Ground motion selection for seismic risk analysis of civil infrastructure - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 83

μ = μ + σ ρ ε [4.3]

ln IM Rup IM , ln IM Rup ln IM Rup ln IM

ln IM ,ln IM Rup

i j i i j

i j

σ = σ 1 − ρ

2

[4.4]

ln IM Rup IM , ln IM Rup

ln IM ,ln IM Rup

i j i

i j

σ ln IM i | Rup are the mean and standard deviation of ln IM i
for a given rupture, which are commonly obtained from ground motion
prediction equations (GMPEs) (e.g. Boore and Atkinson, 2008);

where μ ln IM i | Rup and

ρ ln IM i ,ln IM j | Rup
is the correlation coeffi cient between ln IM i and ln IM j for a given earth-
quake rupture, commonly provided by empirical correlation equations
(e.g. Baker and Jayaram, 2008; Bradley, 2011c); and the parameter, ε ln IM j , is

given by:

ln IM − μ

j ln IM Rup

j

ε = [4.5]

ln IM

j

σ

ln IM Rup

j

Equations (4.1)-(4.5) provide the necessary mathematical details to
compute the conditional marginal distribution of IM i given IM j

im j for all
IM i in IM . The multivariate distribution of IM | Rup , IM j can therefore be
uniquely defi ned by mean and standard deviation vectors of length,
μ ln IM | Rup , IM j and

=

ρ ln IM | Rup , IM j
(Johnson and Wichern, 2007). The i th element of the mean and standard
deviation vectors are given by Equations (4.3) and (4.4). Specifi cally:

σ ln IM | Rup , IM j , respectively; and a correlation matrix,

() =
() =

m
s

i
i

μ
σ

ln IM Rup IM , ln IM Rup IM ,

j i j

[4.6]

ln IM Rup IM , ln IM Rup IM ,

j i j

and the i th row, k th column element of the correlation matrix is given by
(Johnson and Wichern, 2007):

ρρρ
ρ

−

ik ij kj

( ) =

r ln IM Rup IM

j ik

, [4.7]

,

1 −

2

1 − ρ

2

ij kj

where ρ ln IM i ,ln IM k | Rup is shorthand notation for the correlation between

ln IM i and ln IM k for a given rupture, Rup , which can be obtained from
empirical correlation equations (discussed subsequently). Equations (4.3)-
(4.7) provide suffi cient information to uniquely characterise the multivari-
ate distribution IM | Rup , IM j (Johnson and Wichern, 2007).
It is worth noting that in the particular case in which: (i) only a single
causal earthquake is considered; (ii) only spectral accelerations are consid-
ered in the vector, IM , and the conditioning intensity measure, IM j ; and (iii)
only the conditional mean, rather than the conditional distribution is con-
sidered; the above approach is equivalent to the 'conditional mean spectra'
approach proposed initially by Baker and Cornell (2006) and subsequently
discussed by Baker (2011).

ρ ik =