New characteristics of weighted GDOP in multi-GNSS positioning

In positioning, navigation and timing applications of multi-GNSS (global navigation satellite system) constellations, the geometric dilution of precision (GDOP) offers an important index for selecting satellites and evaluating positioning accuracy. However, GDOP assumes that the measurement errors of all the tracked satellites are independent and have the same accuracy level, which is impossible in practice, especially when the tracked satellites are from various constellations. Through introducing a weighted matrix describing the measurement errors of different satellites into a common GDOP, we focus on new characteristics of weighted GDOP (WGDOP) in two aspects. First, we compare the sizes of WGDOP and the common GDOP based on the range of the weights of different satellites, i.e., the diagonal elements of the weighted matrix. In addition, when the weights of different satellites increase, the change of WGDOP with the weights is also derived. Moreover, a closed-form formula for calculating WGDOP is also presented. The theoretical derivations demonstrate that the closed-form can reduce the computation burden effectively. Furthermore, numerical tests verify these analyses.