Uncertainties in ground motion prediction in probabilistic seismic hazard analysis (PSHA) of civil infrastructure - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 37

are assumed to be independent and hence the total model variance can be
computed from the sum of the individual variance components:

( ) +

( )

( ) ≅

var ln Sa var δ var δ [2.4]

Ei , Aij ,

The reason why Equation (2.4) has not been written using an exact equal-
ity is that the inter-event and intra-event variances that are described here
as var(

A , ij ) respectively are not computed from the residuals
in standard regression analyses. Rather, the variances are estimated during
a maximum likelihood procedure. The numerical results are always very
similar to what one obtains from directly computing the variances of the
residuals, but there are important differences that are worth keeping
in mind.
A published ground-motion model will present an equation for the total
standard deviation in the form of Equation (2.5).

δ

E , i ) and var(

δ

2 2 2

=+

σσσ
ln Sa

[2.5]

E A

This overall process of partitioning the observed variability into compo-
nents can be seen in the schematic illustration shown in Fig. 2.4.

Stochastic, or spectral, models

A stochastic-based, or spectral, model is based upon the assumption
that one can compute estimates of response spectral ordinates by fi rst

5

Inter-event residuals → d E,i

2

4

Inter-event residuals → d A,ij

3

d A,1j

1

2 )

N (0, s E

2

d E,1

1

d A,2j 0

d E,2

-0.4 -0.2 0 0.2 0.4

0.2

Inter-event residual d E,i

0.1

1. 5

1

Median prediction (all events)
Median prediction (event 1)
Median prediction (event 2)
Individual records (event 1)
Individual records (event 2)

2 )

N (0, s A

0.5

0.02

0.01 0

1 2 10 20 100 -0.2 0 0.2 0.4

-0.4

Distance (km)

Inter-event residual d A,ij

2.4 Schematic illustration of the process of partitioning the total
observed variability to inter-event and intra-event components.