;************************************************************** WAVETEST ;+ ; NAME: WAVETEST ; ; PURPOSE: Example IDL program for WAVELET, using NINO3 SST dataset ; ; EXECUTION: ; ; IDL> .run wavetest ; ; ; See "http://paos.colorado.edu/research/wavelets/" ; Written January 1998 by C. Torrence ; ;- ;************************************************************** n = 504 sst = FLTARR(n) OPENR,1,'sst_nino3.dat' ; input SST time series READF,1,sst CLOSE,1 ;------------------------------------------------------ Computation ; normalize by standard deviation (not necessary, but makes it easier ; to compare with plot on Interactive Wavelet page, at ; "http://paos.colorado.edu/research/wavelets/plot/" sst = (sst - TOTAL(sst)/n) dt = 0.25 time = FINDGEN(n)*dt + 1871.0 ; construct time array xrange = [1870,2000] ; plotting range pad = 1 s0 = dt ; this says start at a scale of 3 months dj = 0.25 ; this will do 4 sub-octaves per octave j1 = 9./dj ; this says do 9 powers-of-two with dj sub-octaves each mother = 'Morlet' recon_sst = sst ; save an extra copy, so we don't erase original sst ; estimate lag-1 autocorrelation, for red-noise significance tests ; Note that we actually use the global wavelet spectrum (GWS) ; for the significance tests, but if you wanted to use red noise, ; here's how you could calculate it... lag1 = (A_CORRELATE(sst,1) + SQRT(A_CORRELATE(sst,2)))/2. ; Wavelet transform: wave = WAVELET(recon_sst,dt,PERIOD=period,SCALE=scale,S0=s0, \$ PAD=pad,COI=coi,DJ=dj,J=j1,MOTHER=mother,/RECON) power = (ABS(wave))^2 ; compute wavelet power spectrum global_ws = TOTAL(power,1)/n ; global wavelet spectrum (GWS) J = N_ELEMENTS(scale) - 1 ; Significance levels, assuming the GWS as background spectrum: signif = WAVE_SIGNIF(sst,dt,scale,0, \$ GWS=global_ws,SIGLVL=0.90,MOTHER=mother) signif = REBIN(TRANSPOSE(signif),n,J+1) ; expand signif --> (J+1)x(N) array signif = power/signif ; where ratio > 1, power is significant ; GWS significance levels: dof = n - scale ; the -scale corrects for padding at edges global_signif = WAVE_SIGNIF(sst,dt,scale,1, \$ LAG1=0.0,DOF=dof,MOTHER=mother,CDELTA=Cdelta,PSI0=psi0) ; check total variance (Parseval's theorem) [Eqn(14)] scale_avg = REBIN(TRANSPOSE(scale),n,J+1) ; expand scale-->(J+1)x(N) array power_norm = power/scale_avg variance = (MOMENT(sst))(1) recon_variance = dj*dt/(Cdelta*n)*TOTAL(power_norm) ; [Eqn(14)] IF (N_ELEMENTS(recon_sst) GT 1) THEN BEGIN recon_variance = (MOMENT(recon_sst))(1) ; RMS of Reconstruction [Eqn(11)] rms_error = SQRT(TOTAL((sst - recon_sst)^2)/n) PRINT PRINT,' ******** RECONSTRUCTION ********' PRINT,'original variance =',variance,' degC^2' PRINT,'reconstructed var =',FLOAT(recon_variance),' degC^2' PRINT,'Ratio = ',recon_variance/variance PRINT,'root-mean-square error of reconstructed sst = ',rms_error,' degC' PRINT IF (mother EQ 'DOG') THEN BEGIN PRINT,'Note: for better reconstruction with the DOG, you need' PRINT,' to use a very small s0.' ENDIF PRINT ENDIF ; Scale-average between El Nino periods of 2--8 years avg = WHERE((scale GE 2) AND (scale LT 8)) scale_avg = dj*dt/Cdelta*TOTAL(power_norm(*,avg),2) ; [Eqn(24)] scaleavg_signif = WAVE_SIGNIF(sst,dt,scale,2, \$ GWS=global_ws,SIGLVL=0.90,DOF=[2,7.9],MOTHER=mother) ;------------------------------------------------------ Plotting printfile = 0 !P.FONT = -1 !P.CHARSIZE = 1 IF (printfile) THEN BEGIN SET_PLOT,'ps' DEVICE,/PORT,/INCH,XSIZE=6.5,XOFF=1,YSIZE=6,YOFF=3,/COLOR,BITS=8 !P.FONT = 0 !P.CHARSIZE = 0.75 ENDIF ELSE WINDOW,0,XSIZE=600,YSIZE=600 !P.MULTI = 0 !X.STYLE = 1 !Y.STYLE = 1 LOADCT,39 ;--- Plot time series pos1 = [0.1,0.75,0.7,0.95] PLOT,time,sst,XRANGE=xrange, \$ XTITLE='Time (year)',YTITLE='NINO3 SST (!Uo!NC)', \$ TITLE='a) NINO3 Sea Surface Temperature (seasonal)', \$ POSITION=pos1 IF (N_ELEMENTS(recon_sst) GT 1) THEN OPLOT,time,recon_sst,COLOR=144 XYOUTS,0.85,0.9,/NORMAL,ALIGN=0.5, \$ '!5WAVELET ANALYSIS!X'+\$ '!C!CC. Torrence & G.P. Compo'+\$ '!C!Chttp://paos.colorado.edu/!Cresearch/wavelets/' ;--- Contour plot wavelet power spectrum yrange = [64,0.5] ; years levels = [0.5,1,2,4] colors = [64,128,208,254] period2 = FIX(ALOG(period)/ALOG(2)) ; integer powers of 2 in period ytickv = 2.^(period2(UNIQ(period2))) ; unique powers of 2 pos2 = [pos1(0),0.35,pos1(2),0.65] CONTOUR,power,time,period,/NOERASE,POSITION=pos2, \$ XRANGE=xrange,YRANGE=yrange,/YTYPE, \$ YTICKS=N_ELEMENTS(ytickv)-1,YTICKV=ytickv, \$ LEVELS=levels,C_COLORS=colors,/FILL, \$ XTITLE='Time (year)',YTITLE='Period (years)', \$ TITLE='b) Wavelet Power Spectrum (contours at 0.5,1,2,4!Uo!NC!U2!N)' ; significance contour, levels at -99 (fake) and 1 (significant) CONTOUR,signif,time,period,/OVERPLOT,LEVEL=1,THICK=2, \$ C_LABEL=1,C_ANNOT='90%',C_CHARSIZE=1 ; cone-of-influence, anything "below" is dubious x = [time(0),time,MAX(time)] y = [MAX(period),coi,MAX(period)] color = 4 POLYFILL,x,y,ORIEN=+45,SPACING=0.5,COLOR=color,NOCLIP=0,THICK=1 POLYFILL,x,y,ORIEN=-45,SPACING=0.5,COLOR=color,NOCLIP=0,THICK=1 PLOTS,time,coi,COLOR=color,NOCLIP=0,THICK=1 ;--- Plot global wavelet spectrum pos3 = [0.74,pos2(1),0.95,pos2(3)] blank = REPLICATE(' ',29) PLOT,global_ws,period,/NOERASE,POSITION=pos3, \$ THICK=2,XSTYLE=10,YSTYLE=9, \$ YRANGE=yrange,/YTYPE,YTICKLEN=-0.02, \$ XTICKS=2,XMINOR=2, \$ YTICKS=N_ELEMENTS(ytickv)-1,YTICKV=ytickv,YTICKNAME=blank, \$ XTITLE='Power (!Uo!NC!U2!N)',TITLE='c) Global' OPLOT,global_signif,period,LINES=1 XYOUTS,1.7,60,'95%' ;--- Plot 2--8 yr scale-average time series pos4 = [pos1(0),0.05,pos1(2),0.25] PLOT,time,scale_avg,/NOERASE,POSITION=pos4, \$ XRANGE=xrange,YRANGE=[0,MAX(scale_avg)*1.25],THICK=2, \$ XTITLE='Time (year)',YTITLE='Avg variance (!Uo!NC!U2!N)', \$ TITLE='d) 2-8 yr Scale-average Time Series' OPLOT,xrange,scaleavg_signif+[0,0],LINES=1 IF (printfile) THEN DEVICE,/CLOSE END