  The physical position is crucial in location-aware services or protocols
based on geographic information, where localization is performed given a set of
sensor measurements for acquiring the position of an object with respect to a
certain coordinate system. In this paper, we revisit the long-standing
localization methods for locating a radiating source from range difference
measurements, or equivalently, time-difference-of-arrival measurements from the
perspective of least squares (LS). In particular, we focus on the spherical LS
error model, where the error function is defined as the difference between the
squared true distance from a signal receiver (sensor) to the source and its
squared measured value, and the resulting spherical LS estimation problem. This
problem has been known to be challenging due to the non-convex nature of the
hyperbolic measurement model. First of all, we prove that the existence of
least-square solutions is universal and that solutions are bounded under some
assumption on the geometry of the sensor placement. Then a necessary and
sufficient condition is presented for the solution characterization based on
the method of Lagrange multipliers. Next, we derive a characterization for the
uniqueness of the solutions incorporating a second-order optimality condition.
The solution structures for some special cases are also established,
contributing to insights on the effects of the Lagrangian multipliers on global
solutions. These findings establish a comprehensive understanding of the
localizability with range-difference measurements, which are also illustrated
with numerical examples.
