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The monthly files are mean values for the month bounded by the dates in the file name. T20000322000060.L3m_MO_SST_sst_4km.nc would be the mean values for February 2000. (Day of year 32-60)
The monthly climatology files are the means for the same days of the year bounded by the dates in the file name T20000322017059.L3m_MC_SST_sst_4km.nc would be the mean of Februaries between 2000 and 2017