Civil Engineering Reference
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δ
T
,
ij
, is partitioned such that
the inter-event residual is defi ned as in Equation (2.9):
of an event has been made, the total residual,
σ
σσ
δ
2
E
δ
=
[2.9]
Ei
,
Tij
,
2
+
2
EA
Thus far, mixed-effects regression procedures, or variants of this (includ-
ing the approaches discussed herein), have been implemented widely
without necessarily being given the critical evaluation they deserve. For
example, more complex decompositions of the total residual into constitu-
tive parts have been proposed in the literature (Chen & Tsai, 2002; Tsai
et al
., 2006) and a generic framework for referring to these various compo-
nents has also been proposed (Al Atik
et al
., 2010). In the approach of Tsai
et al
. (2006), Equation (2.3) would be rewritten as in Equation (2.10) where
what was the intra-event residual,
δ
A
,
ij
, has now been decomposed into a
path-specifi c term,
δ
P
,
ik
, a site term,
δ
S
,
k
, and a residual error,
ε
ijk
.
(
)
++ ++
ln
Sa
ijk
=
μθ
Z
δ
δ
δ
ε
[2.10]
ijk
E i
,
P ik
,
S k
,
ijk
In this revised formulation, the indices
i
,
j
and
k
correspond to an event,
a recording and a site respectively. Note that the index
j
is required because
of the discrete manner in which different travels paths are defi ned in
practice.
When one works with a large dataset, the decomposition shown by Tsai
et al.
(2006) is preferred over the more commonly implemented approach
of only decomposing the within- and between-event components. However,
even this approach can be signifi cantly enhanced if one makes use of the
full power of mixed-effects modelling procedures. For example, the math-
ematical framework shown in Equation (2.10) implies that the source, path
and site effects all infl uence the scaling of ground motions in the same way,
i.e., they simply shift the median prediction up or down by a constant
amount. However, within a mixed-effects framework there is no reason why
the random effects cannot be attributed to different model coeffi cients, such
as those contained within the vector
of Equation (2.10). The only study
that appears to have recognised this thus far is that of Chiou & Youngs
(2008) in which the nonlinear site response terms include a nonlinear
dependence upon a random effect associated with the source.
θ
2.4 Sensitivity of model components
Thus far the discussions have been focused upon the standard deviation of
a ground-motion model and how this represents the aleatory variability of
the predicted intensity measure. In order to understand the total uncer-
tainty associated with predicted ground-motions we must also give due
attention to epistemic uncertainty. With very few exceptions, in hazard and
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