Fractal analysis of topography and reflectance surfaces

Jiang, Xinxia (1998) Fractal analysis of topography and reflectance surfaces. University of Southampton, Faculty of Science, School of Ocean and Earth Science, Doctoral Thesis , 278pp.


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The definition of a fractal has been successfully deduced from constructing the Koch
curve and the Cantor set. Principles of seven methods (the ruler, box-counting, spectral,
structure function, intersection methods, cube-counting, and triangular prism methods) for
determining the fractal dimensions are illustrated and verified by the Koch curve, Cantor
set, and the simulated 1-dimensional and 2-dimensional ffim samples by comparing the
calculated with the theoretical D values of the theoretical fractal models. The application of
appropriate methods to self-similar or self-affme fractals is essential due to different
theoretical assumptions of the methodologies. The ruler dimension is different from the
spectral dimension. The application of Hanning window to the synthetic fBm samples
(Hanning window weighted) is important to obtain correct fractal dimensions for the
spectral method and structure function methods. The multi-scaling behaviour of a fractal
can be unveiled by revealing the difference between the 1st and 2nd order structure function
methods. The zeroset theory is used to relate the D values of 1-d contour set with 2-d
surface by analyzing the DEM data.
The results of fractal analysing 132 topographic contours digitized from different
Abstract v
scales (1:200,000, 1:50,000, 1:20,000) of maps of the border area between Spain and
Portugal show that contours are self-similar, and have a fractal dimension of about D = 1.23
over length scales ranging from 30 m to 13 km scale (3 orders of magnitude). The thirteen
filed and map profiles from Dorset area of southern England has a D value of 1.03 derived
from the ruler method. The variations in D values are controlled by three geological factors:
erosive processes, lithologies, and fractures. The dominant control is the erosive process
and fractures, and lithologies can either result in significant difference or produce more
subtle variation in D values of coastlines and contours. For example, the river down-cutting
produces higher D value (1.1 ~ 1.5) than the wave action or cliff retreat erosive processes
The results of the fractal analysis of the five TM sub-image of Qatar have shown that
D values of the TM images range from 2.10 to 2.96. The variations in D values are
controlled by different types of surface, band variations, and methodologies. The study area
B of a single rock type has the lowest D value (D is about 2.25) and is significant different
from the other four study areas, whilst the urban area E yields the highest fractal dimension
(about D = 2.6). Band 3 yields the highest fractal dimensions, followed by bands 4, 5, 1,
and 6, and band 2 has the lowest D value. The difference between the D values derived
from the 2nd and 1st order structure function methods for all the six bands of five study areas
is D2s,(q=2) - D2s(q=l) = 0.16 0.13 (the uncertainty is the standard deviation), and suggests
that the TM imagery has a multi-scaling property.

Item Type: Thesis (Doctoral)
Additional Information: Digitized via the E-THOS exercise.
Subjects: G Geography. Anthropology. Recreation > GC Oceanography
Divisions : University Structure - Pre August 2011 > School of Ocean & Earth Science (SOC/SOES)
ePrint ID: 42127
Accepted Date and Publication Date:
August 1998Made publicly available
Date Deposited: 22 Nov 2006
Last Modified: 27 Mar 2014 18:27

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