Covariant Approach to the Geometric Dilution of Precision

A covariant formulation of the Geometric Dilution of Precision (GDOP) matrix is presented in the framework of a Relativistic Positioning System (RPS). By including the receiver–emitter frequency ratios, the Frequency Geometric Dilution of Precision (FGDOP) scalar is computed in terms of observable quantities, the received frequencies and the angular separation between pairs of emitters in view. Some required concepts are first introduced: the FGDOP matrix and the Gram matrix associated to k light-like vectors. From the tensor form of the FGDOP matrix and its trace, a closed form of the FGDOP scalar is obtained, extending previous matrix calculations. Clarifying computations for symmetric emitter configurations are presented. The geometric interpretation of the GDOP scalar in terms of volumes and areas defined by the relative position of the emitters on the unit celestial sphere of the user is also recovered.