384 Seismograms as Signals
Fig. 6.3-12 Illustration, for a boxcar function, that the auto-correlation is
maximum at zero lag and is an even function of the lag.
The auto-correlation is significant in the theory of filtering
because it is related to the amplitude spectrum. To see this,
consider a function f(t) that is zero except between −T/2 and
T/2. The auto-correlation
R L
T
f t f t L dt
T
T
T
( ) lim
( ) (
)
/
/
=
+
→∞
−
1
2
2
Ύ
(28)
can be expanded using the inverse Fourier transform and using
the time shift theorem (Section 6.2.4),
R L
T
f t
F e
d dt
T
T
T
i t L
( ) lim
( )
( )
/
/
(
)
=
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
→∞
−
−∞
∞
+
1
2
2
2
π
ω
ω
ω
Ύ Ύ
=
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
→∞
−∞
∞
−
lim
( )
( )
/
/
T
i L
T
T
i t
T
F e
f t e dt d
1
2
2
2
π
ω
ω
ω
ω
Ύ
Ύ
=
−
→∞
−∞
∞
lim
( ) ( )
T
i L
T
F F
e d
1
2π
ω
ω
ω
ω
Ύ
=
→∞
−∞
∞
lim
( )
,
T
i L
T
F
e d
1
2
2
π
ω
ω
ω
Ύ
|
|
(29)
where the last step uses the fact that F (−ω) = F*(ω). Thus, if
we define the power spectrum, a normalized version of the
amplitude spectrum,
P(ω) = lim
T
T
→∞
1 | F(ω) | 2 ,
(30)
we see that the auto-correlation is the inverse Fourier transform
of the power spectrum:
R(L) =
1
2π Ύ
−∞
∞
| P(ω) |e iωL dω.
(31)
f(t)
T
R(L)
L
T
−T
t
R L
T
f t f t L dt
T
T
T
( ) lim
( ) (
) .
/
/
=
+
→∞
−
1
2
2
Ύ
(27)
Not surprisingly, the auto-correlation is maximum at zero lag
and is an even function of the lag (Figs 6.3-12 and 3.3-30).
When the cross-correlation is used to identify reflections
(Figs 6.3-11 and 3.3-31), it makes the seismogram look like the
auto-correlation of the signal near the reflection.
(a)
(b)
(d)
(c)
S
May 14, 1976
Station KBS
∆ = 73.2°
SS
ScS
50 s
Fig. 6.3-11 Application of the cross-correlation to determine the travel
time difference between direct S and reflected SS phases on a seismogram
(a). The direct S phase (dashed line in (b)) is corrected for attenuation
(solid line in (b)), phase-shifted (c), and then cross-correlated with the rest
of the seismogram (d). The peak in the cross-correlation gives the lag that
measures the arrival time difference between the two phases. (Kuo et al.,
1987. J. Geophys. Res., 92, 6421–36, copyright by the American
Geophysical Union.)