Civil Engineering Reference
In-Depth Information
(
)
∂
μθ
Z
ijk
δ
′
=
Δ
V
+
δ
[2.12]
Sk
,
s
30
S k
,
∂
V
s
30
V
s
30
represent the
difference between the unknown true values of magnitude
M
w
and average
shear-wave velocity
V
s
30
and their estimated values used for the regression
analysis. The regression equation therefore becomes:
In Equations (2.11) and (2.12) the terms
Δ
M
w
and
Δ
(
)
+
ln
Sa
ijk
=
μθ
Z
δ
++
′
δ
δ
+
ε
[2.13]
′
ijk
E i
,
P ik
,
S k
,
ijk
Now, rather than the total variance of the model being defi ned by Equa-
tion (2.14), the variance is given by the expression in Equation (2.15) in
which the terms
σ
M
w
and
2
σ
2
V
s
30
are the variances of the
Δ
M
w
and
Δ
V
s
30
values.
σσσσ
ε
2
=+++
2
2
2
2
[2.14]
ln
Sa
E
P
S
(
)
(
)
2
2
σσ
μθ
=+
∂
Z
+++
∂
μθ
Z
⎡
⎢
⎤
⎥
⎡
⎢
⎤
⎥
ijk
ijk
2
2
σ
2
σσ
2
2
σ
2
+ σ
ε
2
ln
Sa
E
M
P
S
V
∂
M
w
∂
V
s
30
w
s
30
2
2
σ
from Equation
214
.
σ
from Equation 214
.
E
S
[2.15]
When the uncertainties in the independent variables are known, or at
least can be estimated, then the net effect is that the total variance observed
from the regression analysis can be partly explained by the propagated
errors in these variables. From Equation (2.15) it is apparent that in order
for the total
ln
Sa
to be preserved the estimated values of the inter-event
variance and the site-to-site variance must be reduced. It can be demon-
strated that a signifi cant component of what is traditionally regarded as
being the intra-event variance is actually attributable to the site-to-site
variance and that a signifi cant portion of this can again be attributed to the
uncertainty associated with site classifi cation (Gehl
et al.
, 2011).
Naturally, the approaches just outlined can also be extended to account
for the uncertainties in any input variable. The actual reduction in the stan-
dard deviation of the regression model will depend upon the degree of
uncertainty in the independent variables. However, one can obtain roughly
a 10% reduction in the standard deviation due to uncertainties in each of
magnitude, shear-wave velocity or distance (i.e., over 20% total reduction).
While this reduction may not seem to be particularly large, a reduction of
this magnitude can have a signifi cant impact upon computed hazard and
loss curves.
By way of practical example, Fig. 2.5 shows how the computed inter-event
standard deviation can vary depending upon whether or not the individual
magnitude uncertainties are accounted for. The summary of the residuals
presented in this fi gure correspond to models for Arias intensity developed
σ
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