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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:	 x        ;  x [nhx]
Coordinates:

(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.