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| <HTML> | ||
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| <HEAD> | ||
| <TITLE>Wave:Data Analysis</TITLE> | ||
| </HEAD> | ||
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| <BODY BGCOLOR="#FFFFFF"> | ||
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| <CENTER><H1>Geophysical Data Analysis</H1></CENTER> | ||
| <P> | ||
| <BLOCKQUOTE> | ||
| <FONT SIZE=4> | ||
| <UL> | ||
| ¤ To reduce a set of time series to a few parameters that | ||
| capture a large variance percentage, and,<BR> | ||
| ¤ To use these dominant parameters to explain the | ||
| behavior of the time series. | ||
| </UL> | ||
| </FONT> | ||
| </BLOCKQUOTE> | ||
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| <TABLE ALIGN=center CELLPADDING=5> | ||
| <TR ALIGN=center> | ||
| <TD><TD><TD ALIGN=left><FONT SIZE=5><B>Method</B> | ||
| <TD><FONT SIZE=5><B>Single Time Series</B> | ||
| <TD><FONT SIZE=5><B>Multi-series</B> | ||
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| <TR VALIGN=top><TD ROWSPAN=5 ALIGN=left VALIGN=middle><FONT SIZE=5> | ||
| <PRE> | ||
| | | ||
| | | ||
| | | ||
| Increasing | ||
| Complexity | ||
| (more | ||
| variables) | ||
| | | ||
| | | ||
| | | ||
| | | ||
| \|/ | ||
| V | ||
| </PRE> | ||
| </TD> | ||
| <TD> | ||
| <TD><FONT SIZE=5><I><B>Statistics</B></I> | ||
| <TD><FONT SIZE=5><UL><LI>mean, variance, skew, kurtosis | ||
| <LI>curve-fitting | ||
| <LI>autocorrelation | ||
| </UL> | ||
| <TD><FONT SIZE=5><UL><LI>multiple linear regression | ||
| <LI>covariance, cross-correlation | ||
| </UL> | ||
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| <TR VALIGN=top><TD> | ||
| <TD><FONT SIZE=5><I><B>Spectrum</B></I> | ||
| <TD><FONT SIZE=5><UL><LI>Fourier power spectrum (FFT) | ||
| <LI>Maximum Entropy Method (MEM) | ||
| <LI>Multi-Taper Method (MTM) | ||
| </UL> | ||
| <TD><FONT SIZE=5><UL><LI>Cross-spectrum, phase spectrum | ||
| <LI>Coherency | ||
| </UL> | ||
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| <TR BGCOLOR="#FFFF99" VALIGN=top><TD> | ||
| <TD><FONT SIZE=5><I><B>Wavelet Transform</B></I> | ||
| <TD><FONT SIZE=5><UL><LI>Wavelet power spectrum | ||
| <LI>Sliding FFT; evolutionary spectra | ||
| </UL> | ||
| <TD><FONT SIZE=5><UL><LI>Cross-wavelet spectrum, phase | ||
| <LI>Wavelet coherency | ||
| </UL> | ||
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| <TR VALIGN=top><TD> | ||
| <TD><FONT SIZE=5><I><B>Orthogonal Modes</B></I> | ||
| <TD><FONT SIZE=5><UL><LI>Singular Spectrum Analysis (SSA) | ||
| </UL> | ||
| <TD><FONT SIZE=5><UL><LI>Empirical Orthogonal Functions (EOF) | ||
| & Singular Value Decomposition (SVD) | ||
| </UL> | ||
| </TABLE> | ||
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| <!-- | ||
|  <P> | ||
| <HR> | ||
| <A HREF="">back to Wavelet Home Page</A><BR> | ||
| --> | ||
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| </BODY> | ||
| </HTML> |
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| <HTML> | ||
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| <HEAD> | ||
| <TITLE>Wave:Applications</TITLE> | ||
| </HEAD> | ||
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| <BODY BGCOLOR="white"> | ||
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| <H1>Wavelet Applications</H1> | ||
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|  <P> | ||
| <FONT SIZE=5> | ||
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| <UL> | ||
| <LI> Image processing [Mallat (1988)] | ||
| <P> | ||
| <LI> Irregularly-spaced data (stellar activity) | ||
| [Frick et al. (1997)] | ||
| <P> | ||
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| <LI> 2D turbulence prediction; wavelet packet compression [Farge et al. (1992)] | ||
| <P> | ||
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| <LI> Coherent structure detection [Farge (1992), Gamage (1990)] | ||
| <P> | ||
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| <LI> Wave breaking & front detection [Liu (1994), Gamage & Blumen (1993)] | ||
| <P> | ||
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| <LI> Wavelet bases for partial differential equations | ||
| [Farge et al. (1992), Bacry et al. (1993)] | ||
| <P> | ||
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| <LI> Paleoclimate [Bolton et al. (1995), Yiou et al. (1995)] | ||
| </UL> | ||
| </FONT> | ||
|  <P> | ||
| <H1>Summary</H1> | ||
|  <P> | ||
| <FONT SIZE=5> | ||
| <DL COMPACT> | ||
| <DT>¤<DD>Wavelet analysis is a useful addition to methods of | ||
| geophysical data analysis.<BR> | ||
| <DT>¤<DD>Most helpful in situations of locally-coherent signals, | ||
| or non-stationary time series.<BR> | ||
| <DT>¤<DD>More research is needed on the perils & pitfalls, | ||
| choice of wavelet bases, and on statistical significance. | ||
| </DL> | ||
| </FONT> | ||
| <P><!-- | ||
| <HR> | ||
| <A HREF="">back to Wavelet Home Page</A> | ||
| --> | ||
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| </BODY> | ||
| </HTML> |
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| <HTML> | ||
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| <HEAD> | ||
| <TITLE>Wave:History</TITLE> | ||
| </HEAD> | ||
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| <BODY BGCOLOR="white"> | ||
| <H1> | ||
| Sketchy History of Wavelet Analysis | ||
| </H1> | ||
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| <TABLE CELLPADDING=5> | ||
| <TR VALIGN=top><TD><FONT SIZE=5>1800's</FONT><TD><FONT SIZE=5>Joseph Fourier</FONT> | ||
| <TD><FONT SIZE=5>Fourier transform & series; heat diffusion</FONT> | ||
| <TR VALIGN=top><TD> | ||
| <TR VALIGN=top><TD><FONT SIZE=5>1909</FONT><TD><FONT SIZE=5>Alfred Haar</FONT> | ||
| <TD><FONT SIZE=5>rectangular Haar basis functions</FONT> | ||
| <TR VALIGN=top><TD> | ||
| <TR VALIGN=top><TD><FONT SIZE=5>1946</FONT><TD><FONT SIZE=5>Dennis Gabor</FONT> | ||
| <TD><FONT SIZE=5>time-frequency Gabor wavelets; communication theory</FONT> | ||
| <TR VALIGN=top><TD> | ||
| <TR VALIGN=top><TD><FONT SIZE=5>1947</FONT> | ||
| <TD><FONT SIZE=5>Jean Ville & Hermann Wigner</FONT> | ||
| <TD><FONT SIZE=5>Wigner-Ville transform; quantum mechanics</FONT> | ||
| <TR VALIGN=top><TD> | ||
| <TR VALIGN=top><TD><FONT SIZE=5>1965</FONT><TD><FONT SIZE=5>Cooley & Tukey</FONT> | ||
| <TD><FONT SIZE=5>Fast Fourier Transform (FFT)</FONT> | ||
| <TR VALIGN=top><TD> | ||
| <TR VALIGN=top><TD><FONT SIZE=5>1984</FONT><TD><FONT SIZE=5>Grossman & Morlet</FONT> | ||
| <TD><FONT SIZE=5>synthesis of theory; geophysics & earthquakes</FONT> | ||
| <TR VALIGN=top><TD> | ||
| <TR VALIGN=top><TD><FONT SIZE=5>1985</FONT><TD><FONT SIZE=5>Yves Meyer</FONT> | ||
| <TD><FONT SIZE=5>family of wavelets for complex phenomena</FONT> | ||
| <TR VALIGN=top><TD> | ||
| <TR VALIGN=top><TD><FONT SIZE=5>1988-89</FONT> | ||
| <TD><FONT SIZE=5>Stephane Mallat & Ingrid Daubechies</FONT> | ||
| <TD><FONT SIZE=5>discrete wavelet transform</FONT> | ||
| </TABLE> | ||
|  <P> | ||
| <FONT SIZE=5> | ||
| Adapted from ``Signal processing: A historical perspective''<BR> | ||
| (<A HREF="http://www.monash.edu.au/cmcm/wavelet/whistory.htm">http://www.monash.edu.au/cmcm/wavelet/whistory.htm</A>) | ||
| </FONT> | ||
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| <HR><!-----------------------------------------------------------------> | ||
| <!--A HREF="">back to Wavelet Home Page</A> | ||
| --> | ||
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| </BODY> | ||
| </HTML> |
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| FUNCTION psi,k,t,WAVE_PART=wave_part,AMP_PART=amp_part | ||
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| wave_part = COMPLEX(COS(k*t),SIN(k*t)) ; exp(ikt) = cos(kt) + i*sin(kt) | ||
| amp_part = EXP( -ABS(t)^2/2.) ; exp(-|t|^2/2) | ||
| RETURN,wave_part*amp_part ; morlet | ||
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| END | ||
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| k0 = 6.0 | ||
| time = FINDGEN(401)/20. - 10. | ||
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| a = 2 | ||
| b = 0 | ||
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| morlet = (1./SQRT(a))*PSI(k0,(time - b)/a,WAVE=wave,AMP=amp) | ||
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| WINDOW,0,COLOR=256,XSIZE=512,YSIZE=128 | ||
| LOADCT,42 | ||
| !P.BACKGROUND = 255 | ||
| !P.COLOR = 0 | ||
| !P.FONT = -1 | ||
| !P.CHARSIZE = 1.5 | ||
| !X.STYLE = 5 | ||
| !Y.STYLE = 5 | ||
| !P.MULTI = [0,2,1] | ||
| !X.MARGIN = [2,2] | ||
| !Y.MARGIN = [0,0] | ||
| !Y.RANGE = [MIN(FLOAT(wave))*0.95, MAX(amp)*1.05] | ||
| dy = (!Y.CRANGE(1) - !Y.CRANGE(0))/5. | ||
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| PLOT,time,FLOAT(morlet), $ | ||
| THICK=2,COLOR=50 | ||
| XYOUTS,!X.CRANGE(0),!Y.CRANGE(1) - dy,'(a)',CHARSIZE=1.2,ALIGN=0.5 | ||
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| PLOT,time,FLOAT(wave)/SQRT(2), $ | ||
| COLOR=128,THICK=2 | ||
| OPLOT,time,amp,COLOR=208,THICK=2 | ||
| OPLOT,time,FLOAT(morlet),THICK=2,LINES=2,COLOR=50 | ||
| XYOUTS,!X.CRANGE(0),!Y.CRANGE(1) - dy,'(b)',CHARSIZE=1.2,ALIGN=0.5 | ||
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| END |
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| dt = 1 | ||
| n = 128 | ||
| t = FINDGEN(n)*dt | ||
| t1 = 8 | ||
| t2 = 32 | ||
| x = SIN(2*!PI*t/t1) + SIN(2*!PI*t/t2) | ||
| ; x = RANDOMN(s,n) | ||
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| f = SPECTRUM(x,dt,PERIOD=fperiod) | ||
| w = WAVELET(x,dt,PERIOD=wperiod,DJ=0.125) | ||
| gws = TOTAL(ABS(w)^2,1)/n^2 | ||
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| WINDOW,XSIZE=240,YSIZE=350 | ||
| LOADCT,39 | ||
| !P.COLOR = 192 | ||
| !P.MULTI = [0,1,3] | ||
| !X.STYLE = 9 | ||
| !Y.STYLE = 9 | ||
| !X.MARGIN = [8,2] | ||
| !Y.MARGIN = [4,3] | ||
| !Y.MINOR = 1 | ||
| !X.TICKLEN = -0.04 | ||
| !P.CHARSIZE = 1.75 | ||
| xticks = FIX(ALOG(n)/ALOG(2))-1 | ||
| xtickv = 2L^(LINDGEN(xticks+1)+1) | ||
| xrange = [MAX(xtickv),MIN(xtickv)] | ||
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| PLOT,t,x, $ | ||
| XRANGE=[0,n], $ | ||
| XMINOR=1,XTICKS=xticks-2, $ | ||
| XTITLE='Time (days)', $ | ||
| TITLE='Time series with two sine waves' | ||
| OPLOT,t,x,COLOR=254 | ||
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| PLOT,fperiod,f,/XTYPE, $ | ||
| XRANGE=xrange,YRANGE=[0,0.5],YTICKS=2, $ | ||
| XTICKS=xticks,XTICKV=xtickv, $ | ||
| XTITLE='Period (days)',YTITLE='Power', $ | ||
| TITLE='Fourier power spectrum' | ||
| OPLOT,fperiod,f,PSYM=-4,/NOCLIP,COLOR=254,SYMSIZE=0.75 | ||
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| PLOT,wperiod,gws,/XTYPE, $ | ||
| XRANGE=xrange,YRANGE=[0,0.25],YTICKS=2, $ | ||
| XTICKS=xticks,XTICKV=xtickv, $ | ||
| XTITLE='Period (days)',YTITLE='Power', $ | ||
| TITLE='Global wavelet spectrum' | ||
| OPLOT,wperiod,gws,PSYM=-4,/NOCLIP,COLOR=254,SYMSIZE=0.75 | ||
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| TVLCT,r,g,b,/GET | ||
| WRITE_GIF,'wave_bias_faq.gif',TVRD(),r,g,b | ||
| END |
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