Reprinted from MONTHLY WEATHER REVIEW. Vol. 121. No. 10. October 1993~~~Sa.-Iy

An Introduction to Wavelet Analysis in Oceanography and Meteorology:With Application to the Dispersion of Yanai Waves

S. D. MEYERS. B. G. KEU.Y, ANDJ. J. O'BRIEN

2858 MONTHLY WEATHER REVIEW VOLUME 121

An Introduction to Wavelet Analysis in Oceanography and Meteorology:With Application to the Dispersion of Yanai Waves

S. D. MEYERS, B. G. KEu.Y, AND J. J. O'BRIEN

Mesoscole Air-Sea Interaction Group, The Florida StQle Unillersity. Tallahassee. Florida

(Manuscript received 30 ~ 1992, in final form 4 May 1993)

ABSTRACT

Wavdet analysis is a relatively new technique that is an important addition to standard signal analysis methods.Unlike Fourier analysis that yields an average amplitude and phase for each bannonic in a dataset. the wavelettransform produces an "instantaneous" estimate or local value for the amplitude and phase of each bannonic,This allows detailed study of nonstationary spatial or time-dependent signal characteristics.

The wavelet transform is discussed, examples are given, and some methods for preprocessing data for waveletanalysis are compared. By studying the ~on ofYanai waves in a reduced gravity equatorial model, theu~fulness of the transform is demonstrated. The group velocity is measured directly over a finite range ofwavenumbers by examining the time evolution of the transform. The results agree weD with linear theory athigher wavenumber but the measured group velocity is reduced at lower wavenumbers, POSSIoly due to interactionwith the basin boundaries.

1. Introduction

a. The wavelet transform

The wavelet transform has shown promise in a di-versity of scientific fields, but to date it has not beenmuch used in the oceanic and atmospheric sciences.In part, this might be due to a lack of material dis-cussing practical aspects of the technique. This articletherefore includes an introduction to the wavelettransform as a tool for data analysis. For brevity, weshall confine our discussion to the transform of a scalarseriesf(t).

Wavelet analysis is based on the convolution off(t)with a set of functions gab(t) derived from the trans-lations and dilations (and rotations in higher dimen-sions) of a mother wavelet g( t), where

It-b ) ; (1)

This is known as the continuous wavelet transformsince a and b may be varied continuously. Translationparameter b corresponds to position or time if the datais spatial or temporal, respectively. Dilation parametera then corresponds to scale length or temporal period.Equation (2) expands a one-dimensional time seriesinto the two-dimensional parameter space (b, a) andyields a local measure of the relative amplitude of ac-tivity at scale a at time b. This is in contrast to theFourier transform that yields an average amplitude overthe entire dataset. Note, we have avoided the use ofthe words "wavelength" or "frequency" in our de-scription of the WT. Though wavelets have a definitescale, they need not bear any resemblance to Fouriermodes (sines and COSines). However, a co1TeSpondencebetween wavelength and scale a sometimes can beachieved, as discussed in section 3.

To see the limitation of standard Fowier analysisand the incentive for the development of wavelet anal-ysis, consider the time series in Fig. la that changesfrequency halfway through the measurement. Comparethat to the signal in Fig. 1 b that is generated from thesimultaneous presence of both frequencies. These twovery different signals yield similar power spectra, shownin Figs. 1 c,d, both being dominated by the same twopeaks. Without prior knowledge, it would be difficultto know which signal produced which spectrum, sinceinformation on signal evolution is lost during Fourieranalysis. Variations of the Fourier transform have beenused (e.g., Gabor 1946) in attempts to overcome thislimitation, but have met only with qualified success.The WT produces "instantaneous" coefficients and

g",,(t) = L (-:-

Co"esponding author address: Dr. Steven D. Meyers, MesoscaleAir-Sea Interaction Group, The Ronda State University, 8-174, 020Love Building, Tallahassee, FL 32306-3041.

@ 1993 American Meteorolosical Society

al/2 \ a I

a(>O) and b are real. Any set of functions gab(t) con-structed from ( I ) and meeting the conditions outlinedbelow are called wavelets. The convolution off(t) withthe set of wavelets is the wavelet transform (WT)

()cTOBE;R 1993 MEYERS ET AL 2859

.

0 - - a a 1000..

produces the local nature of wavelet analysis, since thecoefficients T g{ b, a) are affected only by the signal inthe cone of influence (COI) about t = b. In practice,the radius of the COI is the point I tl ='c beyond whichgab{x) no longer has significant value. Usually, 'ccx: a, giving rise to conelike structures in the WT incertain cases. The COI of the endpoints is an inlportantconsideration and will be discussed further in sec-tion 2.

( ii) Also, g{ t) must have zero mean. Known as theadmissibility condition, this implies the invertabilityof the WT. That is, the original signal can be obtainedfrom the wavelet coefficients using

f{t) = .!. II T ,{b, a) gab dadbC a2

1

1

:! 1..

1

10'

100

1fT'0.00 o.~ ...

r.Iwhere

FIG. 1. A limitation of standard Fourier analysis is illustrated; verydifferent signals can have very similar power spectra: (a) a mono-chromatic signal that changes fi'equency halfway through the dataset,(b) a signal comprised of two frequencies, (c) the JaW power spectn1Inof (a), and (d) the raw power spectrum of(b).

therefore can yield information on the evolution ofnonstationary processes.

The development of wavelet analysis began relativelyrecently with Morlet (1983). A later collaboration(Grossman and Morlet 1984) produced the continuouswavelet transform in one dimension based on the setof translations and dilations. Meyer ( 1985) extendedthe technique to n dimensions, and Murenzi (1989)included rotation. Three references are used for theintroduction to wavelet analysis: Farge ( 1992), Ruskaiet al. (1992), and Combes et al. (1989). Our goal inthis article is not to give a complete review of the wave-let transform but to aid in its introduction to ocean-ographers and meteorologists. The references containdetails on the history and uses of wavelets in a varietyof scientific applications.

In the following we continue with the introductionto wavelet analysis. In section 2, a few methods forpreprocessing data for wavelet analysis are discussedTwo simple methods from Fourier analysis are triedand shown to be inappropriate as they distort the endregions of the WT. Another method, based on bufferingthe ends of the data with additional points, is shownto yield better results. Section 3 examines the dispersionofYanai waves and demonstrates how quantitative in-formation not available from Fourier methods can beobtained using the WT.

dw

(3)

If one is seeking "chirps" (short segments of linearlyincreasing frequency), the wavelet

g(t) = eikl2!2eicle-t2!2 (4)

would be useful. If one knows the characteristics of thesignal or pattern being sought, the wavelet should bechosen to have that same pattern. Large values of

b. Wavelet selection

To be a mother wavelet both fonnally and in prac-tice, g(t) must have the following properties.

(i) It must be a function centered at zero and in thelimit as I t I - 00, g( t) - 0 rapidly. This condition

~= f

2860 MONTHLY WEATHER REVIEW VOLUME 121

I T 6( b, a) I will then indicate where J( t) has the desiredform.

Not only are there trade-offs in choosing g( t) butthere are considerations when choosing a and b. Forexample, the continuous wavelet transform can yieldredundant information since small changes in a or bare often insignificant. A more efficient choice is tochoose a discrete set of a and b such that the gab(t)constitute an orthogonal basis set. The WT at any pa-rameter value can then be found using a suitable in-terpolation scheme. There is much discussion of thesediscrete wavelets in the literature, as they are useful infields such as image compression and multiresolutionanalysis.

c. Wavelet algorithms

There are several algorithms to implement a WT,and only two will be discussed here. The referencesdetail other methods under development, particularlythose involving discrete wavelets. Here, we discuss thecontinuous transform.

The simplest method is direct numerical integration.Knowingf(t) and g(t), one can compute the transformat arbitrary points in parameter space using a discre-tized form of ( 2 ). The drawback of this technique isthat it is time consuming. If one integrates over 0 < a~ land 0 < b ~ J, the integration time goes as lJ2.

An alternative is to exploit the convolution theoremand do the WT in spectral space:

..Tb(b, a) = aJ/2 eib...g*(acu)/(cu)dcu. (5)

This allows the use of optimized fast Fourier transform(FFT) routines and results in a much faster transform,with CPU time going as IJI~J. To use this method,g( cu) should be known analytically and the data mustbe preprocessed to avoid errors from the FFT algo-rithms. In particular, the discrete form of (5) will pro-duce an artificial periodicity in the WT if f( t) is notperiodic. This is demonstrated in the examples herein,and different methods for dealing with the problem arediscussed Issues of aliasing and bias in FFT routinesare well known and need not be discussed here.

b. The mesh in all the plates indicates the COI of theendpoints. Consider the first endpoint t = O. For b< r c, the wavelet has significant values at t < 0 wherethere is no signal. Hence, there is a steady degradationin the WT as b approaches O.

Regions where I T g( b, a) I are large indicate highcorrelation between the data and the wavelet. In Fig.2, the modulus clearly indicates the abrupt change inthe frequency of the signal by the shift of the largecoefficients to a different scale. The phase indicates thecycling of the signal from -1r to 1r and allows for thelocation of wave crests in the signal. One can countthe number of waves in a simple signal such as the oneshown here. The branchings at larger a are due to thelarger wavelets (large a) detecting more than one cycle.Note the convergence of the phase lines toward higherfrequency (smaller a) in the middle of the transfonn.This indicates when the frequency shifted, and gen-erally occurs at any singularity (sudden changes in fre-quency or phase) in the signal. Interpretation of phaseplots for more complicated signals can be nontrivial.

In the WT phase, and to a lesser extent in the mod-ulus, activity is indicated at many scales, though thesignal is locally monochromatic. Although there is a"best" choice for the scale (a = O(» in the modulus,the nonzero correlations between the wavelets and dataproduce nonzero transfonn values at scales away fromO(). This can lead to confusion when trying to determinewhich scales are present in the data; one method fordealing with this problem is discussed in the next sec-tion.

Note in Fig. 2 the curvature of the modulus at theend regions. The line of maximum wavelet coefficientbends to meet artificial periodic boundary conditions,falsely indicating frequency changes at the beginningand end of the signal. This demonstrates the genericproblem when using ( 5 ) on nonperiodic data-the WTcan induce a periodicity into the transfonn, greatlydistorting the infonnation at the end regions. This isa critical problem when the initial and final signalcharacteristics are very different.

In an attempt to impose periodicity on the signal,method II preprocesses f( t) with a cosine window

/(4) - /(It)[2. Examples of wavelet analysisThe WT of the signal in Fig: la is a classic example

in wavelet literature and will be examined with fourdifferent data preparation methods. All follow (5) anduse the Morletwaveletg(t) = elcte-12/2. Method I doesnot involve any preparation of the data, and the trans-forDl is computed without considering the propertiesof FFr routines. The result is presented in Fig. 2, withthe amplitude and phase shown separately. As in allthe transforDls shown here. the vertical axis is the in-verted scale Q, so the corresponding wavenumber in-creases on the vertical axis. The horizontal axis is time

k = 0, 1, . . " (N - 1); (6)

N is the total number of data points. The resulting WTno longer has the false end regions, because it has ef-fectively lost those areas. as seen in Fig. 3, which isclearly an unacceptable effect. A window with steepcutoffs would not eliminate as much data but wouldproduce only a small correction to the distortion ofthe end regions.

Method III involves detrendingf(tJ and removingthe mean, a standard procedure when computing FFfs.

2861MEYERS ET AL0cr08ER 1993

80~MODULUS

0.0 2.0 ~O 1.0 8.0 10.0 12,00 200 400 600 800 1000

PHASE 140 d

e1OOO~.co2t)DOc.J0~ 3000

0.0 2.0 6.0 1.0 10.0 fZ.o0 200 .00 100 800 1000TIm.

200 d

min max

FIG. 2. Wavelet transform of the signal in Fig. 1a usingf(t) thatwas not ~lIdjtjoned. The cbanp: in ~ueocy is clearly indicatedby the shift in the wavelet modulus maximum. False results are pr0-duced at the end relioDS sin~ the data is not periodic. (a) Modulus:the ooIors ~t val1a from the minimum (Nue) to the maximum(white). (b) Phase: the colors represent values from -or (blue) to...(white). The veItica1 axis is the inverted a scale. The mesh indicatesthe COt (r" = 2a for the Morld wavelet). The color bar is used inthis figure and rig. 8 with the minimum and maximum adjusted foreacbpiot.

0.0 2.0 4.0 a.o 8.0 10.0 _.Jt(!fometers (x 1 oJ)

FK). 8. The modulus of the WI of the heilht fJdds in Fig. 7.Vertical axis is the wavelength X given by Eq. (12), and the horizontalaxis is distance acro8 the model basin. Note the ~on of thetongue from the upper left toward the lower richt in time, com-spondins to the changes in wavelength as the Yanai waves disperseacross the basin: (a) 80. (b) 140. and (c) 200 days. White is themaximum coefficient, and blue is the minimum. The radius of theCOI mesh is r. = 2X/l.2.

'211

The result is shown in Fig. 4. Again, errors in the wave-let coefficients appear as they did with method 1.Though methods II and III are common methods forpreparing data for Fourier analysis, they are inadequatefor wavelet analysis.

Method IV, though developed independently by theauthors, has already been investigated by other re-searchers (Bamier 1992, personal communication).The data f( Ik) is buKered on either side with a tail thatgoes to zero, as indicated in Fig. 5. After the WT iscomplete, the regions corresponding to the tails arediscarded, yielding Fig. 6. This method eliminates theproblem of non periodic data and does not introducesignificant distortions of the end regions. The lengthof the buffers are chosen empirically. When the buffersare too short, distortions as in Fig. 2 are found butlessen as the buffer length is increased. Various forms

of the buff~ can be chosen. Here, we used a verysimple form, but suggest for future applicationsmatching not just the endpoint values but their deriv-atives as well. Even better would be a form that matchesthe endpoint frequency, amplitude, and phase, whichis conceivable for simple signals, and would arguablyeliminate most of the problems ~ted with the COIat the endpoints. Of course, if the end regions are un-important, their WT can be ignored.

A frequent comment about wavelet analysis is thatit is difficult to obtain more than qualitative infor-mation (such as detecting the frequency change in thepreceding example). This may be so in some cases; inothers it is possible to obtain quantitative informationthat would be difficult if not im~ble to obtain by

2862 MONTHLY WEATHER REVIEW VOLUME 121

~

.1100

B

nne

.1100 "

r ' I ,'I

'\

,~

~

, '"",

. ,, ,, ,0

,.,.FIG. 3. ContoUlS of the wavelet traDsfonn of the signa1 in Fig. la

usinaf(t) that was preconditioned using a a>Sine window. The in-formation in the end regions is now lost: (a) modulus; (b) phase.The negative contours are dashed.

traditional Fourier analysis. As a practical illustrationof wavelet analysis we will directly measure the dis-persion ofYanai waves in an ocean model. The resultsindicate general agreement with linear theory exceptnear the eastern boundary where the wave propagationappears to slow. The spectral ocean model used to cre-ate the data is described by Kelly ( 1992) .

3. Dispersion of Yanai waves

Instability waves in the equatorial regions of theworld's oceans have been under study since the mid-1970s. Interest began following the observations made

~

.

.

1100

aX).~ - ~ a 800 1000

Tn.

Dhasef~i f}~I: "tifJ.."..! r, 11",-

j 100 ~/( \- 11)1., I , ' ,. . ' , . ,, -,-

10001"-

FIG. 6. Wavelet transform of the sisnal in Fig. S. There is mumless distortion in the end regions of the data. Buft'er Ienath was 100data points.

FIG. 4. Wavelet transfonn of the signal in Fig. la usingf(t)that was demeaned and detrendeci; (a) and (b) as in Fig. 3.

during the GARP (Global Atmospheric ResearchProject) Atlantic Tropical Experiment ( GATE) in thesummer of 1974 (Duing et al. 1975). Occurring in anarrow frequency band, with periods around 25 daysand zonal wavelengtlis of about 1000 kIn, their struc-ture has been shown to be dynamically similar to Yanaiwaves (mixed Rossby-gravity waves) both in the ob-servations (Weisberg and Horigan 1981; Tsai 1990)and in the numerical models (Cox 1980; Kindle andThompson 1989; Woodberry et al. 1989). The wavespropagate westward and upward with a group velocitythat is eastward and downward ( Weisberg et al. 1979;Cox 1980). The zonal phase and group velocities areabout 33-73 cm s-J and 16 cm S-I, respectively(Weisberg et al. 1979). Similar oscillations were ob-served along the equatorial front in the eastern tropicalPacific using satellite imagery (Legeckis 1977).

As a result of these observations, it was hypothesizedthat they were generated by a meridional shear insta-bility between the westward-flowing South EquatorialCurrent (SEC) and the eastward-flowing North Equa-

2864 MONTHLY WEATHER REVIEW VOLUME 121

~

found in the ocean may be due to the dispersion ofYanai waves; the longer, faster waves leave the westernregion after roughly 100 days.

"".

I ...ml "" ,i

t I .

oL'.'_~_.'J".."'~o~ O.CXE 0.010 0.015 0.- 0.-. (- kin.')FIG. 10. The measured dispersion relatiOD &om the WT. En'Or

ban are based on the standard deviation of the measwm velocity.The theoretical value is shown as the solid curve. Waveuumbers areassumed to be k = 2.1 (2-r)a-' , .

cussed in the Appendix. Such a transform would havea structure as in FIg. 6 but with a single maximum ofthe modulus in a. The line of maximum correlationyields a conversion from wavelet scale to Fourierwavelength. For the Morlet wavelet with c = 5 (usedin the WT in this article), the wavelength is given byA ~ .l.2a.1lUs allows us to compare the measurementsof propagation speed with the theoretical speed at con-ventional wavenumbers.

The results of the measured Cg are compared withtheoretical values in Fig. 10. There is general aareementwith the linear theory in the range &om approximatelyk = -0.005 rad km-1 to k = -0.02 lad kin-I. (Note,we deal only with k < 0.) For smaller k the measuredCg diverges from the theoretical value, possibly due tothe finite size of the basin. When the maxima for allthe days under study are plotted together, one can seea sudden decrease in the propagation speed of wave-lengths 8bove 1600 kin when they near the easternboundary. This scale corresponds to the value of kwhere the drop-off in Cg occurs. To test this we ex.paDded the basin domain from IS 000 to 22 SOO km,while keeping the same number of modes in the spectralmodel. In the longer domain there is no drop-off ofthe measured group velocity ~t lower wavenumbersince the longer waves do not have time to encounterthe eastern boundary during 200 model days.

As mentioned in Kelly ( 1992), in the western regionof the basin Yanai waves exist only in a limited rangeof k. Examining Fig. 9, one can see a gradual flatteningof the curves in the western basin as time progresses,indicating a narrowing of the range of wavenumbers.The waves remaining in the west are several hundredkilometers in wavelength, as observed in the IndianOcean. This suggests that the narrow range of periods

L: f/!{x)dx = 1.The addition of the scaling functions to the set ofwavelets allows for the decomposition of functions thatare not square integrable.

Frequently used with discrete wavelets [followingthe notation of Farge (1992)] \lIifix) that are definedby \lIij(x) = 2j/~2jx - i), the reconstruction equationforf(x) is then

+~f(x) = L < ~ If(x»~(x)

;--~

+~ +~

+ k k (+Ii !/(X»+iJ{x)j-6 i--~

(9)

4. SummaryThe wavelet transfonn promises to be a useful tool

in oceanography and meteorology. For the purpose ofdata analysis, the continuous transfonn in spectralspace is useful and efficient. However, standard meth-ods for preprocessing data for Fourier analysis are in-sufficient for wavelet analysis. The best method ex-amined here is that of buffering the ends of the signalwith points that smoothly go to zero. The region of thetransfonn corresponding to these points is then dis-carded after the transfonn. Without this buffering, asignal whose properties are different near its ends willresult in a WT that has been forced to periodicity atall scales through a distortion (in some cases severe)of the end regions. The greater the aperiodicity of thesignal, the greater the distortion.

We demonstrate the usefulness of the WT by ex-amining the dispersion ofYanai waves. The transfonnmodulus clearly reveals the propagation of the differentwavelengths across the basin. By scanning the modulus,Cg( k) is measured directly and shown to agree withlinear theory, though there is a reduction in propaga-tion speed of the longer wavelengths. The narrowingof the range of wavelengths in the western region ob-served supports the hypothesis that the narrow rangeof frequencies observed in the western equatorialoceans is a consequence of Yanai wave dispersion.These results could not be obtained using standardFourier techniques.

The use of wavelets goes beyond simple data analysis,and a full discussion is beyond the scope of the paper,but a couple of references might suggest possible futureapplications. An extension of wavelet analysis, calledmultiresolution analysis. adds the scaling /unctions

0cr08ER 199 3 MEYERS ET AL

which in this case becomes

I T,(b, a)12 = ae-(ako-c)2. (AI)

To find the scale a of maximum correlation, we setthe derivative of (A I ) with respect to a equal to zeroand obtain

I + 2ckoa - 2k3a2 = O. (A2)Two solutions arise but only one is realistic. The pre.ferred solution is

Qo - ~ [ ..!. + !.:!~£l~ 1ko ko..

Using ko = 2...>..01, the solution becomes a linear re-lation betWeen wavelet scale and Fourier wavelength,

c + (24: C2)1/~]>..0. (A3)

The other solution is a decreasing function of >"0 andis disregarded. Equation (AI) can also be obtained fora real monochromatic signal with any progressivewavelet.

Note, it follows from (2) that any linear superpo-sition of periodic modes will result in separate localmaxima, each described as above. The WT of anyfunction

Of more direct interest to the reader might be recentattempts to model two- and three-dimensional fluidsin wavelet space. A representation of the three-dimen-sional Navier-Stokes equation in wavelet space is givenby Meneveau ( 1991 ), with the emphasis on reducingthe number of degrees of freedom necessary for mod-eling fluid turbulence. Zimin (1981), antecedent tothe formal development of wavelets, studied the Na-vier-Stokes equation in terms of basis functions lo-calized in space and time. In one of the few papersbased on the work of Zimin that has been translatedinto English, Aristov and Frick (1988, 1989) presenta study of convection in a rotating fluid

like previous hot topics in science, the WTshowsmuch promise but solid results have been difficult toachieve. The works of Meneveau (1991) and Zimin( 1981) are far from final results; several more yearsare probably needed for those subjects to mature.However, for applications such as described in section3 of this article, the WT is sufficiently developed thatit can now be used with confidence as a tool in theoceanic and atmospheric sciences.

ao=

Acknowledgments. SDM is supported by the ONRDistinguished Educator Postdoctoral Fellowship to Dr.James O'Brien. The theoretical work of the MesoscaleAir-Sea Interaction Group is CUITently being supportedby the Physical Oceanography Branch of the Office ofNaval Research and the Ocean Processes Branch ofNASA. Discussions on wavelet analysis with Dr. Ber-nard Barnier were useful. The color plates were createdusing the Ferret package, provided by Dr. D. E. Har-rison and Steven Hankin. Thanks also to Mark Ver-schell for his on-site help with the Ferret routines.

f(x) = ~ Aflik]x (A4)j

will have modulus maxima at aj = [c + (2 + C2)1/2]x (2k»-I.

APPENDIX

Wavelet Scales and Fourier Wavelengths

The relation between wavelet scale and the morecommon Fourier wavelength is not necessarilystraightforward. For example, some wavelets are highlyirregular without any dominant periodic Components.In those cases it is probably a meaningless exercise tofmd a relation between the two disparate measures ofdistance. However, in the case of the Morlet wavelet,which is a periodic function enveloped by a Gaussian,it seems more reasonable. Using

2g(x) = eicxe-x 12,we take the transform of eikoX using (5)

Tg(b, a) = al/2 i: dkeiNcH(k)e-(ak-C)2/2o(k - kG).

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