城市规模分布是城镇化的重要产物，对区域生产效率和经济发展质量有着重要的影响，这种影响主要通过资源要素重构和优化配置得以实现，可在一定空间范围内映射其社会经济活力。当前，国内学者大多将行政城市作为研究样本，以人口表征城市规模，探讨城市规模分布的合理性与空间发展趋势。但纵观已有研究，行政城市往往存在统计口径不一致、与城市的实体地域差异较大等问题。本文从实体城市视角出发，基于类DMSP/OLS夜间灯光融合数据和分位数方法，提取实体城市规模，探索了1992年以来京津冀城市群实体城市规模分布及其演化的特征，并比较行政城市和实体城市的Zipf法则拟合优度，得出主要结论如下：

        第一，1992-2020年，京津冀城市群规模扩张迅速，城市的实体地域人口、面积与灯光规模经历了先快后慢的增长过程。北京和天津是京津冀城市群的绝对核心，已处于实体地域人口增长与空间扩展的后期阶段，平均灯光强度稳步提升，属于速度减缓但质量提升的内涵式扩张，而沧州、邢台、衡水、承德、邯郸等一般地级市仍为实体地域的人口、面积快速增加、平均灯光强度缓慢上升的外延式扩张。进一步将京津冀城市群的实体城市划分为五个等级，发现Ⅳ型Ⅴ型大城市的实体地域人口、面积及灯光规模增长迅猛、增速趋于放缓，而Ⅰ型Ⅱ型小城市的实体地域人口、面积及灯光规模增长显著、增速持续加快，证实了京津冀城市群的快速扩张区域已经由核心城市向一般地级市转移，城镇体系的两极分化特征减弱，规模等级结构日趋合理。

        第二，城市首位度与基尼系数的测算结果显示，京津冀城市群的城市体系结构呈现双核特征，核心城市占据主导地位，各实体城市之间未形成合理的发展梯度。1992-2020年，京津冀城市群的实体城市灯光与面积的首位度偏低，次位城市天津市中心城区的灯光与面积远超过理想值，城镇体系分布呈现“双核”特征；而首位城市北京在功能疏解政策背景下，经济社会要素集中程度下降，辐射带动效应不强。尽管高位序城市之间未形成合理的发展梯度，实体城市的人口、面积及灯光基尼系数计算结果仍大于标准值0.6，城市群内部差距分化，核心城市仍居于垄断地位，区域协同发展面临诸多挑战。

        第三，京津冀城市群的实体城市规模分布符合Zipf法则，Zipf指数始终大于1，总体呈下降趋势。1992-2020年，京津冀实体城市规模的累积分布曲线与帕累托分布较为接近，但KS检验结果并不显著，AD检验则显著拒绝了样本服从对数正态分布的原假设，相比之下，京津冀城市群更接近于位序—规模分布。在双对数坐标系中，1992-2020年京津冀实体城市的人口、面积及灯光的位序规模呈直线排列，在所排列区间内未出现明显的间隔，表明各等级实体城市完备，发展相对均衡。不同年份的Zipf法则拟合优度均大于0.9，双尾检验P值低于0.01，属于稳定的位序—规模分布型；从Zipf指数的变化来看，1992-2020年Zipf指数均大于1，总体呈现出下降趋势，低位序城市增长快于高位序城市，城镇体系向分散发展。

        第四，实体城市和夜间灯光数据对于Zipf法则具有很高的适用性。本文加入京津冀城市群行政城市的人口、面积及灯光规模三类主体，与城市群实体城市的人口、面积及灯光规模形成交叉比较，结果显示，实体城市对Zipf法则的拟合优度R²显著高于行政城市，而灯光规模的拟合优度R²则显著高于面积与人口规模，实体城市和夜间灯光数据更符合Zipf分布规律。从变化趋势来看，1990-2020年各城市主体的Zipf指数均呈下滑态势，城市规模分布的集聚性特征减弱，这意味着不论从行政城市或是实体城市视角开展论证，京津冀城市群的城镇体系结构都在向着合理化方向发展。

        本文从实体城市视角出发，探讨京津冀城市群的城市规模分布演进，发现了城市体系趋于扁平化的实证依据，通过行政城市和实体城市的对比，证实以实体城市的灯光值表征城市规模，城市体系分布往往更符合Zipf法则。不可否认，由于DMSP/OLS夜间灯光数据分辨率的限制，本文对实体城市规模的提取精度仍有待提升；此外，在研究深度和广度方面，缺少如局部城市地域的发展特征与形成机理，Zipf法则在其他城市群地区的验证等内容，以上均为后续研究可以重点拓展的方面。

City size distribution is a momentous achievement of urbanization and a space mapping of economic and social vitality. It catalyzes the spatial reconstruction and optimal allocation of resource elements, which has a profound impact on the regional production efficiency as well as the quality of economic development. At present, domestic scholars constantly regard administrative city as a research sample, in their elaboration, city scales are measured by population and built-up areas, and the rationality and distribution trend of city size distribution are further discussed. Most notably, the existing research has many defects and shortcomings, such as the complex statistical caliber, the single sample selection, and the lack of spatial representation, etc. From the perspective of physical cities distribution, this paper extracts the spatial boundary of cities based on the DMSP/OLS nighttime light data and a quantile-based approach, to delve into the scientific law of urban scale distribution and evolution process of Beijing-Tianjin-Hebei urban agglomeration since 1992. As the supplement, we put forward several practical optimization suggestions, and the main conclusions are as following:

Cities in Beijing-Tianjin-Hebei urban agglomeration are expanding rapidly from 1992 to 2020, and both urban areas and DN values of nighttime light results have experienced an inverted “U-shaped” growth process. As the core cities of the Beijing-Tianjin-Hebei urban agglomeration, Beijing and Tianjin have been in the later stage of spatial expansion with slower expansion speed but better quality. In contrast, Xingtai, Cangzhou, Chengde, Hengshui, Handan and other prefecture-level cities or node cities have rapid expansion speed. In addition, we divide city size into five levels from class Ⅰ to class Ⅴ using natural breaks classification. Preliminary results show that the growth rate of class IV and class V tends to slow down, while that of class I and class II continues to accelerate. Proof positive that the swift-expanding urban areas in core cities of Beijing and Tianjin have been gradually diverted to marginal towns of regional prefecture-level cities.

From an urbanological point of view, Beijing-Tianjin-Hebei urban agglomeration has a comparatively mature city size structure, which is consistent with the general rule of swift expansion of fewer large cities but slow-motion recovery of smaller towns. In the past three decades, the city group's primacy index is relatively low. Second city Tianjin has reached a new stage of economic and social development under the Binhai New Area Planning strategy, which leads to a far larger city than the ideal. Ultimately, the urban system presents a “dual core” feature. The concentration degree of economic and social factors of Beijing, the first city, has decreased in the implementation process of non-capital function relief policy. This significant decrease implies primacy distribution feature has weakened and radiation driving effect needs to be strengthened. However, a reasonable development gradient between high ranking cities has not formed yet, Gini coefficient is still high and a certain gap exists between high-order and low-order cities in Beijing-Tianjin-Hebei region. There is still a long way to go to create a synergistic, efficient and orderly urban system.

By using statistical distribution test model, it is found that the city size distribution of Beijing-Tianjin-Hebei urban agglomeration does not obey the model of lognormal distribution. Although actual cumulative curve of physical city size is close to Pareto distribution, KS test was not significant as our previously thought---in contrast, it is subjected to the Rank-Size rule. From 1992 to 2020, cities in Beijing, Tianjin and Hebei are roughly arranged in a continuous and compact straight line in the double logarithmic coordinate system, however there’s no noticeable blank in the arranged interval, demonstrating that the city size distribution at different scales is quantitatively resolved and balanced. Moreover, the related coefficient R² is greater than 0.9, while P value is less than 0.01 as an outstanding symbol of Rank-Size distribution. Overall it's worth noting that, Zipf index since 1992 has been always greater than 1 and shows a downward trend, confirming that the expansion speed of low order cities is faster than that of high ranking cities, and the dispersion power of urban system is larger than that of concentration.

The four main cities are compared by the intersection of administrative and physical city, urban population and Nighttime light data. The results show that Zipf’s law has better goodness of fit with physical city depicted by Nighttime light data. Specifically, R² between rank and scale of physical cities is higher than that of administrative cities as R² of Nighttime light is greater than that of urban population scale. Nevertheless, Zipf index steadily declining over the past 30 years, and the polarization of the urban system has weakened. That is to say, whether from the administrative or physical city perspective, the tendency of urban system of Beijing-Tianjin-Hebei region evolving towards rationalization should not be dismissed.

This study considers that the optimization of city size distribution of Beijing-Tianjin-Hebei urban agglomeration should be based on the non-capital function relief as the key to further implement the Strategic Plan of National City Groups and Beijing, Tianjin and Hebei Coordinated Development. Xiong’an New District and Chongli area are also constructed as the core to optimize urban system of Beijing-Tianjin-Hebei region, and the practice of multi-center strategy positively speeds up the optimization process of urban structure of megacities. Differential development need to be promoted between the integration of Ⅳ&Ⅴ large cities and specialization of Ⅰ&Ⅱ small and medium-sized cities. The barrier-free and efficient flow of all kinds of elements in Beijing, Tianjin and Hebei will be fully encouraged by greater investment in infrastructure development as well as allocating public resources in Hebei province.