Multifractal analysis of the spatial distribution of earthquakes in southern Italy

We evaluate the complete spectrum of the generalized fractal dimension of the spatial pattern of microearthquakes in Southern Italy, revealing a multifractal distribution structure. Our analysis is focused on the dependence of the multifractal distribution on the size of the selected area and the kind of seismicity in the area. As the size of the window varies, we observe that the capacity, information and correlation dimensions vary significantly, while both d _∞ and d _−infin; remain unchanged within their errors limits. We interpret this result in terms of the observation that our data are mainly clustered around a linear fault (the Sisifo fault). When we restrict the selected windows around the fault, clustering around a line (the fault) is highlighted. The capacity dimension changes from about 1.8 to about 1.4 and the correlation dimension decreases because we observe in detail the clustering of the seismicity along the fault, which approximates the maximum intense clustering of the whole data set. Although our results are strongly influenced by the fact that the data are dominated by the epicentres located on the fault, we can conclude that multifractal analysis can be a very useful tool to discriminate the seismicity linked to a particular fault in a given area.     

Stochastic behaviour and scaling laws in geoelectrical signals measured in a seismic area of southern Italy

We explore the inner dynamics of daily geoelectrical time series measured in a seismic area of the southern Apennine chain (southern Italy). Autoregressive models and the Higuchi fractal method are applied to extract maximum quantitative information about the time dynamics from these geoelectrical signals. First, the predictability of the geoelectrical measurements is investigated using autoregressive models. The procedure is based on two forecasting approaches: the global and the local autoregressive approximations. The first views the data as a realization of a linear stochastic process, whereas the second considers the data points as a realization of a deterministic process, which may be non-linear. Comparison of the predictive skills of the two techniques allows discrimination between low-dimensional chaos and stochastic dynamics. Our findings suggest that the physical systems governing electrical phenomena are characterized by a very large number of degrees of freedom and can be described only with statistical laws. Second, we investigate the stochastic properties of the same geoelectrical signals, searching for scaling laws in the power spectrum. The spectrum fits a power law P (  f )∝  f  ^−α , with the scaling exponent α a typical fingerprint of fractional Brownian processes. In this analysis we apply the Higuchi method, which gives a linear relationship between the fractal dimension D _Σ and the spectral power law scaling index α : D _Σ=(3− α )/2. This analysis highlights the stochastic nature of geoelectrical signals recorded in this seismic area of southern Italy.     

城市人口分布空间自相关的功率谱分析

从理论上可以证明标准的城市人口密度负指数距离衰减模型本质上是一种空间相关函数,基于这种思想对Clark模型进行Fourier变换,可以导出城市人口密度的幂次频谱分布,且功率谱指数理应为β=2±。负指数与幂指数的这种变换关系暗示了城市地理系统简单与复杂的辩证关系。借助中国杭州市4年的人口普查资料转换的平均人口密度分布数据对上述推论进行检验,发现β渐进式趋近于2但并不约等于2。将β值进一步换算为人口过程的分维D和Hurst指数H,结果表明：城市人口具有长程负相关作用,但这种空间作用显示明确的局域化倾向。目前的城市形态演化模拟几乎无一例外地引入了长程作用,根据杭州人口分布的局域化特征,有关地理长程作用的假设和应用有必要重新探讨。     

On the practice of estimating fractal dimension

Coastlines epitomize deterministic fractals and fractal (Hausdorff-Besicovitch) dimensions; a divider [compass] method can be used to calculate fractal dimensions for these features. Noise models are used to develop another notion of fractals, a stochastic one. Spectral and variogram methods are used to estimate fractal dimensions for stochastic fractals. When estimating fractal dimension, the objective of the analysis must be consistent with the method chosen for fractal dimension calculation. Spectal and variogram methods yield fractal dimensions which indicate the similarity of the feature under study to noise (e.g., Brownian noise). A divider measurement method yields a fractal dimension which is a measure of complexity of shape.     

Crossover from Nonlinearity Controlled to Heterogeneity Controlled Mixing in Two-Phase Porous Media Flows

We use high resolution Monte Carlo simulations to study the dispersive mixing in two-phase, immiscible, porous media flow that results from the interaction of the nonlinearities in the flow equations with geologic heterogeneity. Our numerical experiments show that distinct dispersive regimes occur depending on the relative strength of nonlinearity and heterogeneity. In particular, for a given degree of multiscale heterogeneity, controlled by the Hurst exponent which characterizes the underlying stochastic model for the heterogeneity, linear and nonlinear flows are essentially identical in their degree of dispersion, if the heterogeneity is strong enough. As the heterogeneity weakens, the dispersion rates cross over from those of linear heterogeneous flows to those typical of nonlinear homogeneous flows.     

Numerical Method for Conditional Simulation of Levy Random Fields

Stochastic simulations of subsurface heterogeneity require accurate statistical models for spatial fluctuations. Incremental values in subsurface properties were shown previously to be approximated accurately by Levy distributions in the center and in the start of the tails of the distribution. New simulation methods utilizing these observations have been developed. Multivariate Levy distributions are used to model the multipoint joint probability density. Explicit bounds on the simulated variables prevent nonphysical extreme values and introduce a cutoff in the tails of the distribution of increments. Long-range spatial dependence is introduced through off-diagonal terms in the Levy association matrix, which is decomposed to yield a maximum likelihood type estimate at unobserved locations. This procedure reduces to a known interpolation formula developed for Gaussian fractal fields in the situation of two control points. The conditional density is not univariate Levy and is not available in closed form, but can be constructed numerically. Sequential simulation algorithms utilizing the numerically constructed conditional density successfully reproduce the desired statistical properties in simulations.