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Some ECMWF data produced by the IFS is stored in GRIB with gridType=sh to indicate that the values are stored as spherical harmonic coefficients.  These are the X(n,m) coefficients in the discrete representation of the data on a grid at time, t, and vertical coordinate, η, by a continuous function expressed as a truncated series of spherical harmonics:

\begin{eqnarray*}
A(\lambda,\mu,\eta,t) & = &  \sum_{n=0m=-\mbox{T}}^{\mbox{T}} \, \sum_{m=-nn=|m|}^{\mbox{nT}}  X(n,m) (\eta,t) \overline{P}_{n}^{m}(\mu) e^{im\lambda}
\end{eqnarray*}

where μ = sinθ with λ  the longitude and θ the latitude of the grid point, T is the triangular spectral truncation number,  e^imλ are the Fourier functions.  P^m_n The

\overline{P}_n^m

are the normalised associated Legendre polynomials of the first kind ,  and  e^imλ are the Fourier functions.  The order of the summations can be interchanged to provide an equivalent expressionof degree n and order m with the normalisation defined by:

\begin{eqnarray*}
A(\lambda,\mu,\eta,t) & = &  \sum_{m=-\mbox{T}}^{\mbox{T}} \frac{1}{2} \int_{-1}^{1}\, \sum_{n=|m|}^{\mbox{T}}  X(n,m) (\eta,t) P\overline{P}_{n}^{m}(\mu) e^{im\lambda}\}^2 \,d\mu = 1
\end{eqnarray*}

In the GRIB binary data section, the complex X(n,m) coefficients are stored for m ≥ 0 as pairs of real numbers Re(X(n,m)) and Im(X(n,m)) ordered with n increasing from m to T, first for m = 0 and then for m = 1, 2, . . . T.  

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