Fractal geometry provides a powerful tool for scale-free spatial modeling and analyses in geography. However, a number of basic concepts are puzzling. The three common fractals, that is, monofractal (unifractal), multifractals, and self-affine fractal, are often misunderstood by students of geography. This article clarifies some confusing fractal concepts for urban fractal modeling and fractal dimension analysis. Using simple mathematical models based on three growing fractals that bear an analogy to urban growth, we can distinguish the three types of common fractal structure. The similarities and differences between monofractal, multifractals, and self-affine fractal are as follows: 1) A monofractal is a simple self-similar fractal that bears only one scaling factor (scaling ratio), and a multifractal object is a complex fractal system that bears at least two scaling factors for different parts. Each scaling factor dominates all different scales and is independent of directions and levels. 2) A self-affine fractal bears different scaling factors in different directions of growth or at different levels of scales. The basic feature of self-affine growing fractal is anisotropy, which differs from the isotropic self-similar growing fractals. 3) Both self-affine fractal and multifractals may possess two scaling factors, but there are essential differences between self-affine fractals and multifractals. A self-affine fractal often takes on the form of bi-fractals, which can be reflected by two scaling ranges on a log-log plot. However, there is only one scaling range for a multifractal pattern. As an example, two-scaling fractal modeling is applied to the rank-size distributions of cities to illustrate the concept of urban multifractals. By comparison with these multifractal models, we can better understand monofractals and self-affine fractals in geographical research.

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。

京津冀地区人水关系矛盾突出,分形可以有效描述城镇体系和水系时空演化特征,从而揭示两者演化关系,为城市问题的解决提供一些理论和经验依据。论文采用分形理论中的网格维数和多分维谱,首先分别刻画了两者的时空演化特征,其次探讨了城镇体系和水系结构之间的时空关系,最后探究了水系结构退化的影响因素。主要结论有：① 1990—2010年,京津冀地区建设用地的网格维数升高、自相似性增强、从集聚向分散转变,意味着建设用地朝着空间填充程度增强、有序、分散的方向发展,而水系反之,证明两者具有不同的时空演化方向;② 21世纪10年代,京津冀的人水关系十分紧张,南水北调虽然缓和了京津冀用水问题,改善了大尺度上的水系结构,但在小尺度上改善有限;③ 越靠近城市中心,建设用地分形形态发育越成熟,结构越有序,越靠近外围越混乱无序;④ 京津冀地区水系退化,由自然和人为两方面因素造成,21世纪以后人为因素的影响较为显著。针对京津冀地区水系退化,提出如下政策建议：在城市建设过程中,一方面科学规划城市水系,重视低等级水系的保护;另一方面节约集约利用水资源,完善水资源管理机制。未来,需要进一步探索城市发展和水系的非线性关系,为城市可持续发展提供依据。

<p>Cities and urban systems both bear scale-free properties, from which no characteristic scales can be found for mathematical modeling and quantitative analysis. Therefore, fractal geometry is useful for making scaling analysis. Complex urban systems cannot be effectively described by simple fractal models, but can be characterized by multifractal theory. This paper is devoted to exploring the spatiotemporal features of urban change in the Beijing-Tianjin-Hebei region from 1995 to 2013. Using multifractal parameter spectrums, we can reveal the spatial dynamics of urbanization from the aspects of urban form and urban system. The main findings are as below: 1) The spectral curves of global fractal dimension are abnormal. If <i>q</i>>1 and <i>q</i>→∞, the spectral lines quickly converge to horizontal lines; when <i>q</i><0 and <i>q</i>→-∞, the spectral lines quickly surpass the upper limit of fractal dimension, 2, which represents Euclidean dimension of embedding space. 2) The spectral curves of local fractal dimension are also not entirely normal. The peaks of the <i>f</i>(<i>α</i>) curves incline to the left, and the left ends of the curves are higher than the right ends. The problem lies in that the curves go beyond the maximum value of 2. 3) Fractal dimension growth curves can be described by the quadratic logistic function. Capacity parameters and inflection points of different fractal dimension growth curves are different. The main conclusions are as follows: 1) The urban fringes are disorderly developed, while the central areas of the main cities in the Beijing-Tianjin-Hebei region is overfilled with construction land, leaving little buffer space. 2) The main mode of urban growth is to expand to the outside region, but there are signs of central agglomeration in the total construction land of Hebei Province. 3) Land use of the main cities is close to saturation, and the speed of land expansion has peaked for Beijing and Tianjin, but not in Hebei Province yet. In short, it is necessary to optimize the land use structure in the Beijing-Tianjin-Hebei region by city planning.</p>

城市形态和城镇体系都具有分形性质,但简单分形模型不能有效揭示城市系统的复杂结构特征及其背后的问题。多分形模型及分析方法是研究城市空间复杂性和描述城市异质性的有效手段。利用城镇建设用地和总建设用地的多分维谱分析,可以发现京津冀城镇体系及主要城市的空间演化问题。主要结果如下：①京津冀总建设用地的全局谱线不正常,代表中心区的谱线收敛过快,而代表边缘区和乡村地区的谱线收敛位置严重越界;②局部谱线单峰左偏,左(趋向中心区)高密、右(趋向边缘区)低疏,且右边数值越界;③多分维增长曲线服从二次logistic函数,但不同区域和城市的增长曲线的拐点位置不同。深入分析谱线特征及其异常根源,得出如下结论：①京津冀主要城市的中心区填充过密,没有太多缓冲空间,而边缘区无序扩展,需要通过规划进行优化;②京津冀城市生长以外延扩展模式为主,但河北省总建设用地有中心集聚迹象;③京津冀地区特别是主要城市用地接近饱和,土地扩展速度高峰已经过去,只有河北省部分区域例外。

Issue of urbanization is a comphrensive subject which is related to harmonious development of each department of national economy and building a harmonious society. New urbanization is the guarantee of healthy and stable development of urbanization. The research work of the thesis has higher academic meaning and practical value. Urbanization promoted social and economic development of China in the past decades. But some problems arised in the process of urbanization in some areas, such as the rapid pace and disordered state of development, the blind expansion on the edge of cities, degradation of land and water resources, destruction of the ecological environment, and many unsafe, uncomfortable problems in urban environment.From the geographical space and natural resource conservation point of view, this article focuses on three theoretical issues. 1) How to have a good understanding of the basic characteristics and realizing route of new urbanization; 2) How to build a innovation model of new urbanization; 3) In the process of implementing new urbanization, how to understand the development law of China's urbanization, and to take a new road of urbanization with Chinese characteristics.

城镇化问题是当代中国社会经济发展的综合性课题,是涉及到国民经济各部门如何协调发展,达到一个新的现代化和谐社会发展的根本问题;新型城镇化是中国城镇化健康稳定发展的基本保证,在当前的新形势下,探索中国新型城镇化理论与实践问题,具有重要的学术价值与实践意义。在过去一阶段,虽然城镇化推动了中国社会经济发展取得了巨大成就,并在城市现代化建设与城乡一体化方面也取得了惊人的发展,但在某个时期或一些地区,城镇化过速发展阶段,出现了无序的发展状态,大中城市边缘盲目扩展,水土资源日渐退化,生态环境遭受破坏,特别是有些政府决策人对城镇化的许多制约因素认识不足,甚至决策失误,导致了城市环境出现许多不安全、不舒适的问题。着重从地理空间与自然资源保护的角度,探索中国新型城镇化3个理论与实践问题：① 如何认知中国新型城镇化的基本特征与新的路径;② 在全球经济一体化形势下,如何构建新型城镇化的创新模式;③ 在新型城镇化实施过程中,如何认识中国城镇化本身的发展规律,走具有中国特色的新型城镇化道路。

The high-quality development of the new urbanization in China is focused on man-earth harmony, people-oriented principle, low-carbon, ecology, innovation, wisdom and safety. It is an organic unity of high quality citizenization, infrastructure, living environment, urban construction, public service and urban management. Promoting the new urbanization is an important way for China to build a moderately prosperous society in all respects and achieve modernization basically, and it is an important way to realize the strategy of rural revitalization. To promote the development of China's new urbanization, we need to follow the four stage rule and gradual development rule. To realize the strategy transformation of new urbanization, it is necessary to make more changes: from the quantitative to the quality-oriented, from the "one step" to "step by step" in realizing our goals, from radical to gradual, from inducing "negative effect" to releasing the "positive energy", from passive to active, from "land-oriented" to "people-oriented". The current situation and background conditions of China's urbanization are very different from one place to another. Thus, we should avoid "one-size-fits-all" in promoting the development of new urbanization. Adhering to the principle of adapting local conditions and appropriate conditions, the paper divides the new urbanization area into five major types, including urban agglomeration region (I), major grain-producing region (II), farming, forestry, and animal husbandry region (III), linked poverty alleviation region (IV), and ethnic autonomous region (V), which are further divided into 47 sub-regions. The urbanization development mode between different regions and different subregions cannot be copied from each other, and we need to implement some diversified and differentiated models. According to the main functions of different types of urbanization regions, the different development policies of urbanization should be formulated, and development goals and priorities should be put forward according to local conditions, and the classification guidance should be proposed according to the appropriate conditions. The key paths to promote the high-quality development of China's new urbanization include: enhance the overall synergy of high-quality development and improve the quality of urban agglomeration development; promote the integrated development of industrial and urban areas and the equalization of basic urban public services, and improve the quality of urban development; propel integrated development between urban and rural areas and accelerate the revitalization of rural areas; highlight the main functions of urbanization quality development in different types of areas in accordance with local conditions; innovate institutional mechanisms and ensure that high-quality development is carried out throughout the trial process of new urbanization; standardize small towns with distinctive features and lay a solid foundation for the high-quality development of new urbanization; strengthen the analysis and regulation of the capacity of resources and environment for the high-quality development of new urbanization.

中国新型城镇化高质量发展是一种人地和谐、高效低碳、生态环保、节约创新、智慧平安的质量提升型城镇化,是高质量的城市建设、高质量的基础设施、高质量的公共服务、高质量的人居环境、高质量的城市管理和高质量的市民化的有机统一。高质量推进新型城镇化发展需要遵循城镇化发展的四阶段性规律和渐进式规律,实现新型城镇化由数量型向质量型、由“一步到位”向“分步到位”、由激进式向渐进式、由诱发“负效应”向释放“正能量”、由被动性向主动型、由“地为本”向“人为本”的战略转型。考虑到中国新型城镇化发展的地域差异显著,新型城镇化高质量发展客观上要因地制宜、因类指导,可将全国新型城镇化高质量发展区域划分为城市群地区（Ⅰ）、粮食主产区（Ⅱ）、农林牧地区（Ⅲ）、连片扶贫区（Ⅳ）、民族自治区（Ⅴ）共5大高质量发展类型区和47个亚区。未来推进中国新型城镇化高质量发展的重点路径包括：增强新型城镇化高质量发展的整体协同性,提高城市群发展质量;推动产城深度融合发展,加快实现基本公共服务均等化,提升城市发展品质与质量;推动城乡深度融合发展,在新型城镇化高质量发展中实现乡村振兴;突出因地制宜,明确不同类型地区城镇化高质量发展的主体功能;创新体制机制,全过程推进城镇化高质量发展;量力而行,以特取胜,规范建设特色小镇,夯实新型城镇化高质量发展的基石;把新型城镇化高质量发展与区域资源环境承载力及高质量保护有机结合起来。

Urban agglomeration is an important space carrier to promote new urbanization, and the spatial structure is the spatial reflection of the degree, stage and process of urban agglomeration. From the 3 aspects of urban agglomeration benefit, distance benefit and spatial connection benefit, this article use ArcGIS and urban interaction intensity model to compare the spatial structure benefits of the Changsha-Zhuzhou-Xiangtan urban agglomeration and Dongting Lake urban agglomeration in the development stage. The results show that Dongting Lake urban agglomeration has a comparative advantage in external scale including the population scale, land area and the city quantity. However, the compactness of urban agglomerations is low, the advantages of urban scale structure are not obvious enough and the spatial connection between important cities is not tight enough. It shows a more obvious "big and scattered" characteristics, and the quality of development and the degree of development are not good. Changsha-Zhuzhou-Xiangtan urban agglomeration is relatively small in terms of number of cities, population size and land use, but its compactness and scale benefits are relatively high, and the scale structure has a relatively high degree of dominance. On this basis, aiming at maximizing the efficiency of urban agglomeration spatial structure, the spatial structure system of “one heart, two circles and three axes” and “1+5” urban agglomeration with Changsha as the center city is proposed, which to provide references for the optimization and development of the urban agglomeration spatial structure.

从城市群规模效益、距离效益和空间联系效益3个方面出发,运用ArcGIS和城市相互作用强度模型对长株潭城市群和环洞庭湖城市群空间结构效益进行评价比较分析。结果表明：环洞庭湖城市群在外在规模上具有相对优势,具体表现为人口规模、空间规模以及城市数量较大,但城市群的紧凑度不高,城市规模等级结构优势不够明显,空间可达性较差,重要城市之间的空间联系不够紧密,呈现“大而散”的特征;相对而言,长株潭城市群城市数量、人口规模及用地较小,但其紧凑度及规模效益较高,城市规模等级结构优势度相对较高,可达性较好,重要城市之间的空间联系较为紧密。在此基础上,以实现城市群空间结构效益最大化为目标,构建了以长沙为中心城市的“一心两圈三轴”结构体系。

随着夜间灯光遥感数据的应用日渐成熟和资源环境研究领域,对空间型社会经济数据的需求增加,利用相关分析和回归分析的方法,首次定量探讨夜间灯光数据与统计型的社会经济数据的空间关系。为提高模型精度,按照我国省级行政边界分区建模,分析全国县级的地区生产总值、第一产业、第二产业、第三产业分别与夜间灯光指数的空间相关关系,最终建立全国的1km GDP密度图。结果表明,全国范围的夜间灯光数据与第一产业的相关性不明显,相关系数0.554,模型拟合效果差,R²为0.306;夜间灯光数据与地区生产总值、第二产业、第三产业均有明显的对数线性关系,尤其是与第二产业和第三产业之和,相关系数为0.824,R²为0.679。利用分区模型估算,生成的GDP密度图能较完整地反映全国社会经济分布详况,以及宏观分布特征。

One of the indicators of regional integration of urban and rural areas is the integration of the structure of systems of cities and towns. This process can be described and evaluated using the concept of scaling. Fractal geometry is one of the powerful tools for scaling analysis. An important parameter of geographic spatial distribution characteristics is fractal dimension. Based on data derived from remote sensing images and census data, this study carried out fractal analysis, rank-size distribution analysis, and allometric scaling analysis of cities and urban system in the Beijing-Tianjin-Hebei region. The aim was to explain the process of urban growth in the region from 1995 to 2013. The results show three characteristics of urban form and growth: 1) Both the spatial structure and the rank-size distribution of the cities in the Beijing-Tianjin-Hebei region are of self-affine pattern, indicating bi-fractal property. 2) The relationship between urban population and urban area of the Beijing-Tianjin-Hebei region indicates a false linear correlation. 3) With the change of the urban system, the self-affine bi-fractal structure evolved gradually into a self-similar fractal structure. The main conclusions are as follows: 1) There is a structural incongruity in the system of cities and towns in the Beijing-Tianjin-Hebei region. The urban hierarchy takes on a dual structure, but the direction of urban change shows a significant trend of internal structure integration. 2) Land use in large cities is not intensive enough. The unordered expansion of urban fringe led to the waste of land resources. Planners and local governments should make use of the characteristics and trends of change of the urban system to formulate planning schemes and management measures.

区域城乡一体化的标志之一是城镇体系结构的一体化,这个过程可以从标度的角度进行描述和评价。分形是标度分析的重要方法,地理空间无尺度分布特征的典型参数是分维。论文以京津冀城镇体系为例,利用遥感图像的解译数据和人口普查数据开展分形分析、位序-规模分布分析和异速标度分析,用以解释1995—2013年间京津冀城镇体系演化的过程。结果表明,京津冀城镇体系及其演化的特征有：①京津冀空间结构和位序-规模分布都表现为自仿射双分形结构;②京津冀区域的城市人口-城区面积异速标度退化为假线性关系;③随着城镇体系的演化,自仿射的双分形结构逐步向自相似分形结构演化。由此得出结论：其一,京津冀城镇体系存在结构性的不协调因素。其空间结构和等级结构具有二元化特征,但演化方向却呈现内在结构一体化的显著趋势。其二,大城市用地不够集约。城市边缘区的无序扩张导致土地利用铺张浪费。地方政府和规划专家可以有意识地利用城镇体系演化的这种特征和趋势制定管理措施和优化规划方案。

Fractal cities and city fractals have been studied for about 30 years, but many basic problems have not yet been fully solved. Among the various basic fractal problems about cities, the most important are how to estimate fractal dimension and how to identify fractal nature of a city as a system or a system of cities in an effective way. Based on urban form and the box-counting method, this article discusses the approaches to calculating fractal dimension and determining fractal property of cities. The theoretical foundation of fractal dimension computation is the concept of perfect covering. In other words, in the process of fractal dimension measurement, a fractal object should be covered with boxes in the best way: nothing more, nothing less. In practice, it is hard to find the optimal way of box covering. Therefore, three rules should be followed. First, the sequence of measuring scales should be consistent with the cascade structure of a fractal city so that the fractal details can be captured in a reasonable way. Second, the operation of obtaining observational data should be simple and convenient to use so that the method can be applied by beginners. Third, the size of the dataset for fractal dimension estimation should be large enough so that the result of fractal parameters is stable. A conclusion can be reached that the geometric scale rather than the arithmetic scale should be employed to make a measurement because the fractal structure can be abstracted as geometric sequences instead of arithmetic sequences, and the measuring scale sequence should comply with the 1/2n rule (that is, 1, 1/2, 1/4, …). After estimating the fractal dimension of a city, the fractal property can be identified by the statistics from fractal dimension calculation. The good way of identifying the fractal nature of urban form is to use confidence statement, which consists of fractal dimension values, margin of error, and level of confidence. Given a level of significance (for example, α=0.05), we can draw an inference that a city's form is of fractal structure because it has a fractal dimension with a determinate level of confidence (for example, (1-α)×100%=95%). Using statistical analysis, however, one can never have full (100%) assurance that a city has a fractal form.

城市形态的分形是城市发育到一定阶段涌现出的有序格局和复杂结构,其基本特征是空间分布的无尺度性质。当研究者基于某个显著性水平推断城市分维存在时,实际上就是基于相应的置信度判断分形特征。虽然分形城市研究已经多年,但大量有关维数测算的基础问题依然悬而未决。本文根据分形几何学的基本思想论证城市形态分维测算的若干问题。分维测量的准则是最佳覆盖——不多不少、恰到好处的覆盖。盒子覆盖是最容易理解的测量方法。采用盒子覆盖法测量城市形态分维时,应考虑三个标准：一是快速逼近,二是简便操作,三是稳定拟合。直观估计分维的办法是利用双对数坐标图。由于城市形态不是严格意义的分形,而是类似于文献中的“前分形”,测量尺度与相应测度的幂律关系通常仅在一定尺度范围内有效,从而形成所谓标度区。本文围绕城市形态的分维测量和分形判断开展一系列讨论,包括尺度选取、标度区识别和统计标准等问题,对今后城市分形研究具有理论启示和方法论的参考价值。