  We consider the centralized optimal estimation problem in spatially
distributed systems. We use the setting of spatially invariant systems as an
idealization for which concrete and detailed results are given. Such estimators
are known to have a degree of spatial localization in the sense that the
estimator gains decay in space, with the spatial decay rates serving as a proxy
for how far measurements need to be shared in an optimal distributed estimator.
In particular, we examine the dependence of spatial decay rates on problem
specifications such as system dynamics, measurement and process noise
variances, as well as their spatial autocorrelations. We propose
non-dimensional parameters that characterize the decay rates as a function of
problem specifications. In particular, we find an interesting matching
condition between the characteristic lengthscale of the dynamics and the
measurement noise correlation lengthscale for which the optimal centralized
estimator is completely decentralized. A new technique - termed the branch
point locus - is introduced to quantify spatial decay rates in terms of
analyticity regions in the complex spatial frequency plane. Our results are
illustrated through two case studies of systems with dynamics modeled by
diffusion and the Swift-Hohenberg equation, respectively.
