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Recent advances in distributed source coding (DSC) for mission-driven wireless sensor networks (WSN) are related to the coding for multiple correlated sensors in applications such as real time target tracking and environment monitoring. The characteristic of these DSC applications provides significant potential opportunities in the associated sensor network to utilize multirate transmissions for enhancing network performance. In this paper, we study a methodology for interplay optimization between routing and DSC in WSN, and propose a novel multirate based routing scheme for mission-driven DSC applications that considerably extends network lifetime. The proposed scheme adopts the rate assignment based on the residual energy, and employs a joint rate and energy scheduling mechanism to meet the end-to-end transmission rate constraint, information precision requirement, and the energy constraints in the network for DSC. Simulation results show that this multirate based routing scheme achieves significantly longer network lifetime compared to other existing research in WSN for DSC applications.