Energy-constrained wireless sensor networks often are designed to measure a spatio-temporal process that is correlated in space, time, or both. The goal of these data-gathering networks is a description of the process that provides the required fidelity with a minimum expenditure of energy. Our approach combines (1) channel coding and estimation/decision processing of coded messages for in-network data censoring with (2) estimation of the encoded and censored information at a fusion center where energy is plentiful. Nodes examine their own messages together with messages from preceding nodes, and compute the fidelity of the estimate at the next node as a function of data censoring proposals. Our algorithm exploits redundancy of two types: intra-message redundancy from channel coding, and inter-message redundancy due to spatio-temporal correlation of the samples. This redundancy is used to alleviate overall energy consumption and message congestion near the fusion center by allowing relaying nodes to censor messages that might otherwise be forwarded, if those messages can be inferred from other messages, given the correlation model. The effect of censoring on fidelity and energy consumption is characterized, and our censoring algorithm shown to provide significant energy savings.