Geoscience Reference
In-Depth Information
where
α
is the realization index,
κ
is spatial wavenumber (2
π
/wavelength) and
φ
α
is the phase angle in realization
α
. We choose the
φ
α
from random numbers
that vary from 0 to 2
π
with uniform probability, giving
u
a different phase in each
realization. Thus
u(x)
is a random variable that is statistically homogeneous in
x
.
The ensemble average of
u
is
N
1
N
u
=
sin
(κx
+
φ)
=
sin
(κx
+
φ
α
).
lim
N
→∞
(2.13)
α
=
1
Expanding sin
(κx
+
φ)
and using the distributive property
(2.5)
of the ensemble
average yields
u
=
sin
κx
cos
φ
+
cos
κx
sin
φ
=
sin
κx
cos
φ
+
cos
κx
sin
φ.
(2.14)
Position
x
is fixed in the ensemble average, so sin
κx
and cos
κx
, being constants
in the averaging process, can be taken out from under the overbar:
u
=
sin
κx
cos
φ
+
cos
κx
sin
φ.
(2.15)
Since the random phase
φ
varies from 0 to 2
π
with uniform probability, we
expect that
0
.
(2.16)
We shall prove this in
Part III
. We
c
onclude from
Eqs. (2.15)
and
(2.16)
that our
test field has zero ensemble mean,
u
cos
φ
=
sin
φ
=
0
.
The same process yields the variance
u
2
:
=
2
1
φ)
1
2
[1
1
sin
2
(κx
u
2
=
+
φ)
=
−
cos 2
(κx
+
φ)
]
=
−
cos 2
(κx
+
2
1
sin 2
κx
sin 2
φ
1
=
−
cos 2
κx
cos 2
φ
+
2
1
sin 2
κx
sin 2
φ
=
1
1
2
.
=
−
cos 2
κx
cos 2
φ
+
(2.17)
The
derivative is
∂u/∂x
=
κ
cos
(κx
+
φ).
The same process shows that
∂u/∂x
0. We can prove this more directly by using property
(2.8)
,the
interchangeability of differentiation and averaging, which implies
=
∂u
∂x
=
∂u
∂x
=
0
,
(2.18)
since in this example
u
=
0
.
