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,n) 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:
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\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 , Pnm and eimλ are the Fourier functions. The normalised associated Legendre polynomials of the first kind , and eimλ are the Fourier functions. The order of the summations can be interchanged to provide an equivalent expressionof degree n and order m are denoted by
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\overline{P}_n^m |
with the normalisation defined by:
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\begin{eqnarray*} A(\lambda,\mu,\eta,t) & = & \sum_{m=-\mbox{T}}^{\mbox{T}} \, \sum_{n=|m|}^{\mbox{T}} X(n,m) (\eta,t) P\frac{1}{2} \int_{-1}^{1}\, \{\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.
Hence, the ecCodes values array contains the coefficients in the order Re(X(0,0)), Im(X(0,0)), Re(X(1,0,1)), Im(X(1,0,1), Re(X(2,0,2)), Im(X(2,0,2)), … , Re(X(T,0,T)), Im(X(T,0,T)), Re(X(1,1)), Im(X(1,1)), Re(X(2,1,2)), Im(X(2,1.2)), … , Re(X(T,1,T)), Im(X(T,1,T)), ... ,Re(X(T,T-1,T)), Im(X(T,T-1,T)), Re(X(T,T)), Im(X(T,T)).
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Only the coefficients for m ≥ 0 are stored. The coefficients for m < 0 are obtained from the complex conjugates of the corresponding coefficient with m > 0 : as X(n,-m)= X*(n,m) / (-1)m |
The spherical harmonic (n, m) wave number space is shown in the figure.
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# Copyright 2020 ECMWF. # # This software is licensed under the terms of the Apache Licence Version 2.0 # which can be obtained at http://www.apache.org/licenses/LICENSE-2.0. # # In applying this licence, ECMWF does not waive the privileges and immunities # granted to it by virtue of its status as an intergovernmental organisation # nor does it submit to any jurisdiction. from __future__ import print_functionsys import traceback import sys from eccodes import * INPUT = "../../data/spherical_model_level.grib2" VERBOSE = 1 # verbose error reporting def example(): f = open(INPUT, "rb") while 1: gid = codes_grib_new_from_file(f) if gid is None: break # Get the values. This will be a numpy array values = codes_get_array(gid,"values") # Store the real and imaginary parts of the coefficients in arrays a and b, respectively # The real parts 'a' stored in every second element starting at element 0 a = values[0::2] # The imaginary parts 'b' stored in every second element starting at element 1 b = values[1::2] codes_release(gid) # Loop through the values and print the m and n indices together with the corresponding # real and imaginary parts of the coefficients m = 0 n = 0 for i in range(len(a)): print("%d\t n=%d\t m=%d %.10f\t%.10f" % (i, n, m, a[i], b[i])) n += 1 if n > T: m += 1 n = m f.close() def main(): try: example() except CodesInternalError as err: if VERBOSE: traceback.print_exc(file=sys.stderr) else: sys.stderr.write(err.msg + "\n") return 1 if __name__ == "__main__": sys.exit(main()) |
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