## 3 Methodology
### The Percentage of Impervious Surface Areas Estimation  
[PERSON] and [PERSON] [10] found that impervious surfaces are likely on or near the line connecting the low albedo and high albedo endmembers in the feature spaces, and most impervious surfaces can be represented by the linear spectral mixture analysis (LSMA) model as follows in Equation (1). \[R_{b}=\sum_{i=1}^{N}f_{i}R_{i,b}+e_{b} \tag{1}\]  
Equation (1) is used to determine the reflection \(R_{b}\) for each band b in the HJ-1B image, where \(N\) is the number of end-members \(i\), \(R_{i,b}\) is the fraction of end-member, \(f_{i}\) is the reflectance of end-member \(i\) in band \(b\), and \(e_{b}\) is the un-modeled residual. \(\sum_{i=1}^{N}f_{i=1}=1\) and \(f_{i}\geq 0\) are required. Initially, the LSMA was adopted to extract ISA fraction. The Maximum Noise Fraction (MNF) was applied to build a statistical model for the images and to transform the noise covariance matrix of the dataset to an identity matrix. The MNF components suggested that the spectral reflectance of the HJ-1B image might best be represented by a three end-member linear mixing model. These end-members are high albedo (e.g., concrete, clouds, and sand), low albedo (e.g., asphalt road, built-up area), and vegetation (e.g., farmland, grass and trees). Based on the end-member extraction function PPI, three types of end-members were identified according to their feature spaces and associated interpretation. The residual term RMS of all image bands was calculated and used to assess the performance of the model fitness [10]. \[RMS=\frac{Me_{b}^{2}}{M^{\frac{1}{2}}} \tag{2}\]  
where \(e_{b}\) is the residual term, \(M\) is the image bands. [PERSON] Land Surface Temperature Estimation  
We initially chose the [PERSON] ([PERSON]'s) algorithm to retrieve the LST [42], and then used the LST results to validate the results of [PERSON]. On the basis of [PERSON]'s law, [PERSON] discovered linear relationship between radiance and temperature, and set up a first order [PERSON]'s approximation formula around a certain temperature value. A Gaussian trigonometric filter was used as a response function for the thermal infrared band channel. According to [PERSON]'s law, the radiance emitted by a blackbody at temperature \(T_{s}\), which can be expressed as  
\[T_{s}=\gamma\Big{[}\varepsilon^{-1}(\psi_{1}L_{sensor}+\psi_{2})+\psi_{3} \Big{]}+\delta \tag{3}\]  
\[\gamma=\left\{\frac{c_{2}L_{sensor}}{T_{sensor}^{2}}\Bigg{[}\frac{\lambda_{ eff}^{4}}{c_{1}}L_{sensor}+\lambda_{eff}^{-1}\Bigg{]}\right\}^{-1}\]  
where  
\[\delta=-\gamma L_{sensor}+T_{sensor} \tag{4}\]  
From Equations (3) and (4), \(L_{sensor}\), as the at-sensor radiance, is given in W m\({}^{-2}\) sr\({}^{-1}\)um\({}^{-1}\), and this data can be obtained from the calibration of satellite data. Tsensor represents the at-sensor brightness temperature, and can be obtained from the lookup table based on Equation (4), given in K. \(\lambda_{eff}\) is the effective central wavelength for the thermal infrared channel, the \(\lambda_{eff}\) of HJ-1B is 11.576 \(\upmu\)m. Three atmospheric parameters \(\psi_{1}\), \(\psi_{2}\), \(\psi_{3}\) in the atmospheric functions, are given by:  
\[\psi_{1}(\lambda,w)\equiv\frac{1}{\tau(\lambda,w)}\]  
\[\psi_{2}(\lambda,w)\equiv-L^{atm\downarrow}(\lambda,w)-\frac{L^{atm \uparrow}(\lambda,w)}{\tau(\lambda,w)}\]  
\[\psi_{3}(\lambda,w)\equiv L^{atm\downarrow}(\lambda,w) \tag{5}\]  
where \(L_{\lambda}^{atm\uparrow}\) is the up-welling atmospheric radiance, \(L_{\lambda}^{atm\downarrow}\) is the down-welling radiance. Overall, depending on the LST retrieval function from Equations (3)-(5), the land surface emissivity (\(\varepsilon\)) and the water vapor content (\(w\)) are the two major variables that need to be calculated to get \(T_{s}\).