Smoothing error dynamics and their use in the solution of smoothing and mapping problems

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 problems. 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 problems 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. Problems of this type arise in a variety of disciplines including geodesy and meteorology [2], [3]. In a previous paper [l] we presented derivations of algorithms for several of the problems we consider here. Unfortunately, the approach in [l] consisted of tedious man ipulations of filtering and smoothing equations which shed no light on the fundamental nature of the problems under investigation. The final forms of the solutions obtained in [l] were simple, suggesting that a more elegant approach must exist which would provide greater insight into problems of mapp ing and which would general ize more readily to problems outside the class considered in Manuscript received January 7, 1985; revised November 6, 1985. This work was supported in part by the MIT Lincoln Laboratory, and in part by the National Science Foundation under Grants ECS-8012668 and ECS-8312921. M. G. Bello was with the Laboratory for Information and Decision Systems and the Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA. He is now with the Analytical Sciences Corporation, 1 Jacob Way, Reading, MA 01867, U.S.A. A. S. Wil lsky and B. C. Levy are with the Laboratory for Information and Decision Systems and the Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, 77 Massachusetts Avenue Cambridge, MA 02139, U.S.A. D. A. Castanon was with the Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA. He is now with Alphatech, Inc., 2 Burlington Executive Center, 111 Middlesex Turnpike, Burlington, MA 01803, U.S.A. IEEE Log Number 8608092. [l]. In this paper we present such an approach and use it to derive solutions to several problems. A variety of different mapp ing problems is of practical interest. The first problem we will consider is that of map updating, in which one wishes to update an existing map (based on previously available measurements) with information contained in a new set of data. We also consider two other problems: the map combining problem, in which we wish to combine two maps over a given region each of which is based on a different set of data; and the map centralization problem, in which we are to produce a single map over a given region given several individual maps of subregions. All three of these problems arise in a variety of applications, including mapp ing of gravitational fields, topographical mapp ing, and the production and updating of meteorological maps. G iven the sizes of the regions being mapped and the large volumes of data to be used to produce the maps, a critical issue in these applications is the development of efficient methods for assimilating new information to produce up-to-date maps incorporating all available data sets. It is the need for efficiency that motivated our research, which had as its goals the development of recursive procedures for updating, combining, and centralization. The basis for our approach comes from viewing a map as an estimate, that is, as the projection of a random quantity onto the space spanned by a set of measurements. At this abstract level the solutions to our mapp ing problems are relatively clear. For example, in the map updating problem our objective is to compute the projection onto the space spanned by the old and new measurements. What we would like to do, however, is to compute this estimate explicitly in terms of the projection onto the old data and the new measurements. As we will explain more precisely in Section III, we can achieve our goal by projecting the error in our old map onto the space spanned by the new information available from the more recent measurement survey. In a similar fashion one can view the other two estimation problems in terms of appropriate projections. The crucial problem then is to find efficient methods for computing these projections. As we will discuss, the key to solving this problem is the construction of a mode l for the error in a given map. 0018-9448/86/0700-0483$01.00 01986 IEEE