## 2 Material and Methods
### Extraction of Total Imperviousness  
Two main steps were used in this research to derive imperviousness from Landsat 8 OLI/TIRS.  
Step 1: Removal of water bodies  
Modified Normalized Difference Water Index (MNDWI) (Equation (1)), as a widely used satellite-derived index, was used to remove water pixels from Landsat images [37].  
\[\text{MNDWI}=\frac{\text{R}_{\text{Green}}-\text{R}_{\text{SWIR1}}}{\text{R}_{ \text{Green}}+\text{R}_{\text{SWIR1}}} \tag{1}\]  
where, R\({}_{\text{Green}}\) and R\({}_{\text{SWIR1}}\) refer to Landsat 8 OLI/TIRS surface reflectance values of consistent bands Green (band 3) and SWIR1 (band 6). MNWDI is a good indicator of liquid water and less sensitive to atmosphere that has been widely used in environmental research [38; 39; 40; 41]. With MNWDI, those pixels with a threshold value of greater than 0 were represented as water bodies and masked from the image.  
Figure 1: Location of the case study and potential development areas (PDAs) in southeast Queensland, Australia.  
Step 2: Sub-pixel classification approach  
The linear spectral mixture analysis (LSMA)-based method was applied in this study to extract total imperviousness from Landsat images [42]. The LSMA approach assumes that the surface reflectance measured by a sensor in each band is a linear combination of the surface reflectance of end members (each end member represents a pure land cover type) within the pixel and that the surface reflectance proportions of the end members represent proportions of the area covered by distinct features on the ground. The linear spectral mixing model is expressed as:  
\[\mathrm{R_{j}=\sum_{i=1}^{N}f_{i}R_{ij}+e_{j}} \tag{2}\]  
\[\mathrm{\sum_{i=1}^{N}f_{i}=1} \tag{3}\]  
where \(\mathrm{R_{j}}\) is the surface reflectance for each band (j) in the Landsat 8 OLI/TIRS image (6 bands), N is the number of end members, \(\mathrm{f_{i}}\) is the fraction of end member i, \(\mathrm{R_{ij}}\) is the surface reflectance by the end member i in band j, and \(\mathrm{e_{j}}\) is the unmodeled residual. Six Landsat 8 OLI/TIRS reflective bands (Visible, NIIR, and SWIR bands) were used to develop fraction images with LSMA. Additionally, constrained condition was imposed based on Equation (3) which means the sum of all fractions must equal 1. Moreover, four endmembers including vegetation, low-albedo (e.g., dark impervious surface materials), high-albedo (e.g., commercial/industrial/transportation areas), and soil, were selected as input to LSMA in this study. Total imperviousness in each pixel was extracted from the sum of high-albedo and low-albedo fraction images [43; 44] (Figure 2). A detailed description of the LSMA method, selected Landsat reflective bands, and end-member selection were found in previous literature [34; 45]. In order to assess the accuracy of extracted imperviousness map, 300 random samples (each sample contains 3 \(\times\) 3 Landsat pixels) were initially selected over the Landsat imperviousness map. Samples were selected over the whole study area with different degrees of urbanization (0 to 100%). This stratified sampling method allowed us to check the performance of the LSMA algorithm for a wide range of imperviousness in the study area. The selected samples over Landsat images were linked to the corresponding Google Earth image. Impervious areas were digitized manually for each sample in the Google Earth image [46; 47]. The impervious areas were then divided by the total sample area to calculate imperviousness in percent for each sample. Both Landsat and Google earth images had similar acquisition dates (June of 2016) that reduce errors related to land use change in the validation process. Values of imperviousness derived from Landsat and Google Earth were compared by widely used statistical criteria including the coefficient of determination (R\({}^{2}\)), the root mean square error (RMSE), and the mean absolute error (MAE).