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Due to a number of reasons, members from ensembles using the "lagged start" approach are bundled together at C3S and labelled with a "nominal" start date (see details here).
The C3S seasonal forecast datasets are based on GRIB files stored at ECMWF MARS archive where that "nominal" start date is used to index the monthly data (monthly statistics and monthly post-processed products like anomalies). Hence, the details about the original initialization date could get lost in those files. Nonetheless, the data at daily and sub-daily time frequencies is always encoded with the real initialization date, as it can be seen in the example below:
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