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Bull Earthquake Eng
DOI 10.1007/s10518-015-9730-5
ORIGINAL RESEARCH PAPER
Exploring the impact of spatial correlations and
uncertainties for portfolio analysis in probabilistic
seismic loss estimation
G. A. Weatherill ·V. Silva ·H. Crowley ·P. Bazzurro
Received: 21 February 2014 / Accepted: 12 January 2015
© Springer Science+Business Media Dordrecht 2015
Abstract The significant potential for human and economic losses arising from earthquakes
affecting urban infrastructure has been demonstrated by many recent events such as, for exam-
ple, L’Aquila (2009), Christchurch (2011) and Tohoku (2012). Within the current practice of
seismic loss estimation in both academic and industry models, the modelling of spatial vari-
ability of the earthquake ground motion input across a region, and its corresponding influence
upon portfolios of heterogeneous building types, may be oversimplified. In particular, the
correlation properties that are well-known in observations of ground motion intensity mea-
sures (IMs) may not always be fully represented within the probabilistic modelling of seismic
loss. Using a case study based on the Tuscany region of Italy, the impacts of including spa-
tially cross-correlated random fields of different ground motion IMs are appraised at varying
spatial resolutions. This case study illustrates the impact on the resulting seismic loss when
considering synthetic aggregated portfolios over different spatial scales. Inclusion of spatial
cross-correlation of IMs into the seismic risk analysis may often result in the likelihood of
observing larger (and in certain cases smaller) losses for a portfolio distributed over a typical
city scale, when compared against simulations in which the cross-correlation is neglected. It
can also be seen that the degree to which the spatial correlations and cross-correlations can
impact upon the loss estimates is sensitive to the conditions of the portfolio, particularly with
respect to the spatial scale, the engineering properties of the different building types within
the portfolio and the heterogeneity of the portfolio with respect to the types.
Keywords Seismic risk ·Spatial correlation ·Cross-correlation ·Heterogeneous portfolios
G. A. Weatherill (B)·H. Crowley
European Centre for Training and Research in Earthquake Engineering (EUCENTRE), Pavia, Italy
e-mail: graeme.weatherill@eucentre.it
V. Silva
University of Aveiro, Aveiro, Portugal
P. Bazzurro
Institute for Advanced Study of Pavia (IUSS-Pavia), Pavia, Italy
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Bull Earthquake Eng
1 Introduction
The process of analysing risk to a spatially distributed portfolio of assets is a central tenet
of catastrophe modelling in both academic research and in industry. A typical portfolio
will often consist of assets that are heterogeneous in terms of structure type, usage, seismic
code design and age (Crowley et al. 2009). The derivation of fragility models for different
structures often defines the ground motion intensity measure (IM) in terms of metrics that
are most efficient in characterising the resulting damage to the structure. This may typically
be the spectral acceleration or displacement at the fundamental elastic period of the structure
(Sa(T0),Sd(T0)). The requirements to define the seismic input in a manner that is most
consistent with the fragility models for each structure type, present challenges to the seismic
risk modeller. The generation of spatial random fields of a ground motion intensity measure
(IM), either for a single scenario event or for a set of stochastic events with a given probability
of occurrence, is a critical step in the loss estimation process.
The analysis of a spatially distributed set of elements, be it a portfolio of buildings, such as
the case here, or an infrastructure system such as utility networks (e.g., water, electrical power,
communications etc.), presents risk modellers with further challenges in the characterisation
of the ground motion. For a probabilistic seismic risk analysis to a single structure, the
aleatory variability in the ground motion is a critical parameter in controlling the losses.
When considering multiple structures it is insufficient to treat the ground motion IMs at
all structures sites as independent variables if the structures themselves can be affected by
the same earthquake and, especially, if they are located closely in space. Therefore, such
probabilistic seismic risk analyses need to consider the spatial correlation properties of the
ground motion IMs.
It is well-established that observations of strong ground motion IMs resulting from the
same earthquake may display correlations over distance (e.g. Boore et al. 2003;Wan g and
Takada 2005;Sokolov et al. 2010), and that the distances over which the correlations may
be significant are generally greater for the long period characteristics of the ground motion
(Jayaram and Baker 2009). In recent years these observations have proceeded to form the
basis for the development of models of spatial cross-correlation to describe the correlation
between different ground motion IMs across sites separated over a distance (e.g. Goda and
Hong 2008;Goda and Atkinson 2009;Loth and Baker 2013). In the analysis considered
here, the focus is primarily upon seismic risk analysis to building portfolios. It should be
recognised, however, that the same requirements to consider spatially correlated and spatially
cross-correlated fields of ground motion also apply, and may indeed be even more critical,
when considering seismic risk to urban infrastructure. Recent developments in the European
SYNER-G project (Franchin et al. 2011) have considered the use of spatial cross-correlation in
the generation of ground motion fields of IMs for use in multi-system infrastructure analysis.
The development of a computationally efficient methodology for spatial co-simulation of
cross-correlated ground motion fields of IMs, and the need to understand its impact in a
seismic risk context has been an important motivation for the current work.
The objective of the current paper is to explore, and illustrate with realistic examples,
the effects of spatial correlation in seismic risk analysis for portfolios of heterogeneous
building types. Assessment of risk to heterogeneous portfolios requires the use of different
IMs to define the vulnerability of different building types. Heterogeneous portfolios can
be seen as a generalisation of the homogeneous case, in which the same IM is used for
all buildings. Therefore„ for heterogeneous portfolios the correlation between the IMs is
more complex than in the homogeneous case. Yet even for heterogeneous portfolios the
incorporation of spatial correlation of IMs into the risk analysis may be done via simulation
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Bull Earthquake Eng
of correlated (in this case jointly normally distributed) random variates. The subsequent
analysis is intended to demonstrate, however, the manner in which the correlation matrix is
constructed using available models of correlation in the published literature, and the means
by which the correlations are simulated in the analysis impact upon the resulting estimates
of seismic losses, and to consider the potential conditions under which such impacts may be
of relevance to the risk modeller.
2 Modelling spatial correlation and cross-correlation of ground motion intensity
measures in seismic risk analysis
Empirical models of both spatial correlation and between-IM cross-correlation of ground
motion IMs generally capture the phenomenon within the aleatory variability of the ground
motion prediction equation (GMPE), which takes the form:
log IM
ij=fMi,Rij,θ
ij+τν
i+σ
ij (1)
where IMij denotes the value of the ground motion IM of interest at site jlocated at a distance
Rij from the source of an earthquake of magnitude Mi. The parameters θij account for terms
related to style-of-faulting and site effects. Aleatory variability in the ground motion model
is separated into an inter-event (τν
i) and an intra-event (σ
ij) term, the inter-event term
representing the variability of the median IM from one earthquake to another earthquake
of the same magnitude and rupture mechanism and the intra-event term representing the
variability of the ground motion value from one site to another at the same distance with the
same site characterisation (Bommer and Crowley 2006). The terms τiand ij are standard
normally distributed random variables, and the constants τand σare the standard deviations
of the inter- and intra-event variability.
Observations of ground motion from densely recorded earthquakes have shown that the
intra-event residual term (ij) is found to be spatially correlated, such that the coefficient
of correlation (ρh) between the intra-event residuals observed at two sites separated by a
distance, h, will decrease with increasing separation distance. In the majority of spatial
correlation models (e.g. Wang and Takada 2005;Goda and Hong 2008;Goda and Atkinson
2009;Jayaram and Baker 2009;Esposito and Iervolino 2011,2012) an exponential function
form has been preferred:
ρh(T)=exp a(T)hb(T)(2)
where a(T)and b(T)are period-dependent coefficients describing the strength of attenuation
of spatial correlation with distance. The distance over which the correlation may be considered
significant, termed “correlation length” in geostatistical analysis, is also known to be greater
for IMs that characterise the lower frequency content of the ground motion (e.g. long period
spectral acceleration/displacement) (e.g. Jayaram and Baker 2009;Esposito and Iervolino
2012). In adopting the exponential model, of the form shown in Eq. 2, as a basis for simulating
spatially correlated fields of ground motions several assumptions are made. The first is that
the ground motion residuals for a set of spatially distributed points can be considered joint
normally distributed, and that therefore the simulation of the residuals need only consider
a multivariate Gaussian distribution. The assumption of multivariate normality in spatially
distributed ground motion residuals has been shown to be appropriate by Jayaram and Baker
(2008). The second assumption underlying this model is that the fields are isotropic and
homogeneous; isotropy indicates that no azimuthal trend is visible in the correlation, as
shown by Jayaram and Baker (2009), while the homogeneity indicates that the mean and
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Bull Earthquake Eng
auto-covariance of the field are not dependent on the location, as shown by Wang and Takada
(2005).
It is also well-established from observed ground motion records, that the intra-event ground
motion residual term for two different periods of spectral acceleration at the same site are
cross-correlated, with the coefficient of correlation decreasing in accordance with an increase
in the spacing between periods (Inoue and Cornell 1990;Baker and Cornell 2006;Jayaram
and Baker 2008). When considering the seismic risk to a heterogeneous type of buildings,
utilising fragility models that are functions of the fundamental period of response of each
building type, it is desirable, even necessary, that the ground motion fields preserve all three
facets of correlation within the ground motion IMs generated by the same earthquake: the
correlation between the intra-event residuals of different ground motion IMs for the same site,
the spatial correlation between the intra-event residuals of the same IM for sites separated by
a distance h, and the spatial cross-correlation between the intra-event residuals of different
ground motion IMs for sites separated by a distance h.
2.1 Simulating spatially correlated ground motion fields
The simplest methodology for simulation of spatially correlated random fields of GMPE
residual values that are not conditioned to any observation comes from the classical decom-
position approach. Under the assumption of joint log-normality in the GMPE residuals, a
multivariate Gaussian distributed random field (Y)isdefinedasasetofy1,y2,...yNground
motion residual values for Nsites, generated from the following function:
Y=μ+LZ (3)
where Zis a vector of independent Gaussian distributed random variates that take on the
values z1,z2,...zN,μ is a zero-valued vector of length N,Lthe lower triangular matrix,
obtained using Cholesky factorisation, such that LLT=C,whereCis the positive-definite
correlation matrix:
C=⎡
⎢
⎢
⎢
⎣
1ρh1,2··· ρh1,N
1··· ρh2,N
....
.
.
sym 1
⎤
⎥
⎥
⎥
⎦
(4)
and ρhi,jis the coefficient of correlation between ground motion residuals for two locations
separated by a distance of hi,j. For spatially distributed portfolios of a homogeneous building
type, requiring only one IM (often a spectral acceleration at a given period), Eq. 3is sufficient
to characterise the spatial correlation of the IM at different sites. To retrieve the resulting
logarithmic ground motion values for the set of sites, the residual values are multiplied by
the aleatory variability term (σ) of the GMPE and added to the expected, or median, ground
motion defined by the GMPE (fMi,Rij,θ
ijin Eq. 1).
For heterogeneous portfolios considering multiple intensity measure types the problem
is more complex as the coefficient of correlation in GMPE residuals between two different
intensity measures ρIM
k,IM
lhijis not equal to unity in the case that IM
k= IM
l.Nev-
ertheless, it is still necessary to co-simulate multiple fields of ground motion IMs whilst
preserving their spatial cross-correlation structure. To undertake this, the several different
approaches are presented below.
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2.2 Conditional hazard (“Markov-type” approach)
The conditional hazard approach (Iervolino et al. 2010) one method by which fields of ground
motion residuals for different IMs can be generated at multiple sites, taking into account the
spatial correlation. Assuming a spatially correlated field of ground motion residuals for IM
1
generated using Eq. 3, and defining ρIM
k,IM
lequivalent to ρIM
k,IM
l(h=0), the distribution
of each additional intensity measure IM
kis described via:
μIM
k|IM
1,M,R=μIM
k|M,R+ρIM
1,IM
kσIM
k
z−μIM
1,M,R
σIM
1
(5)
σIM
k|IM
1=σIM
k1−ρ2
IM
1,IM
k(6)
where μIM
k|IM
1,M,Rand σIM
k|IM
1are the mean and total standard deviation of the logarith-
mic ground motion for intensity measure k,μ
IM
k|M,Rand σIM
kthe corresponding uncon-
ditional mean and standard deviation, and zis the random variate corresponding to the total
aleatory variability term, simulated for the primary IM using Eq. 3. Therefore at each site, the
ground motion residuals at each intensity measure are conditioned upon that of the primary
intensity measure. This approach is equivalent to the “Markov-type” approach described in
Loth and Baker (2013)andJournel (1999).
The conditional hazard approach implicitly assumes that the distribution of the ground
motion residual for each secondary intensity measure IMkis conditional only upon the pri-
mary intensity measure at the site. This makes it an approximate method, by which the spatial
correlations in the secondary intensity measures are inferred from the spatial correlation of
the primary IM. In the case where multiple secondary intensity measures are being consid-
ered, correlations between the secondary IMs are not directly modelled, or are only implicit
via the correlation with the primary IM. Furthermore, the spatial correlation structure of each
the secondary IMs is not faithfully reproduced. As the correlation length of longer period
motion is greater than that of short period motion it is preferable to adopt the longer period
IM as the primary IM in practice (e.g. Goda and Hong 2008) so as to better ensure that
the conditions of the screening hypothesis assumed by the “Markov-type” approach are met
(Journel 1999). The reader is referred to Loth and Baker (2013) for further discussion of this
issue; however, later subsequent analysis in this paper will illustrate the impact of selection
of the primary IM in practice.
2.3 Full-block cross-correlation
A method for co-simulation of multiple cross-correlated fields is demonstrated by Oliver
(2003), who extends the classical matrix decomposition approach to separate the co-
simulation of the random fields Ykand Ylinto the following matrix formulation:
Yk
Yl=μIM
k
μIM
l+LIM
k0
ρIM
k,IM
lLIM
l1−ρ2
IM
k,IM
lLIM
lZIM
k
ZIM
l(7)
where LIM
kLT
IM
k=CIM
k,IM
k,LIM
lLT
IM
l=CIM
l,IM
lare the auto-covariance matrices of
fields YIM
kand YIM
lrespectively. This relatively simple formulation therefore allows for
the definition of the cross-covariance matrix:
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Bull Earthquake Eng
LLT=⎡
⎣
LIM
k0
ρIM
k,IM
lLIM
l1−ρ2
IM
k,IM
lLIM
l⎤
⎦⎡
⎣
LT
IM
kρIM
k,IM
lLT
IM
l
01−ρ2
IM
k,IM
lLT
IM
l⎤
⎦
=LIM
kLT
IM
kρIM
k,IM
lLIM
kLT
IM
l
ρIM
k,IM
lLIM
lLT
IM
kLIM
lLT
IM
l(8)
It follows that this formulation can be extended to an arbitrary number (k)ofIMs in the
following manner:
LLT=⎡
⎢
⎢
⎢
⎢
⎢
⎣
LIM
1LT
IM
1ρIM
1,IM
2LIM
1LT
IM
2··· ρIM
1,IM
kLIM
1LT
IM
k
ρIM
1,IM
2LIM
2LT
IM
1LIM
2LT
IM
2··· ρIM
2,IM
kLIM
2,LT
IM
k
.
.
..
.
.....
.
.
ρIM
1,IM
kLIM
kLT
IM
1ρIM
2,IM
kLIM
kLT
IM
2··· LIM
kLT
IM
k
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(9)
definition of the full cross-correlation matrix, co-simulation of all the corresponding para-
meter fields is simply undertaken in the same manner as in Eq. 3.Itisalsothecasethat
if the spatial correlation matrices for each IM CIM
k,IM
kare each positive-definite, and the
correlation matrix describing the IM to IM correlations is positive-definite, then the resulting
full-block cross-correlation matrix C=LLTwill be positive-definite.
This approach to deriving the cross-correlation matrix ensures that not only are the spatial
correlation properties within each intensity measure preserved. This allows for the distance-
dependent correlation length of each IM to be modelled (as LIM
kLT
IM
lis equivalent to LLT
in the case that k=l), whilst the IM to IM correlation will reduce to the cross-correlation
matrix in the case that N=1, thus rendering the correlation between IMs independent of the
distance scaling of the correlation. This is in contrast to the conditional hazard approach in
which the spatial correlation structures of the ground motion residuals within each IM are not
explicitly retained. It also permits for the construction of the spatial cross-correlation model
using separate correlation models for the spatial correlation and the IM-to-IM correlation,
provided that the correlation matrices for both are positive-definite. This allows for greater
flexibility in adopting correlation models, particularly spatial correlation models, that may
reflect more the local characteristics of a region.
2.4 Linear model of coregionalisation (LMCR)
The third approach to modelling the cross-correlation between fields of ground motion at
different spectral periods is via the linear model of coregionalisation, which is fit to a set of
experimental semi-variograms and cross-variograms of observed GMPE residuals from well-
recorded events by Loth and Baker (2013). The authors propose an appropriate functional
form of the LMCR for this purpose to be:
C(h)=B1exp −3h
20 +B2exp −3h
70 +B3Ih=0(10)
where B1,B2and B3are the coregionalisation matrices for short-range, long-range and
zero-separation, respectively. Ih=0is an indicator function taking the value of 1 for h=0,
and zero otherwise. The coefficients of the coregionalisation matrices are given in Loth and
Baker (2013). As the correlation model is fit directly to the variograms, positive-definiteness
can largely be ensured in the resulting cross-covariance matrix. Therefore the model does
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Bull Earthquake Eng
not separate the spatial correlation and inter-IM correlation models, but instead fits a single
model capturing both characteristics. This may mean that if regional differences, or even
inter-event differences, in the correlation structure exist it may be necessary to determine
different coregionalisation matrices more suitable for local application.
2.5 A note on the residual terms
A new question that arises when transitioning from consideration of single intensity measures
to co-simulation of multiple correlated intensity measures is the role of the inter-event resid-
ual. Both the conditional hazard and LMCR approaches define the spatial correlation and
cross-correlation using the inter-event residual, whilst the full-block cross-correlation models
can be constructed using correlation models from intra-event residuals where such models
are available. In the derivation of existing spatial correlation models, both Park et al. (2007)
and Jayaram and Baker (2009) demonstrate that if the inter-event variability is assumed to be
constant, then the coefficient of correlation for the intra-event residual can be derived from
the total residual term. This assumption is not necessarily valid in many cases that are relevant
here. Recent developments in the modelling of nonlinear site response in modern GMPEs,
such as Abrahamson and Silva (2008), now incorporate magnitude and site dependence into
the inter-event residual term, though notably not in Boore and Atkinson (2008)asusedby
Loth and Baker (2013) for the derivation of the LMCR. This would prevent the inter-event
residual term from being considered constant for a single period. Secondly, and more impor-
tantly for the current analysis, it is known that the coefficient of correlation in the inter-event
residual term between two different IMs is less than unity(Goda and Atkinson 2009), thus
for a given event the inter-event residual is not constant across multiple period. In deriving
the linear model of co-regionalisation, Loth and Baker (2013) use the total residual, thus
aggregating the inter-event correlation into the cross-correlation model. It may be the case,
therefore, that when implementing the LMCR in a co-simulation it may be inconsistent to
separate the inter- and intra-event residuals. The impact of modelling inter-event correlation
versus simply modelling the total residual when applying the LCMR approach is considered
subsequently.
3 Case study application to synthetic portfolios derived from Italian data
To demonstrate the influence of spatial cross-correlation on the risk analysis of a hetero-
geneous portfolio of buildings, we consider a case study from the Tuscany region of Italy.
Seismic risk analysis is carried out for portfolios aggregated over different spatial scales. To
generate the ground motion fields for the Tuscany region, the stochastic seismic hazard analy-
sis is undertaken using the SHARE area source model for Italy (Woessner et al. 2012). In the
current example, 129 uniform area sources are considered within Italy and the surrounding
Mediterranean region, of which 78 are found within a distance of 250 km from Tuscany.
For each source the magnitude frequency distribution is characterised by a doubly-truncated
exponential model, consistent with the Gutenberg-Richter earthquake recurrence model. The
focal mechanism and earthquake depth distributions are defined for each zone allowing for
the characterisation of a finite pseudo-rupture plane for each stochastically generated earth-
quake. In the present example only the Akkar and Bommer (2010) GMPE is considered.
The influence of the choice of GMPE, and other epistemic uncertainties, on the final loss
estimates is not within the scope of this study. As suggested previously the choice of GMPE
may influence upon the results depending on whether the inter- and intra-event components
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Bull Earthquake Eng
of the aleatory variability are homo- or hetero-skedastic, or due to the manner in which soil
nonlinearity is accounted for in the functional form. In this particular case the GMPE in
question assumes a relatively simple functional form in which nonlinear site response is not
considered and inter-event variance is constant for each period. No spatial correlation in the
ground motion residual terms was considered in the fitting of this GMPE to data, though
subsequent analysis of the European strong motion data set by Esposito and Iervolino (2012)
identifies clear evidence of spatial correlation in the strong motion records.
The hazard is rendered for the “rock” site condition (in this case NEHRP class B). Appli-
cation to real portfolios should require detailed microzonation of the exposure region in order
to constrain the spatial distribution of site conditions. In this analysis we simulated the occur-
rence of more than 120,000 earthquakes corresponding to 100,000 realisations of one-year
seismicity in the region. For each earthquake one realisation of the cross-correlated random
fields of the selected ground motion IMs is generated.
3.1 Correlation models
With the exception of the LMCR methodology, the cross-correlation methods described
in Sect. 2require the definition of both a spatial correlation model, and a spectral cross-
correlation model. In the current analysis the spectral cross-correlation model of Baker and
Cornell (2006) is preferred, and the period-dependent spatial correlation model of Jayaram
and Baker (2009) is adopted:
ρ(h)=exp −3h
b(T)where b=⎧
⎪
⎨
⎪
⎩
8.5+17.2T“Case 1” T<1s
40.7−15.0T“Case 2” T<1s
22.0+3.7TT≥1s
(11)
where bdenotes the spatial length scale of the correlation, namely the distance at which the
correlation coefficient is found to fall below 0.05, and is a property of the frequency of the
ground motion IM. The two cases presented in this formulation refer to a model in which no
clustering in site condition (VS30) is expected (“Case 1”) and when clustering is believed to
be observed (“Case 2”).
In this analysis, spectral correlation in the inter-event residual is incorporated using the
model of Goda and Atkinson (2009):
ρτ(T1,T2)=1
31−cos π
2−θ1+θ2ITmin<0.25
×Tmin
Tmax θ3
log10 Tmin
0.25 log10 Tmax
Tmin
+1
31+cos −1.5log
10 Tmin
Tmax (12)
where Tmax and Tmin are the maximum and minimum of the two periods, ITmin<0.25 takes
the value of 1 if Tmin <0.25 and 0 otherwise, and θ1,θ
2and θ3are coefficients taking the
values of 1.374,5.586 and 0.728 respectively. For clarity we note that the inter-event residual
correlation model presented by Goda and Atkinson (2009) is applied to the geometric mean
of the horizontal components, whilst that of Jayaram and Baker (2009) is fit to the rotationally
independent geometric mean.
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Bull Earthquake Eng
Fig. 1 Aggregated value of exposure data for the Tuscany province, with SHARE area sources delineated by
black lines. Florence administrative district and city district are highlighted in pink and yellow, respectively
As discussed in Sect. 2.5, many spatial correlation models available in peer-reviewed liter-
ature, including that selected here, are fit to the total residual term, rather than the inter-event
residual. Whereas in the current simulation method we are separating the simulation of the
aleatory variability into the inter- and intra-event components. This creates an inconsistency
between the manner in which the correlation models are constructed and the manner in which
they are applied. This inconsistency may only be resolved by defining correlation models for
the full-block cross-correlation and LMCR methodologies that explicitly separate the inter-
and inter-event terms. Construction of such models is beyond the scope of this paper, but
differences in the resulting loss curves should be interpreted taking into consideration this
issue.
3.2 Exposure model
An exposure model capable of providing the spatial distribution of each building type, along
with its structural replacement cost, throughout the region of Tuscany has been developed
(Fig. 1), mostly based on information from the Italian Building Census Survey of 1991.
In this source, buildings are organised according to the predominant construction material
[reinforced concrete, masonry, other (i.e., wood and steel) and unknown], age of construction
and number of storeys. For the sake of simplicity buildings from the category “other” and
“unknown” have been ignored as there was insufficient information available regarding their
seismic vulnerability and, moreover, these categories represent only 1 % of the buildings in
the region of interest. To take into account the age of construction, two sub-classes have been
considered: Pre-code (buildings constructed prior to the 1974 design code, which endorsed
the consideration of a horizontal lateral load equal to 2 % of the total weight) and Post-code
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Bull Earthquake Eng
Tab le 1 Characteristics of the building portfolio
Construction
material
Seismic design
level
Height Period of
vibration (s)
Building class Percentage of
assets
Reinforced
concrete
Pre-code Low-rise 0.20 RC LR PC 8.9
Mid-rise 0.85 RC MR PC 4.9
High-rise 1.20 RC HR PC 1.5
Post-code Low-rise 0.20 RC LR C 7.3
Mid-rise 0.50 RC MR C 4.0
High-rise 0.85 RC HR C 0.5
Masonry Pre-code Low-rise 0.20 M LR PC 60.6
Mid-rise 0.50 M MR PC 12.3
(buildings constructed according to the 2003 design code, which establishes a horizontal
lateral load depending on the seismic zone, taken as 7 % of the total weight in our case). All
of the masonry buildings were assumed to have no seismic design. Regarding the number
of storeys, three sub-classes have been considered: low-rise (1–3 storeys), mid-rise (4–6
storeys) and high-rise (7 storeys or greater). This information was available at the level of
the third Italian administrative unit (i.e., “comune”). The spatial extent of these regions can
vary from 5 to 470 km2, with an average area of 80 km2. The organisation of the exposure
model and the percentage of each building type within the building portfolio are described in
Tabl e 1.
The assumption that all the buildings are located at the centroid of the associated “comune”
would not allow for consideration of the variation of the seismic hazard within each region,
and also would essentially induce an artificial perfect correlation of the ground motion IM,
since each building belonging to a given type would experience the same shaking. Fur-
thermore, the distance between the centroids of each “comune” is often larger than typical
correlation lengths, which would impede an adequate evaluation of the various methodolo-
gies described herein. In order to overcome these issues, the original exposure model was
spatially disaggregated according to a 30 arc-second grid (approximately 0.9 km at the lati-
tude considered here). To this end, the LandScan dataset (Dobson et al. 2000) that provides
population count for a grid with the aforementioned resolution was used. For each “comune”,
the number of buildings was distributed throughout the grid cells based on the population
in each cell. Thus, in grid cells with a high population count a large number of buildings
was assigned, whilst in cells with zero population count no buildings were allocated. In this
process, the fraction of each building class was retained, thus not favouring any particular
typology depending on the size of the population. Both the original and disaggregated models
are shown in Fig. 2.
It should be noted that in the current approach there may exist an artificial bias in the
correlation due to the aggregation of the assets from within a cell such that the input ground
motion is taken from a single point at the centroid of the cell (Bazzurro and Luco 2005).
This is demonstrated by Stafford (2012), who propose an adaptation to the conditional sim-
ulation procedure in the case of aggregated portfolios. To demonstrate that the trends seen
in the results can not be attributable to this aggregation bias on the mean loss estimates, a
second set of synthetic portfolios are generated in which a set of heterogeneous assets are
sampled from within each cell (weighted according to the total aggregated value of the cells)
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Fig. 2 Exposure model following an unevenly spaced (commune) resolution (top) and according to the 30
arc-second grid (bottom)
and distributed randomly within the cell. Two sample portfolios, one for the Firenze (Flo-
rence) administrative province, and one for the Firenze (Florence) city district, are shown in
Fig. 3.
The estimation of the structural replacement cost of each building was performed based on
information regarding the average number of dwellings per building type, average building
year per dwelling and average replacement cost per area, taken from the Italian Statistical
Office database.1
1http://www.istat.it/it/censimento- popolazione (last accessed February 2014).
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Fig. 3 Synthetic heterogeneous portfolio of single assets for the Firenze administrative province (main)and
city district (inset)
3.3 Vulnerability model
In order to fully explore the various methodologies for spatial correlation and cross-
correlation modelling of ground motion IMs in seismic risk, the loss estimation for the
different building types was performed using a new set of vulnerability functions (i.e. loss
ratio distribution for a set of intensity measure levels). This vulnerability model was devel-
oped based on a methodology proposed by Silva et al. (2013), and using the material and
geometrical properties for Italian buildings defined in Borzi et al. (2008) for reinforced con-
crete types and in Binda et al. (1999) for the masonry classes. Each vulnerability function is
expressed in terms of spectral acceleration for the yielding period of vibration (see Table 1),
which was calculated using the simplified period-height relationships proposed by Crowley
et al. (2004)andBal et al. (2010).
The vulnerability methodology consists in generating thousands of synthetic buildings
considering the variability in the geometrical and material properties, whose nonlinear capac-
ity is estimated using the displacement-based earthquake loss assessment (DBELA) concept
(Crowley et al. 2004), against a large set of ground motion records compatible with the
seismogenic environment around the region of interest (Barani et al. 2009). The respective
fundamental periods o the buildings remain constant, however. Global limit states are used
to estimate the damage distribution for different levels of ground motion, and a regression
algorithm is applied to derive fragility functions (i.e., probability of exceeding different limit
states for given intensity measure levels). Then, the resulting fragility functions are com-
bined with a damage-to-loss model (Di Pasquale and Goretti 2001) to derive a vulnerability
function for each building type, as illustrated in Fig.4. It is pertinent to mention that this
vulnerability methodology allows for the propagation of a large spectrum of uncertainties
(e.g. variability in the geometric and material properties, uncertainty in the damage defini-
tion, record-to-record variability) into the final vulnerability curves, which may result not in
a single loss ratio per intensity measure level, but rather in a probabilistic distribution of loss
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0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
Spectral acceleration (g) at (T − 0.20 sec)
Loss ratio
RC LR PC
RC LR C
M LR PC
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
Spectral acceleration (g) at (T − 0.50 sec)
Loss ratio
RC MR C
M MR PC
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
Spectral acceleration (g) at (T − 0.85 sec)
Loss ratio
RC MR PC
RC HR C
M HR PC
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
Spectral acceleration (g) at (T − 1.20 sec)
Loss ratio
RC HR PC
Fig. 4 Vulnerability model for the Italian building portfolio. Type acronyms are shown in Table 1
ratio, usually characterised by a lognormal distribution with an associated mean and standard
deviation. However, in order to decrease the computational burden of the calculations and to
reduce the number of variables affecting the final results, a decision was made to consider
only the mean loss ratio at each IM.
4 Impact of Cross-correlation on analyses of seismic loss at an urban scale
4.1 Loss analysis at urban scale
To understand the role that spatial cross-correlation plays in modelling earthquake losses,
the analysis is initially limited to the smallest spatial scale. In this current study for this scale
we considered the Florence (Firenze) city district and the wider Florence administrative
district. To give a clearer perspective on the spatial scale, the greatest site-to-site distance
within the Firenze city district is approximately 15 km, compared to 100 km for the Florence
administrative district and more than 250 km for the Tuscany region as a whole. These
parameters are relevant as it will be seen in due course that the comparative influence of
spatial correlation and cross-correlation on an exposure portfolio is strongly dependent on the
relative density of site-to-site distances within the portfolio. This is intuitive as more spatially
constrained portfolios are more likely affected in their entirety by the same earthquakes
than more widely dispersed ones are. For the Florence city and administrative districts a
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comparison of the annual exceedance probability loss curves is made considering seven IM
correlation modelling options:
1. No spatial correlation or spatial cross-correlation is considered, and inter-event residuals
are sampled independently for each period.
2. Spatial correlation is considered separately for each spectral quantity and the inter-event
residuals are independent for each period.
3. Spatial correlation and cross correlation are modelled using a conditional hazard approach
(Sect. 2.2), with the shortest period represented in the portfolio (Sa(0.2s)) selected as
the primary IM for modelling the spatial correlation.
4. Spatial correlation and cross correlation are modelled using a conditional hazard approach
(Sect. 2.2), with the longest period represented in the portfolio (Sa(1.2s)) selected as the
primary IM for modelling the spatial correlation.
5. Spatial correlation and cross-correlation are included and modelled using the full-block
cross correlation methodology (Sect. 2.3). Spectral correlation in the inter-event residual
is simulated using the model of Goda and Atkinson (2009).
6. Spatial correlation and cross-correlation are included and modelled using the LMCR
methodology (Sect. 2.4). Spectral correlation in the inter-event residual is also simulated
using the model of Goda and Atkinson (2009).
7. Spatial correlation and cross-correlation are included and modelled using the LMCR
methodology (Sect. 2.4) with uncertainty represented using only the total σterm.
To initially verify that spatial correlation is influencing the analysis for a homogenous
portfolio, an initial analysis is undertaken using a single type of building, which in this case
the masonry wall, mid-rise, pre-code type (see Table 1) with a corresponding fundamental
period of 0.5 s. Figure 5demonstrates the impact upon the aggregated loss analysis when
including spatial correlation for the Firenze Administrative Province (with a typical footprint
diameter on the order of approximately 100 km) and for the Firenze City District (with a
typical footprint diameter on the order of 15 km) respectively. The loss curves indicate that
when spatial correlation is included in the model greater losses are observed at lower annual
probabilities of exceedance. Furthermore, the impact that inclusion of spatial correlation has
on the loss analysis is relatively greater for the portfolio with the smaller “footprint”. This
observation is consistent with the trends observed by Park et al. (2007)andSilva et al. (2014).
It can also be observed that for higher annual probabilities the inclusion of spatial correlation
will often reduce the loss estimates. This trend can be explained by the considering how
inclusion of correlation increases variability in the losses for a single scenario. Neglect of
correlation narrows the tails of the distribution, meaning that for each event the probability
of sampling low values in the left tail is reduced, therefore the losses are higher. Equally,
however, the probabilities of sampling the very high values in the right tail are also reduced,
thus leading to lower losses. However, even at the 5 ×10−3annual probability it may be the
case that spatial correlation may still result in higher losses in other portfolios if the assets
are more spatially clustered than is the case here. The noise in the curves at low probabilities
(less than 10−3) is due to the occurrence of rare high-impact events, which due to their very
low likelihood may be under- or over-sampled in the synthetic catalogue with respect to their
long-term occurrence rate. Extension of the synthetic catalogue length would ensure a more
stable sample of the larger events, which control the extreme losses at the lowest probabilities.
The loss analyses for the heterogeneous portfolio (Figs. 6and 7) illustrates the impact of
considering spatial correlation and spatial cross-correlation. For both the Florence city portfo-
lio and the administrative district portfolios, the inclusion of spatial cross-correlation results
in greater losses at lower annual probabilities (typically less than 10−3), with the trend seen
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0 0.5 1 1.5 2 2.5 3 3.5
x 109
10−4
10−3
10−2
10−1
Aggregated Loss (10 9 Euros)
Annual Probability of Being Exceeded
Aggregated Portfolio Firenze (Florence) Administrative Province (M−MR−PC Only)
No Correlation (1)
Spatial Only (2)
0.4 0.6 0.8 1 1.2 1.4 1.6
10−4
10−3
10−2
10−1
Ratio of Loss Curves
Annual Probability of Being Exceeded
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 109
10−4
10−3
10−2
10−1
Aggregated Loss (10 9 Euros)
Annual Probability of Being Exceeded
Aggregated Portfolio Firenze (Florence) City District (M−MR−PC Only)
No Correlation (1)
Spatial Only (2)
0.4 0.6 0.8 1 1.2 1.4 1.6
10−4
10−3
10−2
10−1
Ratio of Loss Curves
Annual Probability of Being Exceeded
Fig. 5 Aggregated Loss Curves for the masonry-wall, mid-rise, low-code type (M-MR-PC) for the Firenze
Administrative Province (top), containing 2591 assets over 1192 locations, and the Firenze (Florence) City
District (bottom), containing 168 assets over 166 locations
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0 2 4 6 8 10 12 14
10−4
10−3
10−2
10−1
Aggregated Loss (10 9 Euros)
Annual Probability of Being Exceeded
Aggregated Portfolio Firenze (Florence) Administrative Province
No Correlation (1)
Spatial Only (2)
Conditional Hazard (0.2s) (3)
Conditional Hazard (1.2s) (4)
Full−Block Cross−Correlation (5)
LMCR (6)
LMCR (Total σ) (7)
0.4 0.6 0.8 1 1.2 1.4
10−4
10−3
10−2
10−1
Ratio of Loss Curves
Annual Probability of Being Exceeded
Fig. 6 Aggregated Loss Curves for Firenze (Florence) Administrative Province for a heterogeneous portfolio
containing 21,151 assets at 1192 locations
more clearly for the smaller scale city portfolio. The full-block cross-correlation methodol-
ogy and the LMCR methodology provides generally similar results for the case when inter-
and intra-event residuals are separated, suggesting that the method by which spatial cross-
correlation is modelled has a relatively small impact on the loss analysis. In comparison, the
similarity in the loss curves when considering spatial correlation for each individual period
without cross-correlation, and when neglecting correlation altogether, shows that for a het-
erogeneous portfolio the neglect of the cross-correlation drastically erodes the total influence
of correlation on the loss estimation. This is to be expected given that neglecting cross-
correlation in a given simulation of random fields of different IMs for a given earthquake
generates completely different spatial patterns of high and low values for each specific IM.It
is the statistical correlation of these patterns of high and low values of different IMs that causes
the unusually high losses (and low as well but of less importance here) that would make the
curve greatly differ at lower annual probabilities for a heterogeneous portfolio. Using this
approach for assessing losses to heterogeneous portfolios, therefore, leads to estimates of
large losses that are not particularly accurate.
The loss curves derived using the conditional hazard methodology require careful inter-
pretation. In the case when the shorter period spectral acceleration (0.2 s) is used as the
primary IM, at the provincial scale the loss curves tend strongly toward the “no-correlation”
case, whilst at the city scale these curves tend more strongly toward the “spatial only” case..
When conditioning on the longer period (1.2 s) spectral acceleration, however, we see the
trend in the loss curves following more closely those of the LMCR and full-block cross-
correlation case, though with a tendency more toward the middle (i.e. with less extremes)
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0 1 2 3 4 5 6 7 8 9 10
10−4
10−3
10−2
10−1
Aggregated Loss (10 9 Euros)
Annual Probability of Being Exceeded
Aggregated Portfolio Firenze (Florence) City District
No Correlation (1)
Spatial Only (2)
Conditional Hazard (0.2s) (3)
Conditional Hazard (1.2s) (4)
Full−Block Cross−Correlation (5)
LMCR (6)
LMCR (Total σ) (7)
0.6 0.8 1 1.2 1.4
10−4
10−3
10−2
10−1
Ratio of Loss Curves
Annual Probability of Being Exceeded
Fig. 7 Aggregated Loss Curves for Firenze (Florence) City District for a heterogeneous portfolio containing
2,082 assets at 168 locations
than in the cases when correlation is fully considered. When using the shorter-period as the
primary IM the correlation length of the spatial field is shorter,thus when combined with
the smaller cross-correlation between longer and shorter period IMs the spatial correlation
for the longer period IMs is significantly underestimated. Conversely, when using the longer
period IM as the conditioning IM, the mid- and long-period IMs the spatial correlation is
being over-estimated. For the short period IMs, however, the effect of overestimation in the
spatial correlation is eroded by the smaller cross-correlation between IMs. Ultimately, the
competing influence of spatial correlation and IM to IM cross-correlation on the resulting
loss curves will depend on the composition of the portfolio, which may be very difficult to
anticipate prior to the analysis. This condition may be one compelling argument against the
use of this particular methodology. Or else, if adopting the conditional hazard approach it
is strongly recommended to use the longer period IM for generating the spatially correlated
fields, as will be pursued in the subsequent comparisons.
Focusing specifically on the full spatial cross-correlation methodologies, all three methods
(full-block cross-correlation, LMCR and LMCR using total σ) result in higher losses at lower
annual probabilities of exceedance, and lower losses at higher annual probabilities, when
compared with those from methods in which spatial cross-correlation is neglected. For the
portfolio with the larger “footprint” (Fig.6) all three methods provide curves that are in close
agreement even at low annual probabilities of exceedance. For the city-district (Fig. 7)itis
relevant to note that whilst the full-block cross-correlation and LMCR methods are in close
agreement, in the case that only the total σterm is modelled with the LMCR the losses
are lower than in the cases where the inter- and intra-event variability are separated. From
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these trends it could be inferred that the means by which the full cross-covariance matrix is
defined, be it by full-block cross-correlation or LMCR, is of less significance than the manner
in which the inter- and inter-event correlations are modelled. In this respect there may be
further work needed to develop an additional LMCR derived specifically for the intra-event
term, thus allowing the correlations in the inter- and intra-event residuals to be separated in
the modelling process.
An additional observation from the loss curves in Figs. 6and 7are the close correspondence
between the losses when only spatial correlation is considered (i.e. the ground motion fields
for each IM are each spatially correlated, but no cross-correlation is considered) and the case
when correlation is neglected. As it is demonstrated in Fig.5that for a single type within
the portfolio the inclusion of spatial correlation increases the losses, it is clear that when
considering a heterogenous portfolio the neglect of spatial cross-correlation can erode the
influence of spatial correlation. This may not be so unexpected if one considers that for a
single realisation of ground motion fields, it is possible, common even, to expect that without
the inclusion of cross-correlation an asset of a particular type my be subject to weaker than
expected ground motion (a strong negative residual) whilst a co-located asset of a different
type may be subject to stronger than expected ground motion (a strong positive residual).
Without the inclusion of spatial cross-correlation to connect the different fields of IMs for
a single realisation then the total effects of the correlation on a heterogenous portfolio are
minimised.
Whilst the impact of the both the spatial correlation and spatial cross-correlation can
be seen, it is also necessary to expand upon the role that the spatial dispersion, or spatial
“footprint” as it is referred to in Park et al. (2007), of the portfolio plays in elucidating the
differences between the methodologies. Recalling that the portfolio itself is rendered here onto
an evenly spaced 30 arc-second grid, the distance between neighbouring assets is the same
regardless of the size of the footprint. For the Florence city portfolio, however, as the assets
are limited to only the city district itself the largest site to site distance is approximately 20 km,
and the majority of site-to-site distances are less than 10 km. Conversely the administrative
district spans over 100km and the typical site-to-site distances are on the order of 20–
30 km, which is beyond the correlation lengths for the spectral accelerations at the periods
under consideration. This drastically reduces the impact of the spatial correlation. It remains
pertinent, however, that for portfolios with a larger footprint the impact of the spatial cross-
correlation remains visible. The portfolio footprint is therefore of great significance. The
impact of spatial correlation and spatial cross-correlation on the loss analysis will depend on
both the degree of spatial clustering within the portfolio, or even just the degree of spatial
clustering of the highest value assets, and on the spatial footprint of the portfolio. This is
not a trivial outcome as many real portfolios will contain assets distributed over a relatively
small spatial scale, such as city or district, which will experience similar levels of ground
shaking in an earthquake, and will therefore likely result in similar levels of damage.
An alternative approach for assessing losses to heterogeneous portfolios can be devised
when cross-correlation of IMs cannot be modelled correctly, as it is in the approach above.
Loss estimation for a heterogeneous portfolio does not necessarily require co-simulation
of spatially cross-correlated ground motions, so long as the risk modeller accepts to derive
fragility models for all structure types within the portfolio using a single intensity measure.
Of course, this alternative approach comes at a price, as the selected IM will likely not be
an efficient predictor of structural response, i.e. an IM that results in a smaller variability
of the structural demand given the ground motion intensity (Luco and Cornell 2007), and
may therefore lead to less accurate building response estimates. This reduction in structural
response efficiency means that greater uncertainty in the vulnerability model will be trans-
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ferred to the loss curve, which may potentially affect the loss estimates. Which one of the
two alternative methods is preferable in these circumstances depends, to a certain extent, on
the degree of heterogeneity in the portfolio.
To illustrate whether the spatial aggregation of the portfolio is biasing the trends, the loss
curves for simpler distributed portfolios of unique assets (Fig. 3)areshowninFig.8.Thesame
trends in the results from the different methodologies are visible as for the fully aggregated
portfolio. In this particular case, however, the impact of the spatial cross-correlation is not
quite so large, indicating that whilst the aggregation may be introducing a degree of artificially
high correlation the overall results are persistent. Of course, it is still important to note that
the degree of divergence between the loss curves derived using methodologies that neglect
spatial cross-correlation and those that include it will depend heavily on the properties of the
portfolio, and in this case in the period range of the IMs being considered in the risk analysis.
4.2 Sensitivity to portfolio weighting
The examples demonstrate that the inclusion of spatial cross-correlation may impact upon
the resulting estimates of seismic losses, it follows that the scale of this impact may depend
not only upon the “footprint” of the portfolio but also upon the range of periods considered
and the weighting of the exposure in different building types. This can be demonstrated by
considering a slightly simplified example in which the spatial distribution of the assets, and
therefore the “footprint”, remain the same as before. A simplification is made in the portfolio
in that building types are limited only to the three low-code reinforced concrete classes:
RC_LR_PC, RC_MR_PC and RC_HR_PC. Figures 9and 10 demonstrate the impact upon
the loss curves when the portfolios are weighted more in the low-rise, mid-rise and high-rise
types, for the province and city areas respectively. In each of the three portfolios two-thirds
of the assets are assigned to the corresponding LR, MR and HR types, and the remaining
third of the assets divided evenly between the other types.
The loss curves from the different methodologies that are displayed in Figs.9and 10
show, to some extent, a greater degree of convergence than those derived for the more evenly
distributed portfolio. As expected, this case demonstrates quite clearly that when weighted
predominantly toward a single asset type, the impact of considering spatial cross-correlation
on the loss curves is relatively minimal. At the level of the city scale (Fig. 10) loss curves
generated using cross-correlation still give noticeably higher values at lower annual proba-
bilities, albeit this effect is somewhat reduced in comparison with those from the more evenly
distributed portfolios.
A second effect that is visible within these analyses is that there is greater convergence
amongst the curves when the portfolio is weighted more with building type that consist of
longer-period (mid-rise and high-rise) structures. This trend is again more evident in the city-
scale portfolio than in the provincial scale portfolio. This would seem to indicate the influence
of the correlation length in the analysis, as in the case of the high-rise building types the
fundamental period is greater and the correlation between sites for these buildings assumes
a greater role in the modelling than the cross-correlation between types. This essentially
minimises the impact of cross-correlation, thus the full-block and LMCR methods produce
similar results to that of the case when cross-correlation is neglected.
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0 0.5 1.0 1.5 2.0 2.5 3.0
10−4
10−3
10−2
10−1
Aggregated Loss (10 7 Euros)
Annual Probability of Being Exceeded
Firenze Provincia
No Correlation (1)
Spatial Only (2)
Conditional Hazard (0.2s) (3)
Conditional Hazard (1.2s) (4)
Full−Block Cross−Correlation (5)
LMCR (6)
LMCR (Total σ) (7)
0.6 0.8 1 1.2 1.4
10−4
10−3
10−2
10−1
Ratio of Loss Curves
Annual Probability of Being Exceeded
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
10−4
10−3
10−2
10−1
Aggregated Loss (10 7 Euros)
Annual Probability of Being Exceeded
Firenze Citta
No Correlation (1)
Spatial Only (2)
Conditional Hazard (0.2s) (3)
Conditional Hazard (1.2s) (4)
Full−Block Cross−Correlation (5)
LMCR (6)
LMCR (Total σ) (7)
0.6 0.8 1 1.2 1.4
10−4
10−3
10−2
10−1
Ratio of Loss Curves
Annual Probability of Being Exceeded
Fig. 8 Aggregated Loss Curves for Firenze (Florence) Administrative Province (top) Firenze (Florence) City
District (bottom) for the sample portfolios shown in Fig. 3
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Fig. 9 Sensitivity of the loss
curves for the Firenze Province to
different weightings of the
portfolio in the low-rise (top),
mid-rise (middle) and high-rise
(bottom) type
0 2 4 6 8 10 12 14 16 18 20
10−4
10−3
10−2
10−1
Aggregated Loss (10 9 Euros)
Annual Probability of Being Exceeded
Weighted in the LR Typology
No Correlation (1)
Spatial Only (2)
Conditional Hazard (Sa 0.8s) (3)
Full−Block Cross−Correlation (5)
LMCR (6)
LMCR (Total σ) (7)
0.8 0.9 1 1.1 1.2
10−4
10−3
10−2
10−1
Ratio of Loss Curves
Annual Probability of Being Exceeded
0 2 4 6 8 10 12 14 16 18 20
10−4
10−3
10−2
10−1
Aggregated Loss (10 9 Euros)
Annual Probability of Being Exceeded
Weighted in the MR Typology
No Correlation (1)
Spatial Only (2)
Conditional Hazard (Sa 0.8s) (3)
Full−Block Cross−Correlation (5)
LMCR (6)
LMCR (Total σ) (7)
0.8 0.9 1 1.1 1.2
10−4
10−3
10−2
10−1
Ratio of Loss Curves
Annual Probability of Being Exceeded
0 2 4 6 8 10 12 14 16 18 20
10−4
10−3
10−2
10−1
Aggregated Loss (10 9 Euros)
Annual Probability of Being Exceeded
Weighted in the HR Typology
No Correlation (1)
Spatial Only (2)
Conditional Hazard (Sa 0.8s) (3)
Full−Block Cross−Correlation (5)
LMCR (6)
LMCR (Total σ) (7)
0.8 0.9 1 1.1 1.2
10−4
10−3
10−2
10−1
Ratio of Loss Curves
Annual Probability of Being Exceeded
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0 1 2 3 4 5 6 7 8 9 10
10−4
10−3
10−2
10−1
Aggregated Loss (109 Euros)
Annual Probability of Being Exceeded
Weighted in the LR Typology
No Correlation (1)
Spatial Only (2)
Conditional Hazard (Sa 0.8s) (3)
Full−Block Cross−Correlation (5)
LMCR (6)
LMCR (Total σ) (7)
0.6 0.8 11.2
10−4
10−3
10−2
10−1
Ratio of Loss Curves
Annual Probability of Being Exceeded
0 1 2 3 4 5 6 7 8 9 10
10−4
10−3
10−2
10−1
Aggregated Loss (109 Euros)
Annual Probability of Being Exceeded
Weighted in the MR Typology
No Correlation (1)
Spatial Only (2)
Conditional Hazard (Sa 0.8s) (3)
Full−Block Cross−Correlation (5)
LMCR (6)
LMCR (Total σ) (7)
0.6 0.8 11.2 1.4
10−4
10−3
10−2
10−1
Ratio of Loss Curves
Annual Probability of Being Exceeded
0 1 2 3 4 5 6 7 8 9 10
10−4
10−3
10−2
10−1
Aggregated Loss (109 Euros)
Annual Probability of Being Exceeded
Weighted in the HR Typology
No Correlation (1)
Spatial Only (2)
Conditional Hazard (Sa 0.8s) (3)
Full−Block Cross−Correlation (5)
LMCR (6)
LMCR (Total σ) (7)
0.6 0.8 11.2 1.4
10−4
10−3
10−2
10−1
Ratio of Loss Curves
Annual Probability of Being Exceeded
Fig. 10 Sensitivity of the loss curves for Firenze City to different weightings of the portfolio in the low-rise
(top), mid-rise (middle) and high-rise (bottom) type
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5 Conclusions
This loss estimation exercise for an aggregated, synthetic, heterogeneous portfolio for the
Tuscany region of Italy demonstrates that the inclusion of spatial cross-correlation is impor-
tant for estimating accurately the likelihood of observing large, infrequent losses. This impact
remains visible, albeit diminished, when considering losses for portfolios spread over larger
spatial scales, even at scales where the influence of ordinary spatial correlation might be
seen to be negligible. It is emphasised, however, that for heterogeneous portfolios of any
spatial scale modelling spatial cross correlation is always important, as loss estimates are
also routinely needed for subsets of the portfolios that are limited geographically (e.g., into
a zipcode or a commune).
The analyses presented herein also highlight the influence of the portfolio composition, in
terms of period range of the IMs and proportions of the building types, when incorporating
spatial cross-correlation of the ground motion variability into the risk analyses. It is evident
from the sensitivity studies that there are many possible conditions under which the impact of
spatial cross-correlation is negligible. Certainly the influence diminishes when the portfolios
are more spatially dispersed (i.e. with a larger “footprint”), or are dominated by a particular
building type.
These results also show, for the first time, a side-by-side comparison of loss curves com-
puted by different methodologies use for the generation of spatially cross-correlated random
fields of ground motion intensity measures. It can be seen clearly that the loss curves may be
sensitive to the choice of methodology, albeit we noted similar (but not identical) losses in
the two methodologies that include both full spatial cross-correlation and inter-event residual
correlation, i.e. the full-block cross-correlation and the LMCR approach. From an implemen-
tation perspective the two methodologies are similar in terms of computational demand and
should both ensure positive-definiteness in the spatial covariance matrix, thus there is not
necessarily a clear case for adopting one over the other in application. It is recommended,
however, that if wishing to represent the full spatial cross-correlation structure of the ground
motion intensity measures that these particular methodologies are adopted in favour of the
others considered within this study.
The potential influence of the spatial cross-correlation is dependent on a balance between
the spatial, spectral (or IM-dependent) and compositional properties of the portfolio, such as
the proportion of different types and/or the form of the vulnerability models, for example.
The balance of the influencing factors may be hard to predict prior to the analysis. Therefore
the most prudent approach to follow when undertaking an analysis of seismic risk to hetero-
geneous portfolios would be one in which the spatial cross-correlation in the ground motion
IMs is included in the modelling process, or at least until it can be established by sensitiv-
ity studies that the effects of the spatial cross-correlation are negligible for the portfolio in
question.
The inferences made from this analysis, and the possible applications in seismic loss
modelling for both research and industry, necessitate further investigation into conditions in
which the spatial cross-correlation may be seen to impact the loss estimates. In particular,
the artificial correlations introduced by aggregating assets within a geospatial region, such
as a grid-cell, zip/postal code or electoral district, need to be compensated for within the risk
analysis. Stafford (2012) describe the process by which this may be undertaken for a single IM
in a post-disaster assessment, demonstrating that when considering the aggregated assets the
standard deviation σof the IM at the aggregation site generated by a given earthquake must
be reduced as a result of the fact that assets are averaged within the spatially extended region.
This can be readily incorporated into analyses of the sort demonstrated here. Similarly, the
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application of spatial cross-correlation to simulations of ground motion conditioned upon a
set of observations would enhance the characterisation of uncertainty in real-time post-event
modelling of earthquake losses.
Acknowledgments We thank the SHARE consortium for providing the seismogenic source model used in
this analysis. The implementation of the methodology has benefitted greatly from discussions with Marco
Pagani and Damiano Monelli. An initial formulation of the ideas presented here began in the FP-7 SYNER-G
project, and we are grateful to Iunio Iervolino and Paolo Franchin for their contributions to these discussions.
The manuscript, and the work as a whole, benefitted greatly from insightful reviews by Peter Stafford and one
anonymous reviewer.
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