Geoscience Reference
In-Depth Information
15
Covariances, autocorrelations, and spectra
15.1 Introduction
In
Chapter 13
we introduced the concept of
moments
of a stochastic, random vari-
able. In this chapter we'll extend that discussion, beginning with scalar functions of
a single variable and the autocorrelation function. We'll then introduce its Fourier
transform, the
power spectral density
or simply
spectrum
. We'll generalize to three
dimensions and vector functions and cover classical material on spectra in isotropic
turbulence. Finallywe'll introduce spectral formulations tailored for use in theABL,
which is inhomogeneous in the vertical but can be homogeneous in the horizontal
plane.
15.2 Scalar functions of a single variable
15.2.1 The autocorrelation function
In
Eq. (13.22)
we defined cross moments
of a stocha
stic function of time, say
u(t)
.
One example is the two-time covariance
u(t
1
)u(t
2
)
, sometimes called the
autoco-
variance.
For a stationary function of time it depends only on the time separation
t
2
−
t
1
.
Its
autocorrelation function
is defined by nondimensionalizing with the
variance:
u(t
1
)u(t
2
)
u
2
=
ρ(t
2
−
t
1
).
(15.1)
ρ(t)
is an e
ven funct
ion
(P
roblem 15.1)
. Schwartz's inequality
(Problem 15.2)
implies that
u(t
1
)u(t
2
)
u
2
, from which it follows that
≤
|
ρ(t)
|≤
ρ(
0
)
=
1
.
(15.2)
For stochastic functions typically
ρ(t)
; we interpret this as
indicating that
u(t)
has a “fading memory” at sufficiently large separation in time.
→
0as
t
→∞
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