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.     

A six-parameter statistical model of the Earth's magnetic field

A six-parameter statistical model of the non-dipole geomagnetic field is fitted to 2597 harmonic coefficients determined by Cain, Holter & Sandee (1990) from MAGSAT data. The model includes sources in the core, sources in the crust, and instrument errors. External fields are included with instrument errors. The core and instrument statistics are invariant under rotation about the centre of the Earth, and one of the six parameters describes the deviation of the crustal statistics from rotational invariance. The model treats the harmonic coefficients as independent random samples drawn from a Gaussian distribution. The statistical model of the core field has a correlation length of about 500 km at the core-mantle boundary, too long to be attributed to a white noise source just below the boundary layers at the top of the core. The estimate of instrument errors obtained from the statistical model is in good agreement with an independent estimate based on tests of the instruments (Langel, Ousley & Berbert 1982).     

城市地理研究中的单分形、多分形和自仿射分形

分形几何学在城市地理研究中具有广泛的应用,然而很多基本概念却让初学者感到迷惑。如何区分单分形、自仿射分形与多分形,是一个基本而重要的问题。简单分形容易理解,而真实的地理现象很少是单分形的。城市生长过程具有自仿射特征,而城市空间格局却具有多分形性质。作者发现,各种分形的共性在于三个方面：标度律、分数维和熵守恒。论文基于标度、分维和熵守恒公式,借助隐喻城市生长的规则分形来区分单分形、多分形和自仿射分形,讨论分形系统演化的机理、分形与空间自相关和空间异质性的联系,同时澄清一些在地理分形研究中的常见错误概念。最后以城市位序-规模分布为例,说明并对比单分形和多分形在城市地理研究中的建模与应用思路。     

On the distribution of anomalous mass within the Earth: forward models of the gravitational potential spectrum using ensembles of discrete mass elements

b
We investigate forward models of the gravitational potential spectrum generated by ensembles of discrete sources of anomalous mass, having radial distributions with different statistical properties. Models with a random distribution of point source locations throughout the volume of the mantle produce spectra similar to that of the Earth only when the (absolute) source magnitudes increase strongly with depth, at least as d ^1.5. The effects of the geographic (latitude-longitude) distribution of source locations are generally unimportant in determining the general degree dependence of the potential spectrum. The dimensions of the sources are also of secondary importance, at least up to an angular diameter of about 40, i.e., continent-sized. Sources of this size confined to the upper mantle do not appear capable of producing the degree dependence of the observed geopotential spectrum; the low harmonics (2-4) in particular appear to require lower mantle sources of considerable strength. Further, at least some of these deep sources must be largely monopolar in nature, (i.e., uncompensated) due to the stronger depth attenuation of dipole (compensated) sources. Because topography on the core-mantle boundary must be either isostatically or dynamically compensated, it may contribute little strength to the observed potential spectrum.     

城市化:相变与自组织临界性

城市化的本质是一种自组织临界相变过程 ,在时间和空间上分别联系着自组织临界性(SOC)和空间复杂性。城市化过程作为SOC有 3个方面的标志 :时间方向的 1/ f涨落、空间方面的分形结构和等级方式的Zipf定律 ,其共性是无标度性。分形中心地景观和位序 -规模分布分别是城市化自组织临界过程在空间和等级两个领域的实证判据。在城市化的临界相变过程中 ,城市体系通常演化为无标度网络 ,而位序 -规模法则乃是空间领域的无标度性在等级领域的结构性“投影”。基于SOC思想研究城市化过程有助于理解自组织城市网络的复杂空间动力学 ,从而为发展城市演化的模拟实验方法提供理论思路     

P- and S-velocity images of the lithosphere–asthenosphere system in the Central Andes from local-source tomographic inversion

About 50 000 P and S arrival times and 25 000 values of t * recorded at seismic arrays operated in the Central Andes between 20°S and 25°S in the time period from 1994 to 1997 have been used for locating more than 1500 deep and crustal earthquakes and creating 3-D P , S velocity and Qp models. The study volume in the reference model is subdivided into three domains: slab, continental crust and mantle wedge. A starting velocity distribution in each domain is set from a priori information: in the crust it is based on the controlled sources seismic studies; in slab and mantle wedge it is defined using relations between P and S velocities, temperature and composition given by mineral physics. Each iteration of tomographic inversion consists of the following steps: (1) absolute location of sources in 3-D velocity model using P and S arrival times; (2) double-difference relocation of the sources and (3) simultaneous determination of P and S velocity anomalies, P and S station corrections and source parameters by inverting one matrix. Velocity parameters are computed in a mesh with the density of nodes proportional to the ray density with double-sided nodes at the domain boundaries. The next iteration is repeated with the updated velocity model and source parameters obtained at the previous step. Different tests aimed at checking the reliability of the obtained velocity models are presented. In addition, we present the results of inversion for Vp and Vp/Vs parameters, which appear to be practically equivalent to Vp and Vs inversion. A separate inversion for Qp has been performed using the ray paths and source locations in the final velocity model. The resulting Vp , Vs and Qp distributions show complicated, essentially 3-D structure in the lithosphere and asthenosphere. P and S velocities appear to be well correlated, suggesting the important role of variations of composition, temperature, water content and degree of partial melting.     

Self-organized complexity in geomorphology: Observations and models

The focus of this paper is to relate fundamental statistical properties of landforms and drainage networks to models that have been developed in statistical physics. Relevant properties and models are reviewed and a general overview is presented. Landforms and drainage networks are clearly complex, but well-defined scaling laws are found. Coastlines, topography contours, and lakes are classic self-similar fractals. The height of topography along a linear track is well approximated as a Brownian walk, a self-affine fractal. This type of behavior has also been found in surface physics, for example the surface roughness of a fracture. An applicable model is the Langevin equation, the heat equation with a stochastic white-noise driver. This model also reproduces the statistics of sediment deposition. Drainage networks were one of the original examples of self-similar fractal trees. An important advance in quantifying the structure of drainage networks is the application of the Tokunaga fractal side-branching statistics. A classic problem in statistical physics is the diffusion-limited aggregation. The resulting tree like structures have been shown to also satisfy the Tokunaga statistics. A modified version of the diffusion-limited aggregation model reproduces the statistics of drainage networks. It is concluded that the models developed in statistical physics have direct applicability to the fundamental problems in geomorphology.     

沙粒级配和沙丘分布的分形分析

研究了沙粒级配和沙丘分布的分形特征,并对其分形维数作了定量计算。结果表明:沙粒级配的粒径分布的分形维数在2.43~2.69之间,沙丘分布的盒分形维数为1.28,这对用数学模型进行风沙动力学的定量研究有重要的参考价值。     

城市范围界定与标度律

标度律作为城市发展的重要规律之一,反映了城市经济活动产值、基础设施数量等要素随城市人口规模的变动情况,在城市研究领域引起了广泛讨论。但由于不同国家的城市统计数据对应的空间范围各不相同,导致标度律系数受城市边界范围选取影响很大。本文通过比较中、美两国统计数据对应的空间范围,并结合普查、城市统计年鉴和遥感数据,计算了不同空间范围对应的标度律系数。结果表明：① 不同空间尺度和数据源得到的标度律系数有较大差异。就空间尺度而言,市辖区比市域范围的数据更符合标度律模型,因为中国城市市域范围内还存在大量的非城市化地区,并不符合标度律模型的适用条件;就数据源而言,遥感数据比城市统计年鉴数据有更好的拟合优度;② 与美国城市相比,中国城市人口集聚带来的经济增长率更高（标度律系数更高）,市辖区人口每增加一倍,经济规模可增加122%,这一数字在美国是111%;而在家庭能源消耗（用水、用电）和土地利用方面,中国城市的效率更低;③ 从中国城市内部对比来看,大城市与中小城市在经济规模、土地利用方面的标度律（集聚效率）明显不同,人口集聚效应带来的大城市经济增长率、工资收入要远高于中小城市;能源消耗方面,中小城市比大城市更有效率。最后,本文还从建立更加有效的统计单元、传统统计数据与大数据结合、模型机制探索3个方面阐述了城市标度律未来可能的研究方向。     

城市分形中心与占优标度区选择

用回转半径法研究城市分形特征时,中心点位置的选择至关重要。以反映城市基础设施情况的兴趣点(point of interest,POI)作为基础数据,分析中心点位置对标度区识别的影响。将POI几何中心、重心、密度中心、中心城区几何中心和重心5种统计中心作为待选中心,给出辐射范围、核心密度与分形维数、中心漂移度等标度区相关指标,提出判断占优标度区的POI数量最大化原则。通过统计中心与地标中心标度区对比,确定城市分形中心和占优标度区。省会城市及直辖市标度区分析结果显示:统计中心各有优劣,占优中心总体上更倾向于POI几何中心和密度中心,地标中心有时优于统计中心;密度中心对应的占优标度区半径维数较低,若半径维数太低则容易陷入局部密度中心;中心漂移度指标表明可能存在更好的分形中心选择方案。     

Scale-dependent analysis of satellite imagery for characterization of glacier surfaces in the Karakoram Himalaya

Information regarding process-structure relationships and change in the Karakoram Himalaya is of great importance in studying glacier hydrollogy, mass balance, and dynamic environmental change. Such information is not readily available. Detailed spatio-temporal assessment requires field investigation coupled with quantitative remote sensing studies. We conducted an investigation of the large Batura Glacier in Pakistan to determine if spectral variability can be quantified and used to characterize glacier surfaces. Specifically, SPOT Panchromatic satellite data were evaluated for differentiating features of glacier structure resulting from ice movement, ablation, and supraglacial fluvial action. Image semivariogram analysis was conducted. for assessing spectral variability patterns and fractal analysis was used to examine scale-dependent variation in the data. Results indicate that spectral variability from fields of ice seracs can exhibit fractal characteristics, although most surface features on the glacier exhibit a change in the fractal dimension over different ranges in scale. The fractal dimension was found to be useful for differentiating between glacier surfaces such as white ice and debris-covered ice. Characteristics of the debris-load and the scale-dependent nature of calculating the fractal dimension ultimately determined the potential of class separability.     

Broad-band power-law spectra of well-log data in Japan

For the purpose of revealing the statistical characteristics of P -wave velocity, S -wave velocity and density in the uppermost part of the crust, we analysed well-log data obtained from five deep wells in different tectonic regions in Japan: three wells through the mainly sedimentary rocks in the Kanto plain and two wells in the Kuju volcano group in Kyushu Island. In the Kanto plain, the power spectral density of fractional fluctuation of P -wave velocity and that of density are proportional to a power of the spatial wavelength from a few metres to 100 m. where the power index (slope of the power spectral density at double logarithmic scale) is 1.1-1.3. At the Kuju volcano group, that of P - and S -wave velocity and density also obey a power law, with a power index of 1.3-1.6 for wavelengths from a few metres to few hundred metres. Correcting the effect of the moving box-car observation window which corresponds to the separation of two receivers of the logging tool, we find that the power-law characteristics hold for wavelengths down to a few tens of centimetres. The 1-D sections of the elastic inhomogeneities follow a kind of band-limited self-affine random process. Comparing the power spectral densities, we find smaller values of the power index in stable areas and larger values in tectonically active areas. The difference in the power index arises from long-wavelength components.     

A model for temperature variations in sedimentary basins due to random radiogenic heat sources

The thermal structure of a sedimentary basin is controlled by its thermal conductivity, its boundary conditions, water flow, rate of sedimentation and erosion and radiogenic heat sources. The radiogenic heat production in the sediments is known to vary over several orders of magnitude, with the lowest values in evaporites and carbonates and the highest values in black shales. Due to a paucity of information available on the existing heat sources, this parameter can be represented with a known mean value and a Gaussian correlation structure rather than a deterministic function. In this paper, the 1-D steady-state thermal structure in a sedimentary basin has been modelled in a stochastic framework with a random radiogenic heat source, and analytical expressions for the first two moments of the temperature field have been obtained. A synthetic example has been examined to quantify the error bounds on the temperature field due to uncertainties in the radiogenic heat sources.     

On the retrieval of pure path velocities from great circle data

Summary. Synthetic great circle Rayleigh wave phase velocities are generated from a 3-D upper mantle model for two great circle distributions: a uniform random distribution and an experimental distribution arising from a real set of sources and receivers. Given a tectonic regionalization, the retrieval of the pure path velocities from these great circle data sets leads to a strongly biased solution when the uniform distribution is used, whereas the results from the experimental distribution are closer to the true values. An analysis of the data importances and the information distribution in the data indicates that the great circles introduce coupling between tectonic regions; thus the degeneracy of the solution inherent in great circle data cannot be completely overcome by introducing a tectonic regionalization. It is shown that a satisfactory solution is obtained by selecting data as pure as possible, i.e. data that over-sample one tectonic region without oversampling the others. This condition is better fulfilled by the experimental distribution than by the uniform distribution.     

Stress resistance strategy in an arid land shrub: interactions between developmental instability and fractal dimension

This paper investigates allocation of energy to mechanisms that generate and preserve architectural forms (i.e. developmental stability, complexity of branching patterns) and productivity (growth and reproduction) in response to environmental disturbances (i.e. grazing and resource availability). The statistical error in translational symmetry was used to detect random intra-individual variability during development. This can be thought of as a measure of developmental instability caused by stress. Additionally, we use changes in fractal complexity and shoot distribution of branch structures as an alternate indicator of stress. These methods were applied to Anthyllis cytisoides L., a semi-arid environment shrub, to ascertain the effect of grazing and slope exposure on developmental traits in a 2×2 factorial design. The results show that A. cytisoides maintains developmental stability at the expense of productivity. Anthyllis cytisoides was developmentally more stable when grazed and when on south-facing, as opposed to north-facing slopes. On the contrary, shoot length, leaf area, fractal dimension and reproductive-to-vegetative allocation ratio were larger in north- than in south-facing slopes. As a consequence, under extreme xeric conditions, shrub mortality increased in north-facing slopes, especially when not grazed. The removal of transpiring area and the reduction of plant competition favoured developmental stability and survival in grazed plants. Differences between grazed and ungrazed plants were most evident in more mesic (north-facing) areas.     

中国国土面积究竟有多大?——标度对称与中国陆地面积的分形分析

海岸线的长度依赖于测量尺度,因此没有确定的测量结果:尺度越小,测量的长度值越大。国境线的测量也具有类似的尺度依赖性。这种性质与分形性质有关。分形的本质是标度对称性,而对称性意味着某种不可观测量。由于标度对称性的存在,一些地理现象的基本测度(如长度、面积)无法准确计算。中国、美国等国家的国土面积大小都没有确切的数据:在版图没有任何变动的情况下,不同时期、不同机构的测量结果不一致,有时甚至大相径庭。本文借助分形模型和标度对称思想探讨这一问题。假定一个国家包括漫长的海岸线,则其国土可以分解两部分:主体部分可以用Koch岛模型刻画,岛屿部分则可以用Pareto分布描述。这两部分叠加的结果是,国土总面积随着测量尺度的减少而逐步增加。因此,中国政府可以正式公布国土面积的最新测量数据,而不必继续担心由此引发的国际政治争端。     

Frequency response modelling of seismic waves using finite difference time domain with phase sensitive detection (TD–PSD)

This paper describes an efficient approach for computing the frequency response of seismic waves propagating in 2- and 3-D earth models within which the magnitude and phase are required at many locations. The approach consists of running an explicit finite difference time domain (TD) code with a time harmonic source out to steady-state. The magnitudes and phases at locations in the model are computed using phase sensitive detection (PSD). PSD does not require storage of time-series (unlike a fast Fourier transform), reducing its memory requirements. Additionally, the response from multiple sources can be obtained from a single finite difference run by encoding each source with a different frequency. For 2-D models with many sources, this time domain phase sensitive detection (TD–PSD) approach has a higher arithmetic complexity than direct solution of the finite difference frequency domain (FD) equations using nested dissection re-ordering (FD–ND). The storage requirements for 2-D finite difference TD–PSD are lower than FD–ND. For 3-D finite difference models, TD–PSD has significantly lower arithmetic complexity and storage requirements than FD–ND, and therefore, may prove useful for computing the frequency response of large 3-D earth models.     

坡面非均匀沙输移机理实验方法

在分析坡面泥沙输移现象的基础上,以室内沙堆模型实验量化其输沙特征,得出其在空间上具有自相似分形的动力学过程,从理论上应用水动力学弥散特性进一步证实,结果符合较好。     

Fractal behavior in space and time in a simplified model of fluvial landform evolution

Two general approaches have been applied to understanding the fractal structure of fluvial topography: (1) deterministic, process-based models, and (2) stochastic partial differential equations (PDE). Deterministic models reproduce the fractal behavior of fluvial topography but have two limitations: they often underestimate the amount of lateral valley and ridge migration that occurs in nature, and the complexity has made it difficult to identify the precise origin of fractal behavior in fluvial landscapes. The simplicity of stochastic PDE models has made them useful for investigating fractal behavior, but they incorrectly suggest that fractal behavior is only possible with stochastic forcing. In this paper I investigate whether simplified, deterministic PDE models of landform evolution also exhibit fractal behavior and other features of complexity (i.e. deterministic chaos). These models are based on the KPZ equation, well known in the physics literature. This equation combines diffusion (i.e. hillslope processes) and nonlinear advection (i.e. bedrock or alluvial channel incision). Two models are considered: (1) a deterministic model with uniform erodibility and random initial topography, and (2) a deterministic model with random erodibility and uniform initial topography. Results illustrate that both of these deterministic models exhibit fractal behavior and deterministic chaos. In this context, chaotic behavior means that valley and ridge migration and nonlinear amplification of small perturbations in these models prevent an ideal steady state landscape from ever developing in the large-system limit. These results suggest that fractal structure and deterministic chaos are intrinsic features of the evolution of fluvial landforms, and that these features result from an inverse cascade of energy from small to large wavelengths in drainage basins. This inverse cascade differs from the direct cascade of three-dimensional turbulence in which energy flows from large to small wavelengths.