Geoscience Reference
In-Depth Information
Recall that the two-dimensional continuous wavelet transform (2D- CWT) is given by a
convolution product of a signal
sx,y
and an analyzing wavelet
g x,y
(Chui, 1992;
Holschneider, 1995):
yb
1
xb
y
x
S( a, b ,b )
s( x , y ) g
,
dx dy
xy
a
a
a
where "
a
" is the scale parameter, "
b
x
" and "
b
y
" are the respective translations according to X-
axis and Y-axis (the symbol "
―
" denotes the complex conjugate).
Alternatively, it can be computed via the Fast Fourier Transform:
1
ˆ
ˆ
S( a ,b ,b )
FFT
s(
,
).
a g a,a
xy
Here, we compute the wavelet coefficients via FFT using the two-dimensional multiple filter
technique (2D MFT). The latter technique is obtained by generalizing the one-dimensional
version (1D MFT), suggested by Dziewonski et
al.
(1969) and improved by Li (1997), to the
two-dimensional case. It consists of filtering a two-dimensional signal using a Gaussian
filter
given by (Gaci, 2011):
G( k ,
,
)
nm
G(k,
,
)
G (k,
)G (k,
)
nm
1
n
2
m
(7)
2
2
k
k
n
n
e
e
n
n
Where
and
are variable center angular frequencies (or wavenumbers) of the
respective filters
G(k,
and
G(k,
)
. The bandwidths
and
of both filters are
1
n
2
m
1
2
calculated as:
k
.Ln(
)
1
2
1
n
m
(8)
k
.Ln(
)
2
2
1
Where is a constant, (
ξ
1
,
ξ
2
) and (
ν
1
,
ν
2
) are respectively the - 3 dB points of the Gaussian
filters
G
1
and
G
2
, respectively.
A fractal surface
sx,y
verifies the self-affinity property (Mandelbrot, 1977, 1982; Feder,
1988) :
H
s(
x ,
y )
.s( x , y )
,
(9)
0
Where
H
is the Hurst exponent (or the self-affinity parameter). The symbol means the
equality of all its finite-dimensional probability distributions.
For sufficiently large values of
k
, the scalogram, defined as the square of the amplitude
spectrum:
2
, can be expressed as:
Pk,x,y
S( k , x , y )
(x,y)
(x,y)
(10)
P( k ,x , y )
P'( x , y ).k
k
Where
x,y
2
H x,y
1
(11)
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