required in the input deﬁnition. In
particular, the model describing the aftershock occurrence has to be speciﬁed, that is the
parameters of Eq.(13), providing the expected number of aftershock in any time interval
given the magnitude of the mainshock. The available models are those of Reasenberg and
Jones (1989, 1994), Lolli and Gasperini (2003) and EberhartPhillips (1998) which refer
to generic California, Italian and New Zealand aftershock sequence, respectively. Such
models, can be selected through a dedicated window (Fig. 5), automatically opened by
REASSESS before running the SPSHA. In the current version of the software, the GMPE
selected for PSHA is also applied to account for the evaluation of aftershock’s IM.
5.1 Finite faults
REASSESS also allows to compute hazard analysis (both PSHA and MSPSHA) in the
case the seismic sources are represented by means of one or more ﬁnite faults. There are
many alternative ways to deﬁne the characteristics of a fault for hazard assessment pur
poses (Scherbaum etal. 2004). In the current version of REASSESS a fault is deﬁned by
means of a point representing its center and the dip, rake, and strikes angles (Aki and Rich
ards 1980). In the case of a ﬁnite fault in REASSESS, PSHA is carried out according to
Eq.(19), which is an adaptation of Eq.(3).
In the equation:
𝜈
is the rate;
{X,Y}
is the position of the center of the rupture with respect
to the center of the fault and its distribution
fX,Y(x,y)
is taken according to Mai et al.
(2005);
fM(m)
is the magnitude distribution that can be deﬁned as G–R or characteristic
(e.g., Convertito etal. 2006);
fAM(am)
is the distribution of the rupture size, conditional
to the magnitude that is modelled according to Wells and Coppersmith (1994); ﬁnally,
fSA(sa)
is the aspect ratio (lengthtowidth ratio) of the rupture and is probabilistically
modeled lognormally according to Iervolino etal. (2016b).3
(19)
𝜆
im =𝜈⋅
∫
X
∫
Y
∫
M
∫
A
∫
S
P
[
IM >imm,x,y
]
⋅fSA(sa)⋅fAM(am)⋅fM(m)⋅fX,Y(x,y)⋅ds ⋅da ⋅dm ⋅dx ⋅
dy
Fig. 5 Graphical interface window for calibration of the aftershock occurrence models
3 The depth of the top of the r upture is assumed to be equal to 5km for all events of magnitude less than
6.5 and one kilometer for events of larger magnitude, following the practice of the U.S. Geological Survey;
however, this constraint is not strictly needed and could be relaxed in updated versions of REASSESS.
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6 Output oftheanalyses
At the end of the analysis, the outputs provided by the software can be consulted via the
GUI in the format of ﬁgure or text ﬁles. Moreover, a compressed folder with all the input
and output (ﬁgures and ﬁles) of the analyses can be saved by the user. In the following sub
sections, the available results are described.
6.1 PSHA andSPSHA results
When the analysis is ﬁnished, the hazard curves are plotted in the singlesite output panel
(see Fig.2). If the analysis is performed for more than one site, the curves for each site of
interest can be selected (via a dropdown menu). The uniform hazard spectrum (UHS) can
be computed, and plotted in a dedicated panel, selecting any return period available (that
depends on the range of IMs deﬁned at the beginning).
In addition, REASSESS is able to provide disaggregation of PSHA, conditional mean
spectrum (CMS; Lin et al. 2013) and conditional hazard (see Sects. 2.1, 2.2). The con
ditional hazard can be computed by REASSESS V2.0 proﬁting of the model of Bradley
(2012), which provides correlation between peak ground velocity (PGV) and spectral
accelerations and the model of Baker and Jayaram (2008), which provides the correlation
among spectral acceleration values at diﬀerent spectral periods. Therefore, the distribution
of PGV or pseudoacceleration response spectra at any vibration period conditional to the
occurrence of any spectral ordinate can be computed.
Results of SPSHA are similar to those for PSHA; however, disaggregation is of two
kinds (see Sect.3.1). The ﬁrst is the joint probability density function of magnitude and
distance of the mainshock conditional to the exceedance, or the occurrence, of a cho
sen hazard threshold during the corresponding cluster (mainshock and subsequent after
shocks). This is equivalent to the classical hazard disaggregation, in terms of magnitude
and distance, but computed in accordance with the approach of the SPSHA, Eq.(15). The
second disaggregation provided represents the probability that, given that exceedance of
im
has been observed during the mainshock–aftershock sequence, it was in fact an aftershock
to cause it, Eq.(16).
6.2 MSPSHA results
MSPSHA can be performed on all or on a subset of the sites deﬁned at the beginning of the
analysis. It is performed through the twosteps procedure described in Sect.4. At the end
of the ﬁrst step, the simulated scenarios of IM realizations at the sites, given the occurrence
of an earthquake on the sources are available. As a reference, these results are also used
to ﬁrst provide singlesite PSHA as per Eq.(2) (in fact, singlesite PSHA can be viewed
as a special case of MSPSHA; Giorgio and Iervolino 2016) and the results are reported in
the singlesite panel. Speciﬁcally referring to MSPSHA, REASSESS V2.0 provides three
kinds of results:
1. the probability of observing an arbitrarily chosen number of exceedances at the sites in
a given time interval;
2. the distribution of the total number of exceedances at the sites in a given time interval;
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3. the distribution of the number of exceedances at the sites given the occurrence of an
earthquake (a timeinvariant results).
Results (1) and (2) are computed by REASSESS for any time interval without repeating the
simulations of the ﬁrst step of the analysis, thus reducing the required time of computation.
Text ﬁles with the GRFs simulated (i.e., realizations) conditional to a generic event and in the
selected time interval are also available at the end of the analyses.
It is to also highlight that, although the vector collecting sites threshold in MSPSHA can
be completely deﬁned by the user, REASSESS allows to deﬁne the threshold vector from the
results of singlesite PSHA. For example, the thresholds can be chosen as the values with the
same exceedance return period at each site according to singlesite PSHA, as illustrated in one
of the examples below.
7 Illustrative examples
Some examples of the analyses REASSESS V2.0 enables are illustrated herein. To this
aim ﬁve sites are considered; incidentally, they correspond to the ﬁve main hospitals of
the health infrastructure for municipality of Naples (Italy): Ospedale del Mare, San Gio
vanni Bosco, Cardarelli, San Paolo and Fatebenefratelli (see Fig.6 in which the sites and
the municipality boundaries are highlighted). The intersite distance ranges between 2 and
13km.
The following sections refer to the results of PSHA, SPSHA and MSPSHA. For all of
them, the Meletti etal. (2008)—Magnitude rates from DPCINGVS1—Branch 921 source
model is used (see Sect.5). For the aim of this paper, all the sites are assumed on rock soil
conditions. In the case of SPSHA, the selected model deﬁning parameters of Eq.(13) is
Lolli and Gasperini (2003). All the data represented in the ﬁgures are taken from the texts
ﬁles automatically saved by REASSESS (to assemble the ﬁgures of the paper, the format of
the plots is slightly diﬀerent from the one of the software).
7.1 Single‑site PSHA
Because the considered sites can be considered close from the seismic hazard assessment
point of view, diﬀerences in terms of singlesite analysis, are minor. Thus, only one of the
locations is considered for PSHA and SPSHA: 14.277°E, 40.873°N. Figure7 summarizes
the result of singlesite PSHA computed for the site.
In Fig.7a it is reported the location of the site (grey triangle) and the twelve seismic
zones (out of the thirtysix in total, numbered from 901 to 936) of the model of Meletti
etal. (2008) contributing to the hazard are plotted (these zones are automatically identiﬁed
by REASSESS among those of the selected database). Figure7b reports the hazard curves
computed for the whole fortyseven spectral periods of the GMPE. In the same plot, the
annual rate of exceedance equal to 0.0021, corresponding to the 475 return period
(
T
R)
of
exceedance, is also plotted (red horizontal line). This return period is the one for which are
computed the UHS’s in Fig.7c (the three soil conditions allowed by the GMPE are con
sidered; i.e., rock, stiﬀ and soft soil). Such spectra have a peak ground acceleration (PGA)
equal to about 0.2g and are representative of a medium–high hazard site in Italy (see Stuc
chi etal. 2011).
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Selecting as the IM the pseudoacceleration response spectral ordinate at 0.5s period,
Sa(0.5 s)
, the occurrence disaggregation for a return period of 475 years is reported in
Fig.7d. Such a disaggregation (for the occurrence of im) is computed as per Eq.(6); how
ever, because RVs are represented in a discretized way assuming bins of 10km distance
and 0.5 magnitude, the PDF,
fM,R,𝜀IM
, is rendered in the plot by the corresponding discre
tized form,
P[m,r,𝜀(IM =im ]
. Disaggregation distribution is bimodal, being the disag
gregated hazard mainly aﬀected by two seismic zones: the one in which the site is enclosed
to (namely zone 928) and the zone 927 that, although is more distant than 928, is able
to generate larger magnitude events and more frequently (see Iervolino etal. 2011, for a
deeper discussion).
The CMS is reported in Fig.7e: conditioning IM is maintained
Sa(0.5 s)
corresponding
to
TR=475
years. Finally, the conditional hazard distribution, Eq. (7), for four pseudo
spectral accelerations at 0 (PGA), 0.2s, 0.6s and 1.0s conditional to the same primary IM
are reported in Fig. 7f. (REASSESS also provides the conditional standard deviation for
any IM.)
Fig. 6 Geographical location of the sites within the municipality of Naples
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7.2 SPSHA
For the same site as Sect.7.1, Fig. 8a shows the UHS’ corresponding to 475years return
period and computed via both SPSHA and PSHA. The latter case corresponds to classical
hazard, while the former includes the eﬀect of aftershocks. Sequence’s eﬀect produces a
maximum increase of UHS from PSHA equal to 12% which corresponds to a vibration
period equal to 0.1s. However, increments over the whole range of analysed periods are
equal or higher than 7%; the minimum value is recorded at 1.5s.
Both kinds of sequencebased disaggregation are also computed. The mainshock magni
tude and distance disaggregation distribution, that is Eq.(15), is shown for the PGA inten
sity measure and 475years exceedance return period (Fig.8b); it is interesting to note that,
accounting for the sequence modiﬁes the proportion between ﬁrst and second modal values
with respect to Fig.7d (Chioccarelli etal. 2018).
Figure 8c provides the aftershock disaggregation, Eq.(16), performed for three IMs:
PGA,
Sa(0.2 s)
and
Sa(0.6 s)
. Aftershock disaggregation is here represented as a function
of the increasing return period even if output text ﬁles provide them as function of both IM
thresholds and return period. All these disaggregation distributions have a nonmonotonic
trend. In fact, they start from zero because it can be veriﬁed that when
im
approaches zero,
results of Eqs.(3) and (11) are equal, i.e., aftershock has no eﬀect. The maximum value
of disaggregation for PGA is 0.26 corresponding to a return period of about 4000years;
maximum disaggregation for
Sa(0.2 s)
is 0.26 and it occurs for a return period of about
(a) (b) (c)
(d) (e) (f)
Fig. 7 a Geographic location of the site and areal sources contributing to the hazard, b hazard curves (grey
lines) computed for all the spectral period provided by the GMPE and annual rate corresponding to the 475
return period (red line), c UHS’ with a 475years return period, d hazard disaggregation distribution for the
occurrence of the
Sa(0.5 s)
with a 475 years return period, e CMS and f conditional hazard distributions
assuming as primary IM the
Sa(0.5 s)
with a 475years return period
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2000years; ﬁnally, maximum disaggregation for
Sa(0.6 s)
is 0.21 and correspond to
TR
of
about 100years. The nonmonotonic trend of the plots indicates that the aftershock contri
bution to the hazard has a variable signiﬁcance with the hazard threshold. Moreover, the
diﬀerent return period to which each disaggregation reaches its maximum suggests that
aftershock eﬀect is also dependent on the considered spectral period.
7.3 MSPSHA
Results of MSPSHA are reported in this section referring to the whole set of the ﬁve sites
introduced above (Sect.7). A set of ﬁve IMs has been selected for each of the site: PGA,
Sa(0.2 s)
,
Sa(0.5 s)
,
Sa(0.6 s)
,
Sa(1.0 s)
. Proﬁting of the REASSESS functionalities dis
cussed in Sect.6.3, the vector of IMs collecting the threshold values for each site, which
is required for MSPSHA, is chosen in a way that the corresponding
TR
, from singlesite
PSHA, are the same among all the sites: the common return period is, arbitrarily, 50years.
The distribution of the number of exceedances at the sites given the occurrence of
the event and the distribution of the number of exceedances collectively observed at the
sites in a time window of 20years are the output here, chosen among those available (see
Sect.6.2). Both types of distribution are computed referring to four diﬀerent cases: in (1)
at each of the ﬁve sites, PGA is the considered IM; in (2) and (3) the considered IM at the
sites is
Sa(0.5 s)
and
Sa(1.0 s)
, respectively; ﬁnally, in (4) a diﬀerent intensity measure is
selected at each site: PGA at site one,
Sa(0.2 s)
at site two,
Sa(0.5 s)
at site three,
Sa(0.6 s)
at site four and
Sa(1.0 s)
at site ﬁve. The MPF of the total number of exceedances given the
occurrence of an earthquake is reported in the ﬁrst line of panels of Fig.9, from (a) to (d)
corresponding to cases from 1 to 4, respectively.
This result is representative of a speciﬁc case scenario which corresponds to the occur
rence of a generic event. It appears that the most probable number of exceedances is zero
while the exceedance probabilities at one, more than one, or all the sites are of the same
order of magnitude. The second line of the ﬁgure, that isplots from (e) to (h), shows the
MPF of the total number of exceedances in 20years.
(a) (b)
(c)
Fig. 8 a Comparison among UHS from PSHA and SPSHA for a 475 years return period, b hazard dis
aggregation distribution of PGA with a 475 years return period, c aftershock disaggregation for PGA,
Sa(0.2 s)
and
Sa(0.6 s)
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8 Final remarks
REASSESS V2.0, a MATLABcoded tool for probabilistic seismic hazard analysis, has
been presented. It is a standalone application which operates via GUI and/or template
based input ﬁles that has been developed to enable classical and advanced probabilis
tic seismic hazard assessment procedures. It is oriented towards earthquake engineering
practice.
In the paper, the basics of probabilistic seismic hazard analyses embedded in the soft
ware have been recalled ﬁrst, then implemented algorithms and the workﬂow of REAS
SESS have been discussed. The software allows the user to deﬁne the input of the analyses
in terms of: site(s) coordinates, GMPEs (selected among an embedded database), inten
sity measures of interest, seismic sources (userdeﬁned threedimensional faults, seismic
sources (areal) zones, or sources selected from embedded databases), and structure of logic
tree, if any.
When singlesite analyses are of concern, REASSESS is able to provide classical results
of PSHA such as hazard curves, even in terms of spectralshapebased (i.e., advanced)
ground motion intensity measures. Moreover, uniform hazard and conditional mean spec
tra, together with disaggregation distributions given the occurrence or the exceedance of
the IM threshold, can be computed. Conditional hazard can also be computed for PGV or
pseudospectral accelerations selected as secondary intensity measures. Moreover, single
site analyses may also be performed accounting for the eﬀect of the aftershocks. With this
type of analysis, named SPSHA, available output is represented by: hazard curves, UHS,
magnitudedistance disaggregation distribution and aftershock disaggregation. PSHA and
SPSHA are implemented taking advantage of the accuracy and low computational demand
allowed by matrix calculus of MATLAB.
For portfolio of sites that can be subjected to the same seismic sources, the software is
able to perform the socalled MSPSHA providing, for a vector of IM thresholds, diﬀerent
probabilistic results all related to the exceedances possibly observed atthe sites. A twostep
0510 15 20
No. exceedances in 20 years
104
103
0.01
0.1
1
ytilibaborp ecnadeecxE
Case 1
0510 15 20
No. exceedances in 20 years
104
103
0.01
0.1
1
Case 2
0510 15 20
No. exceedances in 20 years
104
103
0.01
0.1
1
Case 3
0510 15 20
No. exceedances in 20 years
104
103
0.01
0.1
1
Case 4
012345
No. exceedances given the earthquake
104
103
0.01
0.1
1
ytilibaborp ecnadeecxE
Case 1
012345
No. exceedances given the earthquake
104
103
0.01
0.1
1
Case 2
012345
No. exceedances given the earthquake
104
103
0.01
0.1
1
Case 3
012345
No. exceedances given the earthquake
104
103
0.01
0.1
1
Case 4
(d)(c)(b)(a)
(h)(g)(f)(e)
Fig. 9 MPF of the total number of exceedances at the sites a given the event and b in 20years
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simulation algorithm to carry out MSPSHA, allows to proﬁt of random ﬁeld simulations of
IMs in generic earthquakes.
REASSES was optimized for accuracy of numerical computation, analysis time and
ease of use, which was illustrated herein via a few applications, not exhaustive of the soft
ware capabilities. To this aim it also implements calculation shortcuts and provides a series
of options of input/output management. It is ﬁnally to note that a practical user guide (tuto
rial) can be found online at http://wpage .unina .it/iunie rvo/doc_en/REASS ESS.htm, which
is the same site where the software is available under a Creative Commons license: attribu
tion—noncommercial—non derived.
Acknowledgements The work presented in this paper was developed within the AXADiSt (Dipartimento
di Strutture per l’Ingegneria e l’Architettura, Universita` degli Studi di Napoli Federico II) 2014–2017
research program, funded by AXAMatrix Risk Consultants, Milan, Italy. The H2020MSCARISE2015
research project EXCHANGERisk (Grant Agreement No. 691213) and ReLUIS (Rete dei Laboratori Uni
versitari di Ingegneria Sismica) are also acknowledged.
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