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Martingale decomposition 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 decomposition 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.