Estuarine, Coastal and Shelf Science (2000), 51, 177–200doi:10.1006/ecss.2000.0586, available online at http://www.idealibrary.com on

An Introduction to Wavelet Transform Tidal AnalysisMethods

E. P. Flinchema and D. A. Jayb,c

aUniversity of Washington, Geophysics Program, Box 351650, Seattle, WA 98195-1650, U.S.A.bOregon Graduate Institute, Center for Coastal and Land-Margin Research, P.O. Box 91000, Portland, OR97291-1000, U.S.A.

Received 22 March 1999 and accepted in revised form 18 January 2000

Continuous wavelet transforms (CWTs) provide an approach to understanding the numerous tidal phenomena thatdeviate markedly from an assumed statistical stationarity or exact periodicity inherent in traditional tidal methods. Use ofwavelets allows determination of the degree of non-stationarity present in time series, such as estuarine and shelf currents,usually treated as stationary. Wavelets also provide a consistent analysis of tidal and non-tidal variance, a feature oftenessential for dynamical analyses of non-stationary tides. We summarize basic notions of the wavelet transform, also knownas a perfect reconstruction filter bank or a multire solution analysis, contrast them with those of harmonic analysis andFourier transforms, construct a continuous wavelet transform basis with a scale selection especially adapted to tidalproblems, describe possibilities for analysis of scalar and vector quantities, define a criterion for knowledge ofindependence of process between adjoining scales, and illustrate use of wavelet tools with several examples. In contrastto the nearly periodic barotropic tide typical of coastal stations, this paper analyses processes that are in part tidally drivenbut non-stationary, e.g. baroclinic tidal currents, river tides, continental shelf internal tides, and some kinds of biologicalactivity in the coastal ocean. In all cases, wavelet analysis provides a consistent, linear analysis of tidal and non-tidalvariance and reveals features that harmonic analysis on a Fourier transform approach could not elucidate.

� 2000 Academic Press

Keywords: tides; tidal analysis; wavelet transform; wavelets; non-stationary processes

cCorresponding author: E-mail: djay@ccalmr.ogi.edu

Introduction

The purpose of this contribution is to describesystematically a new method of tidal analysis based oncontinuous wavelet transforms (CWTs). We havesought to adapt wavelets to the special characteristicsof tidal problems without sacrificing the general abilityof wavelets to linearly and optimally extract informa-tion (as defined by the Heisenberg Principle) on timescales limited only by the length of record and theNyquist frequency.

There are several features that distinguish tidalanalysis from most other geophysical applications oftime-series analysis. The most important is themarked contrast between the ‘ tidal daily ’ band(periods of c. 1–25 h), where even non-stationaryflows exhibit a strong dominance by processes occur-ring within narrow frequency ranges (the tidal speciesand the inertial frequency) known from astronomicalconsiderations, and the ‘ sub-tidal ’ band (periods>25 h), where stochastic forcing with a highly variableand broad-band frequency structure is usuallyseen. It is vital to: (a) use information concerning

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astronomical forcing without introducing assumptionsthat obscure non-tidal processes in a record; and(b) provide an internally consistent extraction of tidaland non-tidal variance. A second factor is the verybroad range of tidal time scales, practical analysis ofwhich may involve periods from c. 1 h to c. 19 years(>17 octaves in frequency space). Furthermore, incontrast to many areas of geophysics where analysis offrequency content is the sole objective, tidalists wishto produce forecasts and hindcasts as well as recon-struct the original data. Also, velocity is a vectorquantity that is often: (a) measured at a number ofclosely spaced depths or locations; and (b) influencedby other scalar and vector quantities (e.g. winds, riverflow and density). The requirements of dynamicalanalysis and the volume of data require that all vari-ables and all frequencies be analysed in a consistentmanner, and that a few particularly revealing datapresentations be selected from a large universe ofpossible calculations. After a presentation of thefundamentals of wavelet analysis, several typical non-stationary tidal problems will be used to illustrate thepotential of CWT tidal analysis. It is useful to beginthe discussion with some historical context.

� 2000 Academic Press

178 E. P. Flinchem and D. A. Jay

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Analysis and prediction of coastal tides was one ofthe great triumphs of 19th century mathematicalphysics. Inspired by Newton’s theory of gravitation,the celestial mechanics of Laplace, and the metaphorof a perfect clockwork mechanism in perpetualmotion, physicists endeavoured to predict the tide at agiven station for all past and future time by fitting tothe observed tide a set of coefficients for the amplitudeand phase of a finite sum of sinusoids with theprecisely known orbital periodicities of the earth-moon-sun system. For a record at a typical coastalstation [e.g. Figure 1, 2(a)], the harmonic methodreduces as much as 98% of the variance to a table ofa few dozen numbers. The harmonic approachperforms as well as it does because the tides at coastalstations obey relatively linear dynamics forced bylunisolar motions; they are, almost literally, an earthlymanifestation of the music of the spheres.

Lord Kelvin first proposed a least-squares tidalharmonic analysis (HA) in the 1860s, from whichDarwin (1886, 1891, 1893) formulated a practicallyuseful method. Doodson (1922) perfected Kelvin’stheory by elaborating the formal treatment of theslowest astronomical periodicities; his symbolism isstill in use today to denote the various tidal speciesand constituentes. Munk and Cartwright (1965) andCartwright (1968) reformulated the tidal problem interms of admittances to account for the presence of acontinuous spectrum of background noise (‘ theresponse method ’). This work, however, resulted inlittle change in practical tidal analysis procedures.Aside from the technical refinements introduced byDoodson, HA as it is used in practice has remained

nearly static for c. 100 years. The improvements thathave occurred since 1921 have been related toremoval of the effects of minor constituents thatcannot be determined from a one-year record,inference from one station to another, more precisespecification of astronomical inputs, treatment ofunevenly spaced data, treatment of vector data, anddevelopment of numerically efficient software (e.g.Godin, 1972; Foreman, 1977).

The simpler forms of the analysis/predictionproblem have been solved, but many significant tidalprocesses remain that are quasi-periodic, and there-fore, less amenable to study and prediction by estab-lished methods. Tidal phenomena may be irregulareither because an aperiodic input is competing withtidal forcing or the oceanic response to tidal forcing isbeing modulated by some internal process. Variablewind stress and sea-level pressure are examples of theformer. An example of the latter is modulation offluvial tides by river flow [Figure 2(b)]. Increasingstream flow damps the tidal wave frictionally, therebydecreasing the range of the observed tide (Dronkers,1964; Godin, 1991; Jay & Flinchem, 1997). Morerigorously speaking, however, aperiodic forcing andaperiodic modulation are two aspects of a continuum.Unambiguous examples of the extremes of the con-tinuum are rarely found, the two limits being deeplyintertwined via the nonlinear terms in the Navier–Stokes equations. An intermediate example is the veryunsteady internal tidal activity on continental shelvesand in fjords (Sandstrom, 1991). The origin andpropagation of internal tides depend strongly on thedensity field, which can be altered rapidly by wind

Introduction to wavelet transform tidal analysis methods 179

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F 2. Detail of surface elevation time series for: (a) a fluvial station (BV in Figure 1) and (b) an estuarine station (TPin Figure 1). The time period (3+ months) includes a major winter freshet (c. day 497) and part of the spring freshet (c. day575). Fluvial tides are strongly modulated by river flow. Low-passed river elevation is influenced primarily by tides, butsecondarily by storm surges. Estuarine tides are weakly modulated by river flow. Low-passed elevation in the estuary isstrongly influenced, however, by coastal processes; compare surge effects on c. day 507 with the spring freshet period atc. 570.

mixing, advection, upwelling, and freshwater runoff.Other examples include internal tidal asymmetry, amechanism of baroclinic current generation in strati-fied estuaries that is driven by tidal advection of the

density field (Jay & Musiak, 1994), and the interactionbetween tide and storm surge (Prandle & Wolf 1981).Practical forecasting of such phenomena wouldrequire: (a) an invertible analysis method suitable to

180 E. P. Flinchem and D. A. Jay

non-stationary processes; and (b) a method of fore-casting the stochastic processes that modify the astro-nomical tide. The practicality of the second taskdepends on the problem; we concentrate, therefore,on the CWT as a means of accomplishing thefirst task. As Munk and Cartwright (1965) observed,‘. . . predicting and learning are in a sense orthogonal,and the most interesting effects are those that causethe most trouble with a forecasting . . .’.

The forecasting power of HA derives from theinfinite extent of its basis functions or, equivalently, itsrepresentation of the tide by a line spectrum consist-ing of a finite number of infinitely narrow peaks atfixed, predetermined frequencies, each with a definiteamplitude and phase. A time series with such aspectrum is a stationary signal, i.e. one that may bedivided into multiple, statistically indistinguishablesegments. Of course reality is not so simple, and thepresence of noise must be acknowledged even forstationary tidal processes. For such a signal, however,it is still usually sensible to use Gaussian statistics todetermine the accuracy with which various quantitiesare known. Indeed, careful investigation of theirspectral properties has been, for over a century, amajor factor in improving conceptual models of thetides.

A non-stationary signal has, in contrast, a frequencycontent that evolves over time. If a non-stationarytimes series is dissected, the statistical properties ofthe parts will not all be similar to each other or to thestatistics of the whole, and the usual Gaussian statis-tics based on the whole record will be deceptive. Twocommon examples of non-stationary signals arespeech and music. The meaning of a spoken sentenceor of a musical performance is in the whole and in thesequence of the parts—removing words or notes oraltering their order can change the meaning of thewhole or garble it completely. Therefore, the informa-tion content depends on frequency, duration, andsequence, all three together, as opposed to any onealone. In oceanography, too, non-stationary signalspose intriguing questions. In particular, non-stationary tidal signals provide the opportunity todeepen our understanding of tidal dynamics.

Harmonic analysis and the Fourier transform con-vert the information content in the time domain of asignal, which may be thought of as evolutionary con-tent, into static, averaged frequency information. Thelimitation of analysis methods that yields only a staticpicture of the frequency content of a non-stationarysignal is apparent in a comparison of the powerspectra, plotted in Figure 3(a,b) of the records shownin Figure 2(a,b). The signals evolve quite differently intime, but prima facie, their spectral representations are

nearly indistinguishable. Clearly, such a transforma-tion conceals certain quite striking features visible inthe time domain. Analogously, imagine a piece ofmusic played first forwards and then backwards.Power spectra for the two performances would haveexactly the same shape, though the phases wouldchange. The human ear, which naturally analyseschanges in frequency content through time, wouldimmediately distinguish the two cases and assignthem different meanings. As will be seen, wavelettechniques translate the idea of evolving frequencycontent into mathematics, turning intuitions aboutnon-stationary signals into a useful scientific tool.

The previous paragraphs suggest that a mathemati-cal technique is needed that will transform a one-dimensional input signal (a function of time) into atwo-dimensional field showing the amplitude andphase behaviour of the input as a function of bothfrequency and time. A key link is the Heisenberguncertainty relation:

���t�(4�)�1 (1)

which says that the product of �t (the uncertainty intime; i.e. the duration) and �� (uncertainty in fre-quency) is greater than or equal to a constant. Theseterms have meanings analogous to the concept of thestandard deviation in statistics:

where g(t) describes any envelope imposed on theinput time series, g(�) is the Fourier transform of g(t),�x�2�xx*, and x* is the complex conjugate of x.

For any method of time-frequency analysis,Equation (1) limits the possibility of simultaneous,mathematical knowledge of the time evolution andfrequency content of data. The best known role ofEquation (1) is in quantum mechanics, where ithas given rise to a series of wave-particle dualityparadoxes. But Equation (1) is not an empirical law.Rather, it follows from the definition of frequency (orwavenumber) itself. At issue is not noise or imperfectcalculations, but simply that it is inconsistent for thetwo questions ‘ when ’ and ‘ what frequency? ’ both tohave exact answers with respect to the same data.Infinite precision in frequency implies infinite dura-tion in time. On the other hand, exact time precisionprecludes any knowledge of frequency.

An HA of a long, stationary record represents alimiting case of Equation (1) in which �t (the duration

Introduction to wavelet transform tidal analysis methods 181

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F 3. Power spectra [above, (a) and (b)] for BV (left) and TP (right) surface elevation records. Tidal peaks are clear atboth stations, but tidal monthly peaks are obscured by competing processes—river flow and atmospheric effects. Powerspectra [below, (c) and (d)] of alongshore currents in the CR plume are quite diverse; see (below left) stations N1 (always inplume) and (below right) S3 (intermittently in plume). The two year elevation records provide higher resolution than theshorter current records (3–5 months), but most of the differences arise from physical processes.

of the record) is large and �q is accordingly very small,allowing resolution of closely spaced frequencies. Toexamine evolution of frequency content in a non-stationary record, it is necessary to choose a smaller,variable �t, with the consequence that �� will grow ininverse proportion. Turning attention away fromnatural systems which approach the ‘ clockwork ’model and focusing instead on the numerous devia-tions from that ideal, it is necessary to set aside theline-spectrum concept as the primary intellectualframework because it is an idealization inappropriateto the investigation of non-stationary processes. Thisstance is not a criticism of either the HA or Fouriermethods, merely a recognition of the fact thatEquation (1) precludes, a priori, any one techniquefrom addressing all research questions of interest. ACWT approach is complementary to HA and Fouriermethods. In fact, a power spectrum or HA thatdetermines the average frequency content of a recordis often a good starting point for a CWT analysis.

Two alternative analyses have been developed thatare premised upon the existence of basis functionswith finite time durations, and non-zero spectralwidth. A short-time Fourier transform [STFT; alsoGabor or Weyl-Heisenberg transform (Gabor,1946)] divided a record into segments of equallength; each piece is separately Fourier transformedusing an appropriate window g(t). In the STFT,g(t) fixes �t and ��, with the result that a single scale� set by the duration of g(t) is uniquely privileged inthe analysis. Events with a characteristic period closeto � or a little smaller are well resolved by theanalysis, but bursts of high frequency activity withdurations �� will be poorly resolved. Components ofthe signal at periods long compared to � cannotbe correctly resolved without using longer filters.Tidal studies typically involve periods ranging fromhours to months or longer, a range of scales unsuit-able to the STFT; it will not be considered anyfurther.

182 E. P. Flinchem and D. A. Jay

The second approach is a wavelet transform inwhich a single prototype function, �0(t), similar to abandpass filter, is built into a complete basis set forL2(D), the class of functions whose integrated squaredvalue is finite, by the introduction of scaling andtranslation parameters a and b, such that: �a,b(t)=a�p

�0[(t+b)/a], usually with p=0·5 or 1. The waveletanalysis approach used here relies upon the con-tinuous wavelet transform (CWT), as opposed todiscrete, wavelet transform (DWT). In this context,‘ continuous ’ implies that an arbitrary number ofbasis functions built up from �0(t) may be usedto match tidal frequencies, as is necessary to makeoptimal use of information concerning astronomicalforcing of the tides. The basis functions will not beorthogonal but are still complete. A DWT built upfrom functions with geometrically spaced frequenciescan be both complete and orthogonal, but lacks thenecessary frequency flexibility for tidal analysis. Whenapplied to data, both DWTs and CWTs must bedefined at a set of discrete points in time, as usual fordigital filters. As a general rule, CWTs are betterfor dynamical analyses, while DWTs are preferredfor data compression (Farge, 1992).

Complex demodulation or CM (Bloomfield, 1976)is also sometimes used to calculate time-varying tides,usually for only a few frequencies. Complex demodu-lation is, however, a less systematic form of thetechniques discussed here. Were it to be used toextract a spectrum of information, it would have to beimplemented as either a CWT or a STFT. Given thelimitations of STFT, it is only through CWTs that thetime and frequency uncertainties can be systematicallytreated to provide a useful approach to non-stationarytides; i.e. to give an optimal, self-consistent andapproximately invertible extraction of an entirespectrum of tidal and non-tidal variance.

Wavelet analysis represents a synthesis of develop-ments in mathematics, physics, electrical engineering,and computer science. Some aspects of wavelet theorywere prefigured in papers dating back to Harr (1910),as reviewed by Meyer (1993). But the first expositionof the modern wavelet transform was Morlet et al.(1982). Vetterli and Herley (1992) and Rioul andVetterli (1991) provide broad overviews of the mainconcepts in wavelet analysis with minimal mathemati-cal detail. More formal approaches are presentedby Heil and Walnut (1989), Kaiser (1994) andDaubechies (1988, 1992). Farge (1992) provides asummary of work in turbulence research making useof wavelets. Selected geophysical applications arereviewed by Kumar and Foufoula-Georgiou (1997),mostly using DWTs or wavelet packet analysis.Oceanographic usage of wavelets is not yet common,

and the closest analogue to what follows has beenCWT analysis of wind waves (e.g. Liu, 1994; Donelanet al., 1996). Still, tidal analysis using CWTs remainsdistinctive in its constraints on the selection of fre-quencies and its range of scales, among other factors.

Almost inseparable from the idea of times-seriesanalysis is the issue of reliability of spectral estimates.Analysis of the physics of non-stationary signals isstraightforward only when the dynamically forced partof its variance is substantially greater than the random(e.g. instrumental) part. This is true because thestatistics of distributions are not relevant to a non-stationary time series as a whole, and cannot, takingthe dissection of the time series to its logical andnecessary conclusion, be used to define error boundson the calculated frequency content at any particulartime. A CWT of a time series is essentially a represen-tation of the data on a different set of basis vectors.Individual components thereof no more have reliabil-ity estimates than a single data point viewed in iso-lation. Yet it is these individual estimates that arerequired to analyse the events making up a non-stationary record. Furthermore, familiar time-seriesstatistics are not applicable to processes that do notconform to a Gaussian distribution. Many non-stationary processes, for example the shelf internaltides discussed here, are strongly influenced by large,relatively infrequent events (freshets or storms)imposed on a background of smaller fluctuations innon-tidal forcing. To argue that non-parametricstatistics could be applied to the distributions oftransformed data misses the point—statistics based ondistributions cannot inform us as to the reliability ofindividual, dynamically crucial results in a non-stationary time series (e.g. tidal amplitudes during abrief storm). Suggested below is an alternative point ofview of the reliability of such transformed data, basedon the Heisenberg Principle.

The aim in the remainder of this paper is tosummarize the mathematical basis of CWTs in acontext familiar to oceanographers, to show how theirproperties complement those of HA and the Fouriertransform, to illustrate their strengths with examplecalculations drawn from concrete research problems,and to provide a level of detail sufficient for others tobegin applying the technique to their own research.

Approach to tidal analysis

Harmonic analysis

To motivate the introduction of CWT analysis, it ishelpful to examine the foundations of HA and thecritical assumptions about the tide in the ocean which

Introduction to wavelet transform tidal analysis methods 183

must be satisfied for HA to apply. HA describes thechanging elevation of the sea surface at a point as asum of a finite number of cosine waves with specificamplitudes, frequencies, and phases, H(t)=�Aicos(�i

�+�i). The frequencies �i are given, a priori, as thesmall integer multiples, sums, and differences of sixfundamental periodicities of the earth-moon-sun sys-tem. The Ai and �i are free parameters which arematched to observations by the method of least-squares. Doodson (1922) identified c. 400 terms inthe gravitational potential with magnitudes >10�4

relative to the largest factor, including periods as longas 19 years. Six primary assumptions about the tideand how it is measured need to be satisfied for HA tomeet its full potential; specifically:

(a) The only forcing affecting sea level is theoscillating gravitational potential.

(b) The gravitational potential is descried by a finitenumber of harmonic terms, all of which havebeen identified and have precisely determinedfrequencies.

(c) The sea surface behaves like a damped, drivendynamical system oscillating in a stationary state.There are no transient excitations; only a timeless,particular solution to the inhomogeneous prob-lem is present. This assumption contains thepremise that the structure of the ocean’s densityfield is either constant or irrelevant.

(d) The tide gauge is passive, hence measuring thesystem without disturbing it.

(e) The tide gauge has a known response functionthat may be de-convolved from the data.

(f) The input record is longer than the period of thelowest frequency in the forcing and its samplingrate is more than twice the frequency of the fastestterm in the forcing.

If all six assumptions hold, then HA will yield anexact description of the input time series, and it maybe inverted to reconstruct the input and to predictthe tide for past and future times. Certainly (b), (d),and (e) are all safe or represent technical, not funda-mental, impediments. Darwin (1886) was aware that(a) and (c) are, in practice, questionable. His aimwas to isolate the stationary tide, so he recom-mended avoiding confounding factors: ‘ (t)he heightof the water is subject to considerable perturbationfrom the weather, and the most perfect tide table is. . . when abstraction is made of the disturbingcauses ’. The issues raised by (a) and (c) are morecomplex, however, than this statement indicates: thetidal processes of interest are influenced or modifiedby non-tidal processes.

Harmonic analysis involves, moreover, a non-lineartransformation of the data, and this can cause difficul-ties when dealing with non-stationary data. Godin(1998) demonstrates a variety of unrealistic resultsfrom HA that can result from the nonlinear andmutually dependent behaviour of nearby spectralbands, even when the non-stationarity is mild relativeto the examples considered below. Jay and Flinchem(1999) define the mathematical properties of HA thatlead to such behaviour. Short HA analysis windowscause mixing of information amongst tidal frequenciesand between tidal frequencies and frequencies notincluded in the HA. This issue is of little concern foranalysis of long records of stationary data, because theinteraction terms are inversely proportional to analysiswindow length and become negligible for windowlengths >15–30 days. Conversely, this non-linearityposes a major challenge for analyses of non-stationaryprocesses and short records, where <15 days of data isincluded in an HA window. The linearity of wavelettransforms constitutes a fundamental improvementfor such short records. Point (f) remains problematicfor most oceanographic data no matter what analysismethod is chosen, because of the very broad range oftime scales commonly present in the data. CWTmethods provide an incremental improvement in hisarea, for which there is no real cure except a longerrecord. Specifically, the linearity of CWTs providesrobustness against noise from unresolvable signals andoptimizes extraction of lowest frequencies in an analy-sis, where the period approaches the record length.

Our interest is mainly in natural systems where(a) and (c), the weakest points of HA are stronglyviolated. Such processes include fluvial and internaltides, and biological processes driven in part by tidalforcing. By limiting the form of its answer, HA limitswhat it can tell us about our domains of interest.Paradoxically, jettisoning the assumptions that allowdetailed knowledge of stationary tides and therebyconfronting the limits of the uncertainty relationmakes it possible to learn more about non-stationaryprocesses would be learned otherwise.

Continuous wavelet transforms

The simplest and most intuitive variety of wavelettechnique to develop is the continuous wavelet trans-form or CWT. Definition of a CWT begins with thechoice of an oscillatory prototype function �0(t),which has finite variance, is localized in time near theorigin, and has zero mean:

��0(t)�0*(t) dt>� lim�t����0(t)=O

��0(t)dt=0 (3)

184 E. P. Flinchem and D. A. Jay

where unless otherwise specified, integration is overthe entire real line. These properties guarantee thatthe wavelet is ‘ wavelike ’ (has no zero-frequencyenergy) and localized in time-frequency space. Theconstraints of Equation (5) are quite broad, and thereis wide latitude in the choice of the form of �0(t).From �0(t) a two-parameter family of functionsis defined by scaling and translating the argument(Figure 4). For 0>a>� and ��>b>�:

If a and b are continuous, �a,b(t) forms a completebasis for L2(D), analogous to the Fourier integraltransform basis set [exp(i2��t)] over [��+�]. Set-ting p=0·5 ensures that all basis functions have thesame variance, regardless of scale, i.e.: ���0(t)�2

dt=��a,b(t)�2 dt for all a, b, a>0. Choosing p=1 causes

an input wave with unit amplitude 1 to have unitoutput for all scales.

The forward wavelet transform is a convolutionsimilar to a Fourier transform:

Zya,b=ga,b[Z(t)]��Z(t)�*

a,bdt (5)

(where the inverted hat denotes the transformedquantity) so that Zy is the CWT of Z. Like the Fouriertransform, the CWT has an inverse, or synthetic form:

The completeness of the set {�a,b(t)} means thatEquations (5) and (6) form a reversible transformpair, analogous to Equation (3). Consequently, theCWT shares with the Fourier transform the desirableproperty of conserving variance. The inverse formula,Equation (6), valid in continuous time and scaledomain (scale is inverse to frequency), has ananalogue in discrete, finite implementations, but aboundable error is then incurred (Kaiser, 1994).

F 4. Real ( ) and imaginary ( ) parts of typical Kaiser-filter D1, Dint, D2 and D3 wavelets used for hourly data. Eachfilter has c. 6 cycles; the D1 filter is 145 h long. The scaling is such that a unit input wave at each scale yield unit outputamplitude.

Other important properties of wavelet transforms

The Heisenberg Principle [Equation (1)] defined theminimum product of time and frequency uncertaintiesthat may be achieved by any analysis. It is clear fromdefinitions (2) and (5) that:

�t(�a,b)=a �t(�0) ��(�a,b)=a�1 ��(�0) (7)

Therefore, the CWT maintains the relation���t

=constant, with constant close to the optimumallowed by Equation (1) for all basis vectors across allscales. This is a property unique to wavelet trans-forms, and one of the key facts that make them useful.Another essential property of wavelet transforms istheir linearity; they are additive and distributive:

Introduction to wavelet transform tidal analysis methods 185

for any wavelet transforms gi, functions Z in L2(D)and any complex constant (Holschneider, 1995).These properties guarantee that results in one fre-quency band are independent of those in other bands,so that the frequency responses of a wavelet and of aCWT analysis using a series of wavelet filters arewell-defined functions. This is emphatically not thecase for HA, and short HA windows have frequencyresponses that depend on the details of the dataanalysed and the number of analysis frequencieschosen.

Practical application to tides

A strategy for wavelet tidal analysis will be definedhere. Application of Equations (4) and (5) requiresthat a and b be discretized, with a chosen to matchtidal frequencies. Certain choices of a�2, b and �0

will still form a complete basis. In most waveletapplications, data compression is optimized andredundancy avoided by expressing a and b in geo-metric series, so that the time step bn increases in sizealong with scale an:

an=a0n, 1<a0�2 bn=mb0

n, for n, m=0, 1, 2,. . .(9)

As the step sizes are increased, there is a limit at whicha complete representation becomes impossible. Onlyfor very restricted choices of �0 is it possible to seta0=b0=2 and still have a complete, orthogonal DWTbasis. Redundancy, in the form of several filters (or‘ voices ’) per octave, eases the difficulty in conservingvariance and allows for a wider range of waveletchoices [see Daubechies (1992) or Kaiser (1994) fordetails]. To define a wavelet basis suitable for tidalproblems, it is necessary to sacrifice data compressionand employ a CWT approach that: (a) uses waveletsthat resemble physical waves, (b) has frequenciesselected according to the dictates of astronomy, and(c) retains the property of completeness, so that thetransform may be inverted. In practice, this requiresnon-integral n. Code details are discussed in theAppendix.

The very broad bandwidth of tidal signals alsoaffects data compression and invertibility. Most tidalrecords are short enough relative to the periods of thesignals involved that: (a) data compression possibili-ties are limited, and (b) they fail to capture even onecycle of strong low frequency oscillations present inthe signal. To understand these points, contrast a tidalrecord with music. A 1-year tidal record of hourly datais typical, and consists of 8760 hourly observations.This record cannot be enormously compressed,because it is not over-sampled and contains only

c. 2900 periods of the highest resolvable frequency(D8). Such a record also fails to resolve significantinter-annual, decadal and longer variations. The resultis that no transformation of a tidal record, whetherusing DFT, CWTs, or HA, can perfectly capture thelow-frequency energy, resulting in an imperfect rep-resentation of the data after inversion. On the otherhand, the lowest relevant frequency for recording ofmusic is the lower limit of human hearing at c. 20 Hz.A 10 min recording contains 12 000 periods of thisoscillation, and >10 million oscillations of the highestaudible frequency, allowing precise inversion andgreat compression. Despite its imperfect analysis oflow-frequency variance, CWT methods still possessan important advantage for tidal records—frequencyspacing can be made more or less constant with scale,rather than decreasing at the low-frequency end, aswith an FFT or HA.

A dense, even sampling in scale and time yields onlysmall errors in reproduction of the data after transfor-mation; sparse sampling introduces larger errors. Apartially redundant approach with >1 voice per octaveis useful for tidal analysis, because this: (a) matchesthe expected tidal frequencies, (b) eases restrictionson wavelet form, (c) provides robustness in the face ofnoisy data, and (d) allows a smooth transition betweenthe distinct analysis needs in the tidal and sub-tidalbands. The result is a ‘ snug frame ’ that approxi-mately conserves variance and allows reconstructionof the signal from its CWT. Dynamical studies (asopposed to optimal data compression) also dictatemaintenance of a constant b for all scales, so thattime-series at different scales may be compared toeach other and to external forcing. If one outputper input is produced at each scale, then the analysisis said to have ‘ maximum overlap ’ (Percival &Mojfield, 1997). For tidal analysis, decimation ofhourly data to 6, 12 or 24 h estimates at each fre-quency is typical. This is redundant at low fre-quencies, and only the highest tidal frequencies suffera loss of detail during reconstruction.

The idea that analysis needs for the tidal andsubtidal bands require rather different approachesbears further explanation. In typical wavelets applica-tions (e.g. image processing) there are no preferredfrequencies, and the emphasis is on representation ofthe data as completely and compactly as possible. Buteven non-stationary tidal processes usually show,within the tidal band, a concentration of energy attidal frequencies [e.g. Figure 3(a,c)]. Considering thisand other factors, most investigations of tidal dynam-ics should: (a) employ wavelets that look as much likea linear wave as possible, (b) provide informationconcerning processes occurring at tidal frequencies,

186 E. P. Flinchem and D. A. Jay

1

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(c) allow inversion (and thus, possible prediction),and (d) minimize overlap between tidal frequencies,without presuming that non-tidal processes are absentin the tidal band. The spacing of major tidal speciesD1 (diurnal), D2 (semidiurnal), D3 (terdiurnal), D4

(quarterdiurnal), D6 (sixth-diurnal), D8 (eighth-diurnal) . . . up to the Nyquist frequency gives twovoices (filters) per octave except between D1 and D2.This series is completed by adding filters known as:(a) DInt, centred at the local intertial frequency,conveniently between D1 and D2 in temperate lati-tudes, and (b) DH centred at (1·41 day�1), to matchsmoothly to subtidal filters (Figure 5). This schemegives, a0=√2 in Equation (9), with all values of n closeto integral. �0 is chosen so that overlap between thetidal species is as small as possible. The result is goodcompromise—there is some unevenness in frequencyspacing to accommodate tidal and inertial signals, but(because of finite filter width) little loss of non-tidalvariance present in the tidal band. One may furtheradd species such as D5, D7 . . . if desired, though thiswill increase overlap between frequencies.

There are, in contrast to the tidal band, usually nopreferred frequencies in the subtidal band. Given themore or less continuous frequency distribution, over-lap between frequency bands is desirable, for robust-ness against noise and completeness. Severalapproaches are possible here. One is to continue downthrough the subtidal with two voices per octave

[a0=√2 in Equation (9)] to the lowest frequencyallowed by the length of record. Such a procedure willoptimize reconstruction and maximize consistency inapproach between the tidal and subtidal data. But theshort length of oceanographic time series is frequentlya problem, and the long resulting filters limit thelowest frequency that can be resolved. Anotherchoice, followed here, is to reduce the number of‘ wiggles ’ (decrease ) in the basis function �0 for thesubtidal filters, shortening the subtidal filters whilebroadening their frequency response. It is then poss-ible to represent subtidal frequencies with one voiceper octave [a0=2 in Equation (9)] and simultaneouslyto reach slightly lower frequencies. One can capture(Figure 5) tidal monthly variability with filters centredat 2 days, 3·75 days, 7·5 days, c. 15 days, c. 29·5 days. . . Some variability in is also needed at lowerfrequencies to accommodate 6 month and 1 yearfilters.

It is also necessary to choose a specific wavelet �0(t)to implement. A major factor in this regard is theproblem of energy leakage into the side-lobes. Opti-mal for minimizing side-lobe energy under discreteconvolution are the prolate spheroidal wave functions.Unfortunately, they cannot be expressed in a closedform convenient for computation. Kaiser (1974),however, derived an accurate approximation to theprolate windows in terms of zero-order modifiedBessel functions, I0(t). The Kaiser window also has

Introduction to wavelet transform tidal analysis methods 187

the highly desirable effect of minimizing Equation (1)to the point that ���t

�(4�)�1, in a sense achievingthe best transform theoretically possible. A wavelet isemployed here constructed by windowing a complexexponential with a Kaiser filter:

�a,b(t)=a�1 norm(, a) �0[(t�b)/a] (10b)

where: side-lobe suppression is controlled by �, thenumber of cycles on either side of the central point is, and norm(,a) is a normalization factor that pro-vides unit response for waves of unit amplitude. Iffrequencies are evenly spaced, norm is independentof a.

Filter design always involves a balancing of factors;side-band rejection (i.e. frequency resolution) andfilter length (i.e. time resolution) are the primaryissues here. For a Kaiser filter length specified by ,the frequency response of �a,0 around the frequencyspecified by a is controlled by �. Large values of �broaden the central peak of the Kaiser filter andincrease the rejection of side-band energy. Forexample, �=6·755 gives a filter roll-off such that thefirst side-lobe is diminished by 70 db; �=4·533 yieldsa narrower central peak but a first sidelobe rejectionof only 50 db. A large � is useful for the subtidalband where some overlap of central peaks is desired;this also provides very strong rejection of sideloberesponse. However, � cannot be arbitrarily increased,or a very long filter will be required. Small values of �narrow the central peak at the cost of putting moreenergy into the side bands. Decreasing � modestlyfrom 6·755 allows shorter filters to function well in thetidal band. But sideband energy from D2 and otherlarge peaks can contaminate estimates in other bands,if � is too small. Increasing filter length by increasing tightens the response around the central point buthas no effect on relative side-lobe height, which iscontrolled solely by �.

Choosing =�2–4 and � c. 4·5–6 is compatiblewith a D1 window of 96–192 h (4 to 8 wave periods);other tidal filters scale accordingly. This choice allowsresolution of DH, D1, DInt, D2, D3, D4, D6, D8 . . .,the maximum being determined according to theNyquist criterion. If > c. 4, D5 and D7 can beresolved as well. For a record that is stationary overc. 15 days, =c. 8–10 allows resolution of the threemajor semidiurnal constituents (M2, N2, and S2)within the D2 band. Using =c. 2 and �=6–8 for the

subtidal band provides a desirable overlap betweenspecies in this part of the spectrum and mimimizes thelength of the longer filters. A short data record willoccasionally force =2 for the tidal band; reducing� to 4–5 may then be desirable. However, speciesseparated in frequency by a factor or two overlap forfilters with <�2.

A certain degree of dependence of results onmethod is inherent in the analysis of non-stationarysignals. This is a direct consequence of the trade-off

of time and frequency resolution embodied inEquation (1). Wavelet transforms are, nonetheless,generally linear and robust, so that: (a) smallchanges in analysis method bring about smallchanges in results, and (b) the frequency responsesof a filter and the analysis as a whole are readilycomputed, well-defined functions (e.g. Figure 5).Similar results to those shown below could beachieved using a Morlet or other complex wavelet,but the Kaiser filter more efficiently suppresses side-bands. The dependence of CWT and HA propertieson analysis details is explored in Jay and Flinchem(1999).

Finally, species names are used (e.g. D1 and D2 forthe diurnal and semidiurnal tide) rather than the usualconstituent names (e.g. M2, N2, and S2 within the D2

species) for a reason. In most non-stationary tidalprocesses, the non-tidal forcing varies with sufficientrapidity to render the mathematical construct of thetidal constituent irrelevant. That is, definition of theabove three constituents requires that the tidal processbe stationary over the length of record required todistinguish the three within the constraint ofEquation (1). In practice, this is 10–20 days. Thesituation is worse for the diurnal species—stationarityover >3 months is required to determine the threelargest constituents. Designation of bands with tidalspecies nomenclature (D1, D2, etc.) does not mean,however, that the variance captured by such bands isexclusively tidal in origin.

Discussion of examples

The purpose here is to illustrate CWT analysis proce-dures and to demonstrate the utility of wavelets intidal problems. In both of the two cases discussed,CWT methods lead to questions or hypotheses thatwould have been difficult to develop with other tidalmethods, because least-square analyses cannotprovide accurate results on the necessary time scales(Jay & Flinchem, 1999). It is not, however, the intenthere to present detailed analyses, because this would,in each case, require an entire paper.

188 E. P. Flinchem and D. A. Jay

Internal tides in a buoyant shelf plume

Internal tides are notoriously unsteady (Sandstrom,1991), because their generation and propagation arestrongly dependent on highly variable ambientdensity. Non-linearity of internal tides is recognized inprinciple (Maas & Zimmerman, 1989), but little isknown about it in practice. A common, analyticallydifficult and physically complex example of internaltides arises when a buoyant river plume enters an openshelf environment. For reasons based in the difficultyof obtaining and analysing data concerning internaltides, internal tide analyses usually assume that:(a) there is a spectral separation between tidal andsubtidal processes in both the velocity and densityfield; (b) only the D2 (and at some latitudes D1)motions are important; and (c) horizontal densitygradients are small relative to those in the vertical, sothat the vertical tide may be deduced from tidalexcursions of the density field. Analyses show, in fact,that all three of the above assumptions are violated bythe plume internal tides of interest here.

The marked unsteadiness and non-linear nature ofplume internal tides has several causes:

(a) there are multiple forcing mechanisms, each ofwhich is episodic or non-stationary;

(b) the position of the buoyant plume that supportsthe internal tides is highly variable so that aninstrument may be intermittently within theplume (Hickey et al., 1997)

(c) a buoyant plume often occupies only a smallfraction of the total water shelf column, leading tofinite amplitude non-linearity and to a possiblysupercritical surface layer. Non-stationarity isenhanced because small alterations in layer thick-ness or external forcing can cause large changes inthese very non-linear motions;

(d) non-tidal processes (e.g. wind stress and shelfwaves) may intrude into the tidal band,forcing non-stationary tidal-frequency motions ofnon-tidal origin.

The data analysed here come from a winter 1990–1991 Columbia River plume study. The period ofobservation was stormy, but without major freshetevents [Figure 6(a,b)]. The low-frequency picture forstations that are in the path of the mean plume (e.g.N1, N2 and N5) is that river flow events lead toincreases in the stratification that supports internaltides, whereas storm winds destroyed stratification.Plume position is strongly influenced by winds, andthis means that details of stratification time seriesnorth (the N and K stations) and south (station O5) ofthe estuary entrance differ. But all stratification time

series show a dominant effect of stratification by riverflow and destruction thereof by strong windstressevents.

The spatial variability of the internal tides withinthe plume area can be seen from their time-averagespectral qualities [Figure 3(c,d)]. Station N1 shows avelocity power spectrum reminiscent of estuarinecurrents, with well-defined D1, D2 and overtide peaks.Energy at the intertial peak is insignificant. Station S3,south of the estuary entrance (Figure 1), was onlyintermittently in the plume. This record provides acautionary lesson for any method of tidalanalysis—should we assert, despite the poor spectralseparation of the D1 peak, that the D1 and D2 peaksare both tidal? The signal cannot be described as‘ band-limited ’, and the fact that CWT or HAroutines detect energy at a tidal frequency does notmean that the energy is tidal in origin. Still, examina-tion of the S3 time series show that a qualified answerof ‘ yes ’ to the above question is merited—there aretime periods when the tidal signal is clear and similarto other nearby stations. Note also that CWTmethods are better than HA for records likeFigure 3(d), because of the linearity of CWTs—energy falling outside of the tidal bands has no effecton results for the tidal bands. Finally, there is onecommon feature for stations N1 and S3—the bandfrom c. 4 days to D4 is more energetic than the lowerfrequencies (at least those whose amplitude can bedetermined with the available record).

The tidal flow at N1 is highly non-stationary,despite the reasonably discrete power spectrum inFigure 3(c). An attempt to connect the observedtemporal variability to the forcing functions shown inFigure 6 will help in the development of hypothesesconcerning the processes generating internal tides inthe plume area. A convenient way to examine timeevolution of frequency structure is through a scaleo-gram, where amplitude (or phase) is plotted as afunction of frequency over time. Windstress andstratification scaleograms [Figure 7(a,b)] show a con-tinuity of energy across the tidal and near-subtidalthat is sometimes, but not always, mirrored in thevelocity field major and minor axes (not shown).Tidal-band wind fluctuations arise from the rapidsuccession of fronts characteristic of winter storms inthe area. Little wonder then, that inertial frequencymotion is stronger than D1 (e.g. at Station S3) andeven than D2 at some other locations.

These observations raise two methodologicalpoints. First, any analysis methods used to examineshelf tides should include the local inertial frequency,because of its physical importance and the oftenbroad-band nature of atmospheric forcing. Especially

Introduction to wavelet transform tidal analysis methods 189

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F 6. Relationship of [above, (a)] river flow ( m3 s�1), tidal outflow ( m3 s�1), and negative of total windstress( dyne cm�2) �10 with the low-pass stratification [below, (b)] at stations O5 ( ), K5 (*), N5 ( ), N3 ( ), and N1 ( ),bottom to top in lower panel, all in sigma-t units offset for clarity by 0, 2, 4, 6, 8 units, respectively. River flow provides thebuoyancy that supports plume stratification. Tides have a mixed effect, providing more buoyant outflow and strongeradvection of low density water, but also more vertical turbulent mixing. Negative of the windstress is plotted, because of thedestruction of stratification by wind-induced mixing. Stratification is calculated by subtracting a 2–5 m sigma-t at each stationfrom a reference value taken at 65 m depth south of the entrance. The reference value varies temporally by c.�0·75 unitsfrom its mean. Its temporal structure has little influence on the frequency structure of the stratification, because near-surfacefluctuations are much greater.

if it is not included in a HA, then inertial energy notaccounted for can distort the remainder of the analysis(Jay & Flinchem, 1999). Second, an event in time(formally a mathematical singularity) will show a

vertical ‘ influence cone ’ on a scaleogram that spreads(in time) toward larger scales with longer filters. Inthe present case, separate events have influence conesthat overlap at low frequencies. A stationary process

190 E. P. Flinchem and D. A. Jay

F 7. Amplitude scaleograms for (above) alongshore windstress and (below) for stratification at N5. Both records showsubstantial variation at 2–15 days. There are also sporadic, strong incursions of windstress into the tidal band, and a definitecontinuity of process between the tidal and subtidal bands in the stratification record. Correlation of atmospheric, fluvial andtidal forcing functions (e.g. at 15 days periods) with each other makes unambiguous separation of cause and effect difficultwith regard to tidal fluctuations in the velocity and density.

Introduction to wavelet transform tidal analysis methods 191

will, in contrast, produce horizontal contours on ascaleogram.

The continuous nature of the stratificationspectrum across the tidal band has important impli-cations for tidal analysis. First, linear tidal theoryassumes the existence of a spectral separation betweenmean and tidal components of the velocity and densityfield. Tidal variations in density then arise from tidaladvection, as described by the usual ‘ two-timingassumption ’. The relatively continuous velocity andstratification spectra observed at most stations in theplume do not fit the linear paradigm. These featuresare likely connected to one another, in that irregularadvection of strong horizontal density gradients byfluctuating winds can flatten the velocity and densityspectra. Second, isopycnal motions at individualmoorings cannot be used to define the vertical tide,because they represent an unresolvable mix ofhorizontal and vertical advection. Finally, linearinternal tide theories may also not be very useful insuch a non-linear regime.

The reasons for the variable response of the tidal-band velocity field to wind forcing can be examinedusing a wavelet cross-coherence Cxz defined for anytwo time series X and Z:

where the brackets K L in the denominator denote atime average. The time-varying cross-product in thenumerator of Equation (11) has been normalized bythe product of the time-mean amplitudes for the timeseries (equivalent to the usual practice in the Fourierrealm), rather than the product of the instantaneousamplitudes (as per Liu, 1994). Normalization by theinstantaneous amplitude product yields 0�Cxz�1 (asin the Fourier realm), but the resulting C then conveysno information concerning the absolute amplitude ofthe processes relative to the mean. Normalization bythe mean amplitudes yields Cxz�0 (and sometimes1), with the largest values indicating times when thetwo processes are strong and coherent.

Windstress-velocity Cxz scaleograms [Figure 8(a,b)]tell an interesting story. Windstress encroaches on thetidal band during almost every storm, always forcingthe inertial band, but exciting a variable reaction fromthe internal tides. In particular, the onshore D1 andD2-wind correlations are weak until December. Wehypothesize, therefore, that there is a connectionbetween river flow, background stratification, andwindstress generation of internal tides. The connec-tion may be as follows. Buoyancy input creates strati-fication, while windstress has more complex effects.

Winds destroy stratification by vertical mixing andadvect the plume, but may also excite broad-spectrumoscillations including the tidal band. Perhaps the windevents early in the record (while river flow levels arelow) fail to excite a substantial internal tide responsebecause they reduce the background stratification tooseverely before a substantial tidal response candevelop. Complicating this situation is the subtidalwind-velocity correlation after c. 330 days. This vari-able, broad-band nature of the wind-tide correlationrenders analysis of cause and effect difficult, in com-parison for example, to the clear forcing of D1 internaltides of California by a diurnal sea breeze (Rosenfeld,1990). Still, the atmospheric forcing of internal tidesmight have been totally missed in this case without aCWT analysis of the windstress field.

In summary, internal tidal frequency currents inthe Columbia River plume are non-stationary andnon-linear, and have multiple driving mechanisms.Figures 6–8 suggest that there are two internal tidalmechanisms active in the plume area: (a) propagationfrom the shelf break, and (b) generation by rapid windfluctuations associated with frontal passages. If thedata are probed further here, it is seen that the patternof spatial and temporal overtides suggests a thirdmechanism, non-linear excursions of the interface atthe estuary entrance. It was these complex dynamicsand difficulties in application of HA to this data setthat first caused the authors to investigate CWTs as analternative tidal analysis method. The brief discussionhere also shows that many traditional time-seriesanalysis tools (e.g. the cross-coherence and a rotarypresentation of current species) have analogues in thewavelet world. CWTS provide the flexibility to usesuch tools to examine evolution of the frequencycontent. Once the evolution of frequency content isknown, then it is possible to analyse (only qualitativelyhere) the multiple internal tide forcing processes.Furthermore, analysis of all the fluctuating variance,not just the tidal component thereof, leads to adifferent view of tidal processes than would be thecase with the usual methods.

Tidally modulated microbial productivity in an estuarineturbidity maximum

The second example investigates the possibilities ofexpanding tidal analysis to study tidally influencedbiological processes, and suggests a new form ofsignificance calculation appropriate to non-stationarydata. Time series of biological variables with therequisite sampling interval and duration for tidalanalysis are scarce, largely because of the difficulties inobtaining observations. But time-series analysis has in

192 E. P. Flinchem and D. A. Jay

F 8. Wavelet coherence as a function of scale and time for (above) onshore internal velocity and windstress and (below)alongshore internal velocity and windstress. Vertical structures crossing the tidal band are indicative of a broad-band responseto wind forcing that is often continuous across the tidal band. There is also considerable velocity-windstress coherence at 15days, which renders separation of tidal and atmospheric effects on the velocity field difficult.

principle the same power to elucidate biologicalprocesses as has proven to be the case in fluid mech-anics. Considered here is particle-attached bacterialproductivity (PABP) in an ETM; PABP is smalloutside the ETM. Microbes play a major role indetritally based estuarine food chains, and under-standing controls on their productivity is an importantissue (Baross et al., 1994; Crump et al., 1997).Because microbes are small, their productivity canrespond rapidly to environmental changes. Observa-tions in the Columbia River ETM show that PABPand total bacterial productivity are similar in magni-tude and temporal structure in that system, becausethe after column bacterial productivity is always much

smaller than that associated with particles; only PABPis considered here.

The data employed here come from an c. 9 dayoccupation of station NC (Figure 1) in the ColumbiaRiver North Channel ETM during a moderate springfreshet in May 1995. Observations (every 2 h forbiological parameters, every 0·25–0·5 h for physicalvariables) begin during a weak spring and evolvetoward the neap tide. The diurnal inequality of thetides and location of this station was such that eachgreater ebb removed all salt from the water column,while salt remained near the bed on each lesser ebb.This pattern is fortuitous but very useful in analysingcontrols on PABP, because the time series of PABP

Introduction to wavelet transform tidal analysis methods 193

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F 9. Above: time-series of total ( ), particle attached bacterial production (PABP) ( ) and free ( ) bacterialproduction. PABP is the dominant signal and is closely related to maximum N2 in the water column. Below: tidal elevationon Mean Lower Low Water (MLLW) ( ), depth of maximum N2 (�) and bottom depth ( ) on MLLW. Maximum N2

moves higher in the water column on flood and down toward the bed on ebb, as salt is advected/mixed in and out of thesystem. Maximum N2 is high on each flood and on lesser ebb. Total absence of salt on greater ebb is indicated by intersectionof maximum N2 with the bed. Oscillatory bottom depth is caused by swinging of the sampling vessel.

shows a very striking pattern. There are three maximaand one minimum every tidal day [Figure 9(a)]. Theminimum occurs during the greater ebb when salt isremoved from the near-bed area, bedstress is verystrong, and sand is transported in suspension. There isa maximum on each flood and on the weaker ebb; thelatter is sometimes the strongest peak of the day. Inthe frequency domain, PABP shows very strongdiurnal and terdiurnal responses [Figure 10(b)].

The question naturally arises in attempting to un-derstand controls on PABP, what physical variablesshow a frequency domain signature similar to PABP?In fact, this particular signature is rather exotic anddoes not match any of the usual parameters, e.g.velocity, bedstress, density, and suspended particulatematter (SPM) levels. It does, however, strongly

resemble the time series of near-bed maximum inwater column stability N2=g/�0 )�/)z [Figures 9(b)and 10(a)]. Maximum gradient Richardson numberRig=N2/S2 (where S2=|)UH/)z|2 and UH is the hori-zontal velocity) also shows a similar frequency signa-ture to PABP, but the ADCP data used to determineshear are coarse in terms of spatial resolution, rela-tively noisy and do not always reach close enough tothe bed to reach the level of maximum stratification.We consider, therefore, the relevance of N2, remem-bering that it may be a surrogate for Rig. Shear stressat the bed is necessary to suspend the aggregates[O(1 mm) in diameter in the Columbia ETM] thatsupport PABP, but too much shear disrupts theaggregates, apparently decreasing PABP in the pro-cess. There was sufficient shear during the c. 9 day

194 E. P. Flinchem and D. A. Jay

F 10. Amplitude scaleograms of maximum N2 (above) and PABP (below). Both show dominantly D1 and D3 to D5

energy. The sharpness of overtide peaks seen especially in maximum N2 reflects the reorganization of the water column ongreater ebbs, and is seen in the velocity field was well.

period to allow some re-suspension of aggregates,either locally on the bed at NCA or at locationsupstream. PABP was then highest under relativelystable conditions (N2 high), so that aggregate were notdisrupted.

Several qualifications and comments are in order.First, the above statistical resemblance between N2

and PABP time/frequency domain signatures does notprove the existence of a causal connection between thetwo. The resemblance might be merely a non-causalcoincidence. However, a plausible mechanism is avail-able. And given the present state of knowledge con-cerning physical controls on PABP, the above resultprovides a valuable starting point for experimentation(perhaps in the laboratory) to determine the actualmechanism. Second, more or less energetic tides orlower river flows might reveal other controls. Finally, a

slightly different balance of tidal amplitude, diurnalinequality, fluvial forcing and sampling position mighthave caused salt to be absent twice a day, totallyaltering the frequency structures of N2 and PABP. Ifthese were high either twice or four times a day, thenthey would resemble many other tidal parameters,rendering the search for controls on PABP moredifficult.

This data set also illustrates two methodologicalpoints. The first is the utility of using shorter windowsfor smaller scales, conserving the uncertainty productin Equation (1). Analysis of this short (c. 9 days) dataset required the use of the shortest possible filters, toallow resolution of temporal change. The filters were97 h for D1 [=2 in (10) or four wave cycles at eachscale] and correspondingly shorter for the smallerscales out to D5, the highest possible frequency for the

Introduction to wavelet transform tidal analysis methods 195

F 11. Amplitude scaleograms of D2 (above) and D3 (below) alongchannel currents. The D2 record shows no signs ofperturbation by the change in velocity structure on greater ebb. Over-tides like D3 and D4 show a complete re-organizationonce per tidal day.

sampling interval of 2 h available with the biologicalvariables. Short filters exact a price in frequencyresolution according to Equation (1), the effects ofwhich we examine in the next paragraph. The use ofthese very short filters has, however, a positive sideeffect in terms of time resolution. The alongshorevelocity amplitude scaleogram (Figure 11) shows acurious banded structure of D3 and D4 (filter lengths33 and 25 h, respectively). We hypothesize that once aday on the greater ebb when the velocity profile istotally barotropic for c. 6 h, the overtide structure of

the flow changes drastically. This dynamical effect isnot perfectly resolved even with CWTs, because thefilter windows encompass several overtide waveperiods to achieve the necessary frequency resolution.It would, however, be totally missed in a STFT orHA approach, wherein in all windows have a lengthdetermined by the need to resolve D1 and D2.

The second methodological point concerns theissue of the ‘ validity ’ of CWT estimates at each timeand scale. As discussed above, it is not possible withnon-stationary data to apply the sort of significance

196 E. P. Flinchem and D. A. Jay

estimates used with power spectra, because these areappropriate only to stationary time series. It is poss-ible, however, to define an approach based on theHeisenberg principle. Uncertainties in CWT esti-mates arise from ‘ cross-talk ’ between frequencies,because finite filters have finite-width spectralresponses. Note that this cross-talk is not an error inthe usual sense—the finite breadth of response of afilter may be useful or frustrating, depending on thecontext. An error arises only if one assigns all thevariance captured through application of a broad filter(perhaps encompassing several tidal peaks) to thecentral species for which the filter is named. Still, atidalist normally wishes to separately tally the variancein each species, and it is useful to know the fraction ofvariance in a output CWT band that may be associ-ated with neighbouring frequencies. One way tobound the degree of cross-talk is to use the frequencyresponse function (n,m) of each filter (the response ata scale m of the filter centred scale n) to compute aratio R� of the output amplitude at scale n to themaximum amplitude of the cross-talk from adjacent(smaller and larger) scales (n�1):

An R�1 indicates only a small potential for inter-ference between frequencies. R��O(1) or less indi-cates that a given output is too small to be reliablyseparated from outputs at neighbouring frequencies ata particular time. A small R� value is not, per se, bad;e.g.: (a) if a given frequency is not of interest, as is thecase for inertial variations in PABP in an ETM whereinertial currents are negligible; or (b) in analysis of anevent that is, by definition, a multi-scale process. Onthe other hand, if one wishes to make dynamicalcomparisons between transform outputs and possibleforcing functions, R�>1 is mandatory and R�1preferable.

PABP serves as a good example of the cross-talkproblem. The PABP scaleogram [Figure 10(b)] showsa mix of horizontal contours, indicative of continuityof process, and vertical ‘ event cones ’. To what degreeare the frequencies distinguishable? Figure 12 showsR+ (the cross-talk at a given scale from the next higherscale) and R� (the cross-talk at a given scale from thenext lower scale) for selected PABP scales as anexample. Cross-talk for the dominant D1 and D3

scales in the PABP is negligible, but D2 is sometimesonly marginally significant, especially in the downscaledirection.

In summary, CWTs allow time series analysesmethods to be applied to non-stationary biological

variables, yielding hypotheses that would be difficultto generate otherwise. While spectral significanceestimates (familiar from the realm of power spectra)cannot be applied, it is possible to define a cross-overratio R�. This ratio is a measure of the possibilityof establishing as a function of time, independentknowledge of processes between neighbouringfrequencies.

Summary and conclusions

We have summarized the basic formulation of theCWT, shown how its properties complement those ofHA and Fourier methods, defined a wavelet basis forunevenly spaced tidal frequencies, demonstrated howto adapt CWT tools to concrete problems, describedpossibilities for analysis of scalar and vector quantities,defined a criterion for knowledge of independence ofprocess between adjoining scales, illustrated thestrengths of the method with calculations relevant totwo oceanographic problems, and provided details ofthe algorithms sufficient for the reader to begin apply-ing CWTs to tidal problems. The results show howsubtle features of the evolving frequency content of anon-stationary time series can be brought into sharprelief for comparison to theory. The CWT enables anew mode of interplay between observation andtheory which can only serve to invigorate theoceanographic sciences.

Wavelet techniques are expected to prove invalu-able to future studies of the non-linear, time depen-dent dynamics of estuaries and coastal seas. Likelyapplications include analyses of internal tides, shortinternal wave dynamics, effects of atmospheric forc-ing, and internal tidal asymmetry. Attention has beenfocused on tidal problems because they have a struc-ture perfectly suited to showcasing the strengths of thewavelet transform. We believe, however, that waveletmethods will be applied routinely to a wide variety ofwavelike and unsteady oceanographic processes,regardless of whether they are dynamical, chemical, orbiological in nature.

Finally, a word about prediction of tides with wave-lets is appropriate. The potential utility of CWTanalysis for forecast or hindcast activities depends onthe character of the data. Given a stationary timeseries of sufficient length with a low noise level, all ofthe usual constituents within the tidal species may beresolved with CWT analysis. CWT analysis however,possesses, no particular advantages over HA for pre-dictions in such circumstances. Moving in the direc-tion of more difficult prediction problems, CWTmethods first become competitive with HA for shortstationary, but noisy records where extraction of the

Introduction to wavelet transform tidal analysis methods 197

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F 12. Log10 of D1 (top), D2 (middle) and D3 (bottom) up-scale R+ ( ) and down-scale R� ( ) cross-talk ratios (seetext for definitions) for PABP, estimated for times when outputs are available at the relevant scales. Despite the relativelyshort filters used, there is little cross-over effect on the dominant D1 and D3 scales. An output is plotted for every 2-hourlyinput, so the results are highly redundant.

maximum number of constituents is necessary; e.g.for the problem of extracting the three majorsemidiurnals from <1 month of data. The greateststrength of CWT methods is, however, for non-stationary problems where the tidal signal is modu-lated on short time scales by non-tidal processes,rendering useless the conventional apparatus of tidalconstituents within species.

Strong modulation of tides by non-tidal, stochasticprocesses is also the crux of the prediction problem—

non-stationary tides can be predicted if and only if:(a) the non-tidal forcing can be modelled in some way;and (b) the interaction of the tides with the non-tidalforcing can be deciphered. This contribution hasaddressed methods for point (b), which is seen as akey limiting step in moving toward forecasts of non-stationary tidal processes. The analysis methodsdefined here should assist in improvement in forecastsof some relatively simple non-stationary tidalproblems (e.g. river tides and storm surges) where

198 E. P. Flinchem and D. A. Jay

reasonably accurate numerical models of the non-tidalforcing are available. In more complex cases such asthe shelf internal tides discussed here, predictions maystill be possible in a statistical sense. Thus, climatol-ogy might be used to define shelf circulation scenariosfor a location and season and the frequency of occur-rence of each scenario. Inversion of CWT analyses ofdata for the various scenarios could then be used topredict the distribution of internal tidal responses forthe period, if not the actual sequence of events.

Acknowledgements

Definition of CWT tidal analysis methods was sup-ported primarily by the Office of Naval Researchthrough Grant N00014-94-1-0009, and secondarilyby National Science foundation Grants OCE-8918193 (Columbia River Plume Project) and OCE-9807118 (Columbia River Land-Margin EcosystemResearch Project; LMER). We thank Drs L. J.Pietrafesa of North Carolina State University andB. M. Hickey of the University of Washington forproviding us with Columbia River plume data (col-lected under grant OCE-8918193). Revision of thismanuscript by the second author was supported inpart by Oceanographic and Environmental Charac-terization of Coastal Regions (OECCR; ONR GrantN00014-96-1-089). We are indebted to Dr J. Barrossand B. Crump of the University of Washington forproviding LMER microbial production data. Beavertidal elevation and Columbia River flow data wereprovided by the US Geological Survey. Tongue Pttidal data were provided by the National OceanSurvey.

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Appendix

The purpose of this Appendix is to provide sufficientinformation that readers could themselves write awavelet tidal analysis program. The code describedhere was written in the Mathematica� language,though there is nothing (aside from convenience)about the problem that especially suggests the useof this language. Matlab� or any other higher-levellanguage would, do as well. A code written in C orFortran might be more efficient computationally, butwould be more difficult to implement. Commericalwavelet transform packages do not usually allowunevenly spaced frequencies matched to the tides,‘ maximum overlap ’ output, or different filter proper-ties in the tidal and subtidal bands; these featuresrequire special-purpose software. Sample routines areavailable from U.S. Geological Survey’s SeaMatWeb-site: http://crusty.er.usgs.gov/sea-mat/

Described here is a program that analyses only asingle scalar time series. Vector time series add theadditional issue of rotary calculations based on wave-let transform results (not discussed here), but theinitial transform of each vector component is carriedout as below. ADCPs produce vector time series atmultiple depths, but again, the fundamental analysismethod is the same. Data to be analysed should beevenly spaced in time, with a time interval, �t,suitable to tidal data; typically a few minutes to c1 h. The program described here does not use abso-lute astronomical phases, as would typically bedefined by a harmonic analysis routine. While abso-lute phases could be implemented, experience withresults from non-stationary oceanographic time-series does not suggest that this would be a usefulexercise. Whenever a tidal time-series is sufficientlynon-stationary that an analysis of the sort describedhere is actually necessary, resolution of tidal con-stituents is not possible. The multiple constituentswithin each species then cause considerable phasevariability, which can only be increased by non-tidalprocesses.

The code described here employs a direct, CWTtransformation of the data. It is computationally moreefficient to calculate the wavelet transform indirectlyfrom a fast Fourier transform (FFT) of the data.There are several reasons why we did not pursue thisapproach:

(a) Use of an FFT requires a number of data pointsthat is a power or two, or the data must be assumedperiodic. Given non-stationary data and a desire toachieve a quantitative output, truncation of the dataset to the next lowest power of two is the onlyalternative for an FFT analysis. Most oceanographicdata sets are too short to make this an attractivestrategy.(b) The FFT approach cannot deal with crucial low-frequency information as optimally as the directCWT.(c) The direct transform approach is far more flexiblein choice of frequencies and choice of filters than anFFT. Although the issue is not discussed here, it isalso possible with a CWT (as with HA) to extract tidalinformation from data with gaps. This is not possiblewith a conventional FFT routine.

The code is divided into the following sections: (a)definition of data properties, (b) definition of analysisproperties, (c) convolution, wavelet and filter defi-nitions, (d) the actual transform calculations, and (e)plotting of results. The essential data properties thatmust be specified are: the time-series start time (inlocal time), the �t, the number of data and the stationlatitude (the latter for definition of the inertial fre-quency filter). The data are then loaded into theanalysis vector (a process which is carried out repeat-edly for vector and ADCP analyses). The time basis ofthe program is seconds.

Definition of the analysis properties includes severalsteps:

(a) Base filter lengths are determined for the tidaland subtidal filters. The user sets the length of theD1 filter (an odd number), and the lengths of othertidal filters vary with scale as in Equation (1). Anequivalent D1 length (possibly shorter than thatfor the tidal frequencies) is then set for the subtidalfilters. D1 is obviously not a subtidal filter, but it isconceptually simple to choose the distinct resolutionproperties (determined in part by filter length) of thetidal and subtidal bands using the D1 filter. Separatedecimation factors may also be employed for the tidaland subtidal analyses. Filter frequencies were dis-cussed in the text and include the local inertial fre-quency, thus the necessity of including the latitude asan input.

200 E. P. Flinchem and D. A. Jay

(b) The number of frequencies to be used is set forthe tidal and for the subtidal bands to provide themaximum possible range of scales, as determined byrecord length and Nyquist criterion. The number ofpoints in each filter (an odd number) is calculatedfrom the filter base lengths and �t. Because both therecord length and the �t value varies with the dataset, it is convenient to index the tidal and subtidalfrequencies separately, with the lowest indices per-taining to the lowest tidal and highest subtidalfrequencies.(c) Phase calculations (below) require that the starttime of each filter output be determined from the datastart time and the filter length. One could calculatethe centre times of the outputs, but decimation of theoutputs complicates this approach.(d) Windowing shifts the frequency of a windowed-sinusoid to a frequency slightly higher than the nomi-nal frequency of the sinusoid itself. The astronomicalfrequencies are, therefore, adjusted slightly so that thepeak of each wavelet’s response function will corre-spond to the desired astronomical frequency, afterdigitization and windowing. For a 145 h window, theactual frequency exceeds the nominal frequency by1.04%, for a 97 h window, the difference is 2.33%.

Many aspects of this code would be simpler if thedecimation factors, �t and base filter lengths wereinvariant, but this would rob the code of its generalutility for a large variety of problems.

Convolution, wavelet, and filter definitions includethe following aspects:

(a) The convolution definition is conventional; if abuilt-in definition is available, it may be used.(b) The mother wavelet is defined according toEquation (10), with � chosen separately for the tidaland subtidal filters, as described in the text.(c) Actual (complex) filter values are chosen for eachfilter, based on the filter lengths and �t, note thatthe filters use the complex conjugate of the motherwavelet, as per Equation (5).(d) A normalization constant is determined for eachfilter so that a unit input wave provides a unit outputresponse in each case. Filter plots are optional.

(e) The response of each filter at each other scale isthen determined; this information may be used tocalculate R� but is not otherwise essential.(f) For convenience in further data analyses, wecalculate a lowpass (i.e. subtidal) filter. A low-lowpassfilter matched to the lowest-frequency subtidalwavelet may also be defined.

The application of the wavelet filter bank to the dataproceeds as follows:

(a) The mean of the data is determined and sub-tracted from the input. A somewhat more sophisti-cated approach is to subtract a suitably chosenlowpass. Whatever strategy is adopted here, the intentis to address the problem that short filters inevitablysuffer some leakage of low-frequency energy, becausethey do not (in discrete form) integrate exactly to zero.(b) The lowpass and wavelet filters are convolvedsuccessively with the data (with mean subtracted),resulting in a series of complex output vectors. Theorder of filter application is not important, becausethere is no subtraction of the output variance from theinput time series.(c) An amplitude vector is determined from eachcomplex output, as is the length of each output vector.(d) Phases are rendered ‘ absolute ’, relative to areference time: 00.00h 1/1/1900 (local time), withoutany reference to astronomical properties. This opera-tion requires two types of information: (a) the phaserelative to the time of each output is determined fromeach ‘ raw ’ complex output, and (b) the phase of eachoutput time is determined relative to the referencetime. The primary purpose of ‘ unwrapping ’ thephase is to eliminate the dependence of output phaseon filter central time. A collateral benefit is thatcomparisons of phases between time series (within thesame time zone) may also be made.

Plotting of results can occur in a variety of forms.Time series of amplitudes and phases at each scale isthe obvious first item. Amplitude and phase scaleo-grams are often useful. The details of this process aretoo data and language-dependent to merit furthercomment here.