(1) Index construction

In contrast to soil and vegetation, the impervious layer has similar spectral values in the blue and near-infrared bands. Based on this, Tian Yugang et al.^{11} proposed a method for impervious layer extraction where the blue and near-infrared bands were selected to construct the PII index, as expressed in formula (5):

where *P*_{blue} and *P*_{NIR} are the spectral values in the blue light and near-infrared bands, respectively; *m* and *n* are the coefficients of the blue light and near-infrared bands, respectively; and *C* is a constant. *m*, *n* and *C* can be obtained through the reference line equation. Let the reference line equation of PII be *y=ax+b*, the relative vertical distance from the point of the pixel on the two-dimensional plane to the reference line is expressed by equation (6):

where *a* and *b* denote the slope and intercept of the reference line, respectively; and *x* and *y* are the spectral values of the pixel in the blue and near-infrared bands, respectively. When the coefficient of *PII* takes equation (7), *PII=D*, the slope of its contour is the same as that of the reference line. As shown in Figure 1, when the distance from the sampling point to the reference line is zero (PII=0), the area between the reference line and the *x*-axis is the area when D>0, and the area between the reference line and the *y*-axis is the area when *D<0*.

**Figure 1
**Schematic diagram illustrating the distance from the sampling point to the reference line(2) Parameter estimation

In the study area composed of the five Landsat 8 remote sensing image scenes, we selected 100 bare soil samples and 100 impervious layer samples for obtaining their spectral values in the blue and near-infrared bands. Least square fitting was performed on the two.

As the samples showed varied degrees of dispersion due to complex surface features, a standard deviation was calculated to adjust the fitted soil line and the impervious layer line. The reference line of PII was determined as the angular bisectors of the translated soil line and the impervious layer line.

Let the standard deviation of the bare soil sample, counted as its vertical distance to the soil line, be

. Likewise, let the standard deviation of the impervious layer sample, counted as its vertical distance to the impervious layer line, be

. Translate the lines according to their standard deviations, and the angular bisectors of the two are taken as the reference line of PII.

Suppose the original soil line equation be

*y = a*_{s}x+b_{s}, the impervious layer line equation be

*y = a*_{i}x + b, the adjusted soil line equation be

*y = a*_{s}x + b_{sa}, and the adjusted impervious layer line equation be

*y = a*_{i}x + b_{ia}. Let

and

, and formula (8) can be expressed as follows:

Substitute *a* and *b* into equation (7) to obtain the PII coefficient.

As per the PII method, we first selected a small amount of pure soil samples and impervious layer samples of an even quality in the study area, to obtain the spectral values of each sample in blue and near-infrared bands, the soil and impervious layer fit lines, and also the impervious layer reference line. Then an impervious layer index was constructed mathematically. After that, a band operation was performed to obtain the calculation result of each pixel point on the Landsat image. Finally, a threshold was determined for extracting the impervious layers.