Narrow Results

The scaling ranges of time correlations in the cloud base height records of marine boundary layer stratocumulus are studied applying the Detrended Fluctuation Analysis statistical method. We have found that time dependent variations in the evolution of the α exponent reflect the diurnal dynamics of cloud base height fluctuations in the marine boundary layer. In general, a more stable structure of the boundary layer corresponds to a lower value of the α-indicator, i.e., larger anti-persistence, thus a set of fluctuations tending to induce a greater stability of the stratocumulus. In contrast, during the periods of higher instability in the marine boundary, less anti-persistent (more persistent like) behavior of the system drags it out of equilibrium, corresponding to larger α values. From an analysis of the frequency spectrum, the stratocumulus base height evolution is found to be a nonstationary process with stationary increments. The occurrence of these statistics in cloud base height fluctuations suggests the usefulness of similar studies for the radiation transfer dynamics modeling.

It is the purpose of the present article to show that so-called network models, originally designed to describe static properties of disordered electronic systems, can be easily generalized to quantum-dynamical models, which then allow for an investigation of dynamical and spectral aspects. This concept is exemplified by the Chalker–Coddington model for the quantum Hall effect and a three-dimensional generalization of it. We simulate phase coherent diffusion of wave packets and consider spatial and spectral correlations of network eigenstates as well as the distribution of (quasi-)energy levels. Apart from that, it is demonstrated how network models can be used to determine two-point conductances. Our numerical calculations for the three-dimensional model at the Metal-Insulator transition point delivers, among others, an anomalous diffusion exponent of η=3-D₂=1.7±0.1. The methods presented here in detail have been used partially in earlier work.

We investigate the multifractals of the first passage time on a one-dimensional small-world network with reflecting and absorbing barriers. We analyze numerically the distribution of the first passage time at which the random walker arrives for the first time at an absorbing barrier after starting from an arbitrary initial site. Our simulation is found to estimate the fractal dimension D₀ = 0.920 ∼ 0.930 for the different network sizes and random rewiring fractions. In particular, our simulation results are compared with the numerical computations for regular networks.

Using the S₃-symmetry of Sierpinski fractal resistor networks we determine the current distribution as well as the multifractals spectrum of moments of current distribution by using the real space renormalization group technique based on ([q/4]+1) independent Schure's invariant polynomials of inwards flowing currents.

The complexity of geophysics has been extremely stimulating for developing concepts and techniques to analyze, understand and simulate it. This is particularly true for multifractals and Generalized Scale Invariance. We review the fundamentals, introduced with the help of pedagogical examples, then their abstract generalization is considered. This includes the characterization of multifractals, cascade models, their universality classes, extremes, as well as the necessity to broadly generalize the notion of scale to deal with anisotropy, which is rather ubiquitous in geophysics.

We introduce a new fractional iterative map with two control parameters of β and γ in the range of the fractional value between one and two. Particularly, we present basically chaotic properties and multifractals of this map. The fractal dimension obtained from multifractal measures is taken to be near one as γ→4 and η→2 for fixed values of β, while it is expected to take the minimum value of the fractal dimension 0.538 as γ→γ_∞ for β=1 and η=1. Our result obtained from multifractal measures are also compared with other well-known results.

We used a multifractal approach to characterize scale by scale, the remotely sensed visible and thermal-infrared volcanic field, at Kilauea Volcano, Hawaii, USA. Our results show that (1) the observed fields exhibit a scaling behavior over a resolution range of ~ 2.5 m to 6 km, (2) they show a strong multifractality, (3) the multifractal parameters α, C₁ and H are sensitive to volcanic structural classes such as vent cones, lava ponds and active to inactive lava flows, (4) vegetation area and volcanic gas plumes have a strong effect on the multifractal estimates, and (5) vegetation and cloud-free images show statistical characteristics due to topography related albedo in the visible and predominantly solar heating in the thermal infrared wavelengths.

Usage of a deterministic fractal-multifractal (FM) procedure to model high-resolution rainfall time series, as derived distributions of multifractal measures via fractal interpolating functions, is reported. Four rainfall storm events having distinct geometries, one gathered in Boston and three others observed in Iowa City, are analyzed. Results show that the FM approach captures the main characteristics of these events, as the fitted storms preserve the records' general trends, their autocorrelations and spectra, and their multifractal character.

A fairly large body of observational evidence shows that the hydraulic conductivity is an extremely heterogeneous physical phenomenon that exhibits wide variability over a broad range of horizontal and vertical scales. Stochastic multifractals that result from continuous dynamic cascades are suggested as an appropriate model to capture the scale-invariance and stratification of the Columbus site hydraulic conductivity. Then, observed interrelations between estimates of multifractal parameters, those characterize vertical and horizontal scaling regimes, are interpreted by the unified multifractal model.

The cloud radiances and atmospheric dynamics are strongly nonlinearly coupled, the observed scaling of the former from 1 km to planetary scales is prima facae evidence for scale invariant dynamics. In contrast, the scaling properties of radiances at scales <1 km have not been well studied (contradictory claims have been made) and if a characteristic vertical cloud thickness existed, it could break the scaling of the horizontal radiances. In order to settle this issue, we use ground-based photography to study the cloud radiance field through the range scales where breaks in scaling have been reported (30 m to 500 m). Over the entire range 1 m to 1 km the two-dimensional (2D) energy spectrum (E(k)) of 38 clouds was found to accurately follow the scaling form E(k)≈ k^-β where k is a wave number and β is the spectral exponent. This indirectly shows that there is no characteristic vertical cloud thickness, and that "radiative smoothing" of cloud structures occurs at all scales. We also quantitatively characterize the type of (multifractal) scaling showing that the main difference between transmitted and reflected radiance fields is the (scale-by-scale) non-conservation parameter H. These findings lend support to the unified scaling model of the atmosphere which postulates a single anisotropic scaling regime from planetary down to dissipation scales.

The complexity of the cardiac rhythm is demonstrated to exhibit self-affine multifractal variability. The dynamics of heartbeat interval time series was analyzed by application of the multifractal formalism based on the Cramer theory of large deviations. The continuous multifractal large deviation spectrum uncovers the nonlinear fractal properties in the dynamics of heart rate and presents a useful diagnostic framework for discrimination and classification of patients with cardiac disease, e.g. congestive heart failure. The characteristic multifractal spectral pattern in heart transplant recipients or chronic heart disease highlights the importance of neuroautonomic control mechanisms regulating the fractal dynamics of the cardiac rhythm.

Fractal time series with scaling properties expressed through power laws appear in many contexts. These properties are very important from several viewpoints. For instance, they reveal the nature of the correlations present in the fractal signals. It is common that the scaling properties characterized by means of invariant quantities suffer changes along with the dynamical evolution of the studied systems. One of these changes is a crossover in the scaling properties reflecting an important change in the system dynamical behavior. In this article, we present two cases of crossover behavior corresponding to interbeat and electroseismic time series, we observe the crossovers in time series of experimental data and their corresponding simulation with simple models. We suggest a possible explanation of the observed crossovers in terms of the models considered.

We investigate the multifractals of the first passage time on a one-dimensional small-world network with reflecting and absorbing barriers. The multifractals can be obtained from the distribution of the first passage time at which the random walker arrives for the first time at an absorbing barrier after starting from an arbitrary initial site. Our simulation is found to estimate the fractal dimension D₀ = 0.920 ~ 0.930 for the different network sizes and random rewiring fractions. In particular, the multifractal structure breaks down into a small-world network, when the rewiring fraction p is larger than the critical value p_c = 0.3. Our simulation results are compared with the numerical computations for regular networks.

We consider calculation of the dimensions of self-affine fractals and multifractals that are the attractors of iterated function systems specified in terms of upper-triangular matrices. Using methods from linear algebra, we obtain explicit formulae for the dimensions that are valid in many cases.

A two-phase phenomenon in three financial exchange prices is studied. To understand the underlying mechanism for the formation of market prices, we perform the multifractal analysis and the detrended fluctuation analysis in terms of time series of market prices. We also examine higher order temporal correlations for the market price. Although the multifractal properties of market prices are obtained, it cannot be reproduced the binomial multiplicative process through that was used to understand fully developed turbulence.

Let X = {X(t, x), t ∈ ℝ, x ∈ ℝ^d} be a Lévy-based spatial-temporal random field proposed by Barndorff–Nielsen and Schmiegel¹ for dynamic modeling of turbulence. We describe some fractal geometry for this field, with a view toward a proper non-Gaussian aspect of Mandelbrot's paper.² Recent progress on multifractal scalings of the stationary exponential processes is also reported, and is toward the intermittency fields proposed in Barndorff-Nielsen and Schmiegel.¹

The multifractal spectrum calculated with wavelet transform modulus maxima (WTMM) provides information on the higher moments of market returns distribution and the multiplicative cascade of volatilities. This paper applies a wavelet based methodology for calculation of the multifractal spectrum of financial time series. WTMM methodology provides a better measure of risk changes compared to the structure function approach. It is well founded in applied mathematics and physics with little popularity among finance researchers.

Natural data sets, such as precipitation records, often contain geometries that are too complex to model in their totality with classical stochastic methods. In the past years, we have developed a promising deterministic geometric procedure, the fractal-multifractal (FM) method, capable of generating patterns as projections that share textures and other fine details of individual data sets, in addition to the usual statistics of interest. In this paper, we formulate an extension of the FM method around the concept of "closing the loop" by linking ends of two fractal interpolating functions and then test it on four geometrically distinct rainfall data sets to show that this generalization can provide excellent results.

Lacunarity (L) is a scale (r)-dependent parameter that was developed for quantifying clustering in fractals and has subsequently been employed to characterize various natural patterns. For multifractals it can be shown analytically that L is related to the correlation dimension, D₂, by: dlog(L)/dlog(r) = D₂ - 2. We empirically tested this equation using two-dimensional multifractal grayscale patterns with known correlation dimensions. These patterns were analyzed for their lacunarity using the gliding-box algorithm. D₂ values computed from the dlog(L)/dlog(r) analysis gave a ~1:1 relationship with the known D₂ values. Lacunarity analysis was also employed in discriminating between multifractal grayscale patterns with the same D₂ values, but different degrees of scale-dependent clustering. For this purpose, a new lacunarity parameter, 〈L〉, was formulated based on the weighted mean of the log-transformed lacunarity values at different scales. This approach was further used to evaluate scale-dependent clustering in soil thin section grayscale images that had previously been classified as multifractals based on standard method of moments box-counting. Our results indicate that lacunarity analysis may be a more sensitive indicator of multifractal behavior in natural grayscale patterns than the standard approach. Thus, multifractal behavior can be checked without having to compute the whole spectrum of non-integer dimensions, D_q(-∞ < q < +∞) that typically characterize a multifractal. The new 〈L〉 parameter should be useful to researchers who want to explore the correlative influence of clustering on flow and transport in grayscale representations of soil aggregates and heterogeneous aquifers.

The spatial pattern of the urban–rural regional system is associated with the dynamic process of urbanization. How to characterize the urban–rural terrain using quantitative measurement is a difficult problem remaining to be solved. This paper is devoted to defining urban and rural regions using ideas from fractals. A basic postulate is that human geographical systems are of self-similar patterns correlated with recursive processes. Then multifractal geometry can be employed to describe or define the urban and rural terrain with the level of urbanization. A space-filling index of urban–rural region based on a generalized correlation dimension is presented to reflect the degree of geo-spatial utilization in terms of urbanism. The census data of America and China are used to show how to make empirical analyses of urban–rural multifractals. This work is a normative study rather than a positive study, and it proposes a new way of investigating urban and rural regional systems using fractal theory.