Seismic vulnerability analysis of a complex interconnected civil infrastructure - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 487

assessment of spatially distributed systems. In this framework,
Importance Sampling is used to preferentially sample “important”
ground-motion intensity maps, and K-Means Clustering is used to
identify and combine redundant maps in order to obtain a small
catalogue. The effects of sampling and clustering are accounted for
through a weighting on each remaining map, so that the resulting
catalogue is still a probabilistically correct representation.'

The importance sampling density is the product of shifted Gaussian distri-
butions for both the inter- and intra-event residuals , with shifts of

the mean towards higher values that are given as a function of the system
size and average separation distance amongst sites, and a stratifi ed sampling
density for the magnitude M . Together with the following clustering of the
resulting maps, the overall effort is reduced from the tens of thousands
simulation runs needed with MC to a few hundred runs.
In both cases the typical simulation run consists of sampling x from its
distribution and evaluating the Infrastructure state and all the performance
metrics of interest. Figure 18.10 shows a compact graphical representation
of the variables in x and of their joint distribution , for the simple example
infrastructure shown in Fig. 18.9. Random variables, or vectors, represented
by circles, form the nodes of a graph where directed edges indicate statisti-
cal dependence. Models for fl ow-equations, etc. are pictorially represented
as rectangles. Variables belonging to the same model are grouped with
rounded rectangles. This representation is almost the same as that employed
in BN models (Straub et al. , 2008), with the difference that here no Bayesian
updating is performed and the fi gure simply illustrates the fl ow of the
typical simulation run, from sampling of top (marginal) variables to that of
the bottom (dependent) variables.
For the simple example shown in the fi gure, the run starts with the simu-
lation of a magnitude value from the distribution of M . Conditional on the
obtained magnitude value, the discrete random variable Z for the active
seismogenic zone is sampled in the run (not all zones may be active for a
given magnitude, as shown later in the application), and then the epicentre
location within the zone. This completes the event generation step. The fol-
lowing steps (2 to 3 in Section 18.5.4) lead from the event macro-seismic
characteristics to the local intensity at each site. First the variables

η and ε

η and ε

(vector depends on the standard normal vector u , see Section 18.5.4) are

sampled to produce the primary IM values on the grid points for rock/stiff-
soil conditions, denominated S grj ( j being the index spanning the grid points).
Next these value are used to interpolate the S ri values at each site ( i being
the index spanning the sites), still for rock/stiff-soil conditions. Finally, the
vector S i needed at each site is completed through conditional simulation
of secondary IMs given the primary at the site S ri , and component-wise

ε