Spatial correlation of ground motions in estimating seismic hazards to civil infrastructure - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 60

As can be seen, two approaches for estimating within-earthquake cor-
relation are mentioned above: one is based on the estimation of sample
semi-variogram while another is the direct evaluation of the Pearson's
linear correlation coeffi cient. The former method is based on sample semi-
variogram of normalized residuals

()

2

1
2

σ Δ

⎡
⎢

⎤
⎥

ˆ

() =

d

γ Δ .

σ ε

σ ε is important
here, because it can signifi cantly affect the estimations of within-earthquake
correlation (e.g. Goda and Hong, 2008a; Goda and Atkinson, 2009, 2010).
On one hand,

Evaluation of between-earthquake standard deviation

σ ε for a given seismic event may be assessed by evaluating
the sample semi-variogram for the data pairs with a suffi ciently large sepa-
ration distances , e.g. between 100 and 200 km (Goda and Atkinson, 2009,

2010). It is assumed that between-earthquake residuals are uncorrelated at
suffi ciently long separation distances, and the semi-variogram is expected
to attain a constant plateau level. Alternatively,

Δ

σ ε may be calculated directly
as the event-based standard deviation of regression residuals directly.
However, the value may be underestimated if the residuals are strongly
correlated (Kawakami and Mogi, 2003; Hong et al. , 2009).
The second method, which uses equations 3.10a and 3.10b, implies evalu-
ation of the Pearson's linear correlation coeffi cient for a given separation
distance bin. Goda and Hong (2008a) and Goda and Atkinson (2009) com-
pared the two methods by assessing differences in the estimated

) in
the numerical calculations. Goda and Atkinson (2009) showed that the
method, which is based on the direct evaluation of the correlation coeffi -
cient, tends to result in a slightly more rapid decay of

ρ ε ( Δ

ρ ε ( Δ ). At the same

time, they noted that the possible differences of ) based on two methods

can be largely attributed to the effects of between-earthquake standard
deviation and suggested creating an overall

ρ ε ( Δ

ρ ε ( Δ ) model by averaging the

results of the methods.
In practical calculation, when it is necessary to consider several earth-
quakes, obviously it is not possible to apply in the calculations the event-
dependent between-earthquake standard deviation

σ ε , which was used for
the estimation of within-earthquake correlation based on empirical data.
Thus, to be consistent with the technique used for calculation of spatially
correlation ground motion fi eld, which is described in the next section, it
may be necessary to use generalized between-earthquake standard devia-
tion (e.g. associated with the used ground-motion prediction model) to
estimate site-to-site correlation and to analyze possible differences between
these estimations (e.g. Sokolov et al. , 2012).