# First Evaluation of Topography on GNSS-R: An Empirical Study Based on a Digital Elevation Model
## 3 Data and Methodology  
### Topographic Parameters  
Several topographic parameters were derived from the \(\sim\)250 m GMTED2010 model based on a \(3\times 3\) moving window (Figure 1). This window was composed of the focal cell and its eight surrounding cells. The DEM surface was approximated by a bivariate quadratic equation [26; 27]:  
\[\mathrm{Z}=\mathrm{a}\mathrm{X}^{2}+\mathrm{b}\mathrm{Y}^{2}+\mathrm{c} \mathrm{XY}+\mathrm{d}\mathrm{X}+\mathrm{e}\mathrm{Y}+\mathrm{f}, \tag{8}\]  
where \(\mathrm{Z}\) represents the height of the DEM surface and \(\mathrm{X}\) and \(\mathrm{Y}\) are the horizontal coordinates. The coefficients in Equation (8) were solved within the \(3\times 3\) window using simple combinations of neighboring cells, based on the so-called [PERSON]'s approach [28]. The topographic parameters were used to investigate the impact of different topographic features on \(\left\langle\left|\mathrm{Y}_{\mathrm{r,topo}}(\tau,\mathrm{f})\right|^{2}\right\rangle\). In particular, three heterogeneity variables (topographic position index or TPI, terrain ruggedness index or TRI, vector ruggedness measure or VRM), slope (\(\beta\)), and two curvature variables (tangential (TC) and profile curvature (PC) were used in the parametric study. They have the potential to enable a better understanding on the properties of the Earth's surface than raw DEM data. Topographic heterogeneity is described as the variability of surface elevations within an area. On the other hand, curvature attributes are based on the change of slope in a particular direction. More specifically, the selected topographic parameters (Figure 2) were defined as follows [26; 27; 28]:  
Figure 1: Image of the raster grid. Numbering system is shown. \(\mathrm{Z}\) is the value of the raster. Notation is simplified. \(\mathrm{N}=(\mathrm{n}-1)/2\) is used for any \(\mathrm{n}\times\mathrm{n}\) analysis window, where \(\mathrm{n}\) may be any odd integer smaller than the number of cells in the shortest side of the raster. * TPI is the difference between the elevation of a focal cell and the mean of its eight surrounding cells. Positive and negative values correspond to ridges and valleys, respectively. Zero values correspond to flat areas. It was computed as follows: \[\text{TPI}=\text{Z}_{(0,0)}-\sum_{\text{i=1}}^{\text{m}}Z_{\text{i}}/\text{m},\] (9) where m is total number of surrounding points employed in the evaluation. * TRI is the mean of the absolute differences in elevation between a focal cell and its eight surrounding cells. It quantifies the total elevation change across the \(3\times 3\) moving window. Flat areas have a value of zero and mountain areas with steep ridges have positive values. It was computed as follows: \[\text{TRI}=\frac{\sum_{\text{i=-N}}^{\text{N}}\sum_{\text{j=-N}}^{\text{N}} \bigl{|}Z_{(\text{i,j})}-Z_{(0,0)}\bigr{|}}{(\text{n}^{2}-1)},\] (10) where \(\text{N}=(\text{n}-1)/2\) and n may be any odd integer smaller than the number of cells in the shortest side of the raster. * VRM quantifies the terrain ruggedness based on the measurements of the dispersion of vectors orthogonal to the surface. VRM quantifies the local variation of slope in the terrain more independently than TPI and TRI. VRM values range from 0 over flat areas to 1 over rough areas. It was computed using vector analysis [26]. * \(\beta\) is defined as the rate of change of elevation in magnitude for the steepest descent vector. It influences hydraulic gradients driving any surface flows and also sub-surface flows when the water table has a similar slope to the ground surface. Slope and curvature serve as useful input variables in erosion and hydrological models. It was computed as follows: \[\beta=\arctan(\sqrt{\text{d}^{2}+\text{e}^{2}}).\] (11)
* TC measures the rate of change perpendicular to the slope gradient. It is related to the convergence and divergence of lateral flows across a surface.