Comparing alternative models for multisite probabilistic seismic risk analysis

Abstract

The risk assessment for a building portfolio or a spatially distributed infrastructure requires multi-site probabilistic seismic hazard analysis (MSPSHA). In fact, MSPSHA accounts for the stochastic dependency between the ground motion intensity measures (IMs) at the sites. Multi-site hazard needs to define the correlation structure for the same IM at different sites (spatial correlation), that of different IMs at the same site (cross-correlation) and that of different IMs at different sites (spatial-cross-correlation). Literature shows that such models usually require a significant amount of regional data to be semi-empirically calibrated. An approximated yet simpler-to-model alternative option is the conditional-hazard approach. The latter, originally developed for single-site analyses as an alternative to vector-valued PSHA, allows computing the distribution of a secondary IM given the occurrence or exceedance of a value of a primary IM. Conditional hazard considers the spatial correlation of the primary IM and the cross-correlation at each site for the two IMs, thus, if it is adopted for MSPSHA, the spatial correlation of the secondary IM as well as the spatial-cross-correlation between the two IMs descends from these two models. In the study, the conditional hazard procedure for MSPSHA is discussed and implemented in an illustrative application. Results in terms of distribution of the total number of exceedances of selected thresholds at the sites in a given time interval are compared with the case of complete formulation of MSPSHA and the differences are quantified. It appears that conditional hazard is a solid, yet simpler alternative for MSPSHA, at least in the considered cases.

Full text

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
Comparing alternative models for multisite probabilistic seismic
risk analysis
Pasquale Cito
Post-doctoral researcher, Dept. of Structures for Engineering and Architecture, University of Naples
Federico II, Naples, Italy
Eugenio Chioccarelli
Assistant professor, Pegaso Online University, Naples, Italy
Iunio Iervolino
Professor, Dept. of Structures for Engineering and Architecture, University of Naples Federico II,
Naples, Italy
ABSTRACT: The risk assessment for a building portfolio or a spatially distributed infrastructure
requires multi-site probabilistic seismic hazard analysis (MSPSHA). In fact, MSPSHA accounts for
the stochastic dependency between the ground motion intensity measures (IMs) at the sites. Multi-
site hazard needs to define the correlation structure for the same IM at different sites (spatial
correlation), that of different IMs at the same site (cross-correlation) and that of different IMs at
different sites (spatial-cross-correlation). Literature shows that such models usually require a
significant amount of regional data to be semi-empirically calibrated. An approximated yet simpler-
to-model alternative option is the conditional-hazard approach. The latter, originally developed for
single-site analyses as an alternative to vector-valued PSHA, allows computing the distribution of a
secondary IM given the occurrence or exceedance of a value of a primary IM. Conditional hazard
considers the spatial correlation of the primary IM and the cross-correlation at each site for the two
IMs, thus, if it is adopted for MSPSHA, the spatial correlation of the secondary IM as well as the
spatial-cross-correlation between the two IMs descends from these two models. In the study, the
conditional hazard procedure for MSPSHA is discussed and implemented in an illustrative
application. Results in terms of distribution of the total number of exceedances of selected thresholds
at the sites in a given time interval are compared with the case of complete formulation of MSPSHA
and the differences are quantified. It appears that conditional hazard is a solid, yet simpler alternative
for MSPSHA, at least in the considered cases.
1. INTRODUCTION
Classical probabilistic seismic hazard analysis
(PSHA) allows to compute the exceedance rate
of arbitrary ground motion intensity measure
(IM) thresholds at a site of interest (Cornell,
1968). The rate completely defines the
homogeneous Poisson process (HPP) counting
the occurrences of earthquakes at the site over
time. A number of advancements of PSHA have
been proposed over the years; for example,
vector-valued PSHA (Bazzurro and Cornell,
2002) and conditional hazard (Iervolino et al.,
2010). Both aim at considering multiple IMs in
PSHA. In particular, the latter considers the
distribution of a secondary IM conditional to a
value of the primary for which the hazard is
generally known.
Risk assessment of building portfolios or
spatially distributed infrastructure requires to
assess the exceedance probability, over time, of
different IMs at different sites (e.g., Goda and
Hong, 2009; Esposito et al., 2015). In these
cases, PSHA may be inadequate and the so-
called multi-site PSHA, or MSPSHA, has to be
implemented (e.g., Eguchi, 1991). In fact,
MSPSHA requires to model the correlation
structure between all IMs at all sites.
There are several alternative strategies by
which MSPSHA can be implemented for
computation (Weatherill et al., 2015). In the

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
hypothesis of joint normality of the logarithms
of the IMs at the sites and modelling the whole
correlation structure of IMs at the sites, a full
MSPSHA can be performed. Alternatively, it is
possible to simulate multiple IMs at multiple
sites taking advantage of the concept of
conditional hazard, yet this implies some
approximations. Indeed, as discussed in the
following, it only partly defines the correlation
structure and let the rest descend from the
defined terms.
The study presented herein is intended to
quantify the effect of the approximation
introduced by performing MSPSHA via
conditional hazard, when the exceedance
probability of a given vector of IM thresholds at
multiple sites is of concern.
This paper is structured such that the basics
of MSPSHA, along with the sources of
stochastic dependence between IMs are
introduced first. Subsequently, the conditional
hazard and its implementation for MSPSHA are
illustrated. Finally, an illustrative application is
developed to investigate the implications of
conditional hazard for multi-site seismic hazard
assessment. In particular, the results from the
full approach are compared with the
corresponding conditional hazard counterpart,
with reference to the effect of the number of sites
considered and their spatial configuration (i.e.,
the inter-site distance).
2. MULTI-SITE PSHA
The objective of MSPSHA is to model the
number of exceedances of IM thresholds at
multiple sites. When the sites of interest are all
affected by the same seismic sources, the process
describing the occurrence of earthquakes
causing the exceedance of the thresholds at the
ensemble of the sites is not an HPP. The reason
is in the stochastic dependence between the IMs
that each single earthquake generates at the sites
(e.g., Giorgio and Iervolino, 2016). Hereafter,
without loss of generality, it is assumed that the
IMs of interest are the pseudo-spectral
accelerations at given spectral periods, that is
 
Sa T
.
To deepen how the stochastic dependency
of pseudo-spectral accelerations has to be
accounted for by the so-called ground motion
prediction equations (GMPEs), let the
considered sites be only two, say A and B, and
1
T
and
2
T
the vibration periods of interest at site
A and B, respectively. The threshold of
 
1
Sa T
at site A is identified as
*
1
sa
and the threshold of
 
2
Sa T
at site B is
*
2
sa
. The probability that the
thresholds are both exceeded given the
occurrence of an earthquake
 
E
, that is
   
**
1 1 2 2
P Sa T sa Sa T sa E

  

, is given in
Eq. (1).
   
   
 
**
1 1 2 2
**
1 1 2 2
MZ
M ,Z
P Sa T sa Sa T sa E
P Sa T sa Sa T sa m,z
f m,z dm dz

   


   

  

(1)
In the equation,
   
**
1 1 2 2
P Sa T sa Sa T sa m,z

  

is the
probability of joint exceedance conditional on
the magnitude
 
M
and location
 
Z
of the
earthquake;
 
M ,Z
f m,z
is the joint probability
density function (PDF) of
M
and
Z
. The
integral in the equation is over the domains of
magnitude and earthquake location;
 
M ,Z
f m,z
depends on the characteristics of the seismic
source whereas
   
**
1 1 2 2
P Sa T sa Sa T sa m,z

  

is related
to the probabilistic effects of a common
earthquake at different sites. The latter can be
modelled via (i) GMPEs and (ii) the correlation
structure, which must be defined.
Under the lognormal hypothesis about one
 
Sa T
conditional to earthquake magnitude and
source-to-site distance,
R
(which is a
deterministic function of
Z
), most GMPEs
model the log of
 
Sa T
, at a site j due to
earthquake
i
, according to Eq. (2).
   
 
i i, j i i, j
i,j
log Sa T E log Sa T m ,r ,
  
  
(2)
In the equation,
 
 
i i, j
E log Sa T m ,r ,

is
the mean of
 
i, j
log Sa T
conditional on
parameters such as
M
,
R
and others
 

;
i

,
constant for all the sites in a given earthquake,

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
denotes the inter-event residual, a random
variable (RV) that quantifies how much the
mean of
 
i,j
log Sa T
in the i-th earthquake
differs from
 
 
i i, j
E log Sa T m ,r ,

. On the
other hand,
i , j

represents the intra-event
variability at site
j
in earthquake
i
.
Typically, it is assumed that inter- and intra-
event residuals are stochastically independent
normal RVs with zero mean and standard
deviation equal to
inter

and
intra

, respectively.
The sum of inter- and intra-events residuals
provides the total residual, a Gaussian RV with
zero mean and standard deviation equal to
inter intra
22
  

.
Given magnitude and earthquake location,
it is generally assumed that the logs of
 
Sa T
at
multiple sites form a Gaussian random field
(GRF; e.g., Park et al., 2007). When the same
spectral period is considered at all the sites (say
s
in number), e.g.,
1
T
, the GRF has the mean
vector given by the
 
 
1 i i, j
E log Sa T m ,r ,

terms, one for each site, and the covariance
matrix,

, given by Eq. (3) in which the
matrices have
ss
size:
   
inter intra
intra intra
22
1 1 1,2 1 1 1,s
1 1 1
1 1 1
1 1 1
T ,T ,h T ,T ,h
1
1
sym
1
  




  









(3)
The first matrix on the right-hand side,
accounts for the perfect correlation of inter-event
residuals. The second matrix accounts for the
spatial correlation of intra-event residuals; in
particular, the
 
intra 1 1 k, j
T ,T ,h

denotes the
correlation coefficient between intra-event
residuals of
 
1
Sa T
at sites
k
and
j
, (e.g.,
Esposito and Iervolino, 2012), which tends to
decrease with the increasing of the inter-site
distance,
k , j
h
. Note that the
 
1
Sa T
in one
earthquake are also stochastically dependent
because the means of the GRF in Eq. (2) share
the same event’s magnitude and location (see
Giorgio and Iervolino, 2016, for a discussion).
When different spectral periods are
considered, the lognormal hypothesis is
extended to the joint distribution of all
 
Sa T
(i.e., accelerations for different spectral periods)
at all sites. Thus, an additional correlation of
residuals has to be defined. For example, in the
case of two periods,
1
T
and
2
T
, and
s
sites, the
mean vector is made of
2s
elements (Eq. 2), two
for each site, and the covariance matrix is from
Eq. (4). The matrices in the equation have size
2s 2s
;
inter,1

and
inter,2

are the standard
deviations of the inter-event residuals of
 
1
Sa T
and
 
2
Sa T
, respectively;
intra,1

and
intra,2

are
the standard deviations of the intra-event
residuals of the two
 
Sa T
;
 
inter 12
T ,T

denotes the cross-correlation (or spectral
correlation) coefficient between the inter-event
residuals of the two
 
Sa T
(e.g., Baker and
Jayaram, 2008; Bradley, 2012);
 
intra 1 2 k , j
T ,T ,h

is the spatial-cross-correlation
of the intra-event residuals of
 
1
Sa T
and
 
2
Sa T
for site
k
and
j
(e.g., Loth and Baker,
2013). In fact the covariance matrix of Eq. (4) is
an extension of Eq. (3) to the case of two pseudo-
spectral accelerations.

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
   
   
22
inter,1 inter,1 inter inter,1 inter,2 inter inter,1 inter,2
1 2 1 2
2
inter,1 inter inter,1 inter,2 inter inter,1 inter,2
1 2 1 2
22
inter,2 inter,2
2
inter,2
,,
,,
T T T T
T T T T
sym
       
      



   


   






 
 
 
 
 
22 1 2 1,
1 1 1
intra,1 intra intra,1 intra intra,1 intra,2 intra intra,1 intra,2
12
212
intra,1 intra intra,1 intra,2 intra intra,1 intra,2
12
2
intra,2 int
,,
,, ,
,, ,
s
,s
s,1
T T h
T T h TT
T T h TT
        
      







    
   

 
2
22
ra intra,2
2
intra,2
,,
1,s
T T h
sym













(4)
3. CONDITIONAL HAZARD
The concept of conditional hazard was
introduced by Iervolino et al. (2010) for single-
site applications. It provides the distribution of a
secondary
 
Sa T
given the occurrence (or the
exceedance) of a primary one. If the primary
pseudo-spectral acceleration is
 
1
Sa T
and the
secondary
 
2
Sa T
, the distribution of the
logarithm of
 
2
Sa T
conditional to
 
1
sa T
,
which is the realization of
 
1
Sa T
at the same
site, has conditional mean
   
 
21i i, j
E log Sa T log sa T ,m ,r
and
conditional standard deviation
   
21
logSa T |logsa T

as
per Eq. (5).
   
 
 
 
 
 
 
   
21
2 1 2
11
2 1 2
1
2
2 1 2
1
i i , j i i , j
i i , j
log Sa T log sa T
E log Sa T log sa T ,m ,r E log Sa T m ,r
log sa T E log Sa T m ,r
(T ,T )
(T ,T )
 
  



  


  


(5)
In the equation,
2

and
1

are the standard
deviation of total residuals of
 
2
log Sa T
and
 
1
log Sa T
, respectively, provided by the GMPE
and
12
( , )TT

is the cross-correlation coefficient
between total residuals (Baker and Jayaram,
2008).
Under the hypothesis of bivariate lognormal
distribution of the two spectral ordinates, the
parameters in Eq. (5) are those of a Gaussian
distribution.
When MSPSHA is of concern, and two
 
Sa T
are considered, the concept of
conditional hazard may be used. In practical
terms, after simulating the primary intensity
measure; i.e.,
 
1
Sa T
, at the sites using the mean
and the covariance matrix of Eq. (2) and Eq. (3),
Eq. (5) can be applied to each site to simulate
 
2
Sa T
. This avoids the use of spatial-cross-
correlation models. On the other hand, this
strategy introduces an approximation with
respect to the application of full MSPSHA
because the spatial correlation of
 
2
Sa T
and the
spatial-cross-correlation between
 
1
Sa T
and
 
2
Sa T
are not explicitly modelled, yet they are
consequent to the conditional hazard approach.
More specifically, it is possible to demonstrate
that this procedure corresponds to approximate
the spatial-cross-correlation models of total
residuals as shown in Eq. (6) (e.g., Goda and
Hong, 2008). In other words, Eq. (4) is replaced
by Eq. (7).
1 2 , 1 1 , 1 2
2
2 2 , 1 1 , 1 2
( , , ) ( , , ) ( , )
( , , ) ( , , ) ( , )
k j k j
k j k j
T T h T T h T T
T T h T T h T T
  
  






(6)
It should be noted that the conditional
hazard approach to MSPSHA can be also
applied when secondary
 
Sa T
at the sites are

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
at different periods. Similarly to the previous
case, this implies the simulation of
 
1
Sa T
at the
sites and the application of Eq. (5) to each site
replacing
 
2
Sa T
with the spectral ordinate at
the period of interest. This type of application is
discussed in the following section.
 
 
 
 
 
   
 
 
22
1 1 1 1 1 1
1 1 1 2 1 2
1 2 1 2
21 1 1
1 1 2 1 2
1 2 1 2
2 2 2
1 1 1
22
12
2
2
, , , ,
,,
,, ,,
,, ,
,s ,s
s,
s,
T T h T T h
T T T T
T T h T T T T
T T h TT
sym
         
       
   


     



    







(7)
4. ILLUSTRATIVE APPLICATION
The objective of this section is quantifying, for
an illustrative case, the differences on results
when (i) full MSPSHA and (ii) approximated
conditional hazard procedure (CH) are
implemented. To this aim, a set of one hundred
sites located in the district of Naples (southern
Italy) are considered. The sites are distributed on
a regular grid with inter-site distance equal to 1.5
km and they are assumed to represent the
locations of a hypothetical heterogeneous
building portfolio. It is also assumed that one
intensity measure, one
 
Sa T
, is of interest for
each site, but, due to buildings’ heterogeneity, a
different vibration period is considered for each
site. The spectral ordinate of interest for each site
is set among the following five:
 
0.6sSa
,
 
0.7sSa
,
 
0.8sSa
,
 
0.9sSa
and
 
1sSa
.
The one considered for each site is shown in
Figure 1.
In order to define the threshold values of
interest for risk assessment, a single-site
(classical) PSHA is first performed at each site.
Thus, the threshold values are computed as the
acceleration to which classical single-site PSHA
associates a return period
 
r
T
of 475 years.
Then, the two procedures of MSPSHA are
applied to compute the distribution of the
number of exceedances collectively observed at
the sites in a time interval
 
T

equal to fifty
years. The comparison of the two resulting
distributions is discussed.
Figure 1: Pseudo-spectral acceleration of interest
for each site and location of the seismic source zone
928 from Meletti et al. (2008).
In both PSHA and MSPSHA analyses, the
model adopted to describe the seismic sources is
that of Meletti et al. (2008), which features
thirty-six seismic source zones for the whole
Italy, numbered from 901 to 936. However, for
the purposes of the application, only zone 928 is
considered (Figure 1) for simplicity. The seismic
characterization of the zone is from Barani et al.
(2009), that is, a Gutenberg-Richter type
magnitude distribution (Gutenberg and Richter,
1944) with minimum and maximum magnitude
equal to 4.3 and 5.8, respectively, negative slope
of 1.056, and annual rate of earthquakes of
0.054. The GMPE used herein is that of Akkar
and Bommer (2010). It is applied within its
definition ranges of magnitude (5-7.6) and
Joyner and Boore (
JB
R
; Joyner and Boore, 1981)
distance (0-100 km). Epicentral distance is
converted into
JB
R
according to Montaldo et al.
(2005). According to Meletti et al. (2008), the
normal style of faulting is considered for the
928
Sa(0.6s)
Sa(0.7s)
Sa(0.8s)
Sa(0.9s)
Sa(1s)

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
source. Rock soil condition is assumed for all the
sites.
The two procedures of MSPSHA differ for
the simulation of the GRF of spectral ordinates
at the sites conditional to the magnitude and the
location of the earthquake on the source. Such
simulations, described in the following section,
are repeated for each earthquake occurring on
the source in the fifty years interval. Thus, at the
base of the two procedures there are three-
million seismic histories representing the
earthquakes occurring on the source in fifty
years that have been simulated via a recently
developed software for regional, single-site and
scenario based probabilistic seismic hazard
analysis (REASSESS). The software adopts a
two-step procedure to simulate the earthquake
occurrence over time: the first step is addressed
to simulate and collect the magnitudes and
locations of the earthquakes conditional to the
occurrence of a generic earthquake. Such
seismic scenarios are the input of step two that
consists of simulating the process of earthquakes
affecting the sites in any time interval, that is the
seismic history in
T

. Further details about the
simulation of the seismic histories are in
Chioccarelli et al. (2018).
4.1. Simulation procedures
For each earthquake occurring on the source in
the specific realization of fifty years, the first
strategy (full MSPSHA) simulates a realization
of the GRF made of the
 
Sa T
at all the sites,
via implementation of the full correlation
structure of the type in Eq. (4). More
specifically, the components
 
 
i i, j
E log Sa T m ,r ,

of the mean vector, one
for each site and spectral period of interest, are
computed according to the GMPE. Then, the
vector containing the total residual for each site
and period is sampled from a zero-mean
multivariate normal distribution with covariance
matrix as per Eq. (4). Finally, the realization of
the GRF is obtained by adding the vector of the
residuals’ realization to the mean vector. The
correlation structure of inter-event and spatial-
cross-correlation of intra-event residuals,
implemented in Eq. (4), are those of Baker and
Jayaram (2008) and Loth and Baker (2013),
respectively. The sought distribution of the total
number of exceedances at the sites, in terms of
probability mass function (PMF), has been
carried out through the REASSESS software
(Chioccarelli et al., 2018) and is pictured in
Figure 2.
As alternative strategy, the GRF of
realizations at the sites has been simulated
through the double-step simulation described in
Section 3. First, a primary spectral ordinate to be
simulated at all the site has been selected. It is
known that the higher is the period of the
primary
 
Sa T
, the lower is the approximation
introduced by the conditional hazard approach
(e.g., Goda and Hong, 2008). Thus,
 
1sSa
is
chosen here as primary. Then, for each
 
1sSa
value at each site, obtained from the GRF
simulation, the realization of the secondary
 
Sa T
of interest at each site (see Figure 1) is
sampled from the normal conditional
distribution of Eq. (5), in which the model of
Baker and Jayaram (2008) is used for the
correlation of total residuals. This procedure is
equivalent to sample from a multi-variate
Gaussian distribution with the correlation
structure in Eq. (7). The resulting PMF of the
total number of exceedances observed in fifty
years is also shown in Figure 2.
Figure 2: Distribution of the total number of
exceedances observed at the considered sites in fifty
years obtained through the two approaches.
4.2. Discussion
As expected, the two distributions have the same
mean because the mean is not affected by the
correlation. The mean is the single-site
occurrence rate of the thresholds,
r
1T
,
0 2 4 6 8 10
0.01
0.1
1
Total number of exceedances in 50 years
PMF
Full MSPSHA CH

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
multiplied by the time interval of interest and the
number of sites, that is
0.0021 50 100 10.5  
. It
can be also observed that the PMFs show a very
similar trend; they only slightly differ in terms of
variance
 
VAR
. In order to quantify the
approximation of CH, the relative difference
between variance of the full MSPSHA
 
Full
VAR
and the CH approach
 
CH
VAR
, computed as
Full CH
Full
VAR VAR
VAR

Δ
, is introduced. For the
examined case, it results
1.65%Δ
.
4.3. The effect of the number of sites and the
inter-site distance
In order to study the effect on results of the inter-
site distance and the number of sites, analyses
have been repeated considering the same
portfolio of Figure 1, but with additional inter-
site distances: 0.2 km, 0.5 km and 3 km.
Moreover, for each inter-site distance nine
subsets of sites are considered, from
s4
to
s 100
as shown in the upper panel of Figure 3.
Thus, thirty-six analyses have been performed.
For each combination of inter-site distance,
h
,
and number of sites, s, the investigated PMF has
been computed through full MSPSHA and CH
and the differences of variance of distributions
are reported in terms of
Δ
in Figure 3(b).
Each curve in the figure provides, for a
given inter-site distance, the trend of
Δ
as a
function of the number of sites. It can be noted
first that the inter-site distance has a minor
effect. Then, although a slight increase of
Δ
is
shown for the increasing number of sites up to
s 25
, the general trend of
Δ
is non-monotonic
and the values are always within the range of -
0.5% and 2%.
5. CONCLUSIONS
When the hazard assessment for multiple sites is
of interest, that is MSPSHA, the key issue is to
account for the stochastic dependence of the site-
specific counting processes, which derives from
the correlations of pseudo-spectral accelerations
at the sites. In fact, it was shown that modelling
all the sources of dependency in the
 
Sa T
simulations at multiple sites requires models of
spatial-cross-correlation that have to be fitted on
a relevant amount of data, which can also be
region-dependent.
Figure 3. (a) Configuration of sites for each subset;
(b) Approximation of conditional hazard for
different inter-site distances and number of sites.
On the other hand, conditional hazard
allows to obtain the distribution of a secondary
 
Sa T
, given a primary one, at a site of interest.
Thus, CH can be applied to generate random
fields of a secondary
 
Sa T
at the sites,
conditional to the spatially correlated
realizations of the primary
 
Sa T
. This
procedure can be adopted for MSPSHA and does
not require the covariance structure of different
pseudo-spectral accelerations at different sites to
be modelled. Nevertheless, it introduces some
approximations, which were quantified in an
illustrative application. To this aim, a portfolio
of one-hundred equally spaced sites located in
the southern Italy is considered. The distribution
of the total number of exceedances observed at
the ensemble of the sites in fifty years was
computed through the full and CH approach. The
approximation of CH was evaluated in terms of
relative difference between the variances of the
two distributions. In the case the inter-site
s=4 s=9 s=16 s=25
s=36 s=49 s=64 s=81 s=100
(a)
(b) Sa(0.6s) Sa(0.7s) Sa(0.8s) Sa(0.9s) Sa(1s)
4 9 16 25 36 49 64 81 100
−5
−2.5
0
2.5
5
s
∆ [%]
h=0.2km
h=0.5km
h=1.5km
h=3km

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
distance is 1.5 km, it was found that the relative
difference is equal to 1.65%. In order to
investigate the effect of the inter-site distance
and the number of sites, different spatial
configurations for the portfolio were also
considered. Still with reference to the variance
of the distributions of the total number of
exceedances observed at the sites in fifty years,
it was shown that the relative difference of CH
with respect to the full approach is negligible at
least in the investigated cases.
6. ACKNOWLEDGEMENTS
This paper was developed within the H2020-MSCA-
RISE-2015 research project EXCHANGE-Risk
(Grant Agreement No. 691213).
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