Civil Engineering Reference
In-Depth Information
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[2.8]
In the two-stage approach described by Joyner & Boore (1993), the
'event terms'
δ
E
,
i
are explicitly solved for during the regression analysis. That
is, these terms are effectively coeffi cients of the model with the implication
that statistical tests for signifi cance, etc., could be performed on these terms
(although this is never done). In contrast, in the one-stage approach (often,
although incorrectly, referred to as a random effects approach, Abrahamson
& Youngs, 1992) the variance components are estimated directly rather
than the individual 'random effects'. While in the regression algorithm of
Abrahamson & Youngs (1992) the random effects (the
E
,
i
terms) are com-
puted, these are
a posteriori
estimated values conditioned upon the directly
estimated values of the variance components.
When developing ground-motion models it is common to use the differ-
ent types of residual to evaluate the performance of different parts of the
model. For example, the inter-event residual is related to the relative ampli-
tude of a particular earthquake for a given magnitude, depth and style-of-
faulting, etc. Therefore, when assessing the performance of a fi tted model
one would commonly look at a residual plot of the inter-event residuals
against magnitude in order to assess the performance of a particular func-
tional form. If one were to observe a systematic trend in the inter-event
residuals then one could modify the functional form related to magnitude
scaling in order to correct for this. The result would be a reduction of the
inter-event variance. However, while this is common practice, it must be
kept in mind that the inter-event residuals that are computed are dependent
upon the particular regression approach that is taken. This point is best
illustrated by considering a poorly-recorded event. In this case, the two-
stage regression analysis procedure of Joyner & Boore (1993) would require
the event term for this earthquake to be directly estimated, but this term
would have a large confi dence interval associated with it if the parameter
estimate is based on a small number of records. If this confi dence interval
includes zero then the event-term should be regarded as being statistically
insignifi cant and hence it should be dropped from further consideration.
The implication would be that the intra-event residuals for this event would
be equal to the total residual and that the event term would be zero. This
directly impacts upon the implied scaling of ground-motions with respect
to magnitude if this scaling is based upon inspection of event terms. On the
other hand, in the one-stage maximum likelihood approach of Abrahamson
& Youngs (1992), when an event is not well recorded then the inter-event
residual (the random effect) tends towards a constant fraction of the total
residual that has been observed. In the extreme, when just one recording
δ
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