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that is obtained during the model development. Hence, there is a direct
trade-off between the quality of the metadata used in the model develop-
ment and the value of the standard deviation (and also the median predic-
tions) obtained during the regression procedure. As noted by McGuire
(2004), it has long been realised that one is better off working with a smaller
number of records with high-quality metadata than to simply use as much
data as possible, irrespective of its quality. That said, however, in order to
constrain the scaling of ground-motion amplitudes over a broad range of
scenarios it is important to have a dataset that is as balanced and well dis-
tributed as possible. Here, by balanced the meaning is that the data provides
a reasonable coverage over the magnitude-distance space of greatest rel-
evance for hazard and risk applications.
In principle, the fact that different model developers can select a subset
of recordings from larger available datasets means that some element
of epistemic uncertainty can be accounted for in this way. However, in
reality, the datasets that are used for the development of ground-motion
models are still relatively sparse and this dictates that there are numerous
magnitude-distance combinations where developers do not have the
freedom to pick and choose the records they use. At least, if they were to
be selective in certain magnitude-distance ranges this would have a strong
impact upon their ability to constrain the scaling of their model. This is even
more the case when developers look to be selective about which earth-
quakes should be included, rather than just which recordings are included
(see, for example, the numbers of events above magnitude 7.5 in Fig. 2.1).
2.3.3 Mixed-effects regression analysis
On pages 38-9 the total difference between observed ground-motion ampli-
tudes and model predictions was partitioned into two parts, one referred to
as the inter-event residual,
δ
E
,
i
, and one referred to as the intra-event resid-
ual,
A
,
ij
. Presently, within engineering seismology, two main approaches are
used in order to partition the total residual into these components: 'one-
stage' maximum likelihood (or expectation-maximisation) approaches
(Brillinger & Preisler, 1984, 1985; Abrahamson & Youngs, 1992; Joyner &
Boore, 1993) or 'two-stage' approaches (Joyner & Boore, 1993). While both
approaches appear to solve the same problem and both provide estimates
of the inter-event standard deviation and the intra-event standard deviation
there is one very important difference between the two approaches. The
problem that is addressed is how to maximise the likelihood function
defi ned in Equation (2.8). In this equation, the term
C
is the covariance
matrix and it is this matrix that contains the information regarding how
events recorded from the same earthquake, or how recordings from differ-
ent earthquakes made at the same site, are correlated.
δ
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