7.3 Second-Order Random Functions
159
that VF is derived via a quadratic transformation. It can be shown, therefore [141,
pp. 61-62], that
p VF (α) =
⎧
⎨
⎩
1
2
√
α
p cos φ (
√
α) + p cos φ (−
√
α)
; α ≥ 0
0;
α < 0.
=
⎧
⎨
⎩
1
2
√
α
; 0 < α ≤ 1
0;
otherwise.
,
(7.3)
where the final result follows because p cos φ (α) = 1 for 0 ≤ α ≤ 1 and
vanishes elsewhere. p VF has a weak (integrable) singularity, but it does satisfy the
requirements for a probability density, namely that it is positive, and that its integral
is unity:
1
0
p VF (α)dα =
1
2
1
0
dα
√
α
= 1 .
(7.4)
7.3 Second-Order Random Functions
If we don’t know the a priori probability density for the volume-fractions or
conductivities, then we resort to the next best thing, namely certain second-order
properties of these random functions. The usual property that is either given, or
can be measured, is the covariance function, and this is the point of departure in
calculating properties of random surfaces in [2, 19, 53, 145].
We still need to transform knowledge of the covariance function into probability
densities for the volume fractions in a VIC-3D® model, and we use the KarhunenLoève expansion to do this. Using Loève’s notation [63, pp. 478–479] for a onedimensional problem, the expansion is given by the following
Theorem 7.1 (Proper Orthogonal Decomposition) A random function, X(t),
continuous in quadratic mean (q.m.) on a closed interval, I , has on I an orthogonal
decomposition
X(t) =
λ n ξ n ψ n (t)
(7.5)
with
Eξ m ξ
∗
n = δ mn ,
ψ m (t)ψ
∗
n (t)dt = δ mn ,
(7.6)
if, and only if, the λ n are the proper (eigen-) values and the ψ n (t) are the orthonormalized proper (eigen-) functions of its covariance. Then the series converges in
q.m. uniformly on I .
159
that VF is derived via a quadratic transformation. It can be shown, therefore [141,
pp. 61-62], that
p VF (α) =
⎧
⎨
⎩
1
2
√
α
p cos φ (
√
α) + p cos φ (−
√
α)
; α ≥ 0
0;
α < 0.
=
⎧
⎨
⎩
1
2
√
α
; 0 < α ≤ 1
0;
otherwise.
,
(7.3)
where the final result follows because p cos φ (α) = 1 for 0 ≤ α ≤ 1 and
vanishes elsewhere. p VF has a weak (integrable) singularity, but it does satisfy the
requirements for a probability density, namely that it is positive, and that its integral
is unity:
1
0
p VF (α)dα =
1
2
1
0
dα
√
α
= 1 .
(7.4)
7.3 Second-Order Random Functions
If we don’t know the a priori probability density for the volume-fractions or
conductivities, then we resort to the next best thing, namely certain second-order
properties of these random functions. The usual property that is either given, or
can be measured, is the covariance function, and this is the point of departure in
calculating properties of random surfaces in [2, 19, 53, 145].
We still need to transform knowledge of the covariance function into probability
densities for the volume fractions in a VIC-3D® model, and we use the KarhunenLoève expansion to do this. Using Loève’s notation [63, pp. 478–479] for a onedimensional problem, the expansion is given by the following
Theorem 7.1 (Proper Orthogonal Decomposition) A random function, X(t),
continuous in quadratic mean (q.m.) on a closed interval, I , has on I an orthogonal
decomposition
X(t) =
λ n ξ n ψ n (t)
(7.5)
with
Eξ m ξ
∗
n = δ mn ,
ψ m (t)ψ
∗
n (t)dt = δ mn ,
(7.6)
if, and only if, the λ n are the proper (eigen-) values and the ψ n (t) are the orthonormalized proper (eigen-) functions of its covariance. Then the series converges in
q.m. uniformly on I .