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Single shot gamma-induced positron annihilation spectroscopy (GiPS) based on a laser wakefield accelerator is proposed. In this spectroscopy, a large number of gamma rays generated by laser-wakefield accelerated electrons are injected into a sample in a very short time (<2 ps), and the positron lifetime is detected by measuring the time profile of the annihilation radiation (511 keV gamma ray) from the sample. The processes including gamma-ray production, shielding, annihilation generation, and annihilation detection were simulated by using the Monte Carlo method. The results show that up to 1010 gamma rays with a duration of 1.4 ps can be produced per shot with a 45 TW laser system. A Cherenkov radiator PbF2 was used to detect the annihilation radiation, and its optimal thickness of 15 mm was obtained. A whole process simulation with optimized parameters was conducted. A low statistical fluctuation lifetime spectrum can be produced, which confirms the feasibility of single shot GiPS with a typical 45 TW laser system. [ABSTRACT FROM AUTHOR]