## 2 Materials and Methods
### Models and Methods
#### 2.3.2 Meaning and Characterization of Urban Spatial Morphology Indices  
Urban spatial morphology reflects the processes of external urban expansion and the internal spatial structure of the human economy in this study. In urban planning and urban ecology, urban spatial morphology includes three levels: micro, middle, and macro [44]. The microscale represents the functional space within the city [45] and the middle- and the macroscales are structures of the city cluster system [46]. The urban structure indicators (population density, industrial structure, traffic access, and nighttime remote sensing index) and the urban shape indicators (compactness, fractal dimension, centrality ratio, and circularity ratio) are defined on the middle- and macroscales [47]. Globally, cities, as sites of human activities, export anthropogenic air pollutants annually [48]. Therefore, urban spatial morphology and air pollutants are correlated with emission, dispersion, and purification. The flow of urban spatial morphology and PM\({}_{2.5}\) pollutant-driven impact mechanisms is established in Figure 3.  
In our study, indices such as fractal dimension (FRD), compactness index (CI), circularity ratio (CR), and centrality ratio (CEN) were applied to define urban spatial morphology. FRD reflected the external profile of prefecture-level cities in the \"2 + 26\" cities of BTH. The theoretical range of FRD is 1-2, and the higher the calculated FRD was, the closer the  
Figure 3: The urban spatial morphology and PM\({}_{2.5}\) pollutant-driven impact mechanisms. cities in the \"2 + 26\" cities of BTH were to the traditional urban spatial morphology [49]. CI was the degree of concentration of urban spatial patterns outside the cities of the \"2 + 26\" cities of BTH. The CI was conducive to controlling the rate of urban land expansion and improving public transportation [50]. The circularity ratio, CR, precisely reflected the degree of discrete regional and urban development and measured the circular cities [51]. The centrality ratio or CEN, was the degree of spatial concentration of elements such as population or economic activities in the central area within a city, which showed the grid of the city [49]. Urban transportation is an influential factor that cannot be ignored in the process of urban expansion [49]; this has an impact on the spatial form of cities. Transportation for commuting is an important part of sustainable urban development. The calculation formula for the urban spatial morphology index is obtained using the following formula:  
\[FRD=\frac{2\log_{10}\left(\frac{P}{A}\right)}{\log_{10}A} \tag{4}\]  
\[CompactIndex=\frac{2\sqrt{\pi A}}{P} \tag{5}\]  
\[CR=\frac{4A}{P^{2}} \tag{6}\]  
\[CEN=\frac{\sigma(POP)}{ARV(POP)} \tag{7}\]  
\[TC=\frac{BUS}{A} \tag{8}\]  
where \(A\) is the total area of urban construction land; \(P\) is the perimeter of urban construction land; \(\sigma(POP)\) is the standard deviation of the population density; \(ARV(POP)\) is the average value of the population density; and \(BUS\) denotes the number of buses in the \"2 + 26\" cities of BTH. 3.3 Geo-Detector Modeling to Analyze the Effect of Urban Expansion Factors and PM\({}_{2.5}\) Pollutants  
Based on the core concept of geospatial heterogeneity [52], we used Geo-Detector to construct spatial matrices of urban spatial morphology and PM\({}_{2.5}\) pollutants. Then, we used Geo-Detector to calculate their spatial correlations, which could better reflect the causes and mechanisms behind geographic phenomena than traditional models [53]. Spatial heterogeneity included a significant change in the relationship between certain variables in space with spatial changes; secondly, the statistics (e.g., mean and variance) of certain variables in space changed smoothly or not smoothly with the spatial changes [54]. Our study focused more on the spatial variation of geospatial elements.