Scalable Track Initiation for Optical Space Surveillance

The computational complexity of track initiation, also known as initial orbit determination or IOD, using only angle measurements is polynomial in the number of observations. However, the polynomial degree can be high, always at least cubic and commonly quartic or higher. Therefore, practical implementations require attention to the scalability of the algorithms, when one is dealing with the very large number of observations from large surveillance telescopes. We address two broad categories of algorithms. The first category includes and extends the classical methods of Laplace and Gauss, as well as the more modern method of Gooding, in which one solves explicitly for the apparent range to the target in terms of the given data. We find that the orbit solutions (data association hypotheses) can be ranked by means of a concept we call persistence, in which a simple statistical measure of likelihood is based on the frequency of occurrence of combinations of observations in consistent orbit solutions. However, range-solution methods can be expected to perform poorly if the initial orbit solutions of most interest are not well conditioned. The second category of algorithms addresses this difficulty. Instead of solving for range, these methods attach a set of range hypotheses to each measured line of sight. Then all pair-wise combinations of observations are considered and the family of Lambert problems is solved for each pair. These algorithms also have polynomial complexity, though now the complexity is quadratic in the number of observations and also quadratic in the number of range hypotheses. We offer a novel type of admissible-region analysis, constructing partitions of the orbital element space and deriving rigorous upper and lower bounds on the possible values of the range for each partition. This analysis allows us to parallelize with respect to the element partitions and to reduce the number of range hypotheses that have to be considered in each processor simply by making the partitions smaller. We present numerical results based on simulated data sets.