Quantifying Earth's Topography: Steeper and Larger Than Projected in Digital Terrain Models

Grid- or pixel-based models, used across various scientific disciplines from microscopic to planetary scales, contain an unquantified error that bias our interpretation of the data. The error is produced by projecting 3D data onto a 2D grid. For Digital Terrain Models (DTMs) the projection error affects all slope-dependent topographic metrics, like surface area or slope angle. Due to the proportionality of the error to the cosine of the slope, we can correct for it. We quantify the error and test the correction using synthetic landscapes for which we have analytical solutions of their metrics. Application to real-world landscapes in California, reveal the systematic underestimation of surface area by up to a third, and mean slope angles by up to 10 degrees in steep topography in current DTMs. Correcting projection errors allow for true estimates of surface areas and slope distributions enabling physics-based models of surface processes at any spatial scale. Hiking up a steep mountain slope feels longer than the horizontal distance measured on a map. The slope angle is calculated by taking the height over the horizontal distance. The length of a (mountain) slope is always greater than the horizontal plane. This mistake is also very common in Digital Terrain Models (DTMs). Here landscapes images captured in a bird's eye view are projected onto horizontal gridded surfaces. In the model, each grid cell has the same length and contains the height of the landscape. Due to the view, the slopes are only represented by the horizontal distance, which is shorter. We call this the projection error. Because the error depends on the slope, we use it to fix it. We test the correction on different landscapes with steep and gentle topography. We find that uncorrected models underestimate surface area and length by up to a third. Fixing the projection error shows that mountain slopes are much steeper and longer than usually reported. Knowing the true length of a mountain slope, we can use the topographic data to better understand and maybe predict processes and volumes, like landslides or fluid transport. The projection error in gridded models biases our view of surface topography at all scales Correcting for the projection error allows exploring physics-based erosion and transport laws The correction enables better sourcing of the topographic data we have access to