Geology Reference
In-Depth Information
Again, using the shifting theorem and that the Fourier transform of a convolution is
the product of the transforms (2.269), the Fourier integral representation of
J
m
(τ)
is expressed as
∞
exp (
i
2π
f
τ) exp
i
2π
f
(
b
n
+
m
b
n
)
W
2
(
f
)
df
.
J
m
(τ)
=
−
(2.382)
−∞
For τ
=
0, we then have that
∞
exp
i
2π
f
(
b
n
+
m
b
n
)
W
2
(
f
)
df
J
m
(0)
=
J
m
=
−
−∞
M
/2
−
M
/2
w(
t
)w(
t
−
m
Δ
s
)
dt
.
=
(2.383)
Substituting in expression (2.380), we obtain for the covariance,
cov
S
gg,
n
(
f
),
S
gg,
n
+
m
(
f
)
J
m
I
2
S
2
S
2
=
gg
(
f
)
=
gg
(
f
)ρ(
m
),
(2.384)
J
m
/
I
2
. Since
J
0
where ρ(
m
)
=
=
I
,
cov
S
gg,
n
(
f
),
S
gg,
n
(
f
)
var
S
gg,
n
(
f
)
S
2
=
=
gg
(
f
).
(2.385)
Expression (2.364) for the variance of the final spectral density estimate, taken as
an average of the sample spectral density estimates over κ overlapping segments of
length
M
,thenbecomes
var
S
gg
(
f
)
⎣
κ
ρ(
m
)
⎦
.
2
κ
−
1
S
2
gg
(
f
)
κ
κ
−
m
=
1
+
(2.386)
m
=
1
This formula was quoted and used by Welch (1967) in an analysis using Bartlett
windows with 50% overlap.
Instead, we have adopted the Parzen window (2.284), and we use 75% over-
lap. The Parzen time window is a piecewise cubic, and the integrals,
J
m
, can be
evaluated analytically. We find that
151
560
M
,
397
40
J
2
,
3
224
M
,
1
120
J
2
,
J
0
=
I
=
J
1
=
J
2
=
J
3
=
(2.387)
giving,
397
40
·
2
3
224
·
560
151
ρ(1)
=
=
0.2430131,
3
224
·
2
560
151
ρ(2)
=
=
0.0024670,
(2.388)
1
120
·
2
3
224
·
560
151
ρ(3)
=
=
0.0000002,
ρ(1)
+
ρ(2)
+
ρ(3)
=
0.2454803.
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