Based on various approaches, several different solutions to the smoothing problem have been given. The relationships between these solutions are not immediate, although they solve the same problem. Making use of a certain framework from scattering theory, we derive two families of solutions, with equations evolving forwards and backwards in time, respectively. Within these families three major previous approaches are obtained as special cases, and their relationships are clarified. The set of solutions also contains as a fourth special case a (new) backwards analog of the innovations solution. The Mayne-Fraser two-filter formula belongs to the set of backwards solutions, and within this framework certain difficulties with its interpretation can be resolved.
Nomenclature
~-the left-hand point of the considered time interval t the right-hand point of the considered time interval s a running time variable. Mainly used in equations that are naturally solved from s = ~" cr a running time variable. Mainly used in equations that are naturally solved from tr = t, i.e. backwards in time. tr runs forwards in time, so with the lefthand side of the ODEs written as -d/do, the fighthand side shows the stability properties of the algorithm as it is solved backwards P(r) true initial covariance at ~-II assumed initial covariance at r P(t, r; H) solution to (7) with P(~-, 1-; II) = II e(t, z) = P(t, r; P(r)) Po(t, ~) = e(t, "r; O) P,(r [ t; II) smoothed error covariance, given by, e.g. (12)
with P(~-) = II P,(r I t) = Ps(r I t; e0")) eb( ~" I t) = P,0" I t, co