  This paper derives a \emph{distributed} Kalman filter to estimate a sparsely
connected, large-scale, $n-$dimensional, dynamical system monitored by a
network of $N$ sensors. Local Kalman filters are implemented on the
($n_l-$dimensional, where $n_l\ll n$) sub-systems that are obtained after
spatially decomposing the large-scale system. The resulting sub-systems
overlap, which along with an assimilation procedure on the local Kalman
filters, preserve an $L$th order Gauss-Markovian structure of the centralized
error processes. The information loss due to the $L$th order Gauss-Markovian
approximation is controllable as it can be characterized by a divergence that
decreases as $L\uparrow$. The order of the approximation, $L$, leads to a lower
bound on the dimension of the sub-systems, hence, providing a criterion for
sub-system selection. The assimilation procedure is carried out on the local
error covariances with a distributed iterate collapse inversion (DICI)
algorithm that we introduce. The DICI algorithm computes the (approximated)
centralized Riccati and Lyapunov equations iteratively with only local
communication and low-order computation. We fuse the observations that are
common among the local Kalman filters using bipartite fusion graphs and
consensus averaging algorithms. The proposed algorithm achieves full
distribution of the Kalman filter that is coherent with the centralized Kalman
filter with an $L$th order Gaussian-Markovian structure on the centralized
error processes. Nowhere storage, communication, or computation of
$n-$dimensional vectors and matrices is needed; only $n_l \ll n$ dimensional
vectors and matrices are communicated or used in the computation at the
sensors.
