Ground motion selection for seismic risk analysis of civil infrastructure - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 86

4.5.1 Realizations from the GCIM distributions
As illustrated in the previous section, the construction of the GCIM distri-
butions involves essentially two steps: (i) determining the probability that
if a ground motion was observed with IM j = im j , it was caused by rupture

Rup = rup k , P Rup | IM j ; and (ii) given the observed ground motion with IM j =

im j from rupture rup k , what is the distribution of the other ground motion
intensity measures in IM . While the previous section presented the analyti-
cal formulation of the distribution of IM i | IM j , f IM i | IM j , this distribution can
also be obtained via simulation. In the case of simulation, the two equiva-
lent steps in obtaining f IM i | IM j are: (i) obtain a random earthquake rupture
rup nsim (where the superscript nsim represents the n th simulation) from the
seismic hazard deaggregation probabilities, P Rup | IM j ; and (ii) for this ran-
domly drawn rupture, rup nsim , obtain a random realisation of IM , IM nsim ,
from the GCIM distribution. In order to select ground motions consistent
with the GCIM distribution, use is made of this simulation approach elabo-
rated below.
The fi rst level of the simulation approach involves obtaining a random
rupture, rup nsim , from the deaggregation probability mass function, P Rup | IM j .
This is done by drawing a uniform random number over the interval [0,1],
and then obtaining the corresponding number from the cumulative form of
P Rup | IM j . The second level of the procedure is the classic generation of cor-
related random variables from a multivariate normal distribution. As such,
fi rst an uncorrelated standard normal random vector is simulated, u nsim (i.e.
each of the N IM i elements in u nsim is drawn from a standard normal distribu-
tion independently). Using this vector of uncorrelated random variables, a
correlated vector is obtained from:

v

nsim

= u

nsim

[4.8]

where L is the Cholesky decomposition of the correlation matrix (i.e.
ρ LL T where T is the vector transpose). Using this correlated

random vector, each of the random IM i values in IM nsim can be obtained
from:

lnIM i | IM j , Rup

=

ln IM

i nsim

= μ + σ v

i nsim

[4.9]

ln IMi Rup IM , ln IMi Rup IM ,

j j

where v i nsim = v nsim ( i ) is the i th element of v nsim ; and Rup = rup nsim .

4.5.2 Selecting ground motions using the
IM nsim realizations
The random realizations IM nsim obtained based on the procedure in the
previous section will be consistent with the 'target' GCIM distribution (i.e.
the set of simulated IM nsim vectors will have the same mean, standard devia-