Systematic error is one of the major factors that affect positioning accuracy owing to the changeable and complex nature of seawater environments. Based on a Global Navigation Satellite System-acoustic intelligent buoys system, whose acoustic array consists of a series of surface buoys, a single-difference method for underwater dynamic positioning is proposed to eliminate systematic error. Positioning configuration optimisation was addressed using dilution of precision (DOP). A simulation of DOP proved that for the single-difference method, a radiation network with a centre-difference reference point was superior to a regular polygon network. The positioning experiment showed that the novel method could effectively eliminate systematic error, improving vertical positioning accuracy from a metre- to a decimetre scale.

Code phase Global Navigation Satellite System (GNSS) positioning performance is often described by the Geometric or Position Dilution of Precision (GDOP or PDOP), functions of the number of satellites employed in the solution and their geometry. This paper develops lower bounds to both metrics solely as functions of the number of satellites, effectively removing the added complexity caused by their locations in the sky, to allow users to assess how well their receivers are performing with respect to the best possible performance. Such bounds will be useful as receivers sub-select from the plethora of satellites available with multiple GNSS constellations. The bounds are initially developed for one constellation assuming that the satellites are at or above the horizon. Satellite constellations that essentially achieve the bounds are discussed, again with value toward the problem of satellite selection. The bounds are then extended to a non-zero mask angle and to multiple constellations.

For single point positioning solutions with multi-Global Navigation Satellite System (GNSS) constellations, the Position Dilution Of Precision (PDOP) and Time Dilution Of Precision TDOP) are quality measures to specify the additional multiplicative affect of measurement error on positioning accuracy and the timing accuracy, respectively. Considering the dimension of the unknown vector (including the three-dimensional positioning information of the receiver as well as the receiver clock biases related to these single constellations), this paper theoretically derives new formulae to describe the change of PDOP and TDOP. In addition, the detailed expressions of the variable quantity of the PDOP and TDOP are also obtained. The results show that if the dimension of the unknown vector is invariant in case of removing one or more satellites belonging to the existing tracked multi-GNSS constellations, both the PDOP and TDOP increase. On the other hand, if removing satellites reduces the dimension of the unknown vector, the PDOP neither increases nor decreases. However, the change of the TDOP is different, and it becomes smaller.

For multi-Global Navigation Satellite System (GNSS) constellations, the Geometric Dilution of Precision (GDOP) is an important parameter utilised for the selection of satellites. This paper has derived new formulae to describe the change of GDOP. The result shows that, for GNSS single point positioning solutions, if one more satellite belonging to the existing tracked multi-GNSS constellation used in the single point positioning solution is added, the GDOP always decreases with the number of the added satellites. On the other hand, when the constellation of the added satellite is not from the tracked existing constellations, the different numbers of the added satellites have different influences on the change of GDOP. Generally, adding one satellite from another constellation into the existing multi-GNSS constellations will increase the GDOP, but adding two satellites will decrease the GDOP compared with adding one from another constellation. Additionally, the GDOP also increases in the cases of adding two satellites from two different constellations into the tracked existing constellations.