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wavelet2.html
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<HTML>
<HEAD>
<TITLE>Wave:Wavelets 2</TITLE>
<link rel="icon" href="http://atoc.colorado.edu/research/wavelets/favicon.ico" type="image/x-icon" />
<link rel="shortcut icon" href="http://atoc.colorado.edu/research/wavelets/favicon.ico" type="image/x-icon" />
</HEAD>
<BODY BGCOLOR="white">
<H1>Wavelet Analysis</H1>
<H2>
<UL>
<LI><A HREF="wavelet1.html">Introduction</A>
<LI><FONT COLOR="green">Wavelets</FONT>
<LI><A HREF="wavelet3.html">Algorithms</A>
<LI><A HREF="montecarlo.html">Monte Carlo</A>
<LI><A HREF="references.html">References</A>
</UL>
</H2>
<HR><!----------------------------------------------------------------->
<P>
<H2>
Wavelets
</H2>
      In the <A HREF="wavelet1.html">previous section</A>
we saw how one measure of the stationarity
of a time series was to calculate the running variance
using a fixed-width window. Although we pointed out the disadvantage
of using a fixed-width window, one could repeat
the analysis with a variety of window widths. By smoothly varying the
window width one could then build a picture of the changes in variance
versus both time and window width. The obvious problem with this
technique is the simple "boxcar" shape of the window function, which
introduces edge effects such as ringing. As mentioned earlier, using
such a black-box-car, we still have no information on what is going on
within the box, but only recover the average energy.
<P>
      In Figure 2a we see an example of a wave "packet", of finite duration
and with a specific frequency. One could imagine using such a shape
as our window function for our analysis of variance. This "wavelet"
has the advantage of incorporating a wave of a certain period, as
well as being finite in extent. In fact, the wavelet shown in Figure 2a
(called the Morlet wavelet) is nothing more than a Sine wave
(green curve in Figure 2b) multiplied by a Gaussian envelope (red curve).
<P>
<HR><!----------------------------------------------------------------->
<CENTER>
<IMG SRC="images/morlet.gif" ALT="Morlet" WIDTH=512 HEIGHT=128>
</CENTER>
<BR CLEAR=all>
<DL COMPACT>
<DT>
<B>Fig 2.</B>
<DD>(a) Morlet wavelet of arbitrary width and amplitude,
with time along the x-axis. (b) Construction of the Morlet
wavelet (blue dashed) as a Sine curve (green) modulated by
a Gaussian (red).
</DL>
<HR><!----------------------------------------------------------------->
<P>
      Assuming that the total width of this wavelet is about 15 years,
we can find the correlation
between this curve and the first 15 years
of our time series shown in
<A HREF="wavelet1.html#fig1">Figure 1</A>.
This single number gives us a
measure of the projection of this wave packet on our data during
the 1876-1890 period, i.e. how much [amplitude] does our 15-year
period resemble a Sine wave of this width [frequency]. By sliding this
wavelet along our time series one can then construct a new time series
of the projection amplitude versus time.
<P>
      Finally, one can then vary the "scale" of the wavelet by changing
its width. This is the real advantage of wavelet analysis over
a moving Fourier spectrum.
For a window of a certain width, the sliding FFT
is fitting different numbers of waves, i.e. there can be many
high-frequency waves within a window, while the same window can only
contain a few (or less than one) low-frequency waves. The wavelet analysis
always uses a wavelet of the exact same shape, only the size scales up
or down with the size of the window.
<P>
<A NAME="eqn2.1">
      In addition to the amplitude of any periodic signals,
we would also like information on the phase.
In practice,
the Morlet wavelet shown in Figure 2a is defined as the product
of a complex exponential wave and a Gaussian envelope:
<P>
<FONT SIZE=4 COLOR=00FF00>(2.1)</FONT> . . . . . . . . . . </A>
<IMG SRC="images/wl_eqn2.1.gif" ALIGN=bottom WIDTH=228 HEIGHT=28>
<P>
<A NAME="eqn2.2">
where <I>Psi</I> is the wavelet value at non-dimensional time <I>eta</I>,
and <I>w<SUB>0</SUB></I> is the wavenumber. This is the basic wavelet
function, but we now need some way to change the
overall size as well as slide the entire wavelet along in time.
We thus define the "scaled wavelets" as:
<P>
<FONT SIZE=4 COLOR=00FF00>(2.2)</FONT> . . . . . . . . . . </A>
<IMG SRC="images/wl_eqn2.2.gif" ALIGN=center WIDTH=375 HEIGHT=56>
<P>
where <I>s</I> is the "dilation" parameter used to change the scale,
and <I>n</I> is the translation parameter used to slide in time.
The factor of <I>s</I><SUP>-1/2</SUP> is a normalization to keep the total
energy of the scaled wavelet constant.
<P>
<A NAME="eqn2.3">
      We are given a time series <I>X</I>, with
values of <I>x<SUB>n</SUB></I>, at time index <I>n</I>. Each value is
separated in time by a constant time interval <I>dt</I>.
The wavelet transform <I>W<SUB>n</SUB>(s)</I>
is just the inner product (or convolution) of the
wavelet function with our original timeseries:
<P>
<FONT SIZE=4 COLOR=00FF00>(2.3)</FONT> . . . . . . . . . . </A>
<IMG SRC="images/wl_eqn2.3.gif" ALIGN=center WIDTH=288 HEIGHT=61>
</P>
where the asterisk (*) denotes complex conjugate.
<P>
      The above integral can be evaluated for
various values of the scale <I>s</I>
(usually taken to be multiples of the lowest possible frequency),
as well as all values of <I>n</I> between the start and end dates.
A two-dimensional picture of the variability can then be constructed
by plotting the wavelet amplitude and phase.
<P>
<A NAME="fig3"></A>
      Figure 3 shows the power (absolute value
squared) of the wavelet transform for the NINO3 SST data.
The (absolute value)<SUP>2</SUP> gives information on the relative
power at a certain scale and a certain time.
A plot of the amplitude and phase
would show the actual oscillations of the individual wavelets, rather
than just their magnitude.
<P>
<HR><!----------------------------------------------------------------->
<CENTER>
<IMG SRC="images/nino3_wave.png" ALT="NINO3 Wavelet"
WIDTH=436 HEIGHT=400>
</CENTER>
<BR CLEAR=all>
<DL COMPACT>
<DT>
<B>Fig 3.</B>
<DD>(a) Time series of El Niño sea surface temperature.
(b) The wavelet power spectrum, using the Morlet wavelet.
The <I>x</I>-axis is the wavelet location in time.
The <I>y</I>-axis is the wavelet period in years.
The black contours are the 10% significance regions, using a
red-noise background spectrum.
The red areas indicate that high El Niño activity
occurred during 1880-1920 and 1965-present, while 1920-1960
was relatively calm.
</DL>
<HR><!----------------------------------------------------------------->
<P>
      Comparing Figures <A HREF="wavelet1.html#fig1">1</A> and 3
it is now much clearer that there
was large power in the 2-7 year ENSO period during both the earlier
and latter parts of this century.
In addition we can now see hints of a 16-year
oscillation as well as power at even lower frequencies.
<P>
      The wavelet transform also gives information on changes in frequency
that may have occured. Thus, from 1960-1990
the ENSO time band (2-7 years) seems to have undergone a slow oscillation
in period from a 3-year period between events back in 1965 up to
about a 5-year period in the early 1980s.
<P>
<CENTER>
<H3>
-- <A HREF="wavelet3.html">on to Algorithms</A> --
</H3>
</CENTER>
<HR><!----------------------------------------------------------------->
<A HREF="./">back to Wavelet Home Page</A>
</BODY>
</HTML>