Editing True-range multilateration (section)

=== Hybrid multilateration systems ===

Hybrid multilateration systems – those that are neither true-range nor pseudo range systems – are also possible. For example, in Fig. 1, if the circle centers are shifted to the left so that '''C1''' is at <math>x_1^\prime = - \tfrac{1}{2} U, y_1^\prime = 0</math> and '''C2''' is at <math>x_2^\prime = \tfrac{1}{2} U, y_2^\prime = 0</math> then the point of interest '''P''' is at

: <math> \begin{align}
x^\prime & = \frac { (r_1^\prime + r_2^\prime)(r_1^\prime - r_2^\prime) } {2 U}   \\[4pt]
y^\prime & = \pm \frac { \sqrt{ (r_1^\prime + r_2^\prime)^2 - U^2 } \sqrt{ U^2 - (r_1^\prime - r_2^\prime)^2 } } {2 U}
\end{align} </math>

This form of the solution explicitly depends on the sum and difference of <math>r_1^\prime</math> and <math>r_2^\prime</math> and does not require 'chaining' from the <math>x^\prime</math>-solution to the <math>y^\prime</math>-solution. It could be implemented as a true-range multilateration system by measuring <math>r_1^\prime</math> and <math>r_2^\prime</math>.

However, it could also be implemented as a hybrid multilateration system by measuring <math>r_1^\prime + r_2^\prime</math> and <math>r_1^\prime - r_2^\prime</math> using different equipment – e.g., for surveillance by a [[multistatic radar]] with one transmitter and two receivers (rather than two monostatic [[radar]]s). While eliminating one transmitter  is a benefit, there is a countervailing 'cost': the synchronization tolerance for the two stations becomes dependent on the propagation speed (typically, the speed of light) rather that the speed of point '''P''', in order to accurately measure both <math>r_1^\prime \pm r_2^\prime</math>.

While not implemented operationally, hybrid multilateration systems have been investigated for aircraft surveillance near airports and as a GPS navigation backup system for aviation.<ref name="Narins">[https://www.nap.edu/read/13292/chapter/13 "Alternative Position, Navigation, and Timing: The Need for Robust Radionavigation"]; M.J. Narins, L.V. Eldredge, P. Enge, S.C. Lo, M.J. Harrison, and R. Kenagy; Chapter in ''Global Navigation Satellite Systems''Joint Workshop of the National Academy of Engineering and the Chinese Academy of Engineering (2012).</ref>