Seismic vulnerability analysis of a complex interconnected civil infrastructure - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 498

1.4 1

5.5 6.3

Sampled
6.3<M

≤ M ≤

Sampled by IS
Original f(M)

1.2

7.1
Sampled

≤

0.8

1

Upper
magnitude of
sources 1 & 3

0.6

0.8

0.6

0.4

0.4

0.2

0.2

p 1 = 0

p 3 = 0

0 0

5.5 5.7 5.9 6.1 6.3 6.5 6.7 6.9 7.1 Source 1 Source 2 Source 3

Magnitude M

(a) (b)

0.5 0.5

Sampled
f ( e )
h (

Sampled
f (

Mean shift = 0.5

Mean shift = 0.07 η )

0.4

e )

0.4

h ( η )

0.3 0.3

0.2 0.2

0.1

0.1

0 0

-4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4

Inter-event error h Intra-event error e

(c) (d)

18.14 The regional seismicity: (a) original probability density for the
magnitude versus the histogram of relative frequency of the
magnitudes sampled by IS; (b) conditional probability of seismic zone
given magnitude; (c) original probability density versus histogram of
sampled values for the inter-event error η ; (d) original probability
density versus histogram of sampled values for the intra-event error ε
at a single site.

The plot shows how the density f M has a step corresponding to the
common upper magnitude M U

6.3 of sources 1 and 3. The sampled values
on the other hand have an inverted trend with adequate coverage of the
most important larger magnitude events. Plot (b) compares the target and
sampled conditional distributions of the discrete variable Z (see Fig. 18.10,
where an arrow from M to Z indicates statistical dependence), for values
of M in the 5.5 to 6.3 and the 6.3 to 7.1 ranges. While in the fi rst case any
of the three sources can generate the event, and probabilities refl ect the

=