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We consider a distributed source coding problem of L correlated Gaussian observations Yi, i = 1, 2, ..., L. We assume that the random vector YL = t(Y1, Y2, ..., YL) is an observation of the Gaussian random vector XK = t(X1,X2, ..., XK), having the form YL = AXK + NL, where A is a L × K matrix and NL = t(N1, N2, ..., NL) is a vector of L independent Gaussian random variables also independent of XK. We consider two distortion criterion based on the covariance matrix of the estimation error on XK. One is the criterion called the vector distortion criterion distortion region where each of the the diagonal elements of the covariance matrix is upper bounded by a prescribed level. The other is the criterion called the sum distortion criterion where the trace of the covariance matrix is upper bounded by a prescribed level. For each of the above two distortion criterion we derive explicit inner and outer bounds of the rate distortion region. We also derive an explicit matching condition in the case of the sum distortion criterion.