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(In addition to using grid points, the ECMWF model (IFS) also uses an additional mathematical system, the spectral method, to represent horizontal space. This system uses a set of waves to describe variations in the horizontal. The IFS switches between spectral space and grid point space, in order to perform specific computations.)
ERA5: What is the spatial reference
Vertical
In the vertical, models can use levels, located at discrete points, and/or averages over layers.
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