1

0

1

00

N

'n'n

N

'n'nn s

tn'n*xxsW

WAVELET TRANSFORM

Convolution of time series xn’ 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:

1

0

2'

1ˆN

n

Nknink ex

Nx

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)

1

0

N

'n'nn s

tn'n*xsW

WAVELET TRANSFORM

sˆ Fourier transform of

st

xW kInverse Fourier transform of is Wn (s)

1

0

N

k

tnikkn

kes*ˆxsW

With this relationship and a FFT routine, can calculate the continuous wavelet transform (for each s) at all n simultaneously

22

22

Nk:tNk

Nk:tNk

k

where the angular frequency:

Torrence and Compo (1998)

continuous wavelet transform

1

0

2'

1ˆN

n

Nknink ex

Nx

Discrete Fourier transform of xn

0

0

0

0

0

0

0

0

1

0

N

k

tnikkn

kes*ˆxsW

To ensure direct comparison from one s to the other, need to normalize wavelet function

kk sˆtssˆ

0

212

241 20 ee i

241 20 seH

11

22 m

mmi

!m!mi

sm

mesH

!mm

122

21

2

21

1

edd

mm

mm 22

21

smm

esm

i

120

'd'ˆ

i.e. each unscaled wavelet function has been normalized to have unit energy (daughter wavelets have same energy as mother)

1

0

2N

kk Nsˆ

and at each scale (N is total # of points):

Wavelet transform is weighted by amplitude of Fourier coefficients and not by kx

1

0

N

k

tnikkn

kes*ˆxsW

Wavelet transform Wn(s) is complex when wavelet function is complex

Wn(s) has real and imaginary parts that give the amplitude and phase

and the wavelet power spectrum is |Wn(s)|2

for real (DOG) the imaginary part is zero and there is no phase

for white noise

Nxk

22 22 sWn for all n and s

Normalized wavelet power spectrum is 22 sWn

22kn xNsW Expectation value for

wavelet power spectrum

This is a measure of the power relative to white noise

Seasonal SST averaged over Central Pacific

22 sWn

Power relative to white noise

0

0

0

0

0

0

0

0

Considerations for choice (can be considered arbitrary) of wavelet function:1) Orthogonal or non-orthogonal:Non-orthogonal (like those shown here) are useful for time series analysis. Orthogonal wavelets (for signal processing –discrete blocks of wavelet power)– Haar, Daubechies

2) Complex or real:Complex returns information on amplitude and phase; better adapted for oscillatory behavior. Real returns single component; isolates peaks or discontinuities

3) Width (e-folding time of 0):Narrow function -- good time resolutionBroad function – good frequency resolution

s2

s2

s2

2s

4) Shape:For time series with jumps or steps – use boxcar-like function (Haar)For smoothly varying time series – use a damped cosine (qualitatively similar results of wavelet power spectra).

Seasonal SST averaged over Central Pacific

22 sWn

22 sWn

Relationship between Wavelet Scale and Fourier period

Write scales as fractional powers of 2(this is “convenient”

J,,,j,ss jjj 1020

j

stNlogJ 02

smallest resolvable scale

largest scale

should be chosen so that the equivalent Fourier period is ~2 t j ≤ 0.5 for Morlet wavelet; ≤ 1 for others

N = 506 t = 0.25 yrs0 = 2 t = 0.5 yr

j = 0.125 J = 56

57 scales ranging from 0.5 to 64 yr

Relationship between Wavelet Scale and Fourier period

0

0

0

0

0

0

0

0

Can be derived substituting a cosine wave of a known frequency into

1

0

N

k

tnikkn

kes*ˆxsW

and computing s at which Wn is maximum

200 2

4

s

6031 0 :s.

124

m

s

43961 m:s.

212

m

s

29743 m:s.

64652 m:s.

Seasonal SST averaged over Central Pacific

22 sWn

6031 0 :s.

29743 m:s.

How to determine the COI & significance level?

s2

Cone of Influence

Paul

Morlet & DOG

2s

Because the square of a normally distributed real variable is 2 distributed with 1 DOF

22

2 is xk

22

2 be should sWn

At each point of the wavelet power spectrum, there is a 2

2 distribution

For a real function (Mexican hat) there is a 1

2 distribution

Distribution for the local wavelet power spectrum:

22

2

21

P sW

k2n

Pk is the mean spectrum at Fourier frequency k, corresponding to wavelet scale s

SUMMARY OF WAVELET POWER SPECTRUM PROCEDURES

1) Find Fourier transform of time series (may need to pad it with zeros)

2) Choose wavelet function and a set of scales

3) For each scale, build the normalized wavelet function kk sˆtssˆ

0

212

4) Find wavelet transform at each scale

1

0

N

k

tnikkn

kes*ˆxsW

5) Determine cone of influence and Fourier wavelength at each scale

6) Contour plot wavelet power spectrum

7) Compute and draw 95% significance level contour

Seasonal SST averaged over Central Pacific

22 sWn

Power relative to white noise

WAVELET TRANSFORM