Civil Engineering Reference
In-Depth Information
T
1
()
()
(
)
()
(
)
Cov
Extxt
⎡
⎤
lim
∫
xtxt
dt
(2.22)
τ
=
⋅
+
τ
=
⋅
+
τ
⎣
⎦
xx
1
2
1
2
T
12
T
→∞
0
A normalised version of the cross covariance between the fluctuating parts of the
realisations is defined by the cross covariance coefficient
()
12
Cov
τ
xx
()
(2.23)
ρτ
=
xx
12
σσ
xx
12
where
σ
and
σ
are the standard deviations of the two zero mean time variables.
x
x
2
(
)
Fig. 2.6
Cross covariance of time series at positions
y
k
1,2,....,
N
=
If such cross covariance estimates are taken from a set of simultaneous realisations of
a process distributed in space, e.g. as illustrated in Fig. 2.6 where the
N
realisations of
the process is assumed to be taken at arbitrary positions
y
in the horizontal direction,
then a cross covariance function between realisations at distance
Δ
y
may be defined:
