1. Complex demodulation Windowed Fourier Transform burst data Appropriate for burst data continuous data Questionable for continuous data (may miss important frequencies)

2. Complex demodulation

3. Wavelet Analysis Seasonal SST averaged over Central Pacific (Torrence and Compo, 1998) (Derivative of a Gaussian) “… to analyze time series that contain nonstationary power at different frequencies (Daubechies 1990)”

4. Continuous Wavelet Transform Continuous Wavelet Transform: Convolution of time series x n [N is the # of observations] with a scaled and translated version of a wavelet  0 (  ) [  is non-dimensional time] WAVELET TRANSFORM  t is sampling interval s is scale (frequency) xnxn s The wavelet  0 (  ) must have zero mean and be localized in time and frequency and have a complex conjugate  *

5. H(  ) is the Heaviside step function: H(  ) = 1 @  > 0 H(  ) = 0, otherwise  - scaling factor Real Imaginary Fourier transform of Complex Morlet (in time domain) plane wave modulated by a Gaussian Considerably faster to do the wavelet transform calculation in frequency (or Fourier space)

6. Real Imaginary WAVELET FUNCTIONS m = 6 Fourier transform

7. (derivative of Gauss order 2) m = 2

9. WAVELET FUNCTIONS m = 2 Fourier transform

10. WAVELET FUNCTIONS m = 4 Fourier transform

11. WAVELET FUNCTIONS m = 6 Fourier transform

12. WAVELET TRANSFORM Convolution of time series x n’ with a scaled and translated version of a base function: a wavelet  0 (  ) – continuous function in time and frequency – “mother wavelet” Convolution needs to be effected N (# of points in time series) times for each scale s; n is a translational value Much faster to do the calculation in Fourier space Convolution theorem allows N convolutions to be done simultaneously with the Discrete Fourier Transform: k is the frequency index Convolution theorem: Fourier transform of convolution is the same as the pointwise product of Fourier transforms Torrence and Compo (1998)

13. WAVELET TRANSFORM Fourier transform of Inverse Fourier transform ofis W n (s) With this relationship and a FFT routine, can calculate the continuous wavelet transform (for each s) at all n simultaneously where the angular frequency:

14. EXAMPLE OF WAVELET APPLICATION

15. (Torrence and Compo, 1998)