Spatial correlation of ground motions in estimating seismic hazards to civil infrastructure - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 58

within Δ ± δ /2; (5) all pairs of sites that fall in the bin centered at Δ (i.e. the

sites that are separated by a distance /2)) are used to esti-

mate the variance of the residuals, and then compute the standard deviation
σ d of residuals within each

∈ ( Δ − δ /2, Δ + δ

) is estimated it can be used in
equations (3.3) and (3.5) to calculate estimations of

Δ bin. Once σ d ( Δ

ρ ˆ T ( Δ ) and ρ ˆ ε ( Δ ),

respectively.
In Procedure 2 , Wang and Takada (2005) considered the covariance func-
tion of the values Z ( x 1 ) and Z ( x 2 ), which is described as

[ ] =

{

[ ] [ ]

}

() () () = () − () −

COV Z x , Z x COV Δ E Z x μ Z x μ [3.8]

1 2 1 Z 2 Z

where E [•] is an expectation; Δ is a separation distance between the loca-

tions x 1 and x 2 ; and μ Z is the mean value of Z ( x ). Correlation function is

obtained by normalizing covariance function with the variance

2
Z

σ

()

COV Δ

[ ] =

() () () =

ρ Zx , Zx ρ Δ [3.9]

1 2

σ

2

Z

A discrete version of the equations follows

N

1

all

= ∑

()

μ Z = Zx [3.10a]

j

N

all

j 1

()
∑

N Δ

1

[

() −

] [

( ) −

]

() =

COV Δ Zx μ Zx + Δ μ [3.10b]

j Z j Z

() N Δ

j = 1

where N all is the total number of observation sites; N ( Δ ) is the number of

pairs of sites separated by . Underling assumption of this method is that

Z ( x ) is a homogeneous 2D stochastic fi eld; therefore, the homogeneity of
the residual data should be examined for every considered case (Wang and
Takada, 2005). Homogeneity implies similarity of statistical properties
(mean value and standard deviation) of the residuals for various distances
and magnitudes.
In Procedure 3 , Baker and Jayaram (2008), Jayaram and Baker (2009),
and Esposito and Iervolino (2011) evaluated the within-earthquake correla-
tion in the form of semi-variogram

Δ

) (a measure of average dissimilarity
between the data, e.g. Goovaerts, 1997) as follows:

γ ( Δ

1
2

() = { }

2

⎣ ⎦

γ Δ EZ − Z [3.11]

u u + Δ

where Z u and Z u +Δ are the spatially distributed random functions at site
locations u and u

. In this procedure, second-order stationary is typically
assumed meaning that (1) the expected value of the random variable Z u is
a constant across the space, so the data available over the entire region of
interest can be pooled, and (2) the two-point statistics do not depend on
site locations u and u

+ Δ

+ Δ , but only on their separation distance Δ .