Geoscience Reference
In-Depth Information
2.7. TRANSPORTATION NETWORK RISKCURVE FROM MONTE CARLO
SIMULATION WITHIMPORTANCE SAMPLING
Evaluation of eq. (19.9) that leads to the total risk curve is computationally very expen-
sive. In general, there are three methods to compute eq. (19.9): (a) numerical integra-
tion; (b) conventional Monte Carlo simulations, and (c) Monte Carlos simulation with
importance sampling. Numerical integration considers the full assessment of the equa-
tions describing the risk model. Monte Carlo simulation is an approximate method that
randomlyselectsscenariosovertimeandevaluatesthelossratecurve.Itmustberepeated
manytimestoobtainstableresultsoritneedstoberunoverlongforecastperiodstocap-
ture all possible events. Importance sampling is again a simulation based approach that
selects a combination of scenario events in the region in such a way that the mean and
higher order moment of the risk rate curve are preserved with the minimum number of
scenarios.
Consideringthenatureofthetransportationnetworkproblem,analyticalmethodscannot
beusedfortheriskassessment.Thuswechoosetheimportancesamplingmethodbecause
it minimizes the analyses while preserving important components of the risk curve such
as the mean and at least the second order moment (variance) of the loss rate. Then the
losses fromeach scenario arecombined as follows.
Earthquake events are assumed independent and follow a Poisson process. It is recalled
that an event is defined by its magnitude, rupture length, rupture location, rupture depth,
dip angle and annual rate of occurrence. We denote the probability of a scenario event to
be
P
[
Q
j
]
1,2,...,
N
, where
Q
j
is the
j
th event that is identified as being impor-
tant for the risk curve computation and
N
is the total number of events. If the loss for
each scenario
Q
j
is
L
j
,
j
,
j
=
1,2,3,...,
N
, we order the losses in decreasing order the
probability of exceeding the lossrate inayear is obtained by
=
L
n
>
L
n
−
1
>
···
>
L
k
>
···
>
L
1
(19.15)
Then the probability of exceeding theloss ratein ayear isobtained by
j
=
k
1
Q
j
]
n
P
[
L
k
≥
l
]=
1
−
−
P
[
(19.16)
In Equation 19.16, the assumptions are made that (i) individual losses are independent,
(ii) the system is fully restored after each event, and (iii) only one event occurs at a
time. While this equation is a simplification, it is computationally tractable and provides
additional information over expected value loss estimates as will be demonstrated in the
application section of this paper.
3. Application to the San Francisco Bay Area Transportation Network
The methodology on network seismic risk assessment presented in this paper is applied
to the transportation network in the San Francisco Bay Area. For that purpose, the
data on 2921 state and local bridges were obtained from the California Department of
