Civil Engineering Reference
In-Depth Information
where
I
0
0
⎡
⎤
⎡
ρ
⎤
u
(
)
uu
x
I
=
and
ρ
Δ
=
(7.31)
⎢
⎥
⎢
⎥
v
v
0
I
0
ρ
⎣
⎦
⎣
⎦
w
ww
and thus,
2
2
⎛
⎞
VB
{
}
ρ
ˆ
ˆ
()
()
⎡
2
( )
⎤
TT
()
x
⎠
∫∫
x
x
x
dx dx
Cov
=
G
⋅
B
⋅
I ρ BG
⋅
Δ
⋅
⋅
⎜
⎟
MM
r
⎜
⎟
M
1
q
⎣
v
v
⎦
q
M
2
1
2
B
2
⎝
L
exp
(7.32)
x
and
x
are
The covariance matrix in Eq. 7.32 will be symmetric because
x
xx
interchangeable and
.
In a fully expanded format the variance of the background response components are
given by
ρ
and
ρ
are only functions of the separation
Δ
=−
uu
ww
1
2
⎡
2
⎤
⎡
(
)
⎤
σ
g
x
,
,
x
M
xx
⎢
⎥
2
M
xx
12
⎢
⎥
2
⎛
VB
⎞
ρ
⎢
⎥
2
(
)
∫∫
⎢
⎥
σ
=
g
x
x
dx dx
(7.33)
⎜
⎟
MM
⎜
⎟
MM
12
1 2
⎢
yy
⎥
yy
2
⎢
⎥
⎝
⎠
L
⎢
⎥
exp
(
)
⎢
⎥
2
g
x
,
x
σ
⎣
M
zz
12
⎦
⎢
⎥
⎣
M
zz
B
⎦
where
2
()
(
⎡
)
⎤
2
2
()
(
)
(
)
(
)
g
BG
x G
x
2
C I
x
C I
x
=
ρΔ
+
′
ρ Δ
(7.34)
MM
M
1
M
2
⎢
M u
uu
M w
ww
⎥
xx
x
x
⎣
⎦
⎧
2
⎫
D
⎡
⎤
⎪
2
⎛
⎞
⎪
()
(
)
()
(
)
(
)
g
=
GxGx CI
2
ρΔ
x C
+
′
+
CI
ρ Δ
x
⎨
⎬
⎢
⎜
⎟
⎥
MM
M
1
M
2
Lu
uu
L
D
w
ww
yy
y
y
B
⎣
⎝
⎠
⎦
⎪
⎪
⎩
⎭
(7.35)
()
()
g
=
G
x
G
x
⋅
MM
M
1
M
2
zz
z
z
⎧
2
2
⎫
(7.36)
D
D
⎡
⎤
⎪
⎛
⎞
⎛
⎞
⎪
(
)
(
)
2
CI
ρΔ
x
+
C
′
−
C I
ρ Δ
x
⎨
⎬
⎜
⎟
⎢
⎜
⎟
⎥
Du
uu
D
L
w
ww
B
B
⎝
⎠
⎣
⎝
⎠
⎦
⎪
⎪
⎩
⎭
Similarly, the corresponding covariance between background components may be
expanded into
(
)
⎡
Cov
⎤
⎡
g
x
,
x
⎤
MM
MM
12
xy
2
xy
⎢
⎥
⎢
⎥
2
⎛
⎞
ρ
VB
(
)
⎢
Cov
⎥
∫∫
⎢
g
x
,
x
⎥
dx dx
=
(7.37)
⎜
⎟
MM
⎜
⎟
MM
12
1 2
xz
2
xz
⎢
⎥
⎢
⎥
⎝
⎠
L
(
)
Cov
exp
g
x
,
x
⎢
⎥
⎢
⎥
MM
MM
12
⎣
⎦
⎣
⎦
yz
yz
B
