  We study the problem of the reconstruction of a Gaussian field defined in
[0,1] using N sensors deployed at regular intervals. The goal is to quantify
the total data rate required for the reconstruction of the field with a given
mean square distortion. We consider a class of two-stage mechanisms which a)
send information to allow the reconstruction of the sensor's samples within
sufficient accuracy, and then b) use these reconstructions to estimate the
entire field. To implement the first stage, the heavy correlation between the
sensor samples suggests the use of distributed coding schemes to reduce the
total rate. We demonstrate the existence of a distributed block coding scheme
that achieves, for a given fidelity criterion for the reconstruction of the
field, a total information rate that is bounded by a constant, independent of
the number $N$ of sensors. The constant in general depends on the
autocorrelation function of the field and the desired distortion criterion for
the sensor samples. We then describe a scheme which can be implemented using
only scalar quantizers at the sensors, without any use of distributed source
coding, and which also achieves a total information rate that is a constant,
independent of the number of sensors. While this scheme operates at a rate that
is greater than the rate achievable through distributed coding and entails
greater delay in reconstruction, its simplicity makes it attractive for
implementation in sensor networks.
