  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 longstanding
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
true distance from a signal receiver (sensor) to the source and its 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 Jacobian matrix of the measurement model. 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.
