  Jointly Gaussian memoryless sources are observed at N distinct terminals. The
goal is to efficiently encode the observations in a distributed fashion so as
to enable reconstruction of any one of the observations, say the first one, at
the decoder subject to a quadratic fidelity criterion. Our main result is a
precise characterization of the rate-distortion region when the covariance
matrix of the sources satisfies a "tree-structure" condition. In this
situation, a natural analog-digital separation scheme optimally trades off the
distributed quantization rate tuples and the distortion in the reconstruction:
each encoder consists of a point-to-point Gaussian vector quantizer followed by
a Slepian-Wolf binning encoder. We also provide a partial converse that
suggests that the tree structure condition is fundamental.
