# wavelet

Calculates the wavelet transform of a time series and significance levels.

## Prototype

function wavelet ( y [*] : numeric, mother [1] : integer, dt [1] : numeric, param [1] : numeric, s0 [1] : numeric, dj [1] : numeric, jtot [1] : integer, npad [1] : integer, noise [1] : integer, isigtest [1] : integer, siglvl [1] : numeric, nadof [*] : numeric ) return_val [2][jtot][dimsizes(y)] : float or double

## Arguments

*y*

A one-dimensional array (call the length *N*).

*mother*

An integer giving the mother wavelet to use:

0 = 'Morlet'

1 = 'Paul'

2 = 'DOG' (derivative of Gaussian)

If *mother* < 0 or > 2, then the default is 'Morlet'.
(Most commonly, *mother* = 0.)

*dt*

The amount of time between each *y* value; i.e. the sampling
time. (Most commonly, *dt* = 1.0.)

*param*

The mother wavelet parameter. If *param* < 0, then the
default is used:

For 'Morlet' this is k0 (wavenumber), default is 6.

For 'Paul' this is m (order), default is 4.

For 'DOG' this is m (m-th derivative), default is 2.

*s0*

The smallest scale of the wavelet, which is typically is equal to
2**dt*.

Note: for accurate reconstruction and variance computation, set
*s0* = *dt* for Morlet; *s0* = *dt*/4 for
Paul. (Most commonly, *s0* = 2**dt*.)

*dj*

The spacing between discrete scales, which is typically equal to 0.25. A
smaller value will give better scale resolution, but will be slower.
(Most commonly, *dj* = 0.25.)

*jtot*

The integer number of scales. Scales range from *s0* up to
*s0**2^[(*jtot*-1)**dj*].

Most commonly, *jtot* is equal to:

1 +
**floattointeger**(((**log10**(*N***dt*/*s0*))/*dj*)/**log10**(2.))

*npad*

The total number of points (including padding) to use for the
wavelet transform. Typically, this is some power of 2. It must be
greater or equal to *N*. If *npad* > *N*, then
zeroes are padded onto the end of the time series. (Most commonly,
*npad* = *N* [i.e. no padding].)

*noise*

A value of 0 means use a white noise background for significance
testing. A value of 1 means use a red noise background for
significance testing. (Most commonly, *noise* = 1.)

*isigtest*

A value of 0 means do a regular chi-square test, i.e. Eqn (18) from Torrence and Compo. A value of 1 means do a "time-average" test on the global wavelet spectrum.

*siglvl*

The significance level to use. (Most commonly, *siglvl* =
0.95.)

*nadof*

Currently ignored (set to zero).

## Return value

This function returns a three-dimensional array (call it
*wave*) dimensioned 2 x *jtot* x
*N*. *wave* will contain the real (0,:,:) and
imaginary parts (1,:,:) of the wavelet transform, versus time and
scale. It will be of type double if *y* is double, and float
otherwise.

See the description below for information on attributes of
*wave* that are also returned.

## Description

This function is an interface to the wavelet software written by Christopher Torrence and Gilbert P. Compo, University of Colorado. The original software is available from:

http://paos.colorado.edu/research/wavelets/

This site provides Fortran, IDL and Matlab codes, including examples.

The user should read the following reference:

Torrence, C. and G. P. Compo, 1998: A Practical Guide to Wavelet Analysis.This will clarify the terminology used by the software.Bull. Amer. Meteor. Soc., 79, 61-78. doi: http://dx.doi.org/10.1175/1520-0477(1998)079<0061:APGTWA>2.0.CO;2

**Note**: please acknowledge the use of this software in any
publications:

"Wavelet software was provided by C. Torrence and G. Compo, and is available at the URL: http://paos.colorado.edu/research/wavelets/".The following are returned as attributes of

*wave*:

*wave*@*power*- A one-dimensional array (same type as
*wave*) of length*jtot*N*containing the squared sum of the real and imaginary parts of*wave*. The power spectrum =*wave*(0,:,:)^2 +*wave*(1,:,:)^2The function

**onedtond**should be used to convert to the more logical two-dimensional array. For example:power =

**onedtond**(wave@power, (/jtot,N/) ) *wave*@*phase*- A one-dimensional array (same type as
*wave*) of length*jtot*N*containing the phases (degrees) of wave:phase = atan2(wave(1,:,:),wave(0,:,:)) (*57.29 for degrees)

The function**onedtond**should be used to convert to the more logical two-dimensional array. For example:phase =

**onedtond**(wave@phase, (/jtot,N/) ) *wave*@*mean*- A scalar (same type as
*wave*) containing the mean of the input series. *wave*@*stdev*- A scalar (same type as
*wave*) containing the standard deviation of the input series. *wave*@*lag1*- A scalar (same type as
*wave*) containing the lag-1 autocorrelation of the input series. *wave*@*r1*- A scalar of the same type as
*wave*. If*noise*= 1, this contains the lag-1 autocorrelation of the input series. Otherwise,*wave*@*r1*= 0.0. This is the value used in the significance test. *wave*@*dof*- A one-dimensional array of length
*jtot*(same type as*wave*) containing the degrees-of-freedom for significance test. *wave*@*scale*- A one-dimensional array of length
*jtot*(same type as*wave*) containing the wavelet scales that were used. *wave*@*period*- A one-dimensional array of length
*jtot*(same type as*wave*) containing the "Fourier" periods (in time units) corresponding to "scale". *wave*@*gws*- A one-dimensional array of length
*jtot*(same type as*wave*) containing the global wavelet spectrum. *wave*@*coi*- A one-dimensional array of length
*N*(same type as*wave*) containing the e-folding factor used for the cone of influence. *wave*@*fft_theor*- A one-dimensional array of length
*jtot*(same type as*wave*) containing theoretical red-noise spectrum versus scale. If*isigtest*= 2, then*wave*@*fft_theor*(0) = the average spectrum from S1-->S2, and*wave*@*fft_theor*(1...*jtot*-1) = 0.0. *wave*@*signif*- A one-dimensional array of length
*jtot*(same type as*wave*) containing significance levels versus scale. *wave*@*cdelta*- A scalar (same type as
*wave*) containing the constant "Cdelta" for the mother wavelet (Table 2 of reference). *wave*@*psi0*- A scalar (same type as
*wave*) containing the constant "psi(0)" for the mother wavelet (Table 2 of reference).

A bias-rectified power spectrum as by Liu *et al* (2007) can be obtained using
the returned attributes [courtesy of Eros Albertazzi (CMCC, Italy)]:

power_no_bias = wave@power/conform(power,wave@scale,0) gws_no_bias = wave@gws/wave@scale

**Reference:**

Rectification of the bias in the wavelet power spectrum Liu, Y., X.S. Liang, and R.H. Weisberg, 2007 Journal of Atmospheric and Oceanic Technology, 24(12), 2093-2102. doi: http://dx.doi.org/10.1175/2007JTECHO511.1 see also: http://ocg6.marine.usf.edu/~liu/wavelet.html

## See Also

## Examples

This example reads a time series of seasonal mean sea surface temperatures. It mimics the example provided at the above URL.

load "$NCARG_ROOT/lib/ncarg/nclscripts/csm/gsn_code.ncl" load "$NCARG_ROOT/lib/ncarg/nclscripts/csm/gsn_csm.ncl" begin f =addfile("/fs/cgd/home0/shea/ncld/test/sst_nino3.nc", "r") x = f->SST N =dimsizes(x) ; number of elements [here 504] mother = 0 ; Morlet wavelet param = 6.0 ; common for Morlet dt = 0.25 s0 = 0.25 dj = 0.25 ; 4 sub-octaves per octave jtot = 44 ; =subScale*11 npad = 1024 ; pad with extra zeros nadof = new( 2,float) ; ignored noise = 1 ; test vs red noise siglvl = 0.05 isigtest= 0 w =wavelet(x,mother,dt,param,s0,dj,jtot,npad, \ noise,isigtest,siglvl,nadof) ; create coordinate arrays for plot power =onedtond( w@power, (/jtot,N/) ) power!0 = "period" ; Y axis power&period = w@period power!1 = "time" ; X axis power&time = x&time power@long_name = "Power Spectrum" power@units = "C^2" ; compute significance ( >= 1 is significant) SIG = power ; transfer metadata SIG = power/conform(power,w@signif,0) wks =gsn_open_wks("x11","example") ; PLOT (only up to periods of 64) ; power res = True res@cnFillOn= True res@trYReverse= True plot =gsn_csm_contour(wks,power({0:64},:),res) ; significance RES = True RES@cnLevelSelectionMode= "ManualLevels" ; set manual contour levels RES@cnMinLevelValF= 1.0 ; set min contour level RES@cnMaxLevelValF= 4.0 ; set max contour level RES@cnLevelSpacingF= 3.0 ; set contour spacing RES@trYReverse= True pSIG =gsn_contour(wks,SIG({0:64},:),RES) end