Lessons from a Prototype Geolocation Problem

Abstract

This paper establishes simple and general expressions for the accuracy of geolocation, which may be obtained by optimal filtering of measurements from archival tags. We investigate an idealized geolocation problem where the animal performs a random walk. We derive simple closed-form expressions for the steady-state variance and for the characteristic time scale of the filter, i.e. the smoothing horizon. This leads to temporal and spatial scales defining the limit of resolution and explains the difference between what can be obtained for fast-moving and slow-moving animals. Using frequency-domain methods, we consider the effect of adding additional sensors, and examine the substitution of the random walk model with anomalous diffusion, e.g. a Lévy flight. We also discuss time variations in the accuracy near start and end of the time series, and due to holes in the data stream which e.g. arise in the tidal method for geolocation when the animal is pelagic. Our results are particularly useful to the planning of a tagging study, because our estimates of accuracy can be computed using only three parameters: the swimming speed of the animal, the sample interval, and the variance on the measurement error.