Nonlinear Processes in Geophysics
www.nonlin-processes-geophys.net
1023-5809
1607-7946
11
5/6
2004
10.5194/npg-11-659-2004
http://www.nonlin-processes-geophys.net/11/659/2004/
http://www.nonlin-processes-geophys.net/11/659/2004/npg-11-659-2004.html
http://www.nonlin-processes-geophys.net/11/659/2004/npg-11-659-2004.pdf
659
682
2004-12-15
Influence of a discontinuity on the spectral and fractal analysis of one-dimensional data
R. P. H. Berton
ONERA, Chemin de la Hunière, F-91761 Palaiseau Cedex, France
The analysis of a data area or segment containing steep transitions between
regions with different textures (for example a cloud and its background)
leads to addressing the problem of discontinuities and their impact on
texture analysis. In that purpose, an original one-dimensional analytical
model of spectrum and roughness function has been worked out, with a
discontinuity between two fractal regions, each one specified by its average
µ, standard deviation σ, spectral index β and Hurst
exponent <i>H</i>. This has the advantage of not needing the generation of a
fractal structure with a particular algorithm or random functions and
clearly puts into evidence the role played by the average in generating
spectral poles and side lobes. After validation of the model calibration, a parametric study is carried out
in order to understand the influence of this discontinuity on the estimation
of the spectral index β and the Hurst parameter <i>H</i>. It shows that for
a pure µ-gap, <i>H</i> is well estimated everywhere, though overestimated, and
β is overestimated in the anti-correlation range and saturates in the
correlation range. For a pure σ-gap the retrieval of <i>H</i> is excellent
everywhere and the behaviour of β is better than for a µ-gap,
leading to less overestimation in the anti-correlation range. For a pure
β-gap, saturation degrades measurements in the case of raw data and
the medium with smaller spectral index is predominant in the case of
trend-corrected data. For a pure <i>H</i>-gap, there is also dominance of the
medium with smaller fractal exponent.