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In a passive multisatellite location system with Earth constraint (EC), the localization problem of time differences of arrival (TDOAs) is formulated as a quadratic optimization (QO) problem with two equality constraints: EC and variable constraint (VC). Which constraint is more important? To evaluate the importance of the two constraints, the original QO problem with both EC and VC is relaxed into two individual QO problems with only one equality constraint EC or VC being kept. A convex combination operation is performed on the two optimal solutions associated with the two relaxed QO problems to form a new weighted solution. Numerical simulation results show that the new weighted coefficient can achieve the Cramer-Rao lower bound with EC as the weighted factor of the solution exploiting only VC tends to zero, and increasing the value of this weighted coefficient will gradually degrade the overall localization performance. Thus, we conclude that the EC plays an extremely important role in improving the performance of TDOA-based localization.
Shu et al. (2017) studied this question.