Geoscience Reference
In-Depth Information
consider a generalized fractal model, the two-dimensional multifractional Brownian motion
(2D-mBm), which presents a regularity varying in space.
The mBm model, initially proposed by Peltier and Lévy-Véhél (1995), and Benassi
et al.
(1997), is used in many disciplines: images processing (Bicego and Trudda, 2010), traffic
phenomena (Li
et al.
, 2007), geophysics (Wanliss, 2005; Wanliss and Cersosimo, 2006;
Cersosimo and Wanliss, 2007; Gaci
et al.
, 2010; Gaci and Zaourar, 2010, 2011). For the
estimation of the mBm processes' local regularity, we propose three algorithms based on the
two-dimensional continuous wavelet transform (2D- CWT). The wavelet coefficients are
calculated by Fast Fourier Transform (FFT) using the Morlet wavelet and the Mexican hat
for the first and the second algorithms, respectively. However for the third algorithm, the
coefficients estimation is carried out using the multiple filter technique (2D MFT) that we
generalized to two dimensions (Gaci, 2011), from the one-dimensional case (1D MFT) (Li,
1997; Gaci
et al.
, 2011).
This chapter is organized as follows. First, we give a brief theory on 2D-mBm model and the
wavelet-based estimators of the local regularity. The potential of the suggested algorithms is
then demonstrated on synthetic 2D-mBm paths. The results showed that the 2D MFT
algorithm yields the best Hölder exponent estimates. Next, the suggested regularity analysis
is extended to digitalized image data of a core extracted from an Algerian borehole. It is
shown that the data exhibit a fractal behavior. In addition, the derived regularity maps,
obtained with the 2D MFT algorithm, show a strong correlation with the core
heterogeneities.
2. (Multi)fractional Brownian motion
2.1 Fractional Brownian motion
Fractional Brownian motion (fBm) is one of the most popular stochastic fractal models for
studying rough signals. It was introduced by Kolmogorov (1940) and studied by
Mandelbrot and Van Ness (1968).
A fBm, denoted by
B
H
(
t
), is the zero-mean Gaussian process with stationary increments. It is
parameterized by a constant Hurst parameter
H
. The fBm is
H
-self affine,
i.e.
:
H
Bt
Bt
,
0
(1)
H
H
Where means the equality of all its finite-dimensional probability distributions.
The bidimensional isotropic fractional Brownian motion, or Lévy Brownian fractional field,
with Hurst parameter
H
is a centered Gaussian field
B
with an autocorrelation function
(Kamont, 1996):
2
H
2
H
2
H
EBxBy
x
y
x y
, with 0
H
1
(2)
and . is the usual Euclidian norm.
For
H
=1/2, the fractional Brownian motion is reduced to a Wiener process.
The regularity of the 2D-fBm is measured by the pointwise Hölder exponent
2
where
x y
R
.
x
B
Indeed, it is shown that almost surely:
xH
. Therefore, the higher the
H
value, the
B
smoother the 2D-fBm paths.
Search WWH ::
Custom Search