# Spatio-Temporal Analysis of Surface Urban Heat Island and Canopy Layer Heat Island in Beijing
## 2 Materials and Methods
### Methodology  
To better measure the range and intensity of heat islands and to visualize the changes of heat islands, this paper established a Gaussian model to obtain the intensity and spatial range of urban heat islands through the remote sensing measurements. In this paper, the heat island of Beijing was mainly analyzed over urban and suburban areas. According to the definition of an urban heat island, the pixel threshold of the heat island was taken as the average surface temperature of the suburban area plus 1 \({}^{\circ}\)C [17]. For the measurement of urban heat island parameters, the least squares method was used to approximate the heat island surface to a Gaussian surface [25]. In this paper, IDL (Interactive Data Language) programming was used to build the model with the following Equation (1):  
\[T(x,y)=T_{0}(x,y)+a_{0}\mathrm{exp}\left[-\frac{\left((x-x_{0})cos\mathcal{O}+ (y-y_{0})sin\mathcal{O}\right)^{2}}{2{a_{x}}^{2}}-\frac{\left((x-x_{0})sin \mathcal{O}-(y-y_{0})cos\mathcal{O}\right)^{2}}{2{a_{y}}^{2}}\right] \tag{1}\]  
where \((x,y)\) represents the latitude and longitude at a certain location; \(T(x,y)\) and \(T_{0}(x,y)\) represent the urban temperature and background temperature, respectively; \(a_{0}\) represents the maximum temperature difference between the urban and suburban areas; \((x_{0},y_{0})\) represents the latitude and longitude of the center of the heat island footprint ellipse; \(a_{x}\) and \(a_{y}\) represent the semi-major and semi-minor axes of the heat island footprint ellipse; and \(\varnothing\) is the acute angle between the direction of the major semi-axis of the heat island footprint ellipse and the positive direction of the \(x\)-axis, which represents the heat island direction.  
The Gaussian surface and the background temperature surface form a closed body, and the geometric center of gravity of the closed body is defined as the center of gravity of the heat island. Considering the symmetry of the Gaussian surface, the coordinates of the geometric center of gravity of the closed body projected onto the horizontal plane are \((x_{0},y_{0})\), the \(z\)-axis \(z_{0}(x_{0}\_{-},y_{0}\_)) of the center of gravity of the heat island, and the direction of the heat island is \(\varnothing\). The \(z\)-axis coordinate \(z_{0}\) is not related to the plane coordinates \((x_{0},y_{0})\). Therefore, in order to facilitate the calculation of the \(z\)-axis coordinate \(z_{0}\) of the center of gravity of the heat island, the Gaussian capacity model is simplified to Equation (2), and the calculation equation of \(z_{0}\) is shown in Equation (3) [8]:  
\[z(x,y)=a_{0}exp\left[-\frac{x^{2}}{2{a_{x}}^{2}}-\frac{y^{2}}{2{a_{y}}^{2}}\right] \tag{2}\]  
\[z_{0}=\frac{\iint\!\!\!\int_{\Omega}z\rho dv}{\iint\!\!\!\!\int_{\Omega}\rho dv} \tag{3}\]  
Combined with Formula (1), the parameters to be calculated in the Gaussian capacity model are \(a_{0}\), \(x_{0}\), \(y_{0}\), \(a_{x}\), \(a_{y}\), \(\varnothing\), and \(z_{0}\), which are obtained by least-squares fitting through IDL programming. \(z_{0}\) is defined as the heat island intensity of the Gaussian capacity model, and \((x_{0},y_{0},z_{0})\) represents the center of gravity of the heat island (longitude, latitude, and heat island intensity).