Relations between dilutions of precision and volume of the tetrahedron formed by four satellites

The relationships between the volume of a 4-satellite tetrahedron and the dilutions of precision (GDOP, PDOP, HDOP, VDOP, TDOP) have been frequently quoted and used in technical papers and discussions. The most commonly encountered statement concerning any dilution of precision (DOP) notes that it is approximately inversely proportional to this tetrahedral volume. A closer look at these algebraic, fractional expressions for DOPs shows that this statement is not correct due to the fact that even though volume does appear in the denominator of each DOP, the numerator is not a constant. Thus, some of the factors in the denominator can be cancelled with those same factors that appear in the numerator. The purpose of this paper is to use a sequence of simple arguments to arrive at an exact but complex, set of results which indicate the true relationship(s) between the various DOPs and the tetrahedral volume. Examples for special and limiting cases are also included for clearer illustration as well as to clarify misconceptions