How to distribute sensors in a random field?

We study the optimal way of distributing sensors in a random field to minimize the estimation distortion. We show that this problem is equivalent to certain problems in optimal sampling design and multiterminal source coding. In our model, we partition the field into cells within which the observations are assumed to be the same. A number of identical sensors that can report the observed data to the fusion center at communication rate ΔR are to be allocated to cells so that the distortion achieved at the fusion center is minimized. We find optimal solutions for the Gaussian, binary and simplified Gaussian vector cases.