Bird’s eye view on steep topography

In nadir view, normal to topography, landscapes with gentle slopes and those with steep surfaces look similar. This is due to the projection of a 3D structure onto a 2D plane. In orthophotos and digital elevation models (DEMs) topography is represented in this nadir or bird’s eye perspective. Elevation models of the Earth or planetary surfaces are often represented by gridded cells, each cell assigned with a mean elevation. In geomorphic studies, DEMs are widely used to calculate hillslope angles and surface area. Due to the projection, hillslopes at a steeper angle appear shorter and thus are represented by a smaller fraction of grid cells. Consequently, mean or median hillslope angles are distorted towards gentle slopes. This bias becomes even more obvious when comparing the projected 2D surface area versus the 3D surface area. The ratio by which a 3D surface area is underrepresented in 2D is by the cosine of the slope of the plane. Hence, the degree of area under-representation increases towards steeper slopes. At an angle of 60°, theoretically only half of the 3D surface area would be accounted for in a gridded DEM. And a hillslope at 90° is a no-show in the DEM. But already gentle slopes of 20° to 30° would be under-represented by about 10%. In addition to the under-representation of steep slopes due to the projection, DEM’s spatial resolution amplifies this bias where increasing grid size decreases the representation of steep slopes.

In essence, due to the bird’s eye view, measures of hillslope angle distribution and surface area have a bias disadvantaging steep slopes and skewing our perception towards a (flat) world of gentle slopes. Here we will discuss if and by how much this bias due to the bird’s eye view matters. First, we investigate artificial DEMs of Gaussian hills. We compare slope and surface area values using standard methods of gridded-data analysis to analytical solutions. Second, we investigate the impact of under-representation on a range of natural landscapes. This potential bias favouring gentle landscape elements has several implications for geomorphological interpretation of DEMs, including for example analyses of average erosion rates, landslide distribution or hydrological processes.