  In this manuscript we consider the well-established problem of TDOA-based
source localization and propose a comprehensive analysis of its solutions for
arbitrary sensor measurements and placements. More specifically, we define the
TDOA map from the physical space of source locations to the space of range
measurements (TDOAs), in the specific case of three receivers in 2D space. We
then study the identifiability of the model, giving a complete analytical
characterization of the image of this map and its invertibility. This analysis
has been conducted in a completely mathematical fashion, using many different
tools which make it valid for every sensor configuration. These results are the
first step towards the solution of more general problems involving, for
example, a larger number of sensors, uncertainty in their placement, or lack of
synchronization.
