Geology Reference
In-Depth Information
2.1.5 Properties of wavelets
In general, wavelets have the form
...,0,0,
f
0
↑
,
f
1
,...,
f
j
,...
(2.38)
In practice, the wavelets that arise are
finite wavelets
, which we write as
f
0
↑
,
f
1
,
f
2
,...,
f
n
.
(2.39)
The z-transform of a finite wavelet is the polynomial
=
f
0
+
f
1
z
+
f
2
z
2
+···+
f
n
z
n
F
(
z
)
,
(2.40)
which can be factored to
F
(
z
)
=
f
n
(
z
1
+
z
)(
z
2
+
z
)
···
(
z
n
+
z
).
(2.41)
Since multiplication of z-transforms is equivalent to convolution in the time domain,
we see that the finite wavelet is
f
n
times the successive convolutions of the
dipole
wavelets
z
1
↑
,1
,
z
2
↑
,1
,...,
z
n
↑
,1
.
(2.42)
The
time reverse
of the wavelet (2.39) is the finite sequence
f
n
,
f
n
−
1
,...,
f
0
↑
.
(2.43)
If we shift the
time reverse n
units in the positive time direction, we obtain the
reverse wavelet
to (2.39),
f
n
↑
,
f
n
−
1
,...,
f
0
.
(2.44)
The reverse wavelet has the z-transform
f
n
+
f
n
−
1
z
f
0
z
n
R
(
z
)
=
+···+
.
(2.45)
The time reverse then has the z-transform
R
(
z
)
z
n
.
(2.46)
Dividing the z-transform by
z
n
gives the z-transform of the wavelet shifted
n
time
units in the negative time direction.
Since the autocorrelation of a wavelet is the same as its convolution with its own
time reverse, the z-transform of the autocorrelation of the wavelet (2.39) is
F
(
z
)
R
(
z
)
Φ
(
z
)
=
z
n
.
(2.47)
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