Geology Reference
In-Depth Information
In other words, the advantage of wavelet transform is that the wavelet coef
cient
re
ects in a simple and precise manner the properties of the underlying function.
Wavelet transform theory and its application to multiresolution signal decomposi-
tion has been thoroughly developed and well documented over the past few years
[
19
,
20
]. Daubechies [
19
] introduced the concept of the orthogonal wavelet, gen-
erally referred to as Daubechies wavelet. Indeed, multiresolution analysis and DWT
have a strong potential for exploring various dynamic features of any time-
dependent data.
In an orthonormal multi-resolution analysis, a signal, f
ð
t
Þ e
V
1
, is decomposed
, such that f
ð
t
Þ
¼
P
0
g
i
ð
t
Þ
into an in
rst
level decomposition is done by projecting onto two orthogonal subspaces, V
0
and
W
0
, where V
1
¼
nite series of detail functions, g
i
ð
t
Þ
. The
V
0
W
0
and
are the direct sum operator. The projection
produces f
0
ð
Þ e
ð
Þ
and g
0
ð
Þ e
t
V
0
, a low resolution approximation of f
t
t
W
0
, the
ð
Þ
to f
0
ð
Þ
detail lost in going from f
t
t
. The decomposition continues by projecting
f
0
ð
Þ
t
onto V
1
and W
1
, and so on. The orthonormal bases of V
j
and W
j
are given by
2
j
=
2
2
j
x
w
j
;
k
¼
wð
k
Þ
ð
4
:
32
Þ
2
j
=
2
2
j
x
u
j
;
k
¼
wð
k
Þ
ð
4
:
33
Þ
where
wð
t
Þ
is the mother wavelet and
uð
t
Þ
is the scaling function [
20
] where
Z
wð
x
Þ
dt
¼
0
, wð
0
Þ
¼
0
ð
4
:
34
Þ
Z
uð
x
Þ
dt
¼
1
, uð
1
Þ
¼
1
ð
4
:
35
Þ
where
wð
w
Þ
and
uð
w
Þ
are the Fourier transform of
uð
t
Þ
and
wð
t
Þ
, respectively. The
projection equations are
1
d
k
2
ð
j
=
2
Þ
wð
2
j
t
g
i
ð
t
Þ
¼
k
Þ
ð
4
:
36
Þ
k
1
d
k
¼
h
f
j
1
ð
t
Þ; w
j
;
k
i
ð
4
:
37
Þ
1
c
k
2
ð
j
=
2
Þ
wð
2
j
t
f
i
ð
t
Þ
¼
k
Þ
ð
4
:
38
Þ
k
1
c
k
¼
h
f
j
1
ð
t
Þ; u
j
;
k
i
ð
4
:
39
Þ
where d
k
and c
k
is the L
2
inner
product. The nested sequence of subspaces, V
j
, constitutes the multiresolution
analysis. Conditions for the multiresolution to be orthonormal are (1)
are the projection coef
cients and
h:; :i
w
j
;
k
and
/
j
;
k
