Environmental Engineering Reference
In-Depth Information
of the random field. For a two-dimensional (2D) Gaussian random field, this function is
given as follows:
xx
l
−
yy
l
−
1
2
1
2
2
Cx yxy
[(
,
), (
,
)]
=
σ
exp
−
−
(15.1)
11
22
x
y
in which σ is the standard deviation of the random field, and
l
x
and
l
y
are, respectively, the
horizontal and vertical autocorrelation lengths. An autocorrelation length is the distance
within which the values of the uncertain property are significantly correlated. It is the dis-
tance required for the autocorrelation function to decay from 1 to e
−1
(i.e., 0.3679). Finally,
(
x
1
,
y
1
) and (
x
2
,
y
2
) are the coordinates of two arbitrary points in the space.
Let
R
(
x
,
y
, θ) be a Gaussian random field where
x
and
y
denote, respectively, the horizontal
and vertical coordinates and θ indicates the stochastic character of the random field. The
random field
R
can be approximated by the K-L expansion as follows (Spanos and Ghanem
1989; Ghanem and Spanos 1991):
M
∑
1
Rx y
(, ,
θµ λφ
)
≈+
=
(
xy
,
)
ξθ
()
(15.2)
i
i
i
i
where μ is the mean value of the random field,
M
is the size of the series expansion, λ
i
and
ϕ
i
are the eigenvalues and eigenfunctions of the covariance function, respectively, and ξ
i
(θ)
(for
i
= 1, …,
M
) is a vector of standard uncorrelated random variables. For the exponential
covariance function used in this chapter, Ghanem and Spanos (1991) provided an analytical
solution for the eigenvalues and eigenfunctions as follows:
a. For a one-dimensional (1D) horizontal random field generated in the interval [−
a
x
,
a
x
],
the eigenvalues can be calculated as follows:
2
c
λ
=
(15.3)
i
ω
2
+
c
2
i
where
1
c
=
(15.4)
l
x
and
‡
‰
‰
c
−
ωω
tan(
a
x
)0
=
for oddi
i
i
i
and
(15.5)
ω
+
c
tan(
ω
a
)0
=
for eveni
i
i
i
x
The eigenfunctions are calculated as follows:
cos(
ω
ωω
x
)
i
φ
=
for oddi
i
(15.6)
i
a
+
(sin(
2
a
)
/
2
)
x
i
x
i
sin(
ω
ωω
x
)
i
φ
=
for eveni
i
(15.7)
i
a
−
(sin(
2
a
)
/
2
)
x
i
x
i
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