Two-Dimensional Multifractional Brownian Motion- Based Investigation of Heterogeneities from a Core Image - Advances in Data, Methods, Models and Their Applications in Geoscience - page 118

For each profile, the Fourier amplitude spectrum is computed and represented in a double
logarithmic scale. Then, we estimate the local Hölder function H ( x ) using an algorithm
based on the local growth of the increment process S k,n ( i ) (Peltier and Lévy-Véhél, 1994,
1995; Muniandy et al. , 2001; Li et al. , 2008; Gaci et al. , 2010):

m





   

Si  Xj   1 Xj

, 1<k<n (16)

kn ,

n  1

 

jik ik  2,  2

where n is the signal X length, k is a fixed window size, and m is the largest integer not
exceeding n/k .
The local Hölder function H ( x ) at point

i

x 

(17)

n  1

is given by







log  2

k Si



,



ˆ



Hi 

(18)

 

log n  1

The obtained results corresponding to the horizontal profiles and the vertical profiles are
respectively exposed in figures 6a and 6b.
It can be noted that all the resulted amplitude spectra exhibit an algebraic decay; that
illustrates the fractal properties of the digitalized data. Besides, the analyzed profiles present
a varying regularity with the position according to X- and Y-axis. They can be then regarded
as paths of multifractional Brownian motions (mBms) (Peltier and Lévy-Véhel, 1995). The
variation of H exponent value is related to the local lithological changes of the core
composition.
The next step consists on establishing regularity maps from the digitalized data using the 2D
MFT algorithm. The implementation of the latter algorithm requires the ''reconditioning'' of
the data so that the matrix dimensions corresponding to the digitalized data are a power of
2. For the purpose of processing the digitalized data, and considering the limitations of the
available computer's capacities, we have splited the obtained matrix (3642 x 996) into two
overlapping sub-matrixes whose size is 2048 x 1024. The sub-matrixes are padded by zeros
so that their dimensions following Y -axis, initially equal to 996, reach 1024.
The parameters selected for the 2D MFT are as follow:
- The minimal center wavenumbers of the filter : ξ min =ν min = 2.10 -1 rad/m;

- The maximal center wavenumbers of the filter:

ξ max =ν max = 2/(2 Δx)≈ 25964 rad/m;
The other parameters ( and N ) are similar to those used in the previous section.
The final regularity map is constructed from the H sub-maps related to the two sub-matrixes
(Fig. 7). The H values in the overlapping zone are calculated as the average of the H values
corresponding to the H values in the sub-maps.