Urban land density function: A new method to characterize urban expansion

Landscape and Urban Planning 139 (2015) 26–39

Contents lists available at ScienceDirect

Landscape and Urban Planning journal homepage: www.elsevier.com/locate/landurbplan

Research Paper

Urban land density function: A new method to characterize urban expansion Limin Jiao a,b,∗ a b

School of Resource and Environment Science, Wuhan University, Wuhan, PR China Key Laboratory of Geographic Information System, Ministry of Education, Wuhan University, Wuhan, PR China

h i g h l i g h t s • • • • •

We observe an Inverse S-shape Rule for the variation of urban land density. We propose an inverse S-shaped function to formulate urban land density. We derive an established method for concentric partitioning of urban area. Useful indicators are derived to characterize urban form and urban sprawl. The inverse S-shaped model could be applied to other geographical phenomena.

a r t i c l e

i n f o

Article history: Received 3 March 2014 Received in revised form 10 January 2015 Accepted 24 February 2015 Keywords: Urban land density Inverse S-shape Rule Urban expansion Urban form Urban sprawl

a b s t r a c t Density analysis lies at the core of studies on urban expansion; however, many methods in urban land density analysis are arbitrary and suffer from the lack of an established foundation. We observed an “Inverse S-shape Rule” for urban land density that varies outward from an urban center by investigating 28 major cities in China at three time points. We proposed an inverse S-shaped function to formulate urban land density, which fit well for all of the cities in our sample using a nonlinear least squares fitting method. The parameters of the function explicitly describe the basic properties of an urban form. Based on the fitted functions, we derived an established method for the concentric partitioning of urban area and further proposed indicators to measure the urban compactness, urban expansion rate, and degree of urban sprawl. These indicators are practical for characterizing urban form and urban sprawl for either a single city or for multiple cities. A case study on major Chinese cities from 1990 to 2010 reveals that most of the cities expanded rapidly and became less compact and more dispersed during those two decades. However, most of the cities grew faster and showed more sprawl in the second decade compared to the first one. Discussions show that the model is also applicable for non-monocentric cities and possibly can be applied to many other geographical phenomena. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Cities are dynamic complex systems (Bettencourt & West, 2010), and our understanding of how they evolve is still woefully inadequate (Batty, 2008). A fundamental activity of urban expansion research is the quantitative characterization of city morphology and dynamic growth. The beginning of the 21st century marked a milestone because half of the world’s population has resided

∗ Correspondence to: School of Resource and Environment Science, Wuhan University, 129 Luoyu Road, Wuhan 430079, PR China. Tel.: +86 27 68778381; fax: +86 27 68778893. E-mail address: [email protected] http://dx.doi.org/10.1016/j.landurbplan.2015.02.017 0169-2046/© 2015 Elsevier B.V. All rights reserved.

in cities since then (Fragkias & Seto, 2009). According to United Nations projections, urban population will increase by 1.35 billion by 2030, at which time the urban population in the world will be approximately 5 billion (United Nations, 2012). It is forecasted that global urban land will increase by 1.2 million km2 by 2030, which will be almost triple the global urban land area circa 2000 (Seto, Guneralp, & Hutyra, 2012). The most significant urbanization in the future is expected to occur in Asia, especially in China and India (Angel, Parent, Civco, Blei, & Potere, 2011; Güneralp & Seto, 2008; Seto et al., 2012; Václavík, Lautenbach, Kuemmerle, & Seppelt, 2013). It is important and urgent to quantitatively characterize and evaluate regional or global urban expansion to support urban growth predictions and related decision making, especially for the areas expected to experience rapid urbanization in the future.

L. Jiao / Landscape and Urban Planning 139 (2015) 26–39

Many metrics and statistics have been used to characterize urban form and urban sprawl (Arribas-Bel, Nijkamp, & Scholten, 2011; Bhatta, Saraswati, & Bandyopadhyay, 2010; Jat, Garg, & Khare, 2008; Ji, Ma, Twibell, & Underhill, 2006; Liu et al., 2010; Tavernia & Reed, 2009; Wilson, Hurd, Civco, Prisloe, & Arnold, 2003). These metrics, known as landscape metrics, are numerical measurements of the spatial pattern of land-cover at the patch level, class level, and landscape level. Landscape metrics can be generally referred to as spatial metrics because they have been applied to many research fields other than landscape ecology, in particular, urban areas (Herold, Couclelis, & Clarke, 2005; Herold, Goldstein, & Clarke, 2003). Seto and Fragkias (2005) conducted a comparative analysis of four rapidly developing cities in China using landscape metrics, including the edge density, area weighted mean patch fractal dimension, mean urban patch size, and so on. Tsai (2005) classified the metrics describing the urban form into three categories: density, diversity, and the spatial-structure pattern. Schneider and Woodcock (2008) examined the characteristics of the urban form and growth of 25 mid-sized cities across the world using spatial metrics and statistics, such as the built-up area, built-up land density, patch density, and population density. Taubenböck, Wegmann, Roth, Mehl, and Dech (2009) analyzed the spatiotemporal urban types in India by a combination of statistics (e.g., built-up density) and landscape metrics. Angel, Parent, and Civco (2007) presented five attributes of urban spatial structure to describe urban sprawl that were associated with the urban extent, urban population density, suburbanization, and built-up contiguity. In general, the researchers employed spatial metrics (including the built-up density or population density metrics) to characterize urban form and used the variation of the metrics to evaluate the degree of urban sprawl (Li, Li, & Wu, 2013; Xu & Min, 2013). It can be seen that density analyses are basic and important for urban expansion studies from both the studies mentioned above and many others (Bhatta et al., 2010). However, most density metrics used in previous studies were defined arbitrarily. For example, urban land density or built-up land density were defined by a questionable urban extent or based on a subjectively defined concentric partition of a city. A probe of some basic issues of urban land density is necessary to provide a comprehensive theoretical foundation for density analysis in urban studies. Concentric partitioning of cities was commonly used in studies on urban form and urban sprawl. The variation in spatial metrics or density variables from the city center outward was often discussed (Irwin & Bockstael, 2007). Seto and Fragkias (2005) quantified the spatiotemporal patterns of cities by analyzing the spatial metrics in experientially defined buffer zones, namely, 0–3 km, 3–10 km, and 10–20 km. Taubenböck et al. (2009) analyzed urban structure based on six ring-shaped zones around the main urban center. Schneider and Woodcock (2008) defined the urban core area as a circular area with an urban land density above 50% and divided the remaining landscape into fringe, periphery, and hinterland regions with three 8-km buffers. The buffer zones around the urban center were usually defined by a threshold value for urban land density or given fixed buffer distances that were largely based on the experience of researchers. The researchers found that urban land density generally decreased from the urban center to the outside. Can we identify a universal and explicit rule for the variability of urban land density? Can we quantitatively describe this rule with a reasonable formulation? Researchers have formulated many functions to describe urban population density, such as the negative exponential function, the Gaussian model, and the inverse power function (Batty & Kim, 1992; Chen & Feng, 2012; Clark, 1951). Until recently, there has been no formulation for urban land density. Compared to an urban population distribution, land use data are much easier to accurately acquire using remote sensing data. This provides a perfect

27

data source to precisely investigate changes in urban land density and makes it possible to accurately formulate the trends of changes in urban land density. In this study, with 28 major cities at three time points in China as samples, we will illustrate the general trend of urban land density variation from the urban center to the outside and propose a mathematical formulation of the rule. Then, we will investigate the properties of the function and discuss how to use the function to characterize urban form and the dynamics of urban sprawl. In this study, we observed an “Inverse S-shape Rule” for the variation of urban land density. This is the first report of the “Inverse S-shape Rule” for urban land density, and we also introduce new methods to characterize urban form and measure urban sprawl based on the S-shaped urban land density function. 2. Data 2.1. Sample of cities We examined the urban land density of a total of 28 cities in mainland China in 1990, 2000, and 2010. Most of the cities are provincial capitals or municipalities. Two large cities located in western and southwestern China, Chongqing and Guiyang, were not included in the study due to a lack of available data. These cities are distributed across the Chinese mainland (see Fig. 1). Although most of the cities are capitals or municipalities, they are diverse in scale due to imbalanced regional development. The biggest cities in the sample, such as Shanghai and Beijing, had a population of more than 15 million in 2010 (National Bureau of Statistics of China, 2011), while the smallest cities, such as Haikou and Yinchuan, had a population of 0.37 million and 0.56 million in 1990, respectively (National Bureau of Statistics of China, 1991). 2.2. Image classification We acquired high quality Landsat TM/ETM+ images for the cities for the years 1990, 2000, and 2010. All of the images are cloud-free (cloud < 10%) and were taken in summer months. Google online maps were used as references to acquire ancillary information, such as administrative information and transportation. The steps of the image analysis included preprocessing and classification. Preprocessing the images included geometric correction, geographic registration, and resizing. The ETM+ images were resized to 30-meters to be consistent to the others. Image classification was performed using the Maximum Likelihood Classification method in ENVI 4.5. Four classes were extracted for each image, namely, built-up areas, vegetation, water, and other lands. We assessed the accuracy of each classified image with more than 250 independent samples, which were identified by visual interpretation. We revised the results or reclassified the images according to the accuracy assessment results. Finally, the accuracies for all of the results ranged from 85% to 93%. 3. Urban land density function 3.1. Defining urban land and urban extent There are two problems to be addressed in defining urban land. One is defining what types of land are ‘urban’. The other is determining the spatial extent of a city. Urban land in physical terms refers to a complex of impervious surfaces and urban vegetation. Impervious surfaces, including pavement, roofs, and compacted soil, that are closely associated with a built-up environment dominate an urban area. When using remote sensing images, urban land is best defined by impervious surface area (Arnold & Gibbons,

28

L. Jiao / Landscape and Urban Planning 139 (2015) 26–39

Fig. 1. Cities included in this study.

Fig. 2. The concentric ring partitioning and the outer boundary of a city.

L. Jiao / Landscape and Urban Planning 139 (2015) 26–39

29

Fig. 3. Urban land density measured across space (distance from the center of the city outwards, as indicated on the x axis).

1996). For simplicity and consistence, urban land was defined as impervious surface area in this study. We employed a standardized method for defining the geographical extent of each city on the basis of concentric ring partitioning, which was used in many previous studies (Schneider & Woodcock, 2008; Seto & Fragkias, 2005; Wolman, Galster, & Hanson, 2005). Ring-based analysis is firmly grounded in classic urban theory, and the bias of this method is limited (Burgess, 1967; Schneider & Woodcock, 2008). By considering the central business district (CBD) in 1990 as the center, we created 1-km buffers in an increasing, step-wise manner. Each city was partitioned into a series of 1-km rings extending outward from the center. We selected an outer ring as the final boundary for a city according to two criteria. The first criterion was that the boundary should be large enough to incorporate a continuous functional urban area (Piorr, Ravetz, & Tosics, 2011) as well as the outer ring road of the city. The second criterion was that the city boundary should be constrained to exclude small cities that are far from the city core and do not function as part of the metropolitan area. We defined the spatial extent of a city in terms of both the physical meaning and function instead of using administrative boundaries or other methods. Using administrative boundaries as the spatial extent of a city is unsuitable because they are changeable, include areas that are too large or too small areas, and are not comparable across cities. The concentric ring partitioning and the outer boundary of Zhengzhou city is shown as an example in Fig. 2.

3.2. The inverse S-shaped decrease of urban land density Based on the concentric ring partitioning of a city, we calculated the urban land density (i.e., the area of the urban land divided by the total land area) for each ring. Large water bodies in a city should be excluded in calculation since they will not be used for urban development. To minimize the bias in calculation, the area of water bodies was subtracted from the total area if its influence on the urban land density of any ring is larger than 1%. By examining plots of the variation of urban land density versus the distance to the city center, we can determine the general spatial pattern of urban land variation. Fig. 3 shows plots of the spatial variation of urban land density for three typical cities, which represent different scales of the cities in our sample. Plots of urban land density for the rest of the cities can be found in Fig. 5. We can discover the common rules for urban land density variation by examining scatter plots. First, urban density decreases outward from the city center in general. This is consistent with our traditional understanding of urban land density. For most cities, urban growth moves outward from a central core (Blumenfeld, 1954; Schneider & Woodcock, 2008). The urban land density is high

in the core and decreases from the urban core to inner urban and suburban zones, the urban fringe, the periphery, and the hinterland. Second, the variation of urban land density is not described by any familiar curve that is often used for an urban population density analysis, gravitational modeling, mathematical economics, social physics, and so on. It is neither a linear function, a negative exponential function, nor an inverse power function. The rate of the decline of urban land density from the urban center outward varies with the distance from the center. Graphs of urban land density variations exhibit a distinct inverse S-shape, which implies that the urban land density decreases slowly in the main urban core area, decreases relatively quickly in the inner urban and suburban areas, and then decreases slowly again in the urban fringe and periphery. The rest of this paper will explain our results on the variation of urban land density that verifies this hypothesis, which could be called the “Inverse S-shape Rule” for urban land density. 3.3. Formula for the urban land density decline We propose a modified sigmoid function with an inverse Sshape to describe the spatial variation of urban land density, which is defined as f (r) =

1−c 1 + e˛((2r/D)−1)

+C

(1)

where f is urban land density, r is the distance to the urban center, e is Euler’s number, and ˛, c and D are constants. The graph of the function is shown in Fig. 4(b), with the constants ˛ = 4, c = 0.05, and D = 30. The proposed urban land density function is continuous, monotonically decreasing, and differentiable. It has two horizontal asymptotes, i.e., f = 1 and f = c. When r = 0 (representing the city

Fig. 4. Graphs of the inverse S-shaped model.

30

L. Jiao / Landscape and Urban Planning 139 (2015) 26–39

Fig. 5. Graphs of the fitted urban land density functions.

L. Jiao / Landscape and Urban Planning 139 (2015) 26–39

Fig. 5. (Continued ).

31

32

L. Jiao / Landscape and Urban Planning 139 (2015) 26–39

Table 1 Parameters of the fitted urban land density functions. City

Beijing Shanghai Guangzhou Tianjin Wuhan Shenyang Chengdu Qingdao Xi’an Harbin Jinan Nanjing Shijiazhuang Changchun Nanchang Zhengzhou Hefei Changsha Hangzhou Taiyuan Kunming Hohhot Fuzhou Lanzhou Yinchuan Haikou Xining Nanning

1990

2000

2010

˛

c

D

R2

˛

c

D

R2

˛

c

D

R2

3.788 4.713 2.761 4.448 2.808 3.528 6.652 2.443 4.618 4.679 4.226 1.906 7.856 3.998 4.388 5.361 2.463 3.978 3.256 2.486 3.885 5.221 3.641 3.507 3.316 3.591 3.424 3.600

0.057 0.039 0.033 0.026 0.013 0.010 0.021 0.055 0.022 0.018 0.051 0.009 0.092 0.016 0.035 0.064 0.037 0.011 0.054 0.021 0.026 0.003 0.015 0.046 0.046 0.039 0.021 0.002

22.895 15.763 8.943 14.868 11.638 14.753 11.141 10.304 11.925 12.701 11.801 8.808 12.003 13.626 8.520 11.678 8.037 10.736 10.263 9.479 9.269 8.426 7.663 5.048 2.783 5.190 4.979 8.532

0.994 0.997 0.990 0.996 0.992 0.996 0.998 0.990 0.992 0.991 0.995 0.975 0.996 0.970 0.994 0.996 0.996 0.997 0.992 0.989 0.990 0.988 0.996 0.935 0.967 0.995 0.998 0.998

3.338 4.394 2.476 4.329 3.317 4.362 5.470 2.296 5.008 4.770 4.572 2.485 5.497 4.600 4.480 4.873 3.259 4.144 3.484 2.715 3.896 5.217 3.538 3.815 2.425 2.216 2.943 4.178

0.119 0.110 0.113 0.059 0.025 0.087 0.050 0.078 0.038 0.050 0.065 0.045 0.132 0.032 0.046 0.102 0.050 0.036 0.070 0.046 0.033 0.018 0.048 0.077 0.077 0.026 0.016 0.005

29.715 26.634 13.759 16.826 17.392 20.186 15.166 15.145 14.548 13.891 14.646 13.495 13.929 16.633 10.537 14.871 11.284 13.010 11.702 11.787 11.989 11.130 10.546 5.686 4.179 7.754 6.206 10.201

0.990 0.991 0.981 0.992 0.985 0.995 0.994 0.956 0.991 0.993 0.995 0.979 0.992 0.997 0.994 0.987 0.995 0.995 0.992 0.994 0.988 0.993 0.995 0.957 0.879 0.967 0.995 0.998

3.494 2.556 1.868 3.997 3.200 4.070 3.629 2.497 5.410 3.791 4.386 2.282 4.934 3.643 3.285 4.075 3.011 3.756 1.850 2.753 3.411 3.993 3.105 2.205 0.803 2.649 2.828 3.670

0.220 0.259 0.285 0.255 0.035 0.153 0.101 0.199 0.070 0.095 0.201 0.099 0.197 0.025 0.119 0.119 0.140 0.032 0.232 0.074 0.069 0.019 0.054 0.101 0.034 0.052 0.076 0.016

37.694 44.341 18.182 21.727 27.542 23.232 21.711 28.051 24.518 18.575 18.091 25.903 15.913 21.270 20.296 21.964 14.601 18.703 12.697 14.355 16.619 15.090 12.215 7.267 6.593 16.365 7.907 12.607

0.990 0.985 0.979 0.984 0.979 0.990 0.993 0.946 0.996 0.985 0.993 0.971 0.991 0.987 0.989 0.988 0.982 0.995 0.975 0.996 0.998 0.997 0.993 0.939 0.917 0.975 0.974 0.996

center), f(r) equals (1−c)/(1+e−˛ )+c, which approximates 1 because ˛ is usually between 2 and 6 (see the fitting result in Section 3.4). As r approaches infinity, f(r) approaches c.

4. Characterizing the urban form

boundary distance of the main urban area for a city (including the urban core, inner urban and suburban zones, and the urban fringe), defined by the variation in urban land density. It also can be found that D increases with time for a single city, which describes the expansion of an urban area. Outliers for the interpretation of the parameters exist because the interpretation is based on the assumption that urban expansion is monocentric in form. The interpretations of the parameters may be not accurate for the cities that are linear or multinucleated. These types of cities are represented in our city sample. Lanzhou and Yinchuan are linear cities, and Hangzhou and Guangzhou are polycentric cities. In 1990, these cities were almost monocentric, like other cities. After rapid growth for twenty years, they transformed from their original form and showed clear linear or polycentric characteristics. These transformations are also reflected by the urban density curves in Fig. 5. For these cities, the parameters of the urban land density function represent the corresponding characteristics of the main part of the city where CBD is located but not the characteristics of the entire city. Further discussion of model fitting for non-monocentric cities is provided in Section 6.1.

4.1. Physical meaning of the parameters of the urban land density functions

4.2. Concentric partitioning of cities based on urban land density

The fitted urban land density functions quantitatively describe the declining trend of urban land density from the urban center to the outskirts, and the parameters of the function denote some basic characteristics of the urban form. Because f = c is an asymptote of the function, the constant c represents the background value of built-up land density in the hinterland of the city. Fig. 5 and Table 1 show that c increases with time for almost every city, which indicates that construction on land in the surrounding rural area increases due to the development of the entire region. 1−c When r = D, f (D) = 1+e ˛ + c, which is slightly higher than c and indicates that the constant D denotes the approximate boundary between urban fringe and the periphery or hinterland. D is the

To understand the decrease of urban land density from the urban core to the outside, such as the decreasing rate and its variation, we must examine the derivatives of the fitted curves for urban land density. The first derivative (denoted by f (r)) and the second derivative (denoted by f (r)) of the urban land density function are shown in Fig. 6. The first derivative of the urban land density function describes the decreasing rate of urban land density, and the second derivative describes the rate of the change of the decreasing rate of urban land density. We can achieve a deeper understanding of the “Inverse S-shape Rule” of urban land density from the derivatives. From the urban center to outside, the decreasing rate of urban land density first increases until it achieves the maximum, and then decreases to the

3.4. Fitting of the urban land density functions We employed a non-linear least squares method to fit the proposed urban land density functions for the cities. The basis of the method is to fit a nonlinear function to the observed data by refining the parameters in successive iterations. There are several algorithms that can be used to perform non-linear least squares fitting, such as the Trust-region algorithm, the Levenberg–Marquardt algorithm, and the Gauss–Newton algorithm. The Trust-region algorithm was used in this study. We fit the urban land density functions with Matlab R2010b. The fitted curves and the estimated parameters for urban land density functions are shown in Fig. 5 and Table 1.

L. Jiao / Landscape and Urban Planning 139 (2015) 26–39

33

Table 2 Statistics of c in urban land density functions. Year

Mean

Minimum

Maximum

Standard deviation

1990 2000 2010

0.029 0.055 0.112

0.002 0.005 0.016

0.092 0.132 0.259

0.021 0.033 0.074

the outside (Angel et al., 2007; Schneider & Woodcock, 2008). We partitioned the urban area according to the variation of urban land density, which is a more accurate method based on a more intrinsic rule. Three specific points, p1 , p0 and p2 , can be used to clearly partition the urban area in a concentric manner such that four circular areas are defined from the urban center outward: the urban core, the inner urban area, the suburban area, and the urban fringe (see Fig. 6). By our partitioning, the urban core is the central area of a city, with a high urban land density that is slowly decreasing. The inner urban area and the suburban area both have a rapidly decreasing urban land density, but the former’s urban land density is higher than f0 . The urban fringe surrounds the main urban area and has a low urban land density that decreases relatively slowly. We noticed that f0 , f1 , and f2 are determined only by parameter c. The statistics of c for the urban land density functions are listed in Table 2. The four cities that were not monocentric were not used to derive the statistics. Parameter c for most cities is less than 0.1, although some extremely high values exist. For example, the c value for Beijing in 2010 was 0.220, which is characteristic for metropolises that are located in metropolitan areas that are connected to surrounding cities by almost continuous urban areas. In general, if we set c = 0.05, f1 , f0 , and f2 are approximately equal to 0.8, 0.5, and 0.25, respectively. These values can be used as thresholds for urban land density to concentrically partition urban area for common analyses. 4.3. Measuring the compactness of cities Fig. 6. The derivatives of the urban land density function.

hinterland. The variation in the rate of the decreasing rate of urban land density also changes from the urban core to the outside. By examining the first derivative, we can infer that the coordinates of the fastest decreasing point (denoted by p0 in Fig. 6) in the curve of the urban land density function is r0 =

D , 2

f0 =

1+c 2

(2)

We can see that the second derivative has two extrema, which denote the locations where the rate of decrease changes most greatly. The two corresponding points on the curve of urban land density are denoted by p1 and p2 in Fig. 6. The coordinates of p1 are r1 =

D 2

 −1.316957 ˛



+1 ,

f1 = 0.788675(1 − c) + c

(3)

and the coordinates of p2 are D r2 = 2

 1.316957 ˛



+1 ,

f2 = 0.211325(1 − c) + c

Some researchers described the measurement of the compactness of cities by analyzing their shapes (Angel, Parent, & Civco, 2010; Tsai, 2005). It is difficult to depict the shapes of cities in many cases because the delineation of the urban boundary is often arbitrary. The problem can be overcome when we characterize the compactness of cities using our urban land density functions. The urban land density of a compact city is high in the core area of the city and decreases sharply to a very low value in the area surrounding the core. The curves of urban land density for compact cities are steep, but those for sprawling cities decrease slowly; therefore, the slope of urban land density function reflects the overall density of the urban land. We used two points, p1 and p2 , to define the slope of urban land density function. The slope, denoted by ks , is calculated by ks =

f1 − f2 0.57735(1 − c)˛ = r2 − r1 1.316957D

(6)

(4)

where D, c, and ˛ are the parameters of urban land density function. The ks values of four cities of similar size in 2000 are shown in Table 3. It can be seen that Wuhan has a much lower ks value than

(5)

Table 3 ks and kp of four cities.

Obviously, r1 + r2 = D,

f1 + f2 = 1 + c

The detailed derivation of the derivatives and the coordinates of the points mentioned above can be found in Appendix A. Obviously, r0 , r1 , and r2 are functions of D and ˛, and f0 , f1 , and f2 are functions of c. Urban land density is often used in concentric partitions of the urban area. For example, many researchers use 50% as the threshold value for urban land density between the inner urban area and

City

Wuhan Jinan Zhengzhou Xi’an

ks

kp

1990

2000

2010

1990

2000

2010

0.10 0.15 0.19 0.17

0.08 0.13 0.13 0.15

0.05 0.09 0.07 0.09

0.47 0.31 0.25 0.29

0.40 0.29 0.27 0.26

0.41 0.30 0.32 0.24

34

L. Jiao / Landscape and Urban Planning 139 (2015) 26–39

Fig. 7. kp of cities in 1990, 2000, 2010.

the other three cities, which indicates that Wuhan has a lower overall urban land density and possibly experienced more expansive urban growth. Although ks can provide an intuitionistic estimate of the overall urban land density, it must be mentioned that the index can lead to a biased interpretation when we examine the index across cities with diverse sizes because ks is determined by three parameters in which D is a changeable variable. A higher D leads to a lower ks . The ks value for a city tends to decrease with urban growth, and large cities tend to have lower ks values. Thus ks is useful in the comparison of urban densities only when it is used among cities of the same or similar scales. A second index was proposed to avoid bias, which is designed as the proportion of the rapidly decreasing part of the curve, i.e., the part denoting the inner urban and suburban areas. The index, represented by kp , can be written as kp =

r2 − r1 1.316957 = D ˛

(7)

where ˛ is the parameter that controls the slope of the curve of urban land density function. A compact city has a high urban land density in the urban core area, which decreases quickly to a very low value outside the urban core. A compact city has a narrow area that covers the inner urban area and the suburban area and thus has a small kp value. On the other hand, a city with a large kp value is a low-density or dispersed city. The kp value is only related to ˛ and is suitable as an index for the degree of dispersion of diverse cities with time. The kp values of the four cities in Tables 2 and 3. Wuhan was much less compact than the other three cities in 1990, 2000, and 2010. We also can see that ks decreases with time while kp is not influenced by urban growth, and it is consistent with our former hypothesis that kp is more robust. Fig. 7 shows the change of the kp value for the 24 cities and three time points. In 1990, Nanjing, Qingdao, Hefei, Taiyuan and Wuhan were the top five dispersed cities, and Shijiazhuang and Chengdu were the two most compact cities. Over the next twenty years, only a few cities became slightly more compact, such as Nanjing, Hefei, Taiyuan, Shenyang and Xi’an. Most of the other cities became more dispersed. Particularly, the kp values for Shanghai, Chengdu, Nanchang, and Haikou increased significantly, which implied that these cities experienced dispersed urban growth for two decades. Figure 7 shows that the former three cities became more sprawling in the latter decade. The statistics of kp for the 24 cities are listed in Table 4. The average kp in 2010 was higher than those in 1990 and 2000, and the standard deviation in 2010 was lower, which shows that, in general, the cities are more dispersed in 2010 than earlier.

Table 4 Statistics of kp . Year

Mean

Minimum

Maximum

Standard deviation

1990 2000 2010

0.36 0.35 0.39

0.17 0.24 0.24

0.69 0.59 0.58

0.12 0.12 0.08

In other words, prevalent urban sprawling occurred during those two decades. The kp value is a continuous measurement of urban dispersion at a specific time point in terms of urban land density. 5. Measuring the dynamics of urban expansion 5.1. Measuring the rate of urban expansion The rate of the urban expansion of a city is reflected by the growth rate of its size. As for the fitted urban land density function, parameter D is an estimation of the radius of the main urban area of a city. By examining the variation of the parameter, we can compare the growth rates of various cities. Fig. 8 shows the growth rate of D values for the 24 cities over the two decades. Most of the cities increased their radii by 1.1- to 1.5-fold from 1990 to 2000, except for Shanghai (1.7 times), but seven cities increased their radii by more than 1.5-fold from 2000 to 2010. The growth rates over the second decade for all cities except Hefei, Fuzhou, and Shenyang were almost equal to or higher than the growth rates over the first decade. Particularly, Nanjing, Haikou, Qingdao, and Nanchang experienced tremendous growth during the second decade. In general, it can be inferred that the Chinese cities grew more quickly during the second decade than earlier. The observed growth rates of D implies significant urban growth because the area increased exponentially. 5.2. Measuring the degree of urban sprawl It is a general consensus that urban sprawl is characterized by an uneven growth pattern in a low-density manner, leading to inefficient land resource utilization (Bhatta et al., 2010). Many researchers analyzed urban sprawl using land cover data in conjunction with census data, for example, to examine the variation of urban population density, which can be well characterized using spatial metrics or other indices representing land cover patterns (Bhatta et al., 2010; Schneider & Woodcock, 2008). The basic characteristic of urban sprawl is low-density expansion. We can clearly characterize the process of urban expansion in terms of urban

L. Jiao / Landscape and Urban Planning 139 (2015) 26–39

35

Fig. 8. Growth rate of the radius of the main urban area of cities.

density using our urban land density functions, and then, we can estimate the degree of urban sprawl. Based on the quantitative description of the inverse-S decline of urban land density, we can define the urban core area, the inner urban and suburban zones, and the urban fringe. The basic of our assumption is as follows: if the growth rate of suburban area exceeds that of the urban core, then the density of the city tends to be decreasing and the city is sprawling. For the concentric partitioning of cities based on urban land density, we defined the radius of urban core as r1 and the radius of suburban area as r2 (see Eqs. (3) and (4)). For a city in a given time span, if r2 increases faster than r1 , we can infer that urban expansion was dominated by dispersed growth instead of infilling. Thus, the ratio of the growth rate of r2 to that of r1 can be used as an indicator of the degree of urban sprawl, which can be defined as Sr =

(r i − r2i−1 )r1i−1 ır2 = 2 ır1 r2i−1 (r1i − r1i−1 )

(8)

where Sr represents the degree of urban sprawl; ır1 and ır2 represent the growth rates of r1 and r2 , respectively; r1i and r2i are the radii

of the urban core and the suburban zone of a city at time i, respectively; and r1i−1 and r2i−1 are the radii at a previous time-point. It should be noted that Sr may produce unreasonable values in very few cases when ır1 approximates zero or ır1 and ır2 have negative values. Fig. 9 shows the 2-dimensional plots that are used to illustrate the index for the cities. A few cities were not included in the figure because their coordinates exceeded the range of the axes, and their coordinates are listed in a box in each plot. Obviously, in the plots, the index Sr for a city can be represented by the slope of the line from the origin to the point corresponding to the city. A higher slope indicates a higher degree of urban sprawl. The Sr values of the cities in the upper left area of the plots are larger than 1, and the corresponding cities can be considered to be sprawling. At the same time, the plots can also reflect the growth rates of the cities. Particularly, it can be inferred that the cities with high Sr values have largely increased the boundaries of their fringes. The cities that are distant from the origin and closest to the upper left corner are the most rapidly growing and sprawling, and the cities that are far from the origin and closest to the lower right corner have rapid-growing core areas and remain compact. The cities closest

Fig. 9. Plots of the comparison of the growth rates of r2 and r1 of the cities.

36

L. Jiao / Landscape and Urban Planning 139 (2015) 26–39

Fig. 10. Plots of the comparison of the kp values of cities.

to the diagonal are neither sprawling nor compact, but those far from the origin grow much faster than those close to the origin. The sprawling properties reflected by the plots only refer to the expansion over the corresponding time span and are not an overall estimation of the urban expansion for the entire history of the city. It can be seen clearly from Fig. 9 that most of the cities grew much more expansively from 2000 to 2010 than during the previous decade. Most were located in the upper left area in the plot of 2000–2010, which implied that, in general, the cities were sprawling during that decade. In the preceding decade, Beijing, Chengdu, Qingdao and Haikou were sprawling and fast-growing. Shijiazhuang and Xining also had high Sr values but did not grow so quickly. Although Shanghai did not show much sprawl, it did grow rapidly because its ır2 value was very high. In the second decade, Shanghai, Chengdu, Nanjing, Changchun, Harbin, Hohhot, Nanchang and several other cities were growing rapidly and sprawling. Xi’an greatly expanded but remained compact during that period. Because the kp index introduced in Section 4.3 represents the degree of dispersion of cities, we also can evaluate the degree of sprawling by comparing the kp values at two time-points. An increase in the kp value of a city means that the compactness of a city is less and it possibly sprawls. Fig. 10 shows a plot of the kp values of the cities, comparing urban dispersion at different time-points. It can be seen that the sprawling of the cities was accompanied with a decrease in compactness. Compared to the previous decade,

more cities decreased their compactness from 2000 to 2010, which indicated that these cities sprawled more greatly in the second decade. It should be noted that although the cities located close to the diagonal in the plots in Fig. 10 are not classified as sprawling, in general, those close to the origin are more compact than those distant from the origin. For example, Xi’an and Taiyuan did not show sprawling from 2000 to 2010, but Xi’an was much more compact than Taiyuan. Comparing Fig. 10 with Fig. 9, we can see that we may derive similar conclusions on urban sprawling properties, but Fig. 9 simultaneously shows the urban growth rate while Fig. 10 simultaneously indicates urban compactness. 6. Discussion 6.1. Model fitting for non-monocentric cities The proposed model was formulated for monocentric cities or nearly monocentric cities. By adjusting the buffer partitioning of cities, we can make the model applicable to non-monocentric cities, such as polycentric or linear cities. Fig. 11 shows examples of buffer partitioning for three semi-monocentric or non-monocentric cities, Wuhan, Hangzhou and Lanzhou. The Yangtze River and Han River converge in central Wuhan and split it into three comparable parts, Hankou, Hanyang and Wuchang. The city can be seen as polycentric because each part has

Fig. 11. Buffer partitioning of non-monocentric cities.

L. Jiao / Landscape and Urban Planning 139 (2015) 26–39

37

Fig. 12. Fittings of urban land density for non-monocentric cities.

its own center. Wuhan can also be seen as generally monocentric because the three centers form an equilateral triangle and are close to each other. The fitting result of the urban land density function of Wuhan under the monocentric hypothesis can be found in Fig. 5 and Table 1, and the fitting result under the polycentric hypothesis is shown in Fig. 12. The two fits both have high coefficients of determination, but the latter is better. Hangzhou city has a CBD and some subcenters. We performed the buffer partitioning of Hangzhou in three steps. First, we chose two centers, for example, the CBD and a subcenter, and found the location where urban land density was lowest along the belt between CBD and the subcenter. Second, we buffered the two centers separately and merged the subsequent buffers from the two buffers that met at the location with the lowest urban land density (Fig. 11). Third, step two was repeated for the remaining pairs of centers, the buffers were numbered from the CBD center to the outside, and the numbers of the buffers generated from subcenters were retrieved from the merged buffers (Fig. 11). Lanzhou is a linear city (Fig. 11). We can perform buffer partitioning of the city from its central line, which is mainly composed of a set of segments of main roads. Meanwhile, Lanzhou can also be roughly treated as a polycentric city with four subcenters (Fig. 11). We can see that the fitting results based on non-concentric portioning are better than those based on concentric partitioning for these non-monocentric cities from Fig. 12 and Table 1, which implies that we can apply the inverse s-shaped function to fit the distribution of urban land density for most cities if proper buffer partitioning is employed. Customized buffer partitioning improves the fitting performance and can provide more reasonable spatial partitioning for the temporal analysis of the urban expansion of a non-monocentric city. It should be noted that customized buffer partitioning could not be used to compare the

characteristics derived from the fitted functions between different cities. 6.2. The generalization of the “Inverse S-shape Rule” It will be interesting to investigate whether the proposed inverse s-shaped model is applicable to other geographical phenomena. Due to limited available data, we can only test some examples in urban areas. Built-up land is highly associated with many phenomena in urban areas, such as urban population, road density, and commercial facilities. Urban land density is easy to precisely measure compared with urban population density, and this facilitates a quantitative formulation of its distribution. Fig. 13 shows the results of model fitting for the densities of selected phenomena in urban areas, including urban roads in Shenyang, Kunming, and Hohhot in 2013 (Fig. 13(a)); the urban population in Wuhan in 2010 (Fig. 13(c)); and public facilities (restrooms, book stores, newsstands, kiosks, and emergency shelter), Chinese restaurants, and convenience stores in Beijing in 2012 (Fig. 13(b)). Due to the differences in the data range, Eq. (1) was slightly revised to f (r) =

m−c 1 + e˛((2r/D)−1)

+c

(9)

where f = m and f = c are the two horizontal asymptotes. All of these distributions are well fitted with the inverse S-shaped model. On the basis of the above analyses, we can conclude that the “Inverse S-shape Rule” is applicable for the characterization of urban land density in China and that it is probably applicable for the description of other geographical phenomena in urban areas. A more general hypothesis should be tested in the future: Spatial aggregation is ubiquitous, and the density of a geographical

38

L. Jiao / Landscape and Urban Planning 139 (2015) 26–39

Fig. 13. Model fittings for different phenomena in an urban area.

phenomenon usually decreases from center to outside in an inverse S-shaped manner. 7. Conclusions Density analysis lies at the center of studies on urban form and urban sprawl. This study attempted to address the lack of universal and quantitative methods to describe urban land density. With 28 major cities in China as samples, we observed an “Inverse-S shape Rule” in the variation of urban land density from urban center outward. We proposed an urban land density function by modifying an ordinary sigmoid function. The urban land density functions for all of the cities in our sample were fitted well by non-linear least squares fitting. The parameters of the urban land density functions were shown to explicit physical analogs concerning urban forms. Based on the fitted urban land density functions, we derived a method to partition urban areas concentrically, which avoids the subjectivity of previous research. We further proposed indicators to measure the urban compactness, urban expansion rate, and degree of urban sprawl. These indicators can be used to compare cities at different time points and different cities at the same time point. Plots of the indicators can illustrate the properties of urban sprawl very well. The proposed model is based on the “Inverse S-shape Rule”, which is different from other models used in urban population density analysis that are based on various hypotheses, such as the exponential model, the Tanner-Sherratt model, and Newling’s model (Wang, 2006, chap. 6). Our model’s parameters have clear physical meanings, and the derived variables are useful for describing urban structure and characterizing urban expansion. The model does not generate unreasonable values in model fitting, for example, for urban land density values above 1. Our analysis on the expansion of major Chinese cities over two decades showed that most of the cities decreased their compactness and became more dispersed. Most of the cities increased their boundary radius by 1.2- to 1.6-fold during each decade and generally grew more quickly in the second decade compared to the previous one. As for the sprawling properties, most of the cities expanded more from 2000 to 2010 than during the previous decade. Haikou, Beijing, Fuzhou, and Chengdu were most sprawling and rapidly growing during the first decade, and Nanchang, Shanghai, Nanjing, Chengdu, Harbin, Changchun, Hohhot, and several other cities had the highest growth rates and degrees of sprawling during the second decade. This research identified a common fact that urban land density decreases from the urban center outward according to the inverse S-shape rule. For polycentric or linear cities, the model fitting can be improved by using appropriate non-concentric buffer partitioning. In many cases, a polycentric or linear city can be treated as a monocentric one when we apply a coarse-grained concentric partition. More experimental case studies elsewhere in the world are needed to test our findings, which were based on a sample of

Chinese cities. A city is a complex conglomeration representing the common phenomenon of human-induced spatial aggregation. We suppose that the “Inverse S-shape Rule” could be ubiquitous for many spatially aggregated phenomena, which is a hypothesis that should be confirmed by studies in other related areas. Other future studies could include applications of the proposed methods to other related areas in urban studies, such as the heterogeneity of urban structure, the consequences of urban growth, and the simulation of urban expansion. Acknowledgements This research was funded by the National Natural Science Foundation of China (41171312). Appendix A. The derivatives of urban land density function and the coordinates of p0 , p1 , and p2 The first derivative of the urban land density function (Eq. (1)) is: f = −

2˛ e˛((2r/D)−1) · 2 D(1 − c) 1 + e˛((2r/D)−1)

(A.1)

From Eq. (1), we have the following transformations: f −c 1 = 1−c 1 + e˛((2r/D)−1)

(A.2)

and 1−

f −c e˛((2r/D)−1) = f −c 1 + e˛((2r/D)−1)

(A.3)

Thus, the first derivative can be rewritten as: f = − =−

2˛ 1 e˛((2r/D)−1) · · D(1 − c) 1 + e˛((2r/D)−1) 1 + e˛((2r/D)−1)



2˛ f −c f −c · 1− · 1−c D(1 − c) 1 − c



(A.4)

Let g=

f −c 1−c

(A.5)

Then, f = −

2˛ g(1 − g). D(1 − c)

(A.6)

Note that −(2˛/D(1 − c)) is a negative constant. At g = 0.5, f has its lowest negative value, indicating the point where urban land density declines most rapidly. By Eqs. (A.2) and (A.5), we can calculate the coordinates of the most rapidly decreasing point (denoted by p0 ) on the curve of the urban land density function, i.e., r0 = D/2, f0 = (1 + c)/2.

L. Jiao / Landscape and Urban Planning 139 (2015) 26–39

The second derivative of the urban land density function is: f  = − =− =

f

2˛ 2˛ (1 − 2g)g  = − (1 − 2g) 1−c D(1 − c) D(1 − c) 2˛ 1 (1 − 2g) · 1−c D(1 − c)

4˛2 (1 − 2g)(1 − g)g D2 (1 − c)3

=



−

2˛ D(1 − c)

4˛2 D2 (1 − c)2



g(1 − g)

(2g 3 − 3g 2 + g)

(A.7)

Note that df 

=−

dg

4˛2 D2 (1 − c)3

(6g 2 − 6g + 1)

(A.8)

√ Let df /dg = 0, we have g = 1/2 ± 1/ 12. When g = 1/2 ± √ 1/ 12, i.e., g1 ≈ 0.788675, g2 ≈ 0.211325 |f | has a maximum value where f changes most rapidly. By Eqs. (A.2) and (A.5): r=

D 2

 ln(1 − g) − ln g ˛



+1 ,

f = g(1 − c) + c.

(A.9)

Thus the coordinates of p1 and p2 in Fig. 6 can be expressed as follows: r1 =

D 2

and r2 =

D 2

 −1.316957 ˛

 1.316957 ˛



+1 ,



+1 ,

f1 = 0.788675(1 − c) + c

f2 = 0.211325(1 − c) + c

(A.10)

(A.11)

Obviously, r1 + r2 = D,

f1 + f2 = 1 + c

(A.12)

References Angel, S., Parent, J., & Civco, D. L. (2007). Urban sprawl metrics: An analysis of global urban expansion using GIS. In Proceedings of ASPRS 2007 annual conference Tampa, FL, May 7–11. Angel, S., Parent, J., & Civco, D. L. (2010). Ten compactness properties of circles: Measuring shape in geography. Canadian Geographer, 54(4), 441–461. Angel, S., Parent, J., Civco, D. L., Blei, A., & Potere, D. (2011). The dimensions of global urban expansion: Estimates and projections for all countries, 2000–2050. Progress in Planning, 75(2), 53–107. Arnold, C. L., & Gibbons, C. J. (1996). Impervious surface coverage: The emergence of a key environmental indicator. Journal of the American Planning Association, 62, 243–258. Arribas-Bel, D., Nijkamp, P., & Scholten, H. (2011). Multidimensional urban sprawl in Europe: A self-organizing map approach. Computers, Environment and Urban Systems, 35(4), 263–275. Batty, M. (2008). The size, scale, and shape of cities. Science, 319(5864), 769–771. Batty, M., & Kim, K. S. (1992). Form follows function: Reformulating urban population density functions. Urban Studies, 29(7), 1043–1069. Bettencourt, L., & West, G. (2010). A unified theory of urban living. Nature, 467, 912–913. Bhatta, B., Saraswati, S., & Bandyopadhyay, D. (2010). Urban sprawl measurement from remote sensing data. Applied Geography, 30(4), 731–740. Blumenfeld, H. (1954). The tidal wave of metropolitan expansion. Journal of the American Institute of Planners, 20, 3–14. Burgess, E. W. (1967). The growth of the city: An introduction to a research project. In R. E. Park, E. W. Burgess, & R. D. McKenzie (Eds.), The city (pp. 47–62). Chicago, IL: University of Chicago Press.

39

Chen, Y., & Feng, J. (2012). Fractal-based exponential distribution of urban density and self-affine fractal forms of cities. Chaos, Solitons & Fractals, 45(11), 1404–1416. Clark, C. (1951). Urban population densities. Journal of the Royal Statistical Society, 114(4), 490–496. Fragkias, M., & Seto, K. C. (2009). Evolving rank-size distributions of intrametropolitan urban clusters in South China. Computers, Environment and Urban Systems, 33, 189–199. Güneralp, B., & Seto, K. C. (2008). Environmental impacts of urban growth from an integrated dynamic perspective: A case study of Shenzhen, South China. Global Environmental Change, 18(4), 720–735. Herold, M., Couclelis, H., & Clarke, K. C. (2005). The role of spatial metrics in the analysis and modeling of urban land use change. Computers, Environment and Urban Systems, 29(4), 369–399. Herold, M., Goldstein, N. C., & Clarke, K. C. (2003). The spatiotemporal form of urban growth: Measurement, analysis and modeling. Remote Sensing of Environment, 86(3), 286–302. Irwin, E. G., & Bockstael, N. E. (2007). The evolution of urban sprawl: Evidence of spatial heterogeneity and increasing land fragmentation. Proceedings of the National Academy of Sciences of the United States of America, 104(52), 20672–20677. Jat, M. K., Garg, P. K., & Khare, D. (2008). Modelling of urban growth using spatial analysis techniques: A case study of Ajmer city (India). International Journal of Remote Sensing, 29(2), 543–567. Ji, W., Ma, J., Twibell, R. W., & Underhill, K. (2006). Characterizing urban sprawl using multi-stage remote sensing images and landscape metrics. Computers, Environment and Urban Systems, 30(6), 861–879. Li, C., Li, J., & Wu, J. (2013). Quantifying the speed, growth modes, and landscape pattern changes of urbanization: A hierarchical patch dynamics approach. Landscape Ecology, 28(10), 1875–1888. Liu, X., Li, X., Chen, Y., Tan, Z., Li, S., & Ai, B. (2010). A new landscape index for quantifying urban expansion using multi-temporal remotely sensed data. Landscape Ecology, 25, 671–682. National Bureau of Statistics of China. (1991). China city statistical yearbook. Beijing: China Statistics Press. National Bureau of Statistics of China. (2011). China city statistical yearbook. Beijing: China Statistics Press. Piorr, A., Ravetz, J., & Tosics, I. (2011). Peri-urbanisation in Europe: Towards a European policy to sustain urban–rural futures (synthesis report of PLUREL project). Copenhagen: Copenhagen, Denmark – PLUREL consortium. Retrieved from http://www.plurel.net/images/Peri Urbanisation in Europe printversion.pdf Schneider, A., & Woodcock, C. E. (2008). Compact, dispersed, fragmented, extensive? A comparison of urban growth in twenty-five global cities using remotely sensed data, pattern metrics and census information. Urban Studies, 45(3), 659–692. Seto, K. C., & Fragkias, M. (2005). Quantifying spatiotemporal patterns of urban landuse change in four cities of China with time series landscape metrics. Landscape Ecology, 20(7), 871–888. Seto, K. C., Guneralp, B., & Hutyra, L. R. (2012). Global forecasts of urban expansion to 2030 and direct impacts on biodiversity and carbon pools. Proceedings of the National Academy of Sciences of the United States of America, 109(40), 16083–16088. Taubenböck, H., Wegmann, M., Roth, A., Mehl, H., & Dech, S. (2009). Urbanization in India – Spatiotemporal analysis using remote sensing data. Computers, Environment and Urban Systems, 33(3), 179–188. Tavernia, B. G., & Reed, J. M. (2009). Spatial extent and habitat context influence the nature and strength of relationships between urbanization measures. Landscape and Urban Planning, 92(1), 47–52. Tsai, Y.-H. (2005). Quantifying urban form: Compactness versus ‘sprawl’. Urban Studies, 42(1), 141–161. United Nations. (2012). World urbanization prospects, the 2011 revision. Available at:. http://esa.un.org/unpd/wup/index.htm. Accessed 16.06.13 Václavík, T., Lautenbach, S., Kuemmerle, T., & Seppelt, R. (2013). Mapping global land system archetypes. Global Environmental Change, 23(6), 1637–1647. Wang, F. (2006). Quantitative methods and applications in GIS. Boca Raton, FL: CRC Press. Wilson, E. H., Hurd, J. D., Civco, D. L., Prisloe, M. P., & Arnold, C. (2003). Development of a geospatial model to quantify, describe and map urban growth. Remote Sensing of Environment, 86(3), 275–285. Wolman, H., Galster, G., & Hanson, R. (2005). The fundamental challenge in measuring sprawl: Which land should be considered? The Professional Geographer, 57, 94–105. Xu, X., & Min, X. (2013). Quantifying spatiotemporal patterns of urban expansion in China using remote sensing data. Cities, 35, 104–113.