Smoothing Error Dynamics and The ir Use in the Solution o f Smoothing and Mapp ing Problems

Abstract-Martingale decomposit ion techniques are used to derive Markovian models for the error in smoothed estimates of processes described by linear models driven by white noise. These models, together with some simple Hilbert space decomposit ion ideas, provide a simple unified framework for examining a variety of problems involving the efficient assimilation of spatial data, which we refer to as mapping prob-lems. Algorithms for several different mapping problems are derived. A specific example of map updating for a two-dimensional random field is included. I. INTR~DU~TION I N THIS PAPER we consider several estimation prob- lems motivated by the subject of mapp ing. Our work is directed toward problems in which the objective is to obtain an efficient procedure for producing a map of a random field which combines the information contained in several other maps and/or sets of measurements. Problem