Regularity Analysis of Airborne Natural Gamma Ray Data Measured in the Hoggar Area (Algeria) - Advances in Data, Methods, Models and Their Applications in Geoscience - page 98

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  (5)

n  1

is given by (Peltier and Lévy-Véhél, 1994, 1995; Muniandy et al. , 2001 ; Li et al. , 2007, 2008;
Gaci et al. , 2010):



 

log  2

k, Si

 

ˆ Hi



 (6)

 

log n  1

From figure 3, it can be seen that all the calculated amplitude spectra, represented in a log-
log plan, decay algebraically, the analyzed data exhibit then a fractal behavior. Moreover,
the latter is described by a Hölder exponent varying with the latitude of the measure. Hence
the data can be considered as paths of multifractional Brownian motions (mBms) (Peltier
and Lévy-Véhel, 1995; Gaci et al. , 2011).
A significant result deserves to be noted is the fact that the spectra obtained from the ''raw'',
''corrected'' and ''corrected and interpolated'' measurements display a similar form. That is
the applied operations (corrections and interpolations) do not affect the fractal aspect of the
raw data.

a)

Fig. 3. (Continued)