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Efficient target localization is critical to many marine applications in the Internet of Underwater Things (IoUT). Doppler effect becomes more predominant in the underwater environment when the relative speed of the moving object, i.e., the observer, to the signal propagation speed in water is much larger than that in the air. This along with the observer-target geometry will bring in a notable impact on the localization performance. In this paper, we derive the Cram´er-Rao lower bound (CRLB) and formulate the A-optimality criterion-based observer trajectory optimization problem to improve the localization performance based on time delay and Doppler shift measurements. We show that in the worst-case scenario, there will be a conflict between the non-accessible zone constraint and the observer dynamics constraint, which will lead to an erroneous result. To address this problem, we propose a warning zone-based augmented Lagrange multiplier method (ALMM) where the non-accessible zone constraint is relaxed to resolve the conflict and ensure the non-accessible requirement of the targeted zone is maintained. Performance evaluations are conducted through extensive simulations for different scenarios, and the results are compared to other methods with or without trajectory optimization. We demonstrate that trajectory optimization using time delays and Doppler shifts can greatly improve the target localization accuracy in underwater networks.