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How to improve the data persistence, i.e., the availability of all source data, is an important issue in wireless sensor networks. The issue requires that each source node can disseminate its data (packet) to a subset of nodes in the network for effective storage. In this paper, a distributed scheme based on LT(Luby Transform)-codes, named LTSIDP, is proposed. LT codes are the first rateless erasure codes that are very efficient as the amount of data grows. In LTSIDP, each node uses overhearing to get information whether a packet has been transmitted by one of its neighbors. When a node needs to transmit a packet, it randomly chooses one of its neighbors that does not transmit the packet as receiver. On the other hand, each node can compute a key parameter of LT codes by using some properties of the packet transmission mechanism, and then store the data accordingly. After the process of storage is finished, a collector (e.g., a motor vehicle) can recover all data by visiting a small subset of nodes. To the best of our knowledge, LTSIDP is the first scheme that uses overhearing to improve the data persistence. Theoretical analyses and simulations show that LTSIDP can achieve higher data persistence and energy efficiency than existing schemes.