Geoscience Reference
In-Depth Information
Equation (15.20)
says that the quadrature spectrum is zero if
C
uv
is even. If the
maximum correlation between
u
and
v
occurs at some nonzero time difference, then
C
uv
is not even. For example,
u
and
v
might be a conserved scalar at two points
separated in the streamwise direction; the maximum correlation would occur at a
lag corresponding to the transport time between the points.
If
v(t)
is simply
u(t)
delayed by a time interval
t
, we say that
v(t)
has a “phase
shift” of
θ
=
ωt
and we can write
+∞
+∞
e
iωt
dZ(ω),
e
i(ωt
−
θ)
dZ(ω).
u(t)
=
v(t)
=
(15.23)
−∞
−∞
The cross covariance is
+∞
e
i(ωτ
−
θ)
φ(ω)dω,
C
uv
(τ )
=
(15.24)
−∞
since
dZ(ω)dZ
∗
(ω)
=
φ(ω)dω
. Thus we have
e
−
iθ
φ(ω)
φ
uv
(ω)
=
=
cos
θφ(ω)
−
i
sin
θφ(ω),
Co
uv
(ω)
=
cos
θφ(ω), Q
uv
(ω)
=
sin
θφ(ω),
Q
uv
(ω)
Co
uv
(ω)
.
tan
θ
=
(15.25)
A time lag
t (ω)
at each frequency is defined by
Q
uv
(ω)
Co
uv
(ω)
.
tan [
ωt(ω)
]
=
(15.26)
Another dimensionless quantity is the coherence, the square of the spectral
correlation, or the normalized covariance:
2
dZ
u
(ω) dZ
v
(ω)
|
|
Coh
uv
(ω)
=
dZ
u
(ω) dZ
u
(ω) dZ
v
(ω) dZ
v
(ω)
=
|
(15.27)
2
φ
u
φ
v
=
Co
uv
(ω)
Q
uv
(ω)
φ
u
(ω) φ
v
(ω)
φ
uv
|
+
.
By Schwartz's inequality this cannot exceed 1; that value occurs when the Fourier
components of
u
and
v
at frequency
ω
are proportional. The example of
(15.23)
to
(15.25)
has a coherence of unity at all frequencies.
