Civil Engineering Reference
In-Depth Information
Magnitude recurrence model, either a characteristic or a truncated
Gutenberg-Richter model:
e
−
β
m
−
−
e
−
β
m
i
iui
,
[
]
=
()
=
()
λ
m
λ
1
−
F
m
λ
[18.3]
i
0
,
i
M
0
,
i
−
β
m
−
β
m
e
e
ili
,
iui
,
where
0,
i
is the mean annual rate of all events with magnitude between
the lower bound magnitude
m
l,i
and the upper bound magnitude
m
u,i
,
and
λ
i
is the rate of decay of the mean annual frequency with
magnitude.
Localisation model, either a uniform distribution for a point-source
model within a polygonal seismic source, or a more refi ned model that
predicts the rupture area and position over the fault plane, as a function
of magnitude.
β
The model for the primary IM consists of a ground-motion prediction
equation (the abstract class
GMPE
in Fig. 18.2), which in general has
the form:
(
)
++
ln
s
=
μ
MR
,
,
F V
,
ε σ
η τ
[18.4]
i
ln
s
i
i
i
i
i
i
where
μ
ln
s
(
M
,
R
i
,
F
,
V
i
) is the mean of the logarithm, or log-mean, of the
intensity as a function of magnitude
M
, distance
R
, faulting style
F
and local
site conditions, as expressed by the shear wave velocity
V
. Total variability
of the intensity around the predicted log-mean is split in two terms, the
intra-event error
ε
i
σ
i
which varies from site to site, for a given earthquake,
and the inter-event error
η
τ
i
which varies from event to event but is con-
i
stant for all sites (with
η
i
standard Gaussian variables).
Simulation of (spatially) correlated intra-event residuals is carried out
according to:
ε
i
and
Lu
[18.5]
e =
where
u
is a vector of independent standard Gaussian variables and
LLT
R
is the Cholesky decomposition of the correlation matrix
R
whose terms
are obtained from an exponential correlation model of decay with distance
of the form:
=
−
bh
r
()
=
ρ
ε
ij
h
e
[18.6]
where
b
and
r
(the so-called 'range', or correlation distance) are the model
parameters. Figure 18.7 shows one simulated
shake-fi eld
of the primary IM,
which has been obtained from the models employed for the application in
Section 18.8. The fi gure shows four plots. Plots (a) to (c) are maps on the
study region of the application. In each map the three rectangular areas
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