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NCL: Simple Fourier Analysis of Climate Data

Global atmospheric data are periodic in longitude (0-360) and climatological data are periodic in time (here, 12 months). The Fourier Analysis in the following examples uses a climatological data set derived from ERA-Interim data spanning 1989-2005.

The fourier_info, ezfftf and ezfftb can be used to perform variations of Fourier Analysis. Using these functions on a variable with longitude as the rightmost dimension performs spatial analysis. To examine temporal harmonics, the input series must be reordered so the dimension 'time' is the rightmost dimension.

fanal_1.ncl: A variable (here, gropotential height at 500 hPa) dimensioned (time,lat,lon) is examined to determine the amplitude, phase (1st maximum) and percent variance explained. Harmonics 1 and 2 are displayed.

fanal_2.ncl: Two similar variables TREFHT (temperature at 2m) and TSKIN (temperature at actual surface) are compared via harmonic analysis.
fanal_3.ncl: A forward fast Fourier transform (ezfftf) performs a 'Fourier Analysis'. Selected coefficents are set to zero to isolate different waves. A backward fast Fourier transform (ezfftb) is used to perform a 'Fourier Synthesis'.
fanal_4.ncl: Create three simple sine waves (blue, red, green) and combine the waves (superposition; black). Use fourier_info on the combined series to derive the amplitudes, phases and per-cent variances explained by each harmonic. Use a polymarker to mark the derived locations of the harmonic phase. The (edited) printed output is:

     Variable: finfo
     Type: float
     Total Size: 36 bytes
                 9 values
     Number of Dimensions: 2
     Dimensions and sizes:	[3] x [3]       ; [3] x [nhx]
     (0,0)	20         <=== Amplitudes
     (0,1)	10
     (0,2)	 5
     (1,0)	72         <=== Phases: location of first maximum
     (1,1)	36
     (1,2)	12
     (2,0)	76.19048   <=== % variance
     (2,1)	19.04762
     (2,2)	4.761904

fanal_5.ncl: Similar to fanal_3 and fanal_4 except that spherical harmonic analysis and synthesis are also used.