Multifractal Analysis of Geographical Structures and Processes: Concepts and Applications

Following its development by Benoit Mandelbrot, fractal has been used in a large number of studies of a wide variety of geographical phenomena exhibiting complexity. Fractal makes the study of highly irregular and complex structures and processes that defy traditional mathematical analysis feasible. One of the most popular uses of the fractal is to draw attention to the variance of structures and processes across multiple scales (e.g. scaling behavior). The concept of a power-law, which is expressed as a straight line in a double-logarithmic plot, is the most characteristic scaling behavior of fractal to measure self-similarity invariant across multiple scales. However, the single fractal dimension falls short in capturing scale-invariant geographical phenomena for the characterization of their non-linear variation across a wide range of scales. The concept of multifractal was, therefore, introduced to give a more complete description. The fractal dimension was extended to the generalized fractal dimensions. The main objective of this thesis is to improve the theoretical formulation of Multifractal Analysis (MFA) along a number of directions, and to make applications to investigate the self-similarity, long-range correlation or multifractality of some real-life geographical phenomena for substantiation.;Detrended Fluctuation Analysis (DFA) and Multifractal Detrended Fluctuation Analysis (MF-DFA) have become the most popularly used because of their effectiveness and easy implementation. Actually, MF-DFA is based on DFA, which is designed to calculate the Hurst exponent, H, through the power-law between the square of fluctuations and the corresponding scales. H aims at quantifying the long-range correlation of a process and can be related to the fractal dimension. The generalized Hurst exponents h(q) can be obtained by studying the scaling behavior of the qth moment of the fluctuations. On the conceptual level, MF-DFA and DFA (the base of MF-DFA) are explored in this thesis.;Two problems of DFA and MF-DFA in this study are: firstly, oscillations in the fluctuation function and significant errors in the crossover locations; and secondly, the negative influence of periodic trend on the scaling behavior in DFA. The Multifractal Moving-Window Detrended Fluctuation Analysis (MF-MWDFA) and the more general Multifractal Temporally Weighted Detrended Fluctuation Analysis (MF-TWDFA) are formulated as a solution of the first problem. The second problem is solved by a pre-detrending method on the basis of Empirical Mode Decomposition (EMD) for the elimination of the effect of the periodic trend in DFA. Furthermore, some classical relationships of the exponents in MF-DFA are revisited. This study will rectify the incorrectness of existing results found under some situations, and propose modified relationships to obtain the appropriate characterizations.;In terms of applications, the efficacy of the improved DFA is shown by two real-life examples, namely: temperature variations and sunspot activities. A substantial systematic analysis of the temporal and spatial patterns of the earthquake process is studied at length. As a complement to the inter-event spatial and temporal distance, the epicenter motion direction is investigated by DFA. The scaling behaviors under different conditions (e.g. the threshold magnitudes, boundary effect, random removal of some events, and different seismic zones) are also investigated. At the small scale, there is a general scaling behavior indicating the random process and independence of the different sensitive testing conditions. In the large scaling range, the long-range correlation appears. Furthermore, the behavior on the dependence of different conditions is uncovered.;This thesis, therefore, gives a rigorous and systematic study of geographical phenomena in multiple temporal and spatial scales.