The fractal nature of maps and mapping

A fractal can be simply understood as a set or pattern in which there are far more small things than large ones, for example, far more small geographic features than large ones on the earth surface, or far more large-scale maps than small-scale maps for a geographic region. This article attempts to argue and provide evidence for the fractal nature of maps and mapping. It is the underlying fractal structure of geographic features, either natural or man-made, that make reality mappable, large-scale maps generalizable, and cities imageable. The fractal nature is also what underlies the beauty of maps. After introducing some key fractal concepts such as recursion, self-similarity, scaling ratio, and scaling exponent, this article demonstrates that fractal thought is rooted in long-standing map-making practices such as series maps subdivision, visual hierarchy, and Töpfer’s radical law. Drawing on previous studies on head/tail breaks, mapping can be considered a head/tail breaks process; that is to divide things around an average, according to their geometric, topological and/or semantic properties, into the head (for those above the average) and the tail (for those below the average), and recursively continue the dividing process for the head for map generalization, statistical mapping, and cognitive mapping. Given the fractal nature of maps and mapping, cartography should be considered a perfect combination of science and art, and scaling must be formulated as a law of cartography or that of geography in general.     

分形理论在地理学中的应用研究进展

美国数学家Mandelbrot提出的分形理论为解决复杂性问题提供了有效方法。地理学的复杂性问题已成为分形理论的重要实证研究领域。本文根据国内已有的文献,在概括分形理论主要内容的基础上,对地理学各分支领域的应用研究现状及有关问题进行了评述,并就地理学分形研究的前景作了展望。     

Multicriteria generalization (MCG): a decision-making framework for formalizing multiscale environmental data reduction

Environmental simulation models need automated geographic data reduction methods to optimize the use of high-resolution data in complex environmental models. Advanced map generalization methods have been developed for multiscale geographic data representation. In the case of map generalization, positional, geometric and topological constraints are focused on to improve map legibility and communication of geographic semantics. In the context of environmental modelling, in addition to the spatial criteria, domain criteria and constraints also need to be considered. Currently, due to the absence of domain-specific generalization methods, modellers resort to ad hoc methods of manual digitization or use cartographic methods available in off-the-shelf software. Such manual methods are not feasible solutions when large data sets are to be processed, thus limiting modellers to the single-scale representations. Automated map generalization methods can rarely be used with confidence because simplified data sets may violate domain semantics and may also result in suboptimal model performance. For best modelling results, it is necessary to prioritize domain criteria and constraints during data generalization. Modellers should also be able to automate the generalization techniques and explore the trade-off between model efficiency and model simulation quality for alternative versions of input geographic data at different geographic scales. Based on our long-term research with experts in the analytic element method of groundwater modelling, we developed the multicriteria generalization (MCG) framework as a constraint-based approach to automated geographic data reduction. The MCG framework is based on the spatial multicriteria decision-making paradigm since multiscale data modelling is too complex to be fully automated and should be driven by modellers at each stage. Apart from a detailed discussion of the theoretical aspects of the MCG framework, we discuss two groundwater data modelling experiments that demonstrate how MCG is not just a framework for automated data reduction, but an approach for systematically exploring model performance at multiple geographic scales. Experimental results clearly indicate the benefits of MCG-based data reduction and encourage us to continue expanding the scope of and implement MCG for multiple application domains.     

基于CE-1数据的数字月海分形刻画及其动力学意义

月海是月球表面的主要地理单元。对月海地貌特征的研究能够认识月表起伏及其动力学成因。依据CE-1 影像图和全月球数字高程模型数据,提取了南北半球不同纬度地区、不同填充岩性的典型月海范围,运用盒维数法、立方体覆盖法和多重分形q-Dq语言对月海边界及其地形表面的地貌特征进行了线和面的描述和研究。结果表明：① 分形方法可以有效地描述月球地貌的复杂程度。月海边界的分维值介于1.065~1.100 之间,平均值为1.086,标准差为0.014,在双对数坐标下5 个月海区边界复杂度拟合的直线都近似平行。② 月海地形表面的分维值介于2.704~2.852 之间,5 个月海地形表面的分维值标准差为0.051,ln δ -lnN（δ）图也近似平行,但截距的离散程度相较于月海边界的高。③ 月海地貌多重分析的q-Dq曲线图总体呈现非递增趋势,Dq值在q=-10 和q=0 时基本达到饱和,但差异较小。④ 从线、面单分形以及多重分形可见,5 个月海均具有分形特征,但彼此之间差异较小。其动力学意义在于它们在形成之后,内动力作用基本停止或很小,与分形特征基本一致;而外力主要由陨石撞击作用少量地改变着月表的面貌,导致在分形特征上表现出差异性。     

A scale-explicit model for checking directional consistency in multi-resolution spatial data

Multi-resolution spatial data always contain the inconsistencies of topological, directional, and metric relations due to measurement methods, data acquisition approaches, and map generalization algorithms. Therefore, checking these inconsistencies is critical for maintaining the integrity of multi-resolution or multi-source spatial data. To date, research has focused on the topological consistency, while the directional consistency at different resolutions has been largely overlooked. In this study we developed computation methods to derive the direction relations between coarse spatial objects from the relations between detailed objects. Then, the consistency of direction relations at different resolutions can be evaluated by checking whether the derived relations are compatible with the relations computed from the coarse objects in multi-resolution spatial data. The methods in this study modeled explicitly the scale effects of direction relations induced by the map generalization operator – merging, thus they are efficient for evaluating consistency. The directional consistency is an essential complement to topological and object-based consistencies.     

土地利用类型结构的多尺度转换特征

尺度一直是LUCC到GLP研究所关注的重要问题。以不同比例尺的辽宁省土地利用数据为例,以贵州省土地利用数据进行验证,系统探讨了土地利用类型结构在不同尺度间可能存在的转换特征。结论如下:(1)辽宁省土地利用的分形结构跨1 ∶ 10万、1 ∶ 50万、1 ∶ 100万这3个比例尺客观存在;(2)对不同比例尺的同一区域不同土地利用类型而言,各土地利用类型分维在不同比例尺之间存在的响应关系并不一致,某些土地利用类型的分维随着比例尺由1 ∶ 10万向1 ∶ 50万再向1 ∶ 100万的变小,分维呈现变大趋势,而某些土地利用类型的分维随比例尺的变化并不如此,所以,对于不同的土地利用类型而言存在一定的差别,不可一概而论;(3)不同尺度间的土地利用类型结构只能够在两两连续间隔的两种比例尺土地分维之间建立起关系显著的定量关系模型。所得结论对科学认知土地结构空间配置的多尺度特征以及建立相应尺度转换模型具有积极的参考意义。     

塔里木河干流两岸土壤水分特征曲线的分形模拟

应用Van Genuchten提出的土壤水分特征曲线公式,推导出了非饱和土壤水分特征曲线的分形模型。通过对塔里木河干流英巴扎断面6种不同土壤样本利用中子水分仪和负压计实测的水分特征曲线资料反求得到相应的分形维数,分析了分形维数与土壤质地之间的关系,结果表明随着土壤砂粒含量的减少或粘粒含量的增加,分形维数逐渐增大。此外,基于土壤颗粒的重量与粒径分布求出了塔里木河干流两岸沙漠地风沙土壤粒径的分形维数。通过对土壤水分特征曲线的分形维数与土壤粒径的分形维数的对比,得知它们之间存在着良好的线性关系。根据此关系,利用易测得的土壤粒径结合所推导的分形模型,对土壤水分特征曲线进行了预测,结果表明土壤水分特征曲线的预测值与模拟值比较接近,模型的预测结果很好地吻合了实测的土壤水分特征曲线。这一结果对于实际工作中根据风沙土土壤颗粒大小分布的分形维数来预测沙漠风沙土土壤水分特征曲线具有一定的指导意义。     

智能化制图综合在GIS环境下的实现方法研究

自动制图综合的研究长期以来未能取得突破性进展,原因之一是人们对制图综合的知识处理的深度不够。本文探讨在ＧＩＳ环境下实现智能化制图综合的方法。方法要点是：①采用知识推理＋数学模型的方法,即将基于专家知识的智能化推理与数学模型处理结合起来,用知识推理引导和操纵数学模型处理,以期达到较理想的结果;②在专家知识规则中贯穿面向地理特征的思想,即包括面向地理要素、面向地物对象－类,面向特殊地理区域等方面的知识法则。本文首先阐述在ＧＩＳ环境下实现智能化制图综合功能的策略,其次描述知识推理控制下的数学模型操作流程,以及制图综合专家知识库的组织形式;最后阐述各种面向地理特征的制图综合知识规则。     

旅游系统网络空间分形研究的科学展望

大自然千奇百怪, 大自然用其"无形之手"造就了奇妙无比的分形世界, 在自然形胜之处, 形成了无数的自然- 人工分形复合体, 它们逐渐吸引人们的关注, 渐进由旅游景物- 景点- 景群- 景区—风景区以及风景区域等而形成不同规模、不同层次的旅游网络系统。旅游网络系统的最基 本要素之一是景区(点), 不同等级、不同层次、不同规模的景区(点) 由各种旅游线路和廊道( 交通 线路) 相串通、连接, 从而形成旅游系统的网络空间结构。文中首先明确指出了将分形理论应用于 旅游地理学研究的科学意义, 系统提出了旅游系统网络空间结构在旅游景点、旅游线路与旅游交 通三方面的分形科学问题; 并进而以北京市旅游景点为例, 定量给出了旅游网络系统空间结构的 分形证据; 最后, 还指出了积极支持开展分形旅游研究对促进我国旅游地理学理论研究水平的提 升和在国际旅游学领域尽快占据分形旅游研究前沿之地的积极意义。     

Fractal analysis of tidal channels in the Bahía Blanca Estuary (Argentina)

The fractal dimension (D) was estimated for nine tidal channels depicted in thematic mapper (TM) Landsat-5 imagery to derive information about the degree of geomorphological control on a tidal channel network characteristic of the Bahía Blanca Estuary (Argentina). Two methods, box counting and contiguity, were used to estimate fractal dimensions for each tidal channel. All channels produced D values close to 1, meaning that they are self-affine fractal features. However, these fractal dimensions do not represent the meandering pattern complexity characteristic of the tidal channels analysed. Although both methods allowed for estimation of D, the contiguity method showed that three of the channels actually are not fractal but have sinusoidal characteristics, a condition that was not detected by the former method.     

Unique identification of vector geographic objects by location and geometry

This article presents a method to uniquely identify vector geographic objects based on their spatial position and geometric characteristics. The method comprises three steps. First, a grid-based spatial indexing (Quadtree grid) is automatically generated according to location, shape and size of the geographic objects. This allows for a simple Quadtree grid generation (the level, row and column number) for every geographic object by three grid parameters (origin, start and end size). Second, a set of algorithms have been developed for extracting geometric characteristics (e.g. representative points, outline and orientation) of geographic objects corresponding to the grid. Third, a strategy for the unique identification of geographic objects is proposed in order to integrate the Quadtree grid and the geometric information. The application of the proposed method is presented in a case study conducted in Jiangsu Province, China. The experiment indicates that this method is more efficient than existing methods at positioning, searching and querying. The unique identifier is also shown to be useful as a technical method to support geographic information systems analytical applications, such as the matching, conflating and sharing of spatial objects.     

Recommending attractive thematic regions by semantic community detection with multi-sourced VGI data

Attractive regions can be detected and recommended by investigating users’ online footprints. However, social media data suffers from short noisy text and lack of a-priori knowledge, impeding the usefulness of traditional semantic modelling methods. Another challenge is the need for an effective strategy for the selection/recommendation of candidate regions. To address these challenges, we propose a comprehensive workflow which combines semantic and location information of social media data to recommend thematic urban regions to users with specific interests. This workflow is novel in: (1) developing a data-driven geographic topic modelling method which utilizes the co-occurrence patterns of self-explanatory semantic information to detect semantic communities; (2) proposing a new recommendation strategy with the consideration of region’s spatial scale. The workflow was implemented using a real-world dataset and evaluation conducted at three different levels: semantic representativeness, topic identification and recommendation desirability. The evaluation showed that the semantic communities detected were internally consistent and externally differentiable and that the recommended regions had a high degree of desirability. The work has demonstrated the effectiveness of self-explanatory semantic information for geographic topic modelling and highlighted the importance of including region spatial scale into the model for an effective region recommending strategy.     

城镇体系空间结构的分形维数及其测算方法

总结了刻画城镇体系空间结构的三种分形维数:聚集维数(半径维数)、网格维数和空间关联维数,阐释了各种维数的地理意义,然后以实例说明了它们的测算方法。     

泥石流的分形特征及其意义

分析了泥石流流体和堆积物粒度分布的分形结构特征，其分维值介于２￣３之间，平均值分别为２．７１和２．６５左右。研究了泥石流沟水系的分形特征及其在不同观察尺度下分维的变化特征，结果表明，分维可作为描述泥石流粒度成分和泥石流沟水系的参数，反映了泥石流活动的形成演化特征，在此基础上，论述了分维的研究意义。     

珠江河网分形机理研究

以不同时期珠江网河历史地图及遥感影像为基础,进行了河网分形计算,并从河道发育、人为活动等多个角度分析了珠江网河分形机理。在发育过程中,1883-2001年间河网分维值逐渐减小;在空间上,正干水道中潮道分维值大于分汊河道。分析认为,人为活动是导致珠江网河分形变化的主要因素:联围筑闸消灭横向支汊、口门整治及河道固化减小了河道冲决的几率、河道挖沙引起的水沙分流比变化改变了河道冲淤特性。     

Influences of different surveying and mapping methods on fractal characteristics of gully-head shoulder lines

Abstract

Accurate quantification of gully shoulder lines (gully borderlines) will help better understand gully formation and evolution. Surveying and mapping are the most important ways to obtain precise morphology. To evaluate the influences of different steps of surveying and of curve-fitting methods of mapping on the morphology of the shoulder line characterized by fractal dimensions, 13 shoulder lines at gully heads were surveyed using a total station and then mapped with different methods of curve fitting, with the fractal dimensions calculated by maps compared with those measured in the field. Fractal dimensions by field measurement ranged from 1.185 to 1.456. Compared with field measurements, the average absolute errors of polygonal line, quadratic B-spline, and arc-fitting methods are 0.045, 0.040, and 0.046, respectively; the average relative errors are 3.48, 3.13, and 3.59%. Therefore, the quadratic B-spline method has a higher accuracy. The standard error of the fractal dimension tends to be larger as average step length increases. The error is ~5% when the step length is 0.7 m, which is advisable for field surveying. This study will help promote the efficiency of field surveying and mapping, and thus promote the accuracy and credibility of gully morphology.     

中国大陆山系、断层系与水系的空间维数及其关系

根据分形理论,应用网格法系统地探讨了中国大陆山系、断层系与水系的空间分形结构,并计算出中国大陆以及三大自然区山系、断层系与水系的空间维数,在此基础上还探讨了山系、断层系与水系空间维数在三大自然区的变化情况。得出的基本结论为：(1)中国大陆山系、断层系与水系的空间分形结构是客观存在的;(2)中国大陆山系的空间维数值为1．1513,中国大陆断层系的空间维数值为1．2507,中国大陆水系的空间维数值为1．4169;(3)在三大自然区中,山系、断层系空间维数值均以青藏高原区相对为最大,而以西北干旱区相对为最小,山系与断层系空间维数值的变化在空间上有着一致的对应关系,山系空间维数值较大,则断层系空间维数值亦较大,而水系空间维数值的大小依次为东部季风区水系、青藏高原区水系与西北干旱区水系;(4)中国大陆山系、断层系空间维数响应的关系为DF＝0．6884DM 0．4446(式中：DM为山系的空间维数,DF为断层系的空间维数)。该式不仅为山系、断层系关系的定量化研究提供了全新的手段,而且还为山系、断层系内在密切关系的证实提供了全新的分形理论的证据;(5)整个中国大陆山系、断层系与水系的空间维数值并不等于三大自然区相应要素空间级数值的简单加和平均值;(6)进行山系等地貌现象的分形研究,应遵循相同的分维计算方法,否则将失去进行分维数横纵向比较的意义。     

基于分形的城市体系经济规模等级演变研究

利用区域城市体系经济规模等级分形理论,选取长江三角洲1995~2009期间4个年份的县级以上行政单位经济发展数据,计算了经济规模等级分形维数。以该地区城市体系经济规模等级分形维数的演变特征为载体,探讨了长江三角洲地区城市体系经济规模等级演变规律。结果表明:①虽然当前长三角地区各城市的经济社会发展水平差异较大,中心城市垄断性仍然较强,但其均匀化趋势明显。②经济规模等级的空间梯度特征明显,经济规模等级的均匀性随着距离区域核心城市距离的扩大而降低。③产业空间布局层次化逐步分明,中心城市的第三产业比重不断提升,区域控制能力不断增强,与以第二产业为主导的中小城市形成有机整体,区域经济竞争力整体得到提升。     

虚拟地理环境的多维数据模型与地理过程表达

该文通过剖析虚拟地理环境所表达的地球表层空间与其在计算机空间的映射之间的关系,定义了一维、二维、三维及多维虚拟地理环境;指出虚拟地理环境与传统地图、GIS的区别及联系。阐述近年来虚拟地理环境研究对地理学乃至地球科学研究方法和手段的影响以及对地理科学思维方式和重点研究领域的影响。