Localizing Multiple Events Using Times of Arrival: a Parallelized, Hierarchical Approach to the Association Problem

A fundamental problem in localizing multiple events based on Times of Arrival (ToAs) at a number of sensors is that of associating ToAs with events. We consider this problem in the context of acoustic sensors monitoring events that are closely spaced in time. Due to the relatively low speed of propagation of sound, the order in which the events arrive at a sensor need not be the same as the order in which they occur, potentially creating fundamental ambiguities. We first explore such ambiguities in an idealized setting with two events and noiseless observations, showing that it is possible to localize both events with nine or more sensors (as long as degenerate sensor placement is avoided), but that we can construct examples with six sensors for which unambiguous space-time localization is not possible. We then show that these potential ambiguities are not a bottleneck in typical practical settings, proposing and evaluating an algorithm that successfully localizes multiple events using noisy observations. The algorithm employs parallelism and hierarchical processing to avoid the excessive complexity of naïvely trying all possible associations of events with ToAs. We use discretization of hypothesized event times to enable us to efficiently generate a set of candidate event locations, which contain noisy versions of true events as well as phantom events. We refine these estimates iteratively, discarding “obvious” phantoms, and then solve a linear programming formulation for matching true events to ToAs, while identifying outliers and misses. Simulation results indicate excellent performance that is comparable to a genie-based algorithm which is given the correct association between ToAs and events.