Editing True-range multilateration (section)

=== Redundant range measurements ===

When there are more range measurements available than there are problem dimensions, either from the same '''C1''' and '''C2''' (or '''C1''', '''C2''' and '''C3''') stations, or from additional stations, at least these benefits accrue: 
* 'Bad' measurements can be identified and rejected
* Ambiguous solutions can be identified automatically (i.e., without human involvement) -- requires an additional station
* Errors in 'good' measurements can be averaged, reducing their effect.

The iterative [[Gauss–Newton algorithm]] for solving [[non-linear least squares]] (NLLS) problems is generally preferred when there are more 'good' measurements than the minimum necessary. An important advantage of the Gauss–Newton method over many closed-form algorithms is that it treats range errors linearly, which is often their nature, thereby reducing the effect of range errors by averaging.<ref name="Sirola" /> The Gauss–Newton method may also be used with the minimum number of measured ranges. Since it is iterative, the Gauss–Newton method requires an initial solution estimate.

In 3-D Cartesian space, a fourth sphere eliminates the ambiguous solution that occurs with three ranges, provided its center is not co-planar with the first three. In 2-D Cartesian or spherical space, a third circle eliminates the ambiguous solution that occurs with two ranges, provided its center is not co-linear with the first two.