# Flood Inundation Mapping by Combining GNSS-R Signals with Topographical Information
## 3 Methodology
Furthermore, for wide rectangular channels, (\(B\gg H\)) \(R\) simplifies to \(BH^{5/3}\), where \(B\) is the channel width and \(H\) is the discharge height. If the value of \(B/\eta\) is assumed constant (where \(\eta\) denotes the roughness coefficient of the riverbed), it can be concluded that for a given discharge, the water level in the stream is inversely related to the slope of channel pixel given as \(H\propto S^{-3/10}\), and therefore,  
\[\text{P(F)}\propto\frac{1}{\text{SND}^{0.3}} \tag{3}\]  
Figure 2: (**a**) Hydrologically inconsistent sample digital elevation model (DEM) with hill and sink indicated; (**b**) with DEM corrections undertaken (green); (**c**) D8 flow direction applied on corrected DEM; (**d**) flow accumulation calculated; (**e**) sample contributing area threshold of 3 applied to obtain drainage pixels; (**f**) nearest drainage pixel mapped for every other pixel in study region; (**g**) height above nearest drainage (HAND) value calculated with respect to nearest drainage pixel; (**h**) slope of nearest drainage (SND) assigned to corresponding floodplain pixels. In order to characterize the static potential of flooding, the following points are required: (a) the height of a given pixel relative to the nearest drainage pixel height; and (b) the height of the water level in the channel, which is inversely related to the slope of the channel pixel, as shown above. This characterization is in contrast to [24], which considers the slope of a floodplain pixel relative to the nearest drainage pixel as an extension of HAND. Thus, combining (1), (2), and (3), the dynamic (CYGNSS) and static (HAND and SND) dataset available for the study region under consideration was formalized as follows:  
\[\mathrm{P(F)}\ \propto\ \frac{\mathrm{MAX(SNR)}|_{w}}{\left(\mathrm{HAND}\times \mathrm{SND}^{0.3}\right)} \tag{4}\]  
The product of HAND and SND can result in null values in the case of pixels located within channels, leading to undefined model output. We consider that \(1+\left(\mathrm{HAND}\times\mathrm{SND}^{0.3}\right)\equiv\left(\mathrm{HAND }\times\mathrm{SND}^{0.3}\right)\), and thus the model variable \(F\) from (4) was further simplified without considering any additional parameters in the following form:  
\[\mathrm{F}\ =\ \frac{\mathrm{MAX(SNR)}|_{w}}{1+\left(\mathrm{HAND}\times \mathrm{SND}^{0.3}\right)} \tag{5}\]  
In addition, the proposed model output raster was subjected to different model output threshold values to obtain the respective maximum extent of inundation across the flood duration at 30 m resolution. The model was compared against co-located Sentinel-1 GRD VV-polarized images. Those images were pre-processed by applying standard steps, including (a) applying the orbit file; (b) sub-setting of the images; (c) thermal noise removal; (d) border noise removal; (e) radiometric calibration; (f) speckle filtering using a Lee filter of \(7\times 7\) window size; and (g) range-Doppler terrain correction, followed by image binarization using [PERSON]'s method [33]. The model sensitivity analysis was performed by introducing parameter \(\beta\) to (5), yielding  
\[\mathrm{F}\ =\ \frac{\mathrm{MAX(SNR)}|_{w}}{1+\left(\mathrm{HAND}\times \mathrm{SND}^{0.3}\right)^{\beta}} \tag{6}\]  
and analyzing the flood accuracy (FA) and critical success index (CSI) parameters for different threshold values. These parameters are given by  
\[\mathrm{FA}\ =\ \frac{\mathrm{h}}{\mathrm{h}+\mathrm{m}} \tag{7}\]  
Figure 1:0  
where h represents the number of pixels which have an agreement (hit) between model and reference pixels, m represents the number of pixels that the model has failed (miss) to identify as inundated when compared with the reference inundation map, and z represents the number of pixels that the model has overestimated as inundated when compared to the reference.