Fix Probabilities from LOP Geometry

Abstract

A simple scheme is presented for mapping the 2D probability density for an observer's position, defined by any number of lines of position (LOPs) on the surface of the Earth, assuming that the LOPs result from uncorrelated observations that have normally distributed errors. Although the mapping can be used to determine the position fix corresponding to the LOPs (which is consistent with other methods), its intended use is computing the total probability that the observer is located within (or outside) some specified area of interest, such as a zone of avoidance around a navigational hazard. Numerical experiments with areas where the average total interior probability is known, such as the triangles and polygons formed by nearly convergent LOPs, show that the method provides correct answers. The numerical experiments also revealed that theoretical probabilities associated with commonly used error ellipses are overstated for navigational solutions based on small numbers of LOPs.