Characterizing Urban Expansion Combining Concentric-Ring and Grid-Based Analysis for Latin American Cities

Spatio-temporal characterization of urban expansion is the first step towards understanding how cities grow in space. We summarize two approaches used in urban expansion measurement, namely, concentric-ring analysis and grid-based analysis. Concentric-ring analysis divides urban areas into a series of rings, which is used to quantify the distance decay of urban elements from city centers. Grid-based analysis partitions a city into regular grids that are used to interpret local dynamics of urban growth. We combined these two approaches to characterize the urban expansion between 2000–2014 for five large Latin American cities (São Paulo, Brazil; Mexico City, Mexico; Buenos Aires, Argentina; Bogotá, Columbia; Santiago, Chile). Results show that the urban land (builtup area) density in concentric rings decreases from city centers to urban fringe, which can be well fitted by an inverse S curve. Parameters of fitting curves reflect disparities of urban extents and urban form among these five cities over time. Grid-based analysis presents the transformation of population from central to suburban areas, where new urban land mostly expands. In the global context, urban expansion in Latin America is far less rapid than countries or regions that are experiencing fast urbanization, such as Asia and Africa. Urban form of Latin American cities is particularly compact because of their rugged topographies with natural limitations.


Introduction
More and more people are now living in cities, driving the persistent expansion of urban land across the world [1]. Rapid expansion of urban land swallows cultivated land and natural land, threatening biodiversity and exacerbating environmental degradation [2][3][4]. Numerous studies have investigated spatio-temporal characteristics of urban land expansion and its environmental consequences [5,6]. Remote sensing and geographical information science (GIS) have greatly contributed to quantifying land use changes [7]. Remotely sensed imagery provides the first-hand data used to monitor urban growth in space [8]. GIS-based technologies provide a wealth of tools for urban expansion measurement [9].
The first step to measure urban expansion is its speed and intensity, for instance, the annual growth rate of built-up areas [10,11]. The intensity of urban expansion refers to proportions of urban land use changes to the total land area in a defined region [10,12]. Beyond the statistical description of urban land use changes, the spatial perspective is employed to answer the spatial pattern of urban form and where the urban expansion happens [13]. Thirdly, the temporal process cannot be separated when we characterize

Study Area
New York University, United Nations-Habitat, and Lincoln Institute of Land Policy have published The Atlas of Urban Expansion (2016 Edition) of 200 cities around the world, which includes 26 cities in Latin America and the Caribbean [46,47]. Considering the representativeness of large cities and the evenness of their spatial distributions, we selected five large cities with a population of more than five million in 2014 from 26 cities, namely, São Paulo (Brazil), Mexico City (Mexico), Buenos Aires (Argentina), Bogotá (Colombia), and Santiago (Chile) (Figure 1, Table 1). São Paulo is the largest city in Brazil and the economic, cultural, and technological center. Nearly 20 million people lived in São Paulo in 2014 with the total built-up area being around 1700 km 2 . Mexico City is the capital of Mexico and is located in the valley of the plateau of central Mexico, at an altitude over 2000 m. It is the political, economic, cultural and transportation center of Mexico. Buenos Aires is the capital and largest city of Argentina and it is a coastal city on the southern bank of the La Plata River. Bogotá is the capital and largest city in Colombia and it is located in the center of Colombia, on a high plateau known as the Bogotá savanna. Due to the eastern mountains, Bogotá can only expand to the west (Figure 1c). Santiago is the capital and largest city in Chile, which is entirely located in the country's central valley. We identified city centers for the five cities using road maps and high resolution images from Google Map ( Figure 1, Table 1).  being around 1700 km 2 . Mexico City is the capital of Mexico and is located in the valley of the plateau of central Mexico, at an altitude over 2000 m. It is the political, economic, cultural and transportation center of Mexico. Buenos Aires is the capital and largest city of Argentina and it is a coastal city on the southern bank of the La Plata River. Bogotá is the capital and largest city in Colombia and it is located in the center of Colombia, on a high plateau known as the Bogotá savanna. Due to the eastern mountains, Bogotá can only expand to the west (Figure 1c). Santiago is the capital and largest city in Chile, which is entirely located in the country's central valley. We identified city centers for the five cities using road maps and high resolution images from Google Map ( Figure 1, Table 1).

Land Use and Population Data
We collected urban land use maps of five Latin American cities in 2000 and 2014 from The Atlas of Urban Expansion (2016 Edition), which were interpreted using Landsat imagery (30 m resolution) [46]. They classified land use into three categories, namely, built-up area, open space, and water bodies, using the unsupervised classification method (clustering analysis) with the ISODATA (Iterative Self-Organizing Data Analysis Technique) algorithm. The classification result has a high accuracy with verification using Google Earth high resolution imagery. The user's accuracy for the built-up area is 91% and the producer's accuracy is 89.3% [46]. Detailed information on the land use classification procedures can be found in Hurd [48]. The data from The Atlas of Urban Expansion (2016 Edition) has been successfully used in related studies on urban expansion and urban form [49][50][51]. The spatial distributions of urban land in 2000 and 2014 and open space and water bodies in 2014 of five Latin American cities are shown in Figure 1.
The population data in Table 1 also comes from The Atlas of Urban Expansion (2016 Edition), but it is statistical data without information on its spatial distribution. We further collected LandScan population data for the five sample cities in 2000 and 2014. LandScan is a community standard for global population distribution data at approximately 1 km (30" × 30") spatial resolution, and it represents an ambient population (average over 24 h) distribution [52][53][54]. The spatial distribution of population data supports the correlation analysis between urban expansion and population growth using grid-based analysis.

Concentric Ring and Grid-Based Analysis
We generate a series of 1-km equidistant buffer rings to cover almost the entire built-up areas in 2014 ( Figure 2a). Previous studies calculated landscape metrics in concentric rings and analyze their spatial variations [15]. In this study, we calculate the urban land density (built-up density) in each concentric ring, and then fit its distance decay using the inverse S curve (See Section 2.4). The urban land density is defined as the proportion of urban land to the area of buildable land in each ring (Equation (1)).
where Dens is the urban land density in a concentric ring (doughnut). S urban land is the area of urban land (built-up area) in a ring, and S buidable land is the area of buildable land in a ring. The buildable land is the total land area excluding water bodies and mountains in a ring. The concentric rings are developed and urban land density in each ring are calculated in ArcGIS 10.6 (ESRI, Inc., Redwoodds, CA, USA). The unit of grid-based analysis is 1 km, which is consistent with the resolution of LandScan population data. We first clip the LandScan population data in 2000 and 2014 for each city using the spatial extent of land use map of the same city ( Figure 2b). We convert the population raster file to a vector file in 1 km × 1 km grid segments. We calculate the area of built-up area in each grid for two years, 2000 and 2014, using zonal statistics, and then calculate the change of built-up area in each grid from 2000 to 2014. Finally, we use map algebra to calculate the change in population from 2000 to 2014 in each grid of each city and convert the result to a vector file. We combine the changes in built-up areas and where Dens is the urban land density in a concentric ring (doughnut). Surban land is the area of urban land (built-up area) in a ring, and Sbuidable land is the area of buildable land in a ring. The buildable land is the total land area excluding water bodies and mountains in a ring. The concentric rings are developed and urban land density in each ring are calculated in ArcGIS 10.6 (ESRI, Inc., Redwoodds, CA, USA). The unit of grid-based analysis is 1 km, which is consistent with the resolution of LandScan population data. We first clip the LandScan population data in 2000 and 2014 for each city using the spatial extent of land use map of the same city ( Figure 2b). We convert the population raster file to a vector file in 1 km × 1 km grid segments. We calculate the area of built-up area in each grid for two years, 2000 and 2014, using zonal statistics, and then calculate the change of built-up area in each grid from 2000 to 2014. Finally, we use map algebra to calculate the change in population from 2000 to 2014 in each grid of each city and convert the result to a vector file. We combine the changes in built-up areas and population for the 2000-2014 period based on the ID of each grid. All of these spatial analytics, such as format conversion, zonal statistics, and map algebra, are conducted in ArcGIS 10.6 (ESRI, Inc., Redwoodds, CA, USA).

Inverse S Curve
Jiao (2015) proposed a function with an inverse S curve ( Figure 3) to characterize the spatial distribution of the urban land density within a city, as shown in Equation (2) [19]: where f is the urban land density (built-up density) in each concentric ring that is calculated using Equation (1), r is the distance to the city center, e is the Euler's number, and α, c and D are parameters, which vary across cities and over time.
Parameters in the inverse S curve have explicit physical meanings. The α parameter controls the form of the fitting curve and a higher α indicates a more compact urban form ( Figure 3). The D parameter denotes the approximate boundary of an urban extent. The c parameter represents the background value of the urban land density. The D parameter generally increases over time because the urban extent expands. Detailed explanations of these three parameters can be found in Jiao (2015) [19].
The inverse S curve has been an effective and quantitative tool in urban studies. It can be used to not only fit the distance decay of urban land density but also can be applied to other urban indicators, such as the population density, road density, and even land

Inverse S Curve
Jiao (2015) proposed a function with an inverse S curve ( Figure 3) to characterize the spatial distribution of the urban land density within a city, as shown in Equation (2) [19]: where f is the urban land density (built-up density) in each concentric ring that is calculated using Equation (1), r is the distance to the city center, e is the Euler's number, and α, c and D are parameters, which vary across cities and over time.
Land 2021, 10, x FOR PEER REVIEW surface temperature (LST) [19,55,56]. The inverse S curve was first proposed usin nese cities and then it has been applied to cities in other countries across the wor 60].

Distance Decay of Urban Land Density
We calculate the urban land density in each concentric ring using formula (1) all, the urban land density decreases slowly around the city center, and then de quickly to a relatively low level, and finally decreases slowly again to the back parameter represents the background value of the urban land density. The D parameter generally increases over time because the urban extent expands. Detailed explanations of these three parameters can be found in Jiao (2015) [19].
The inverse S curve has been an effective and quantitative tool in urban studies. It can be used to not only fit the distance decay of urban land density but also can be applied to other urban indicators, such as the population density, road density, and even land surface temperature (LST) [19,55,56]. The inverse S curve was first proposed using Chinese cities and then it has been applied to cities in other countries across the world [57][58][59][60].

Distance Decay of Urban Land Density
We calculate the urban land density in each concentric ring using Formula (1). Overall, the urban land density decreases slowly around the city center, and then decreases quickly to a relatively low level, and finally decreases slowly again to the background level of urban land density, showing an inverse S-shape ( Figure 4). The inverse S curve can fit the distance decay of urban land density very well in all cities in both 2000 and 2014. Parameters for fitted curves are shown in Table 2    The α parameter reflects the compactness of urban form and a higher α means a more compact urban form. Bogotá has the highest value of α (=4.49) in 2014, indicating the most compact urban form among the five cities. The urban population density in Bogotá in 2014 is also the highest among the five cities (24,460 person/km 2 , Table 1), further supporting its compact form. Generally, the α parameter increases from 2000 to 2014 in four cities except for Santiago, which has a small decline. Cross-sectionally, Buenos Aires has the lowest value of α, indicating that it has the most relatively dispersed urban form. The urban population density in Buenos Aires is also the lowest among the five cities (9421 person/km 2 , Table 1). The gap between the two curves of the same city for 2000 and 2014 reflects the dynamics of the urban expansion, which varies among the five cities. Mexico City has the largest gap between the two curves, indicating the most obvious urban expansion in this city. Comparatively, the two curves of São Paulo nearly overlap, implying a limited urban expansion in this city. The gap between the two curves also reflects the spatial disparities in urban expansion. The new urban land is mainly added around 10 km from the city center in Bogotá from 2000 to 2014, while in Santiago the urban expansion took place in its outskirts. This is why the urban form of Santiago has become more dispersed during 2000-2014 ( Table 2). The new urban land is evenly distributed in space in Buenos Aires (Figure 4).

Directional Heterogeneity in Urban Form
Urban expansion is generally not homogeneous, but there is usually a directional heterogeneity due to the influence of natural conditions or urban planning [15,61]. Taking São Paulo as an example, we divided the concentric ring into four directions, namely East, South, West, and North, as shown in Figure 5a. Due to the limited urban expansion from 2000 to 2014 in São Paulo (Figure 4), we only take the land use information in 2014 to present the heterogeneity of urban form in different directions. Overall, São Paulo has an urban form with a longer-extent in the east-west direction but a more limited extent in a north-south direction, which is restricted by the macro physical geography (mountains in the north and water bodies in the south). The built-up area in the easterly direction is 717 km 2 , which is the largest among the four directions, taking 42% of the whole built-up area (1724 km 2 , Table 1). The built-up area in the northern direction is only 248 km 2 .
We calculate the urban land density in four directions in São Paulo using concentric rings (Figure 5a). There are obvious disparities in the distance decay of urban land density and fitted curves in the four directions (Figure 5b). The parameters for the inverse S curve in the four directions are shown in Table 3. The fitted urban extents (parameter D) are 55 km, 47 km, 47 km, and 27 km in the East, South, West, and North, respectively. Unsurprisingly, the urban extent in the east is the largest, which is more than twice of that in the north. Remember that the α parameter indicates the compactness of urban form. Urban expansion in the south is restricted by water bodies, resulting in the most compact form in this direction (α = 8.81), while the urban form is relatively dispersed in the west (α = 4.37). There is an obvious bump of the urban land density in the north at around 26 km from the city center, which is caused by leapfrogging urban patches in the north of the city center (Figure 5b). We calculate the urban land density in four directions in São Paulo using concentric rings (Figure 5a). There are obvious disparities in the distance decay of urban land density and fitted curves in the four directions (Figure 5b). The parameters for the inverse S curve in the four directions are shown in Table 3. The fitted urban extents (parameter D) are 55 km, 47 km, 47 km, and 27 km in the East, South, West, and North, respectively. Unsurprisingly, the urban extent in the east is the largest, which is more than twice of that in the north. Remember that the α parameter indicates the compactness of urban form. Urban expansion in the south is restricted by water bodies, resulting in the most compact form in this direction (α = 8.81), while the urban form is relatively dispersed in the west (α = 4.37). There is an obvious bump of the urban land density in the north at around 26 km from the city center, which is caused by leapfrogging urban patches in the north of the city center (Figure 5b).

Correlations Between Land Expansion and Population Growth
We use grid-based analysis to investigate urban land expansion, population growth and their correlations. Changes in population and built-up areas in 1 km grid squares for 2000 and 2014 are presented in Figure 6. There are apparent disparities in space for both land expansion and population growth. Population increases in some places while it decreases in other locations. Taking São Paulo as an example, the urban population decreased around the city center and urban core area, while it increased in the urban fringe and suburban areas (Figure 6a), reflecting the process of suburbanization [62]. Mexico City, Buenos Aires, and Santiago also experienced the transformation of population from the city center to suburban areas, but with different characteristics (Figure 6b,c,h). In Mexico City, urban population transferred from the east to the west during 2000-2014 ( Figure  6b). Although the central part of Buenos Aires had an increased population, the urban population obviously decreased near the city center, forming a doughnut with a reduced population around the center (Figure 6c). The urban population change in Santiago shows the strongest heterogeneity, which declined in the main corridors passing through the city

Correlations between Land Expansion and Population Growth
We use grid-based analysis to investigate urban land expansion, population growth and their correlations. Changes in population and built-up areas in 1 km grid squares for 2000 and 2014 are presented in Figure 6. There are apparent disparities in space for both land expansion and population growth. Population increases in some places while it decreases in other locations. Taking São Paulo as an example, the urban population decreased around the city center and urban core area, while it increased in the urban fringe and suburban areas (Figure 6a), reflecting the process of suburbanization [62]. Mexico City, Buenos Aires, and Santiago also experienced the transformation of population from the city center to suburban areas, but with different characteristics (Figure 6b,c,h). In Mexico City, urban population transferred from the east to the west during 2000-2014 (Figure 6b). Although the central part of Buenos Aires had an increased population, the urban population obviously decreased near the city center, forming a doughnut with a reduced population around the center (Figure 6c). The urban population change in Santiago shows the strongest heterogeneity, which declined in the main corridors passing through the city center. Different from the other four cities, there is no obvious decline in population around the city center in Bogotá (Figure 6g).
Generally, the expansion of urban land shares a similar pattern as seen in the population growth. The new built-up areas mainly occurred at the urban fringe and suburban areas, while a limited newly built-up area was found around the city center (Figure 6d-f,i,j). In São Paulo, the urban expansion hotspots are located in the east and west from 2000 to 2014 (Figure 6b). The urban land spread evenly around the city center in Mexico City, particularly in the north and southeast (Figure 6e). As a coastal city, the new urban land can only spread to the inland side of Buenos Aires and most of these lie along main roads (Figure 6f). Bogotá shows a massive urban expansion northwest of the city center (Figure 6i). The urban expansion surrounded the city center of Santiago with more new urban land located in the northwest of the city (Figure 6j). center. Different from the other four cities, there is no obvious decline in population around the city center in Bogotá (Figure 6g). Generally, the expansion of urban land shares a similar pattern as seen in the population growth. The new built-up areas mainly occurred at the urban fringe and suburban areas, while a limited newly built-up area was found around the city center (Figure 6df,i,j). In São Paulo, the urban expansion hotspots are located in the east and west from 2000 to 2014 (Figure 6b). The urban land spread evenly around the city center in Mexico City, ( Figure 6f). Bogotá shows a massive urban expansion northwest of the city center ( Figure  6i). The urban expansion surrounded the city center of Santiago with more new urban land located in the northwest of the city (Figure 6j).
The frequency distributions of changes in population and built-up area from 2000 to 2014 in 1 km grid squares are shown in Figure 7a

Discussion
The city is composed of different spatial units. In addition to administrative divisions, the two most commonly used units in urban studies are concentric-rings and grids. Concentric-ring analysis, also called gradient analysis, is developed based on the monocentric model of cities, which does well in measuring macro pattern (urban-rural gra-

Discussion
The city is composed of different spatial units. In addition to administrative divisions, the two most commonly used units in urban studies are concentric-rings and grids. Concentric-ring analysis, also called gradient analysis, is developed based on the monocentric model of cities, which does well in measuring macro pattern (urban-rural gradient) of urban expansion and urban form. Grid-based analysis has advantages in portraying the local microscopic dynamics of urban expansion. Grid-based analysis is usually used to build correlations between urban land and population changes. Urban expansion is a process from micro dynamics to macro patterns. Every land use change occurred in a local area, but the accumulation of local micro-processes shaped the macro pattern of urban expansion. The combined use of concentric-ring and grid-based analysis can measure urban expansion from both macro and micro perspectives. Although the concentric-ring analysis and grid-based analysis have been used in previous studies, they are all used in a single way. We propose for the first time that these are two different strategies for measuring urban expansion.
We combined these two approaches to quantify urban expansion for five major Latin America cities from 2000 to 2014. The urban land density (built-up density) in concentric rings decreases from the city center to suburban areas, presenting an inverse S-shape ( Figure 4). Partitioning concentric rings into different sectors can effectively quantify the directional heterogeneity of urban expansion ( Figure 5). The grid-based analysis particularly presents the process of suburbanization where population transferred from central areas to suburban areas from 2000 to 2014 ( Figure 6). Meanwhile, the newly added urban land mainly expands in the urban fringe and suburban areas to accommodate the increasing population there ( Figure 6).
Parameters of the inverse S curve reveal the urban extents and compactness of urban form ( Figure 4, Table 2). Although São Paulo has the largest built-up area, Buenos Aires has a larger urban extent than São Paulo. Buenos Aires experienced the most dispersed urban form, while the urban form is the most compact in Bogotá. The compactness of the urban form quantified by the inverse S curve is consistent with urban population densities that is the highest in Bogotá but the lowest in Buenos Aires ( Table 1). The calculation of population change and built-up change using the grid-based analysis also confirms that the compact growth in Bogotá, where the urban population averagely increase by 944 person at a cost of 1.96 hm 2 for each 1 km grid during 2000-2014 ( Figure 7).
In the global context, the speed of urban land expansion in Latin American cities is relatively low. The intensity of the urban expansion is reflected by the gaps between the two fitting curves using the inverse S curve. The gaps between the two curves in 2000 and 2014 are very close and even overlapped in the five major Latin American cities (Figure 4). Not to mention the fast urban expansion in China and other countries in Asia and Africa, even the expansion of urban land in the United States and Europe is greater than in Latin America [19,44,45,63]. On the other hand, the urban form of Latin American cities is more compact than for cities in other regions, which is quantified by the α parameter of the inverse S curve. The α parameter for the five major Latin American cities varies in 3.39-4.49, while that for Chinese and European cities is around 2.5, and is less than 2 for cities in the United States for the same two years (2000 and 2014) [6,20,44]. Many Latin American cities grow in rugged topographies which constrains expansion, resulting in the overall compact urban form [43].
The distance-decay of urban population densities from the city center has long been revealed in urban geography and urban economics [64]. Classical models include the Clark's negative exponential model [64], the Batty's inverse power function [65] and many other models [66]. The urban land density, defined as the proportion of built-up areas to the buildable land in concentric rings, also declines with distance to the city center, but there are limited quantitative models to measure this process [28]. Urban land density declines slowly around the city center followed by a quick decline outward, and then declines slowly again in the urban periphery, showing an inverse S shape as a whole [19]. Inspired by the widely used sigmoid function with an S curve, Jiao (2015) proposed an inverse S curve to quantify the distance-decay of urban land density from the city center [19]. The urban land density function (inverse S curve) has been not only used to quantify past urban expansion but also to guide the spatial allocation of future urban land, which has potential applications in urban planning [67]. Recently, a geographic micro-process model was proposed to explain why the distance-decay of urban land density presents an inverse S shape [20].
This study also has limitations. This study only quantifies the growth of cities from a spatial perspective, neglecting the underlying driving forces of urban transformation [25]. Secondly, we chose the city center based on the road network and we only identified one city center for each city. With the development of cities, those large cities tend to be polycentric. There are slight bumps of urban land densities around 12 km from the city center in Buenos Aires (Figure 4). These bumps are closely related to the polycentric structure of cities, which can be controlled using polycentric scenarios in concentric ring partitioning [19,45]. In addition, the intervals (1 km, 2 km, and so on) in concentric-ring analysis and the size of grids (1 × 1 km, 2 × 2 km, and so on) have an effect of scale [68], which is not further discussed in this study.

Conclusions
This study contributes to the characterization of urban expansion by summarizing two approaches: Concentric-ring analysis and grid-based analysis. Concentric-ring analysis is developed based on the monocentric model of cities and it is easy to present the distance decay of urban elements from the city center to suburban areas. Grid-based analysis partitions a city into regular grids, which reflects local dynamics of urban growth.
Taking five major cities in Latin America as examples, this study combines concentricring analysis and grid-based analysis to quantify their urban expansion and population growth from 2000 to 2014. Urban land density in concentric rings declines with the distance from the city center. The inverse S curve not only can well fit the distance decay of densities but also reflects disparities in the urban extent and compactness of the urban form. Grid-based analysis characterizes the transformation of population from central areas to suburban areas. It is suggested to combine concentric-ring and grid-based analysis to fully quantify urban expansion from both perspectives of macro patterns and micro dynamics in other global cities.