Civil Engineering Reference
In-Depth Information
(
)
(
)
(
)
Cov
Δ=
y
,
τ
E x yt
⎡
,
⋅
x y
Δ+
yt
,
τ
⎤
⎣
⎦
xx
T
(2.24)
1
(
)
(
)
lim
∫
x yt
,
x y
yt
,
dt
=
⋅
+ Δ
+
τ
T
T
→∞
0
(
)
2
Cov
y
0,
0
Obviously,
Δ= = = . In wind engineering such covariance estimates
will in general be a decaying function with increasing
τ
σ
xx
x
Δ
τ
or spatial separation
,
sxy z
, or
, as illustrated in Fig. 2.7. The covariance function may attain negative
values at large values of
=
Δ
or
τ
.
Fig. 2.7
Typical spatial separation and time lag covariance function
As previously indicated, the statistical properties defined above may also be applied
to functions that are obtained from realisations of two different processes. Then, by
simple arithmetic, the variance of the sum of two zero mean variables,
()
x
t
and
()
x
t
, is given by
2
(
)
(
) (
)
Var x
x
E
x
x
x
x
+=
⎡
+⋅
+
⎤
⎣
⎦
1
2
1
2
1
2
(2.25)
()
()
(
)
Var x
Var x
2
Cov x
x
=
+
+
⋅
⋅
1
2
1
2
()
x
t
, is given by
Similarly, the variance of the sum of
N
different variables,
i
N
⎛ ⎞
(
)
∑
(
)
⎡
⎤
Var
x
=
E
x
+ ++ ++
x
...
x
...
x
⋅
x
+ ++ ++
x
...
x
...
x
⎜ ⎟
i
⎣
1
2
i
N
1
2
j
N
⎦
⎝ ⎠
i
1
=
