Study area
The Beijing-Tianjin-Hebei region, in the north of North China Plain, is the largest urban metropolitan areas of China (Fig. 1). This region occupies an administrative area of 216,600 km2 (Dong et al. 2008), with more than 100 million people. During the past decades, this region experienced rapid growth of population and economy, resulting in a notable sprawl of urban extent. As two primary engines of this region, Beijing and Tianjin lead the development in this area with almost an exponential growth of urban areas over past decades (Chai and Li 2018; Li et al. 2015a). This rapid urbanization is raising public concerns on water scarcity (Li et al. 2018a), energy consumption (Wang and Chen 2016), and air pollution (Liu et al. 2018). Therefore, modeling of historical urban sprawl and predicting of future growth are urgently needed in this region.
Data collection
We collected seven spatial proxies in the Beijing-Tianjin-Hebei region in the urban sprawl modeling (Table S1). These proxies consist of spatial features and images such as terrain, land cover, traffic, and location. Except for proxies in the category of terrain (i.e., elevation and slope), all other proxies were processed as the distance to city centers, roads, and specific land cover types (Fig. 2). These spatial proxies were used to determine the suitability of urban sprawl of each pixel, according to their biophysical and socioeconomic conditions (Li et al. 2014).
We used the annual urban extent data derived from nighttime light (NTL) and Landsat satellite data in the model calibration and evaluation. These two annual urban extent datasets were used not only for providing the temporal trend of urban sprawl but also for evaluating the robustness of the improved urban CA model. The NTL-derived urban extent maps (Fig. 2h) span from 1992 to 2013, with a medium resolution of 1 km. The mean accuracy of this developed dataset is about 89% in China (Zhou et al. 2018). The Landsat-derived urban extent maps (Fig. S1) have a longer temporal interval (1985–2015) than the NTL-derived results, with a fine spatial resolution of 30 m. Based on the long-term Landsat time series data, this Landsat-derived dataset was generated by a temporal segmentation approach (Li et al. 2018b). The overall accuracy of the detected urbanized year in this dataset is around 83% in the Beijing-Tianjin-Hebei region. Besides, both of these two urban dynamic datasets follow the logic of urban development. That is, this development is a monotonic conversion from non-urban to urban (Li et al. 2015a).
Framework
We developed a Logistic-Trend-CA model and assessed its performance to relevant factors in urban CA model (Fig. 3). First, we proposed a trend-adjusted neighborhood with the consideration of the historical pathway of urban sprawl and developed a Logistic-Trend-CA model. Second, we analyzed the performance of the proposed model to key elements in the urban CA model, including the start year of modeling, the suitability surface, and the neighborhood size. The improved urban CA model was applied at 1-km and 30-m spatial resolutions to explore its capability in cross-scale modeling. Details of each step are given in the following sections.
The logistic-trend-CA model
The urban CA model is a grid-based self-evolution system to simulate the dynamics of urban land (Batty and Xie 1999). In this system, the status (i.e., urban and non-urban) of each grid is determined by its surrounding neighbors. A non-urban grid is more likely to change to urban in the near future if there are more urban grids surrounded. Evolution of massive grids using this rule simultaneously can simulate the change of complex urban landscapes. With the consideration of additional spatial factors such as traffic networks and land covers, the urban CA model can be used to simulate the dynamic of urban land with a high degree of reliability.
We built our Logistic-Trend-CA model on the Logistic-CA model as it has been widely used in urban sprawl modeling due to its clear explanation of spatial proxies and ease of implementation (Hu and Lo 2007; Wu 2002). The logistic regression function is the key of the Logistic-CA model. Its output is a spatially explicit suitability surface, which indicates the suitability for development under considerations of different spatial proxies. Assuming there are n spatial proxies [x1, x2, …xn], the logistic regression function can be expressed as Eqs. (1 and 2).
$$ z={b}_1{x}_1+\cdots +{b}_n{x}_n $$
(1)
$$ {P}_{\mathrm{suit}}=1/\left(1+{\exp}^{-z}\right) $$
(2)
where Psuit is the obtained suitability of development from the biophysical and socioeconomic conditions and bi and xi are the ith coefficient and spatial proxy, respectively.
We improved the neighborhood of the Logistic-CA model by considering the historical pathway of urban sprawl. The neighborhood is a crucial component in the urban CA model because it is a basic driver of urban dynamics modeling (Kocabas and Dragicevic 2006). The configuration of the neighborhood is closely related to its size, shape, and surrounding land cover types. Here, we developed the trend-adjusted neighborhood by incorporating the historical pathway of urban sprawl as a weighting factor, based on the widely used Moore configuration (Eqs. (3 and 4)).
$$ {W}_{ij}^{ts}=1-\frac{N_{ij}^u}{N} $$
(3)
$$ \Omega =\frac{\Sigma_{m\times m}\mathrm{Con}\left({S}_{ij}=\mathrm{urban}\right)\times {W}_{ij}^{ts}}{m\times m-1} $$
(4)
where Ω is the influence of neighborhood that considers the historical pathway of urban sprawl using a weighting factor of \( {W}_{ij}^{ts} \). \( {N}_{ij}^u \) is the accumulated year of cell (i, j) with the status as urban from the annual urban time series data with a temporal interval of N. As a result, for a potential cell, urban neighbors that were developed in more recent years have larger impacts than those developed in earlier years. m is the window size, and Con() is a conditional function, which returns 1 when the status of cell (i, j) is urban.
Compared with traditional neighborhoods, the developed trend-adjusted neighborhood can result in a sprawl pattern following the historical pathway, as illustrated in Fig. 4. Urban sprawl has an inertia of development as it generally follows the temporal trend of historical development (Liu et al. 2017), i.e., there is a relatively higher development probability around those newly developed urban areas. As a result, the weighting factors of \( {W}_{ij}^{ts} \) of urban pixels developed in more recent years are higher than those developed in earlier years. Assuming the weighting factors of urbanized pixels across years are different as illustrated in Fig. 4 (a), thus, pixels 1 and 2 have the same neighborhood influence if using the traditional neighborhood. However, if taking into account of the historical pathway of urban sprawl, pixel 2 has a higher neighborhood influence because its surrounding pixels were developed more recent compared to pixel 1 (Fig. 4 (b)), although they have the same number of urban neighbors for pixels 1 and 2. Such smaller difference regarding the neighborhood influence would result in a notable different sprawl pattern after several iterations (Fig. 4 (c)), as most urbanized pixels were developed following the historical pathway if the neighborhood was adjusted by the temporal trend.
We also included land constraint and stochastic perturbation in the developed Logistic-Trend-CA model. Restricted lands, such as water and protected areas, were not considered for development in our model; thus, they were represented as a land constraint term as Land = 0 (Li et al. 2014). In addition, we used the stochastic perturbation SP to represent unconsidered factors (e.g., policy) in the modeling (White and Engelen 1993), as expressed in Eq. (5).
$$ SP=1+{\left(-1\mathrm{n}\lambda \right)}^{\alpha } $$
(5)
where SP is the stochastic perturbation, λ is a random value [0, 1], and α is a parameter to determine the degree of perturbation.
The development probability was calculated based on the suitability surface, neighborhood, land constrain, and stochastic perturbation. For urban time series data derived from NTL and Landsat, we determined their development probabilities using Eqs. (6) and (7), respectively. The SP is not considered in the modeling with NTL-derived urban extent maps, due to the homogeneity of urban land within the boundary (Zhou et al. 2018).
$$ {P}_{\mathrm{dev}}={P}_{\mathrm{suit}}\times \Omega \times Land $$
(6)
$$ {P}_{\mathrm{dev}}={P}_{\mathrm{suit}}\times \Omega \times Land\times SP $$
(7)
Model validation
We assessed the performance of the developed Logistic-Trend-CA model to key factors in urban CA model. Two quantitative metrics were used for the assessment, namely the overall accuracy (OA) and the figure of merit (FOM). The OA was directly calculated as the percentage of consistent pixels to all pixels in the entire map, while the FOM indicates the consistency between modeled and observed maps on those changed pixels. The FOM has been widely used in many studies of urban CA models since it can provide a relatively comprehensive evaluation of the model performance (Chen et al. 2014; Li et al. 2014; Pontius et al. 2007). The FOM can be expressed as Eq. (8) (Pontius et al. 2008).
$$ \mathrm{FOM}=B/\left(A+B+C\right)\times 100\% $$
(8)
where FOM is the figure of merit, B is the number of observed urban pixels that were simulated as urban, A is the number of observed urban pixels that were simulated as non-urban, and C is the number of observed non-urban pixels that were simulated as urban.
We evaluated the model performance by exploring sensitivities of three key factors in our urban CA model: the start year of modeling, the suitability surface, and the neighborhood size. The influences of these three key factors can be quantitatively evaluated with clear physical meanings in the CA model. Although there are other factors that may also influence the model performance, they are considered in the selected elements in the model. For example, the urban spatial configuration can be captured by the neighborhood and stochastic disturbance in the CA model. Other factors (e.g., the degree of urban development) are more related to regional urban demand compared to the spatial allocation of increased urban demand (Li et al. 2019a). First, we examined the modeling capability of the Logistic-Trend-CA model over a long temporal span through changing the start year of modeling, which is closely related to the iterations and error propagation in the modeling (Li et al. 2014). Second, we investigated the model performance to suitability surfaces using calibrated results from different periods. The suitability surface characterizes the likelihood of urban development from the biogeophysical (e.g., terrain and land cover) and socioeconomic (e.g., traffic networks) aspects. In addition, we evaluated the derived suitability surface using the receiver operating characteristic (ROC) approach (Liu et al. 2017; Pontius Jr et al. 2001; Wu et al. 2009). The ROC curve was calculated by dividing the continuous suitability surface into binary maps using different thresholds and then comparing the derived binary map with the reference map. Finally, we compared the model performance by varying neighborhood sizes. The neighborhood is a crucial component to drive the self-evolution of urban land system, and the neighborhood size reflects the degree of local impact from neighbors (Kocabas and Dragicevic 2006). Many urban CA models have been developed for particular applications with different structures, functions, and data requirements (Li and Gong 2016). Quantitative indicators such as the FOM have been used for comparing urban CA models. We evaluated our model performance based on the FOM and compared it with previous studies. In addition, we compared our model with the similar Logistic-CA model, which has been widely used in previous urban CA studies and can serve as a benchmarking model, for several key factors. The Logistic-CA model also has the same structure as our proposed model except for the consideration of the neighborhood.
Setting of the urban CA model
The inputs of urban CA model are the urban extent map in the beginning year associated with a variety of spatial proxies (Fig. 2) and a set of parameters (Li et al. 2017a), and the output of our model is the urban extent map in the target year. In our study, the neighborhood size (m) was set as 3 and 5 in calculating the influence of neighborhood (Ω) using Eq. (4), for urban sprawl modeling with medium (1 km) and fine (30 m) resolutions, respectively. The degree of stochastic perturbation (α) was set as 3 as suggested for modeling at a 30-m resolution using Eq. (5) (Li et al., 2014). Also, we set the restricted conversion type as water in our study to avoid the conversion from water to urban. Finally, the modeled results were compared with the observed urban extent map from remote sensing observations in the same year.