Geometry and First-Order Error Statistics for Three-and-Four-Station Hyperbolic Fixes on a Spherical Earth

摘要

Certain fundamental geometric properties of hyperbolas and ellipses on the surface of a sphere are developed. A first-order theory is then presented for the random errors in the location of a point on a spherical earth, by the intersection of spherical hyperbolas physically determined by a time difference technique in which the individual errors are 'normally' distributed and uncorrelated. Two cases are treated: (a) Four time-measuring stations working as two independent pairs, and (b) Three time-measuring stations working on a common time-base. In both cases, the contours of equal error probability density are shown to be ellipses and their properties are discussed. In both cases the distribution of radial error without regard to direction is given by a generalized distribution function which contains the Rayleigh distribution as a special case. Using the general form, the most probable mean, root mean square, and median radial errors are obtained. The results are adaptable to include errors in ordinary direction finders as well as those of the hyperbolic type. Numerical examples are included. (Author)