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Recent advances in wireless sensor networks have led to renewed interests in the problem of source localization. Source localization has broad range of applications such as emergency rescue, asset inventory, and resource management. Among various measurement models, one important and practical source signal measurement is the received signal time of arrival (TOA) at a group of collaborative wireless sensors. Without time-stamp at the transmitter, in traditional approaches, these received TOA measurements are subtracted pairwise to form time-difference of arrival (TDOA) data for source localization, thereby leading to a 3-dB loss in signal-to-noise ratio (SNR). We take a different approach by directly applying the original measurement model without the subtraction preprocessing. We present two new methods that utilize semidefinite programming (SDP) relaxation for direct source localization. We further address the issue of robust estimation given measurement errors and inaccuracy in the locations of receiving sensors. Our results demonstrate some potential advantages of source localization based on the direct TOA data over time-difference preprocessing.