  We consider the distributed source coding system of $L$ correlated Gaussian
sources $Y_i,i=1,2,...,L$ which are noisy observations of correlated Gaussian
remote sources $X_k, k=1,2,...,K$. We assume that $Y^{L}={}^{\rm t}(Y_1,Y_2,$
$..., Y_L)$ is an observation of the source vector $X^K={}^{\rm t}(X_1,X_2,...,
X_K)$, having the form $Y^L=AX^K+N^L$, where $A$ is a $L\times K$ matrix and
$N^L={}^{\rm t}(N_1,N_2,...,N_L)$ is a vector of $L$ independent Gaussian
random variables also independent of $X^K$. In this system $L$ correlated
Gaussian observations are separately compressed by $L$ encoders and sent to the
information processing center. We study the remote source coding problem where
the decoder at the center attempts to reconstruct the remote source $X^K$. We
consider three distortion criteria based on the covariance matrix of the
estimation error on $X^K$. For each of those three criteria we derive explicit
inner and outer bounds of the rate distortion region. Next, in the case of
$K=L$ and $A=I_L$, we study the multiterminal source coding problem where the
decoder wishes to reconstruct the observation $Y^L=X^L+N^L$. To investigate
this problem we shall establish a result which provides a strong connection
between the remote source coding problem and the multiterminal source coding
problem. Using this result, we drive several new partial solutions to the
multiterminal source coding problem.
