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Database-driven Dynamic Spectrum Sharing (DSS) is the de-facto technical paradigm adopted by Federal Communications Commission for increasing spectrum efficiency, which allows licensed spectrum to be opportunistically used by secondary users. In database-driven DSS, a geo-location database administrator (DBA) maintains spectrum availability information over its service region in the form of a Radio Environment Map (REM), where the received signal strength from the primary user at every location is either directly measured via spectrum sensing or estimated via statistical spatial interpolation. Crowdsourcing-based spectrum sensing is a promising approach for periodically collecting spectrum measurements over a large geographic area but is unfortunately vulnerable to false spectrum measurements. Despite a large body of prior work on secure cooperative spectrum sensing, how to construct an accurate REM in the presence of false measurements remains an open challenge. In this paper, we introduce ST-REM, a novel spatiotemporal approach for securely constructing an REM in the presence of false spectrum measurements. Inspired by the self-label techniques developed for semi-supervised learning, ST-REM iteratively constructs an REM from a small number of spectrum measurements from trusted anchor sensors and many more measurements from mobile users. During each iteration, the DBA evaluates the trustworthiness of each measurement by jointly considering its spatial fitness with other trusted measurements and the mobile user's long-term behavior. By gradually incorporating the most trustworthy spectrum measurements, the DBA is able to construct a REM with high accuracy. Extensive simulation studies using a real spectrum measurement dataset confirm the efficacy and efficiency of ST-REM.