COSINE-100 is a direct detection dark matter search experiment that uses a 106 kg array of eight NaI(Tl) crystals that are kept underground at the Yangyang Underground Laboratory to avoid cosmogenic activation of radioisotopes by cosmic rays. Even though the cosmogenic activity is declining with time, there are still significant background rates from the remnant nuclides. In this paper, we report measurements of cosmogenic isotope contaminations with less than one year half-lives that are based on extrapolations of the time dependent activities of their characteristic energy peaks to activity rates at the time the crystals were deployed underground. For longer-lived $^{109}$Cd ($T_{1/2}=1.6$ y) and $^{22}$Na ($T_{1/2}=2.6$ y), we investigate time correlations of characteristic $gamma$/X-ray peaks. The inferred sea-level production rates are compared with caluclations based on the ACTIVIA and MENDL-2 model calculations and experimental data. For $^{3}$H, which has a long, 12.3 year half-life, we evaluated the activity levels from the exposure times and determined a cosmogenic activation rate that is consistent with other measurements.
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