Geoscience Reference
In-Depth Information
(
2
π)
2
+∞
1
e
−
i(κ
1
r
1
+
κ
2
r
2
)
θ
2
(x
3
, t)ρ(r
1
,r
2
;
φ(κ
1
,κ
2
;
x
3
,t)
=
x
3
,t)dr
1
dr
2
.
−∞
(15.32)
That is, both
x
3
and
t
are parameters, correlations being made in the
x
1
-
x
2
plane at
time
t
. Correlations between two different values of
x
3
or
t
are treated as co- and
quadrature spectra and cross covariances were treated.
If a scalar random function depends on three space coordinates and time, and
is stationary but inhomogeneous in all three directions, then the analysis of
Sec-
tion 15.1
applies, but with an additional parameter denoting the space point. Joint
statistics between two space points are handled as a cross covariance.
If the function were homogeneous in all three directions but nonstationary, then
the analysis of
Eqs. (15.28
)to
(15.30)
applies, with an additional parameter of time.
Statistics at two times would be handled by cross covariances. If the function were
both stationary and homogeneous we would have
+∞
e
i
κ
·
x
+
iωt
dZ(
κ
,ω),
θ(
x
,t)
=
−∞
0
,
κ
=
κ
,ω
ω
,
=
dZ(
κ
,ω)dZ
∗
(
κ
,ω
)
=
(15.33)
κ
=
κ
,ω
ω
,
φ(
κ
,ω)d
κ
dω,
=
+∞
e
i
κ
·
ξ
+
iωτ
φ(
κ
,ω)d
κ
dω,
θ(
x
,t)θ(
x
+
ξ
,t
+
τ)
=
θ
2
ρ(
ξ
,τ)
=
−∞
(
2
π)
4
+∞
1
e
−
i
κ
·
ξ
−
iωτ
θ
2
ρ(
ξ
,τ)d
ξ
dτ,
φ(
κ
,ω)
=
−∞
where
ρ
is called a space-time correlation and
φ
is a space-time spectrum.
15.3.2 Application to typical measurements
Say we are measuring a turbulent scalar
θ
along the
x
1
coordinate - e.g., on an
aircraft, with
x
1
the flight-path direction, or by using Taylor's hypothesis with a
stationary sensor at a fixed point. Then in homogeneous turbulence the autocorre-
lation function, spectrum relation of
Eq. (15.30)
becomes (hereafter not explicitly
indicating dependence on parameters such as
t
)
+∞
e
iκ
1
ξ
1
φ(
κ
)d
κ
θ
2
ρ(ξ
1
,
0
,
0
)
=
−∞
+∞
e
iκ
1
ξ
1
+∞
−∞
φ(
κ
)dκ
2
dκ
3
dκ
1
=
−∞
