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The proliferation of advanced mobile terminals opened up a new crowdsourcing avenue, spatial crowdsourcing, to utilize the crowd potential to perform real-world tasks. In this work, we study a new type of spatial crowdsourcing, called time-continuous spatial crowdsourcing (TCSC in short). It supports broad applications for long-term continuous spatial data acquisition, ranging from environmental monitoring to traffic surveillance in citizen science and crowdsourcing projects. However, due to limited budgets and limited availability of workers in practice, the data collected is often incomplete, incurring data deficiency problem. To tackle that, in this work, we first propose an entropy-based quality metric, which captures the joint effects of incompletion in data acquisition and the imprecision in data interpolation. Based on that, we investigate quality-aware task assignment methods for both single- and multi-task scenarios. We show the NP-hardness of the single-task case, and design polynomial-time algorithms with guaranteed approximation ratios. We study novel indexing and pruning techniques for further enhancing the performance in practice. Then, we extend the solution to multi-task scenarios and devise a parallel framework for speeding up the process of optimization. We conduct extensive experiments on both real and synthetic datasets to show the effectiveness of our proposals.