This paper introduces the Multiplicative Measurement Model for observed vectors where the error is modeled as a rigid rotation of the true direction about an axis orthogonal to it. Motivation comes from the fact that the linear additive model, b = Cr + v, where v N{0, R), is in contradiction with nonlinear rotational mechanics. The resulting true- to-nature multiplicative error model highlights a small effect (due to the spherical geometry) affecting any vector observation in space. The consequence of this effect is that the angle between unbiased measurements is biased. This three-dimensional effect, which increases when the sensor accuracy decreases, is here quantified in two direction-based observation systems: a) attitude estimation and b) positioning estimation systems. The multiplicative measurement covariance matrix is derived. Preliminary calculations suggest that the problem of attitude estimation is affected to a small extent because attitude optimality indirectly compensates for the 3D effect. On the other hand, this 3D effect is shown to affect positioning estimation from direction measurements. A comparison between the precision provided by multiplicative and additive models is given for a simple two-measurement case.
