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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.

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

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)

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

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

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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