Head/Tail Breaks: A New Classification Scheme for Data with a Heavy-Tailed Distribution

This article introduces a new classification scheme—head/tail breaks—to find groupings or hierarchy for data with a heavy-tailed distribution. The heavy-tailed distributions are heavily right skewed, with a minority of large values in the head and a majority of small values in the tail, commonly characterized by a power law, a lognormal, or an exponential function. For example, a country's population is often distributed in such a heavy-tailed manner, with a minority of people (e.g., 20 percent) in the countryside and the vast majority (e.g., 80 percent) in urban areas. This new classification scheme partitions all of the data values around the mean into two parts and continues the process iteratively for the values (above the mean) in the head until the head part values are no longer heavy-tailed distributed. Thus, the number of classes and the class intervals are both naturally determined. I therefore claim that the new classification scheme is more natural than the natural breaks in finding the groupings or hierarchy for data with a heavy-tailed distribution. I demonstrate the advantages of the head/tail breaks method over Jenks's natural breaks in capturing the underlying hierarchy of the data.     

城市紧凑性测度指标研究及典型城市分析——以南京、苏州建设用地紧凑度为例

量化表征城市紧凑性是当前地理学界和规划领域共同关注的热点问题,明确紧凑性指标反映的实际意义与适用范围,成为构建测度体系的关键.当前已有众多研究提出数量相当可观的量化指标,本文结合国内外城市紧凑性定量研究进展,根据表征紧凑内涵的不同将其归纳为形状、规模、密度、结构、功能和过程6 类,分析了不同类型指标的适用范围,并以结构类指标为例,采用南京和苏州的实际建设用地数据,根据指标自身稳定特性识别的实用性和局限性,达到评判和筛选指标的目的.结果显示：指标能有效指示建设用地的实际空间结构特征;4 个指标受尺度变化的影响不强烈,具有良好的应用性能,其中Gini 系数和Moran’s I 在应用中,前提条件局限小,适用广泛;而连续度和向心度较直观,但在进行城市比较研究时限制条件较多.该研究结果能够为城市紧凑性量化研究提供一定的借鉴,并指导城市规划与管理政策制定.     

Scale Invariances in the Morphology and Evolution of Braided Rivers

This paper presents an overview and synthesis of an extensive research effort to characterize and quantify scale invariances in the morphology and evolution of braided rivers. Braided rivers were shown to exhibit anisotropic spatial scaling (self-affinity) in their morphology, implying a statistical scale invariance under appropriate rescaling of the axes along and perpendicular to the main direction of flow. The scaling exponents were found similar in rivers of diverse flow regimes, slopes, types of bed material and braid plain widths, indicating the presence of universal features in the underlying mechanisms responsible for the formation of their spatial structure. In regions where predominant geologic controls or predominant flow paths were present, no spatial scaling was found. Regarding their spatiotemporal evolution, braided rivers were found to exhibit dynamic scaling, implying that a smaller part of a braided river evolves identically to a larger one provided that space and time are appropriately normalized. Based on these findings, and some additional analysis of experimental rivers as they approach equilibrium, it was concluded that the mechanism bringing braided rivers to a state where they show spatial and temporal scaling is self-organized criticality and inferences about the physical mechanisms of self-organization were suggested.     

Power-laws and applications to earthquake populations

In this paper we study the rooted tree model applying the concepts of probability to obtain results of importance in understanding power-law distributions in pure populations and also in an ensemble of pure populations. The well-known Gutenberg-Richter relation, which is an empirical relation providing the number of earthquakes whose magnitude exceeds a given value, is shown to be an asymptotic form of survivor function of earthquake magnitudes. The implications of this model are briefly discussed in relation to other branches of sciences where power-law distributions are encountered.     

Fractures: Finite-size scaling and multifractals

The distributions of contact area and void space in single fractures in granite rock have been determined experimentally by making metal casts of the void spaces between the fracture surfaces under normal loads. The resulting metal casts on 52 cm diameter core samples show a complex geometry for the flow paths through the fracture. This geometry is analyzed using finite-size scaling. The spanning probabilities and percolation probabilities of the metal casts are calculted as functions of observation scale. Under the highest stresses of 33 MPa and 85 MPa there is a significant size-dependence of the geometric flow properties for observation scales smaller than 2 mm. Based on this data, the macroscopic percolation properties of the extended fracture can be well represented by relatively small core samples, even under normal stresses larger than 33 MPa. The metal casts also have rich multifractal structure that changes with changing stress.     

Research on scale effect of histogram

To describe the spatial relationship among the earth objects compactly, in this paper, we raised the concept of histo-variogram based on the analysis of the characteristics of other spatial analyzing methods such as variogram, information entropy. And we also raised a new spatial analysis method of histogram decomposition based on the definition of standing pixel and contour pixel. At the end of this paper, we demonstrated the characteristics of histo-variogram by two experiments, one for spatial analysis, the other for image fusion.     

Can Damage Mechanics Explain Temporal Scaling Laws in Brittle Fracture and Seismicity?

Time delays associated with processes leading to a failure or stress relaxation in materials and earthquakes are studied in terms of continuum damage mechanics. Damage mechanics is a quasi-empirical approach that describes inelastic irreversible phenomena in the deformation of solids. When a rock sample is loaded, there is generally a time delay before the rock fails. This period is characterized by the occurrence and coalescence of microcracks which radiate acoustic signals of broad amplitudes. These acoustic emission events have been shown to exhibit power-law scaling as they increase in intensity prior to a rupture. In case of seismogenic processes in the Earth's brittle crust, all earthquakes are followed by an aftershock sequence. A universal feature of aftershocks is that their rate decays in time according to the modified Omori's law, a power-law decay. In this paper a model of continuum damage mechanics in which damage (microcracking) starts to develop when the applied stress exceeds a prescribed yield stress (a material parameter) is introduced to explain both laboratory experiments and systematic temporal variations in seismicity.     

Spectral scaling in Mars topography: effect of craters

Extensive computer simulations aimed at testing a hypothesis that impact craters may explain the scaling behaviour of surface spectra are presented. The simulations show that indeed crater effects alone may explain the spectral scaling of Mars’ topography revealed from MOLA data. The range of the scaling exponents obtained for a wide range of simulation parameters does not exceed the limits observed for Mars. The simulations suggest that the shape of large craters is the key factor leading to the two scaling ranges in the surface spectra. Particular values of the scaling exponents may additionally depend on the depth-diameter relationship and the crater size distribution.     

环境污染及保护对策

环境保护中一个非常重要的工作,是广泛深入地开展环境地质工作。在赣南、江西乃至全国,环境地质工作的主要内容是防治水土流失,严禁废渣、废水随意排放,控制人口激增,保护有限耕地;为保护生态平衡,必须加强环境立法与执法,提高环境保护的重要性和必要性的认识,大规模切实地保护耕地,因地制宜植树造林。