Geology Reference
In-Depth Information
We recognise this expression as the infinite Fourier series expansion of the periodic
function of frequency,
S
gg
(
f
). The power spectral density of discrete, equispaced
data is thus periodic, with period 2
f
N
t
, where
f
N
is the Nyquist frequency.
The corresponding sequence of autocorrelations can be recovered from the Fourier
series (2.336) as the Euler coe
=
1/
Δ
cients
f
N
S
gg
(
f
)
e
i
2π
fj
Δ
t
df
.
φ
gg
(
j
)
=
(2.337)
−
f
N
2.5.2 Multiple discrete segment estimate
The variance of a power spectral density estimate is reduced by averaging over
multiple individual estimates, based on discrete segments of the record.
For a windowed data segment of length
M
,wehaveforanalysis,
=
w(
t
)g(
t
),
h
(
t
)
(2.338)
where
w(
t
)
=
0,
|
t
|
>
M
/2.
(2.339)
We window the data with a window extending over
−
M
/2<
t
<
M
/2. The sample
autocorrelation estimator for this data segment is
M
/2
1
M
˜
h
(
t
)
h
∗
(
t
φ
hh
(τ)
=
−
τ)
dt
.
(2.340)
−
M
/2
To determine what this estimator gives, on average, we find its expected value
across an infinite ensemble of realisations of the process generatingg(
t
). The expec-
ted value is
E
˜
=
E
1
M
M
/2
h
(
t
)
h
∗
(
t
−
τ)
dt
φ
hh
(τ)
−
M
/2
=
E
1
M
−
M
/2
w(
t
)g(
t
)w
∗
(
t
−
τ)g
∗
(
t
−
τ)
dt
M
/2
.
(2.341)
The window, w(
t
), is deterministic, real and even, and both integration and taking
the expected value are linear operations, giving
E
˜
φ
hh
(τ)
M
/2
−
M
/2
w(
t
)w
∗
(
t
−
τ)
E
g(
t
)g
∗
(
t
−
τ)
dt
1
M
=
M
/2
−
M
/2
w(
t
)w(τ
−
1
M
=
t
)φ
gg
(τ)
dt
M
φ
gg
(τ)
∞
1
=
w(
t
)w(τ
−
t
)
dt
.
(2.342)
−∞
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