Introduction

More than half of the world’s population is now residing in cities, and this proportion is continuing to increase in the near future (Angel, 2012). Urbanization is a crucial social transformation across the world in this century, closely related to our environment, climate, and global sustainability (Duarte, Álvarez, 2019; Seto et al., 2010). Monitoring urban land-cover changes and characterizing urban expansion is a common concern shared by environmentalists, urban planners, geographers, and other scientists (Reba, Seto, 2020; Xu et al., 2019).

Spatial agglomeration is an essential feature that distinguishes the city from the countryside (Scott, Storper, 2015). Despite the constant outward expansion of cities, industrial and economic activities in cities are excessively concentrated in places with abundant resources, convenient transportation, and favorable locations (Geddes, 1915). These places are most likely to be the centers of urban development. Central Place Theory explains the hierarchical organization of cities (Christaller, 1966). The city center serves as a core area with concentrated functions such as commerce, politics, culture, and transportation (Harris, Ullman, 1945). Urban population is mainly concentrated in the core areas, resulting in positive feedback leading to the agglomeration of other elements. Thus, the urban population gradually declines with the distance from the city centers and various urban elements also show a similar spatial agglomeration and gradient (Li et al., 2021). In the process of urban expansion from the center to the periphery, there exist concentric zones with unique social and economic characteristics according to concentric zone theory (Burgess, 2008). Concentric partitioning has been commonly used to measure urban elements from the city center outward quantitively (Seto, Fragkias, 2005; Taubenböck et al., 2009).

The process of urban growth is different in physical and social spaces, especially in megacities. The impervious surface acquired from remotely sensed data is the direct representation of urban expansion (Patino, Duque, 2013; Van de Voorde et al., 2011), but it neglects the urban dynamics of socioeconomic activities. The nighttime light satellite imagery can inspect the intensity and spatial disparity of urban dynamics (Bagan, Yamagata, 2015; Li, et al., 2015), which has been successfully applied to monitor and characterize urban growth, detect socioeconomic dynamics, and estimate energy consumption (Bennett, Smith, 2017; Letu et al., 2010; Levin et al., 2020). As an appropriate proxy to measure the intensity of socioeconomic activities, nighttime light is a comprehensive indicator that conveys information from spatial distribution, economy, and other factors influencing urban development (Bennett, Smith, 2017; Mård et al., 2018). Nighttime light shows a solid ability to assess urban development in terms of both spatial and socioeconomic status (Elvidge et al., 2023) and can provide insights into the identification of urban structure and the state of urban development (Kyba et al., 2017; McCallum et al., 2022; Ren et al., 2022). The Visible Infrared Imaging Radiometer Suite (VIIRS) nighttime light imagery acquired by the Suomi National Polar-orbiting Partnership (NPP) Satellite is widely used for regional economy and urban studies because of its high spatial resolution and high sensitivity of visible light (Li et al., 2022b).

The difference between socioeconomic activities and land use development in cities is often caused by excessive urban sprawl (Xu et al., 2022; Yasi et al., 2020). The rapid expansion of urban land resulted in an increase in land take per person across the globe in recent decades (Li et al., 2022a). Previous studies on the spatial distribution of urban elements mainly focus on population and urban land (Akbar et al., 2019; Batty, Sik Kim, 1992; Kroll, Haase, 2010; Song, Yu, 2019). In terms of quantitative models describing urban land density spatial patterns, previous studies proposed a linear model with two stages, inverse-S function, and Geographic Micro-Process (GMP) models to describe the spatial distribution of urban land (Guérois, Pumain, 2008; Jiao, 2015; Jiao et al., 2021). The spatial distribution of other elements except urban land also plays a key role in the analysis of urban expansion (Cai et al., 2021; Chen et al., 2022, Yang et al., 2023). The spatial distribution of different urban elements determines the internal structure of each city, and the mismatch between them hinders the sustainable development of cities (Xiao et al., 2021).

The growth of cities is uneven among different-sized cities (Shukla et al., 2021). More and more people transfer to large cities, making large cities grow into megacities with more than ten million residents (Huang et al., 2017). For example, the total urban population in developed countries has not increased in recent decades, but their large cities have been expanding and growing (Zhang, 2016). The world’s largest urbanized region, Tokyo, Japan, has accommodated more than 35 million urban residents (Bagan, Yamagata, 2012). Studying the urban expansion and urban spatial structure of megacities has a positive significance for guiding urban development, especially for the development of small and medium-sized cities (Novotný et al., 2022).

In this study, we first choose 30 megacities with prominent political and economic status across the world as sample cities. We use the land use/cover data to reflect the physical expansion of cities and nighttime light to represent the intensity of urban dynamics in social space. We analyze the spatial gradients of nighttime light and urban land density to fully quantify the pattern and dynamics of urban expansion in megacities and provide policy implications for global sustainable development.

Study area and data

Study area

We selected 30 cities around the world, including 21 megacities with a population of more than 10 million and 9 regional central cities with prominent political and economic status, such as Berlin, Rome, Madrid, and so on. Population is not the only criterion for selection because most cities with huge populations are located in East Asia, while some essential European and African cities have relatively small populations due to the constraints of terrain, economy, and other conditions. Therefore, to ensure the representativeness of sample cities on a global scale, we select 30 cities that can reflect the current urban physical and social spatial relationships around the world. With the rapid urban expansion in the past few decades, these megacities are the central cities of various countries, which can provide a reliable theoretical basis for the comparison of global urban expansion (Fig. 1).

Fig. 1: Spatial distribution of 30 sample cities.
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(Demographic data is from https://worldpopulationreview.com/).

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Data and pre-processing

Land use and nighttime light data were used to compare the differences between physical and social expansions of cities. We used land use data in 2020 derived from the ESRI 10 m Global Land Cover products (https://livingatlas.arcgis.com/landcover/), which is mapped based on Sentinel-2 scene collection. The production provides a 9-class map of the surface, including vegetation types, bare surface, water, cultivated land, trees, grass, scrub, built-up areas, and snow/ice. As water bodies are deemed incapable of conversion to urban land, the land use products were reclassified into three types: built-up areas, water bodies, and open space. Open space covers other types of land, which may have the potential to be converted into urban land. The yearly composite of NPP-VIIRS nighttime light of 2020 was applicated to characterize the urban forms in social space (https://eogdata.mines.edu/nighttime_light/annual/v20/). Administrative boundaries data was used to clip the images and extract the selected study area of the 30 megacities, which are from the Database of Global Administrative Areas (https://gadm.org/).

Methods

Concentric ring analysis

Concentric ring analysis is used to explore the spatial pattern of urban elements. A series of equidistant concentric rings are built outward from the centers of the cities (Schneider, Woodcock, 2008). A buffer ring is a basic unit for calculating urban land density and nighttime light intensity. The scale effects caused by the differences in the resolution of land use and nighttime light data can be eliminated after calculation. Using Google high-resolution imagery, we define the central business district (CBD) or the birthplace of a city as its city center (Burgess, 2008; Jiao, 2015; Yu et al., 2021). The spatial extents of urban land in 2020 determine urban boundaries. And the extent of the series of equidistant buffer rings should ensure that the buffers cover the main area of the cities. Taking Beijing (China) as an example, the concentric rings are shown in Fig. 2a.

Fig. 2: Concentric ring analysis using 1-km series buffers in Beijing, China.
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a Urban land density. b Nighttime light intensity. The unit of nighttime light radiance is nW⋅cm−2⋅sr−1.

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Urban land density is defined as the proportion of built-up areas to buildable areas in a buffer ring (Jiao, 2015; Xu et al., 2019). The urban land density of a buffer ring is calculated using formula (1):

$$Dens = \frac{{S_{{{{\mathrm{urban}}}}\,{{{\mathrm{land}}}}}}}{{S - S_{{{{\mathrm{water}}}}}}}$$
(1)

where Dens is the urban land density of a ring. S is the total land area in a ring. Surban land and Swater are the areas of built-up areas and water bodies in a ring, respectively.

The nighttime light intensity in concentric rings is defined as the ratio of total digital number values of nighttime light to the area of a concentric ring. The nighttime light intensity is standardized, that is, linearly scaled to [0, 1]. The normalization formula of nighttime light data is:

$$DN_i^\prime = \frac{{DN_i - DN_{{\mathrm{min}}}}}{{DN_{{\mathrm{max}}} - DN_{\mathrm{min}}}}$$
(2)

where \(DN_i^\prime\) is the mean value of the NLI of the ith ring after standardized treatment, DNi is the mean value of the NLI of the ith ring from the city center. DNmin and DNmax are the minimum and maximum value of the NLI of all the rings.

Inverse-S function

The spatial distribution of urban land density presents an inverse-S shape. The density of urban land is the highest in the urban center, and with increasing distance from the city center, it slowly decays in the core areas. Then, it decreases relatively fast in the inner and suburban areas, and finally slowly decays in the periphery of the city. Jiao (2015) proposed a modified function with an inverse-S shape (inverse-S function) to characterize the spatial gradients of urban land density. The inverse-S function is defined in formula (3):

$$f\left( r \right) = \frac{{1 - c}}{{1 + e^{\alpha (\frac{{2r}}{D} - 1)}}} + c$$
(3)

where f is the urban elements calculated using formula (1) and formula (2), r is the radius from the city center to a buffer ring, e is Euler’s number, and α, c and D are parameters.

The parameters can characterize the urban form. Parameter c represents the background density of urban land or nighttime light in the periphery of a city and parameter D is the city radius. Since parameter α is negatively correlated with the proportion of the rapidly decreasing part of the curve, it is regarded as the representation of the compactness of a city (see detailed explanations in Jiao (2015)).

The urban land density function has been widely used, covering many countries and cities around the world (Keeratikasikorn, 2018; Shukla et al., 2021; Xu et al., 2019). Urban elements generally show the nature of spatial agglomeration and have similar spatial distribution patterns. Thus, in addition to the urban land density, the distribution of other geographical elements also follows an “inverse-S shape rule”. Previous studies have applied the urban land density function to population density, road density, and land surface temperature (Bonafoni, Keeratikasikorn 2018; Govind, Ramesh 2020; Li et al., 2021).

Concentration degree index

To compare the spatial agglomeration scope of nighttime light and urban land, we adopted the ratio of parameter D between nighttime light intensity and urban land density, which was named concentration degree index (CDI) (Li et al., 2021). CDI laterally reflects the relationship between nighttime light intensity and urban land density of each city and builds a bridge between urban social space and physical space. The index can be written as:

$$CDI = \frac{{D_{{{{\mathrm{NLI}}}}}}}{{D_{{{{\mathrm{ULD}}}}}}}$$
(4)

where DNLI and DULD are fitted parameter D for the nighttime light intensity and urban land density, respectively. For a city with a larger CDI, its spatial size in social space is closer to its physical size.

Results

Spatial patterns of urban land and nighttime light

The spatial distribution of urban land and nighttime light of the 30 megacities in 2020 are shown in Figs. 3 and 4, respectively. The diversity of spatial distribution of various elements in cities can be attributed to factors such as geographical location, policy planning, and development direction. Overall, both land use and nighttime light exhibit a circular spatial pattern with a center-periphery structure, indicating a certain degree of similarity. Figure 3 primarily displays the cities’ land use, which shows a high density of urban land near the city centers, gradually decreasing towards the cities’ periphery due to land expansion. The distribution patterns of nighttime light and urban land are also similar, with higher nighttime light intensity observed in the urban core area and reduced intensity towards the urban edge area (Fig. 4). However, the main scope of the cities in social space is much smaller than the size of cities in physical space.

Fig. 3: Urban land use in three types of built-up areas, open space, and water bodies in 30 megacities in 2020.
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Two buffers with radii of 10 km and 20 km show the extents of sample cities. (Data Source: https://livingatlas.arcgis.com/landcover/).

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Fig. 4: Spatial distribution of nighttime light of 30 global megacities in 2020.
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Two buffers with radii of 10 km and 20 km show extents of sample cities. (Data Source: https://eogdata.mines.edu/nighttime_light/annual/v20/).

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Nighttime light helps to identify whether urban land exists and to highlight the intensity of social and economic activities. In recent years, the urban expansion mode of megacities has changed from "extension" to "infill" (Xu et al., 2020). Built-up areas in central areas of cities are contiguous with relatively scattered forms, showing compactness in the images, and boundaries between various areas in cities are not obvious. In contrast, high-intensity patches in nighttime light images are more dispersed and interconnected, but easily distinguishable. Greater socioeconomic activities are more likely to be identified. In addition, Figs. 3 and 4 reveal that although megacities have experienced drastic urban expansion and the urban land density has increased significantly, socioeconomic activities are not active in urban fringe areas because of the higher rate of land expansion than the rate of population growth (Seto et al., 2011).

Spatial gradients of urban land density and nighttime light intensity

Urban land density and nighttime light intensity have been calculated in each concentric ring (Fig. 2). The distance decay of the two elements with the city center is shown in Fig. 5. We use the inverse-S function to fit spatial gradients of the urban land density and nighttime light intensity. The parameters of fitting curves are shown in Table 1. For urban land density, the fitting results show the radius and compactness of cities in physical space. NLI indicates the agglomeration of urban socioeconomic activities and the scale of cities in social space. The fitting curves of the two urban elements in each city both show an obvious inverse-S shape, that is, slow decay around the central areas, then fast decay, and finally slow decay again in the hinterland of the cities. Urban land density in most cities showed a more complete inverse-S shape spatial gradient. However, NLI decays rapidly in the core areas of the cities, and the first slow decay interval is not obvious. The spatial gradient of NLI is more agglomerate. The gap between the fitting curves of the two urban elements is too large at the same distance from the city center. On the one hand, the clustering effect of nighttime light is more obvious, so the intensity attenuation is more rapid. On the other hand, after normalization, NLI at the city edge is close to 0, which is different from the actual density of urban land.

Fig. 5: Spatial gradients of urban land density and nighttime light intensity.
figure 5

ULD and NLI are calculated in a series of 1-km buffer rings with concentric ring analysis and NLI of each city has been normalized.

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Table 1 Parameters of inverse-S function for urban land density and nighttime light intensity.
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The inverse-S function has good fitting results for both types of urban elements, with R2 of all cities above 0.85 (Table 1). Parameter α controls the slope of the fitting curve of the inverse-S function, which is related to the compactness of the cities. Larger values of α indicate that the cities are more compact. The values of α of the urban land density are generally larger than that of nighttime light intensity. The values of α fitted by urban land density in sample cities range from 1.54 to 5.92 with an average of 3.49. The parameters α of the nighttime light intensity ranges from 0.68 to 3.73 with an average of 2.04. The constant c is the value of the density of urban elements in the hinterland of the cities, and generally, the value of c is less than 0.3. In some cities such as Shanghai, New York, and Kolkata, as the megacities are expanding to saturation, there is still a large density value on the outskirt of the cities, so the c value fitted by the density of urban land may be rather large. The parameter D represents the radius of the main area of a city. Larger values of D fitted by urban land density indicate larger urban entities. Likewise, a larger value of D of NLI means a larger urban scale in social space, which is consistent with Figs. 2 and 3. The urban radii fitted by the density of built-up areas are generally larger than that of nighttime light intensity.

Comparisons of urban form and urban extent

The comparison of parameters of the inverse-S function between urban land density and nighttime light intensity is shown in Fig. 6. In the case of a compact urban form, the ULD curve is characterized by a higher density in the urban core area that decreases rapidly in the surrounding area, leading to a steep curve (Jiao, 2015; Li et al., 2021). On the contrary, the urban land density curve of the sprawling urban form decreases more slowly. Jiao (2015) defines compactness as the proportion of the rapidly decaying region to the overall part of the curve, which is linearly related to the parameter α that controls the slope of the curve. Therefore, parameter α represents urban compactness, and a larger value of α indicates a more compact urban form. It clearly shows that nighttime light intensity declines more rapidly outward from city centers, which made the fitting curves of NLI exhibit a steeper slope but also a longer interval of rapid decay. The socioeconomic activities represented by nighttime light show stronger spatial agglomeration. The intensity of nighttime light is the highest in urban core areas and then decreases rapidly. In a fitting curve, the first decay interval is small, and the second decay interval accounts for a larger proportion, which means a more sprawling urban form in comparison. Except for Dhaka, Ho Chi Minh City, and Kolkata, the α values fitted by NLI of most cities are smaller than that of ULD. These cities are more compact in the physical space represented by urban land, while the urban form in the social space symbolized by nighttime light is more sprawling (Fig. 6a). The values of parameter α of NLI in Dhaka, Ho Chi Minh City, and Kolkata are greater than that of ULD. The aggregation scope of NLI in these cities is smaller than urban entities. And these cities have more compact forms in social space, showing a more significant influence of city centers on socioeconomic activities.

Fig. 6: Scatter plots of parameters of inverse-S function for urban land density and nighttime light intensity in concentric rings.
figure 6

a Parameter α. b Parameter D.

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The rapid expansion of megacities in physical space makes the density of urban land almost reach saturation. The population inside the cities is concentrated, and the gaps between urban centers and peripheries are too large. It results in an unbalanced population distribution, which is especially obvious in Tokyo and Shanghai. Similarly, this also causes the aggregation radii of NLI to be substantially smaller than the physical radii of the cities. In Fig. 5, the fitting curves of the two elements in Madrid are relatively similar, and the values of parameter D of Madrid are also relatively close. While in Fig. 6b, Tokyo, Shanghai, Toronto, New York, etc. with relatively large differences in fitted curves are mainly distributed in the upper left corner. The significant gap between the radii of nighttime light aggregation and urban entities highlights the disorderly expansion of cities and the consequent widening gap between physical and social space.

Cities are generally more or less sprawling and smaller in social space. Concentration degree index visualizes these differences across the cities in Fig. 7. The value of CDI is between 0.098 and 0.904, 0.474 on average. Toronto has the lowest value of CDI, but this does not mean that Toronto has a small urban radius in social space. As shown in Fig. 5, Toronto, Shanghai, Tokyo, and New York have very large urban entities, and NLI of these cities around city centers is so high that socioeconomic activities seem clustered only in the core area. As a result, these cities appear to have small radii in social space and urban forms that are not compact.

Fig. 7: Concentration degree index (CDI) of 30 cities.
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The CDI is defined as the ratio of radii fitted by nighttime light intensity and urban land density.

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Urban structure of poly-centric cities

With the rapid urban expansion, many megacities have gradually developed an urban distribution pattern with a single main center and multiple sub-centers. For example, in Fig. 5 the scatter plots of urban elements of Ho Chi Minh City and Seoul show slight fluctuations in the overall decay trend. These fluctuations are caused by the agglomeration of elements of other sub-centers. The Inverse-S function is also suitable for poly-centric cities by adjusting the buffer partitioning of cities (Jiao, 2015). Based on the fluctuations in the fitting curves, we chose Ho Chi Minh City and Seoul as poly-centric cities to adjust their buffer partitioning (Fig. 8).

Fig. 8: Spatial gradients of urban land density and nighttime light intensity of poly-centric cities.
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a Buffer partitioning. b Spatial gradients.

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According to the spatial distribution of nighttime light, other clustering places of nighttime light except for the CBD of cities can be determined. And the specific location of sub-centers of cities can be confirmed through the distance between the small and medium fluctuations of the fitting curves and the city centers. The buffer with the lowest density of urban land or the lowest intensity of nighttime light between the main center and a subcenter of a city is considered as the boundary of the area that the two centers can radiate. Figure 8 shows the spatial distribution of the main center and sub-centers of each city as well as their buffer partitioning.

From the parameters of fitting curves in Table 2, it demonstrates that the inverse-S function can be still applicated to ULD and NLI well when using customized poly-centric buffers, and the fluctuations in fitting curves caused by sub-centers are significantly improved. By choosing proper city sub-centers and constructing reasonable poly-centric buffer zones, the inverse-S function can be applied to most cities, even if their spatial structure is varied. While it should be noted that spatial patterns of urban elements across different cities can be compared only with single-concentric buffers (Jiao, 2015).

Table 2 Parameters of inverse-S function for urban land density and nighttime light intensity of poly-centric cities.
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Discussion

In this study, we compare spatial gradients of urban land density and nighttime light intensity using the inverse-S function, which represents the physical and socioeconomic space of cities, respectively. Overall, both the urban land density and nighttime light intensity in concentric rings decrease with the distance to the city centers (Figs. 3 and 4). They first decrease slowly around city centers followed by a quick decay to suburban areas and then decrease slowly again until the background level, showing an obvious inverse-S shape (Fig. 5). This spatial gradient can be well-fitted by the inverse-S function. This study first proves that the inverse-S function can also be applied to nighttime light intensity, which was originally proposed to characterize the spatial gradient of the urban land density.

Spatial agglomeration is a common characteristic of urban elements while urban land density and nighttime light intensity show different spatial gradients. The inverse-S function provides a more intuitive and reliable way of revealing the spatial distribution laws of different urban elements. Generally speaking, the parameters fitted by the inverse-S function of night light intensity and urban land density are quite different. The parameters α and D of NLI are both smaller than those of urban land density, and the aggregation radius of a city in social space is markedly smaller than the radius of the physical entity. From remote sensing images, some megacities also show high urban land density in the suburbs and urban fringes. As the areas within the cities are adjacent to each other, the urban form is more compact with the urban functions more complete (Bay, Lehmann, 2017). However, socioeconomic activities are more fragmented in the hinterland and more spatial agglomeration is shown.

Building inclusive, safe, resilient, and sustainable cities and communities is one of the United Nations’ sustainable development goals (Fritz et al., 2019). A balanced relationship between urban expansion and population growth will promote sustainable urban development (Xu et al., 2019). For many megacities with high populations, they are still expanding outwards and the upper limit of the population that they can accommodate is still rising (Angel et al., 2021). But the population density in urban core areas keeps high relatively and population distribution inside the cities may not match land use. It results in the increment of the cost and pressure of development in the central areas and the extravagance and waste caused by excessive expansion of the city boundary is also a major problem that needs to be solved urgently. Specific measures should be taken to enhance the coordinated and sustainable development of megacities in both physical and social spaces. The government needs to strengthen infrastructure construction in the suburbs and new urban areas to attract population migration from city centers and reduce the burden of resources in urban core areas (Xu et al., 2022).

In this study, nighttime light was used to represent the intensity of social and economic activities, and the density of urban land represented the actual development situation of cities. However, nighttime light cannot express the vitality of cities at daytime and urban land represents the level of development in two-dimensional space, ignoring the total available living space per unit area on a three-dimensional level. In further research, multi-source spatiotemporal data such as population density, road density, and urban three-dimensional building information can be added to understand and analyze the similarities and differences between physical and social space in the process of urbanization from multiple perspectives. In addition, temporal and spatial data can be used to conduct time series experiments to explore the distribution trends and relationships among various urban elements in the process of urban expansion.

Conclusion

This study uncovers different spatial patterns of the agglomeration of urban physical and socioeconomic elements. Taking 30 global megacities as examples, we use urban land and nighttime light to represent urban physical and social space, respectively. It is the spatial agglomeration of different types of urban elements that jointly determine the urban spatial structure and forms a complex urban system. Urban land density and nighttime light intensity share an overall trend of spatial gradients but with different details.

The urban land density function is also applicable for nighttime light intensity, and both urban land density and nighttime light intensity exhibit similar distribution patterns and “inverse-S” spatial gradients. The nighttime light intensity largely depends on the density of urban land. Higher density of built-up areas indicates greater nighttime light intensity, while the spatial gradient of the urban land density is flatter than that of nighttime light. In general, compared with the more fully developed urban land, the intensity of socioeconomic activities in megacities still has a large room for development. Areas of the cities in social space have a smaller coverage than those in physical space. The socioeconomic development of these megacities may not match the disorderly expansion of cities. How to mitigate or exploit this mismatch is worth paying attention to. The potential practical application of the proposed function is to compare the coordination of urban development in different spatial dimensions and to provide policy guidance for the sustainable development of megacities. In addition, the applicability of the inverse-S function can also be tested against other urban elements to explore a more complete mechanism of their spatial gradients.