Accounting for Earth's curvature and elevation in animal movement modeling

Animals move in three dimensions, yet movement models do not capture the third dimension when data are collected as longitude and latitude coordinates (only) and models are developed in two dimensions. Here, we ask the question to what extent topography and the shape of the Earth-its ellipsoidal shape-affect the quantification of animal movement. To answer this question, we draw from the fields of ecology and geodesy: geodesy provides the mathematics to solve the problem on the curved surface of the Earth exactly, as opposed to using a map projection, because all map projections impart a scale distortion that systematically enlarges or shrinks the distances between points in space. We provide mathematical derivations to solve this problem in two different ways and show that they are equivalent. We also provide analyses of data from pumas (Puma concolor) and humpback whales (Megaptera novaeangliae) to discuss the impact of not accounting for the macroscopic, ellipsoidal shape of the Earth nor for topography when quantifying animal movements. In short, if the vertical extent of the movement is small compared to the horizontal extent, then the difference is de minimis. Conversely, if an animal moves vertically as much, or more, than how far it moves horizontally, then the difference is not negligible. Using map-projection coordinates without corrections systematically increases and/or decreases distances and, therefore, speeds.