Geoscience Reference
In-Depth Information
By taking the scale “
a
” inversely proportional to the wavenumber
k
:
a
1
k
, the wavelet
coefficients will be expressed in (
k
,
b
x
,
b
y
) plane.
The scalogram can be defined as the square of the amplitude spectrum:
2
Pkxy
,,
Skxy
(,,)
. For large values of
k
, it can be expressed as:
(,)
x y
(,)
x y
Pkxy
(,,)
P xy k
(, .
k
(8)
Where
xy
,
2
H xy
,
1
(9)
is the local spectral exponent which is related to the local Hurst (or Hölder) exponent,
. The spectral exponent
in each point
x y
is computed as the slope of the
scalogram versus the wavenumber in the log-log plan, the
Hxy
,
x y
Hxy
,
value is then derived
using the equation (9).
3.2 Used analyzing wavelets
The analyzing wavelets used in this application are the Morlet wavelet and the Mexican hat
(Fig. 1). This choice is motivated by their adequate properties for the regularity analysis.
-
The Morlet wavelet:
1
2
2
2
x
y
i
x
y
gxy
,
e
0
x
0
y
.
e
1
2
2
2
; with
12
0
x
0
y
2
2
g
ˆ (,)
e
5
(10)
0
x
0
y
-
The Mexican hat:
22
2
22
2
e
xy
2
2
2
2
;
ˆ
g
,
.
e
(11)
gxy
,
2
x
y
3.3 Wavelet-based estimators of the local regularity
As explained earlier, the computation of the local Hölder exponents
H
(
x
,
y
) requires to
calculate the two-dimensional wavelet continuous transform. Here, we suggest three
algorithms for the implementation of the 2D- CWT, which are:
3.3.1 FFT-based algorithms
These algorithms are based on the property that wavelet coefficients, expressed by the
equation (4), can be performed via the Fourier transform using the Morlet wavelet and the
Mexican hat:
1
ˆ
ˆ
Sab b
(, , )
FFT
s
( , ).
ag a
,
a
(12)
xy
These algorithms are accurate but slow, and the signal length must be a power of 2.
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