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A Practical Guide to Wavelet Analysis

A Practical Guide to Wavelet Analysis

A Practical Guide to Analysis

Christopher Torrence and Gilbert P. Compo Program in Atmospheric and Oceanic Sciences, University of Colorado, Boulder, Colorado

ABSTRACT

A practical step-by-step guide to wavelet analysis is given, with examples taken from of the El Niño– Southern Oscillation (ENSO). The guide includes a comparison to the windowed , the choice of an appropriate wavelet , edge effects due to finite-length time series, and the relationship between wavelet scale and Fourier . New tests for wavelet power spectra are developed by deriving theo- retical wavelet spectra for white and red processes and using these to establish significance levels and confidence intervals. It is shown that smoothing in time or scale can be used to increase the confidence of the wavelet spectrum. Empirical formulas are given for the effect of smoothing on significance levels and confidence intervals. Extensions to wavelet analysis such as filtering, the power Hovmöller, cross-wavelet spectra, and are described. The statistical significance tests are used to give a quantitative measure of changes in ENSO on interdecadal timescales. Using new datasets that extend back to 1871, the Niño3 sea surface temperature and the Southern Oscilla- tion index show significantly higher power during 1880–1920 and 1960–90, and lower power during 1920–60, as well as a possible 15-yr modulation of variance. The power Hovmöller of sea level pressure shows significant variations in 2–8-yr wavelet power in both longitude and time.

1. Introduction complete description of geophysical applications can be found in Foufoula-Georgiou and Kumar (1995), Wavelet analysis is becoming a common tool for while a theoretical treatment of wavelet analysis is analyzing localized variations of power within a time given in Daubechies (1992). series. By decomposing a time series into time–fre- Unfortunately, many studies using wavelet analy- quency space, one is able to determine both the domi- sis have suffered from an apparent lack of quantita- nant modes of variability and how those modes vary tive results. The has been regarded in time. The wavelet transform has been used for nu- by many as an interesting diversion that produces col- merous studies in geophysics, including tropical con- orful pictures, yet purely qualitative results. This mis- vection (Weng and Lau 1994), the El Niño–Southern conception is in some sense the fault of wavelet analy- Oscillation (ENSO; Gu and Philander 1995; Wang and sis itself, as it involves a transform from a one-dimen- Wang 1996), atmospheric cold fronts (Gamage and sional time series (or frequency spectrum) to a diffuse Blumen 1993), central England temperature (Baliunas two-dimensional time–frequency image. This diffuse- et al. 1997), the dispersion of ocean (Meyers et ness has been exacerbated by the use of arbitrary nor- al. 1993), growth and breaking (Liu 1994), and malizations and the lack of statistical significance tests. coherent structures in turbulent flows (Farge 1992). A In Lau and Weng (1995), an excellent introduction to wavelet analysis is provided. Their paper, however, did not provide all of the essential details necessary Corresponding author address: Dr. Christopher Torrence, Ad- for wavelet analysis and avoided the issue of statisti- vanced Study Program, National Center for Atmospheric Re- cal significance. search, P.O. Box 3000, Boulder, CO 80307-3000. E-mail: [email protected] The purpose of this paper is to provide an easy-to- In final form 20 October 1997. use wavelet analysis toolkit, including statistical sig- ©1998 American Meteorological Society nificance testing. The consistent use of examples of

Bulletin of the American Meteorological Society 61 ENSO provides a substantive addition to a. NINO3 SST 3 the ENSO literature. In particular, the 2 1

statistical significance testing allows C) o greater confidence in the previous wave- ( 0 -1 let-based ENSO results of Wang and -2 Wang (1996). The use of new datasets 1880 1900 1920 1940 1960 1980 2000 with longer time series permits a more b. Morlet 0.5 robust classification of interdecadal 1 1.0 changes in ENSO variance. 2 2.0

The first section describes the datasets 4 4.0 used for the examples. Section 3 de- 8 8.0 Scale (years) scribes the method of wavelet analysis Period (years) 16 16.0 using discrete notation. This includes a 32 32.0 discussion of the inherent limitations of 64 64.0 1880 1900 1920 1940 1960 1980 2000 the windowed Fourier transform (WFT), Time (year) the definition of the wavelet transform, c. DOG the choice of a wavelet basis function, 0.125 1 0.250 edge effects due to finite-length time se-

2 0.500 ) ries, the relationship between wavelet

4 1.000 ears scale and Fourier period, and time series (y 8 2.000 Scale reconstruction. Section 4 presents the Period (years) 16 4.000 theoretical wavelet spectra for both 32 8.000 white-noise and red-noise processes. 16.000 1880 1900 1920 1940 1960 1980 2000 These theoretical spectra are compared to Time (year) Monte Carlo results and are used to es- FIG. 1. (a) The Niño3 SST time series used for the wavelet analysis. (b) The tablish significance levels and confi- local wavelet power spectrum of (a) using the Morlet wavelet, normalized by 1/ dence intervals for the wavelet power σ2 (σ2 = 0.54°C2). The left axis is the Fourier period (in yr) corresponding to the spectrum. Section 5 describes time or wavelet scale on the right axis. The bottom axis is time (yr). The shaded contours scale averaging to increase significance are at normalized of 1, 2, 5, and 10. The thick contour encloses regions levels and confidence intervals. Section of greater than 95% confidence for a red-noise process with a lag-1 coefficient of 6 describes other wavelet applications 0.72. Cross-hatched regions on either end indicate the “cone of influence,” where edge effects become important. (c) Same as (b) but using the real-valued Mexican such as filtering, the power Hovmöller, hat wavelet (derivative of a Gaussian; DOG m = 2). The shaded contour is at cross-wavelet spectra, and wavelet co- normalized variance of 2.0. herence. The summary contains a step- by-step guide to wavelet analysis. entire record have been removed to define an anomaly time series. The Niño3 SST is shown in the top plot 2. of Fig. 1a. Gridded sea level pressure (SLP) data is from the Several time series will be used for examples of UKMO/CSIRO historical GMSLP2.1f (courtesy of D. wavelet analysis. These include the Niño3 sea surface Parker and T. Basnett, Hadley Centre for Climate Pre- temperature (SST) used as a measure of the amplitude diction and Research, UKMO). The data is on a 5° of the El Niño–Southern Oscillation (ENSO). The global grid, with monthly resolution from January Niño3 SST index is defined as the seasonal SST av- 1871 to December 1994. Anomaly time series have eraged over the central Pacific (5°S–5°N, 90°– been constructed by removing the first three harmon- 150°W). Data for 1871–1996 are from an area aver- ics of the annual cycle (periods of 365.25, 182.625, and age of the U.K. Meteorological Office GISST2.3 121.75 days) using a least-squares fit. (Rayner et al. 1996), while data for January–June 1997 The Southern Oscillation index is derived from the are from the Climate Center (CPC) opti- GMSLP2.1f and is defined as the seasonally averaged mally interpolated Niño3 SST index (courtesy of D. pressure difference between the eastern Pacific (20°S, Garrett at CPC, NOAA). The seasonal for the 150°W) and the western Pacific (10°S, 130°E).

62 Vol. 79, No. 1, January 1998 3. Wavelet analysis a. Windowed Fourier transform The WFT represents one analysis tool for extract- This section describes the method of wavelet analy- ing local-frequency information from a . The sis, includes a discussion of different wavelet func- Fourier transform is performed on a sliding segment tions, and gives details for the analysis of the wavelet of length T from a time series of time step δt and total power spectrum. Results in this section are adapted to length Nδt, thus returning from T−1 to discrete notation from the continuous formulas given (2δt)−1 at each time step. The segments can be win- in Daubechies (1990). Practical details in applying dowed with an arbitrary function such as a boxcar (no wavelet analysis are taken from Farge (1992), Weng smoothing) or a Gaussian window (Kaiser 1994). and Lau (1994), and Meyers et al. (1993). Each sec- As discussed by Kaiser (1994), the WFT represents tion is illustrated with examples using the Niño3 SST. an inaccurate and inefficient method of time–fre- quency localization, as it imposes a scale or “response ” T into the analysis. The inaccuracy arises ψ ψ ^ ω from the aliasing of high- and low-frequency compo- (t / s) (s ) nents that do not fall within the frequency of the window. The inefficiency comes from the T/(2δt) fre- a. Morlet quencies, which must be analyzed at each time step, 0.3 6 regardless of the window size or the dominant frequen- 4 cies present. In addition, several window lengths must 0.0 2 usually be analyzed to determine the most appropri- -0.3 0 ate choice. For analyses where a predetermined scal- -4 -2 0 2 4 -2 -1 0 1 2 ing may not be appropriate because of a wide range of dominant frequencies, a method of time–frequency b. Paul (m=4) localization that is scale independent, such as wave- 6 0.3 let analysis, should be employed. 4 0.0 2 b. Wavelet transform The wavelet transform can be used to analyze time -0.3 0 series that contain nonstationary power at many dif- -4 -2 0 2 4 -2 -1 0 1 2 ferent frequencies (Daubechies 1990). Assume that c. DOG (m=2) δ one has a time series, xn, with equal time spacing t 0.3 6 and n = 0 … N − 1. Also assume that one has a wave- 4 ψ η let function, 0( ), that depends on a nondimensional 0.0 “time” parameter η 2 . To be “admissible” as a wavelet, this function must have zero and be localized in -0.3 0 both time and frequency space (Farge 1992). An ex- -4 -2 0 2 4 -2 -1 0 1 2 ample is the Morlet wavelet, consisting of a plane d. DOG (m=6) wave modulated by a Gaussian: 0.3 6

4 −−14iωη η2 2 ψ (ηπ) = ee0 , (1) 0.0 0 2 -0.3 0 ω where 0 is the nondimensional frequency, here taken -4 -2 0 2 4 -2 -1 0 1 2 to be 6 to satisfy the admissibility condition (Farge t / s s ω / (2π) 1992). This wavelet is shown in Fig. 2a. The term “wavelet function” is used generically to FIG. 2. Four different wavelet bases, from Table 1. The plots refer to either orthogonal or nonorthogonal . on the left give the real part (solid) and imaginary part (dashed) The term “wavelet basis” refers only to an orthogo- for the wavelets in the . The plots on the right give the corresponding wavelets in the . For plotting nal set of functions. The use of an orthogonal basis purposes, the scale was chosen to be s = 10δt. (a) Morlet, (b) Paul implies the use of the discrete wavelet transform, (m = 4), (c) Mexican hat (DOG m = 2), and (d) DOG (m = 6). while a nonorthogonal wavelet function can be used

Bulletin of the American Meteorological Society 63 with either the discrete or the continuous wavelet ⎧ 2πk N transform (Farge 1992). In this paper, only the con- ⎪ : k ≤ ⎪ Ntδ 2 tinuous transform is used, although all of the results ω = ⎨ k 2πk N . (5) for significance testing, smoothing in time and scale, ⎪−>: k and cross wavelets are applicable to the discrete wave- ⎩⎪ Ntδ 2 let transform. The continuous wavelet transform of a discrete se- Using (4) and a standard Fourier transform routine, one quence x is defined as the of x with a n n can calculate the continuous wavelet transform (for a scaled and translated version of ψ (η): 0 given s) at all n simultaneously and efficiently.

N−1 ⎡( ′ − )δ ⎤ c. Normalization ( ) =∗ψ nnt Wsn ∑ xn′ ⎢ ⎥ , (2) To ensure that the wavelet transforms (4) at each ⎣ s ⎦ n′=0 scale s are directly comparable to each other and to the transforms of other time series, the wavelet function where the (*) indicates the complex conjugate. By at each scale s is normalized to have unit energy: varying the wavelet scale s and translating along the localized time index n, one can construct a picture 12 ⎛ 2πs⎞ showing both the amplitude of any features versus the ψˆˆ(sω ) = ⎜ ⎟ ψω(s ) kk⎝ δ ⎠ 0 . (6) scale and how this amplitude varies with time. The t subscript 0 on ψ has been dropped to indicate that this ψ has also been normalized (see next section). Al- Examples of different wavelet functions are given in though it is possible to calculate the wavelet transform Table 1 and illustrated in Fig. 2. Each of the unscaled ψ$ using (2), it is considerably faster to do the calcula- 0 are defined in Table 1 to have tions in Fourier space. +∞ To approximate the continuous wavelet transform, 2 ψωˆ ( ′) d ω′ = 1; the convolution (2) should be done N times for each ∫−∞ 0 scale, where N is the number of points in the time se- ries (Kaiser 1994). (The choice of doing all N convo- that is, they have been normalized to have unit energy. lutions is arbitrary, and one could choose a smaller Using these normalizations, at each scale s one has number, say by skipping every other point in n.) By choosing N points, the convolution theorem allows us N−1 do all N simultaneously in Fourier space ψωˆ ( ) 2 = ∑ sNk , (7) using a discrete Fourier transform (DFT). The DFT k=0 of xn is where N is the number of points. Thus, the wavelet N−1 1 − π transform is weighted only by the amplitude of the ˆ = 2 ikn N xk ∑ xen , (3) Fourier coefficients x$ and not by the wavelet function. N = k n 0 If one is using the convolution formula (2), the nor- malization is where k = 0 … N − 1 is the frequency index. In the continuous limit, the Fourier transform of a function ⎡( ′ − )δ ⎤ ⎛ δ ⎞ 12 ⎡( ′ − )δ ⎤ ψ(t/s) is given by ψ$ (sω). By the convolution theorem, ψ nnt= t ψ nnt ⎢ ⎥ ⎜ ⎟ 0 ⎢ ⎥, (8) the wavelet transform is the inverse Fourier transform ⎣ s ⎦ ⎝ s ⎠ ⎣ s ⎦ of the product: ψ η where 0( ) is normalized to have unit energy. N−1 ωδ ( ) =∗ψωˆ ( ) intk Wsn ∑ xˆkk s e , (4) d. Wavelet power spectrum k=0 Because the wavelet function ψ(η) is in general

complex, the wavelet transform Wn(s) is also complex. where the angular frequency is defined as The transform can then be divided into the real part,

64 Vol. 79, No. 1, January 1998 $ ABLE ψ ψ T 1. Three wavelet basis functions and their properties. Constant factors for 0 and 0 ensure a total energy of unity.

e-folding Fourier ψ η ψ∧ ω τ λ Name 0( ) 0(s ) time s wavelength

4πs 2 −−ωη η2 −14 −−()sωω 2 14i 0 2 πω 0 2 Morlet π ee He( ) 2s ωω++2 ω 00 ( 0 = frequency)

mm m 2 im! ( − η)−+()m 1 2 (ωω)( )m −sω 4πs Paul 1 i Hse s 2 π(2m)! mm(21− )! 21m + (m = order)

m+1 m m − 2 (−1) d −η2 i m −()sω 2 2πs (e 2 ) (seω) dηm ⎛ ⎞ DOG Γ⎛ + 1⎞ Γ+1 + 1 ⎝m ⎠ ⎝m ⎠ 2s m (m = derivative) 2 2 2

H(ω) = Heaviside step function, H(ω) = 1 if ω > 0, H(ω) = 0 otherwise. DOG = derivative of a Gaussian; m = 2 is the Marr or Mexican hat wavelet.

ℜ ℑ {Wn(s)}, and imaginary part, {Wn(s)}, or ampli- ies (Trenberth 1976) and is also seen in the Fourier | | −1 ℑ ℜ tude, Wn(s) , and phase, tan [ {Wn(s)}/ {Wn(s)}]. Fi- spectrum in Fig. 3. With wavelet analysis, one can see nally, one can define the wavelet power spectrum as variations in the frequency of occurrence and ampli- | |2 Wn(s) . For real-valued wavelet functions such as the tude of El Niño (warm) and La Niña (cold) events. DOGs (derivatives of a Gaussian) the imaginary part During 1875–1920 and 1960–90 there were many is zero and the phase is undefined. warm and cold events of large amplitude, while dur- To make it easier to compare different wavelet ing 1920–60 there were few events (Torrence and power spectra, it is desirable to find a common nor- Webster 1997). From 1875–1910, there was a slight malization for the wavelet spectrum. Using the nor- shift from a period near 4 yr to a period closer to 2 yr, malization in (6), and referring to (4), the expectation while from 1960–90 the shift is from shorter to longer | |2 value for Wn(s) is equal to N times the expectation periods. |$ |2 value for xk . For a white-noise time series, this ex- These results are similar to those of Wang and pectation value is σ2/N, where σ2 is the variance. Thus, Wang (1996), who used both wavelet and waveform for a white-noise process, the expectation value for the analysis on ENSO indices derived from the Compre- | |2 σ2 wavelet transform is Wn(s) = at all n and s. hensive Ocean–Atmosphere Data Set (COADS) Figure 1b shows the normalized wavelet power dataset. Wang and Wang’s analysis showed reduced | |2/σ2 spectrum, Wn(s) , for the Niño3 SST time series. wavelet power before 1950, especially 1875–1920. The The normalization by 1/σ2 gives a measure of the reduced power is possibly due to the sparseness and de- power relative to . In Fig. 1b, most of the creased reliability of the pre-1950 COADS data (Folland power is concentrated within the ENSO band of 2–8 et al. 1984). With the GISST2.3 data, the wavelet trans- yr, although there is appreciable power at longer peri- form of Niño3 SST in Fig. 1b shows that the pre-1920 ods. The 2–8-yr band for ENSO agrees with other stud- period has equal power to the post-1960 period.

Bulletin of the American Meteorological Society 65 and is better adapted for capturing oscillatory be- 14 havior. A real wavelet function returns only a single component and can be used to isolate peaks 12 or discontinuities. 3) Width. For concreteness, the width of a wavelet ) 10 2

σ function is defined here as the e-folding time of the 8 wavelet amplitude. The resolution of a wavelet function is determined by the balance between the 6 width in real space and the width in Fourier space. Variance ( A narrow (in time) function will have good time 4 resolution but poor frequency resolution, while a 2 broad function will have poor time resolution, yet good frequency resolution. 0 4) Shape. The wavelet function should reflect the type 64 32 16 8 4 2 1 0.5 of features present in the time series. For time se- Period (years) ries with sharp jumps or steps, one would choose a boxcar-like function such as the Harr, while for smoothly varying time series one would choose a FIG. 3. Fourier power spectrum of Niño3 SST (solid), smooth function such as a damped cosine. If one / σ2 normalized by N (2 ). The lower dashed line is the mean red- is primarily interested in wavelet power spectra, noise spectrum from (16) assuming a lag-1 of α = 0.72. The upper dashed line is the 95% confidence spectrum. then the choice of wavelet function is not critical, and one function will give the same qualitative results as another (see discussion of Fig. 1 below). e. Wavelet functions Four common nonorthogonal wavelet functions are One criticism of wavelet analysis is the arbitrary given in Table 1. The Morlet and Paul wavelets are ψ η choice of the wavelet function, 0( ). (It should be both complex, while the DOGs are real valued. Pic- noted that the same arbitrary choice is made in using tures of these wavelet in both the time and frequency one of the more traditional transforms such as the Fou- domain are shown in Fig. 2. Many other types of wave- rier, Bessel, Legendre, etc.) In choosing the wavelet lets exist, such as the Haar and Daubechies, most of function, there are several factors which should be which are used for orthogonal wavelet analysis (e.g., considered (for more discussion see Farge 1992). Weng and Lau 1994; Mak 1995; Lindsay et al. 1996). For more examples of wavelet bases and functions, see 1) Orthogonal or nonorthogonal. In orthogonal Kaiser (1994). wavelet analysis, the number of convolutions at For comparison, Fig. 1c shows the same analysis each scale is proportional to the width of the wave- as in 1b but using the Mexican hat wavelet (DOG, let basis at that scale. This produces a wavelet spec- m = 2) rather than the Morlet. The most noticeable dif- trum that contains discrete “blocks” of wavelet ference is the fine scale structure using the Mexican power and is useful for as it gives hat. This is because the Mexican hat is real valued and the most compact representation of the signal. Un- captures both the positive and negative oscillations of fortunately for time series analysis, an aperiodic the time series as separate peaks in wavelet power. The shift in the time series produces a different wave- Morlet wavelet is both complex and contains more let spectrum. Conversely, a nonorthogonal analy- oscillations than the Mexican hat, and hence the wave- sis (such as used in this study) is highly redundant let power combines both positive and negative peaks at large scales, where the wavelet spectrum at ad- into a single broad peak. A plot of the real or imagi-

jacent times is highly correlated. The nonorthog- nary part of Wn(s) using the Morlet would produce a onal transform is useful for time series analysis, plot similar to Fig. 1c. Overall, the same features ap- where smooth, continuous variations in wavelet pear in both plots, approximately at the same locations, amplitude are expected. and with the same power. Comparing Figs. 2a and 2c, 2) Complex or real. A complex wavelet function will the Mexican hat is narrower in time-space, yet broader return information about both amplitude and phase in spectral-space than the Morlet. Thus, in Fig. 1c, the

66 Vol. 79, No. 1, January 1998 peaks appear very sharp in the time direction, yet are more elongated in the scale direction. Finally, the re- TABLE 2. Empirically derived factors for four wavelet bases. lationship between wavelet scale and Fourier period γδψ is very different for the two functions (see section 3h). Name Cδ j0 0(0)

ω π − 1/4 f. Choice of scales Morlet ( 0 = 6) 0.776 2.32 0.60 Once a wavelet function is chosen, it is necessary Paul (m = 4) 1.132 1.17 1.5 1.079 to choose a set of scales s to use in the wavelet trans- form (4). For an orthogonal wavelet, one is limited to Marr (DOG m = 2) 3.541 1.43 1.4 0.867 a discrete set of scales as given by Farge (1992). For nonorthogonal wavelet analysis, one can use an arbi- DOG (m = 6) 1.966 1.37 0.97 0.884 trary set of scales to build up a more complete picture.

It is convenient to write the scales as fractional pow- Cδ = reconstruction factor. ers of two: γ = decorrelation factor for time averaging. δ j0 = factor for scale averaging. ==jjδ K ssj 0 201,,,, j J (9) region of the wavelet spectrum in which edge effects = δδ−1 ( ) Jjlog20 Nts, (10) become important and is defined here as the e-fold- ing time for the of wavelet power at where s0 is the smallest resolvable scale and J deter- each scale (see Table 1). This e-folding time is cho- mines the largest scale. The s0 should be chosen so that sen so that the wavelet power for a discontinuity at the the equivalent Fourier period (see section 3h) is ap- edge drops by a factor e−2 and ensures that the edge proximately 2δt. The choice of a sufficiently small δj effects are negligible beyond this point. For cyclic depends on the width in spectral-space of the wavelet series (such as a longitudinal strip at a fixed latitude), function. For the Morlet wavelet, a δj of about 0.5 is there is no need to pad with zeroes, and there is no COI. the largest value that still gives adequate in The size of the COI at each scale also gives a mea- scale, while for the other wavelet functions, a larger sure of the decorrelation time for a single spike in the value can be used. Smaller values of δj give finer reso- time series. By comparing the width of a peak in the lution. wavelet power spectrum with this decorrelation time, δ / δ δ In Fig. 1b, N = 506, t = 1 4 yr, s0 = 2 t, j = 0.125, one can distinguish between a spike in the data (pos- and J = 56, giving a total of 57 scales ranging from sibly due to random noise) and a component 0.5 yr up to 64 yr. This value of δj appears adequate at the equivalent Fourier frequency. to provide a smooth picture of wavelet power. The COI is indicated in Figs. 1b and 1c by the cross- hatched regions. The peaks within these regions have g. Cone of influence presumably been reduced in magnitude due to the zero Because one is dealing with finite-length time se- padding. Thus, it is unclear whether the decrease in 2– ries, errors will occur at the beginning and end of the 8-yr power after 1990 is a true decrease in variance or wavelet power spectrum, as the Fourier transform in an artifact of the padding. Note that the much narrower (4) assumes the data is cyclic. One solution is to pad Mexican hat wavelet in Fig. 1c has a much smaller the end of the time series with zeroes before doing the COI and is thus less affected by edge effects. wavelet transform and then remove them afterward [for other possibilities such as cosine damping, see h. Wavelet scale and Fourier frequency Meyers et al. (1993)]. In this study, the time series is An examination of the wavelets in Fig. 2 shows that padded with sufficient zeroes to bring the total length the peak in ψ$ (sω) does not necessarily occur at a fre- N up to the next-higher power of two, thus limiting quency of s−1. Following the method of Meyers et al. the edge effects and speeding up the Fourier transform. (1993), the relationship between the equivalent Fou- Padding with zeroes introduces discontinuities at rier period and the wavelet scale can be derived ana- the endpoints and, as one goes to larger scales, de- lytically for a particular wavelet function by substitut- creases the amplitude near the edges as more zeroes ing a cosine wave of a known frequency into (4) and enter the analysis. The cone of influence (COI) is the computing the scale s at which the wavelet power spec-

Bulletin of the American Meteorological Society 67 trum reaches its maximum. For the Morlet wavelet − 1 N 1 ω λ λ ( ) =∗ψωˆ ( ) with 0 = 6, this gives a value of = 1.03s, where is Wsδ ∑ s k . (12) the Fourier period, indicating that for the Morlet wave- N k=0 let the wavelet scale is almost equal to the Fourier period. Formulas for other wavelet functions are given The reconstruction (11) then gives in Table 1, while Fig. 2 gives a graphical representation. In Figs. 1b,c, the ratio of Fourier period to wavelet 12 J ℜ δδjt {}Wsδ ( j ) scale can be seen by a comparison of the left and right C = ∑ δ ψ ( ) 12 . (13) axes. For the Morlet, the two are nearly identical, while 0 0 j=0 s j for the Mexican hat, the Fourier period is four times larger than the scale. This ratio has no special signifi- The C is scale independent and is a constant for each cance and is due solely to the functional form of each δ wavelet function. wavelet function. However, one should certainly con- The total energy is conserved under the wavelet vert from scale to Fourier period before plotting, as transform, and the equivalent of Parseval’s theorem presumably one is interested in equating wavelet for wavelet analysis is power at a certain time and scale with a (possibly short- lived) Fourier at the equivalent Fourier period. 2 N−1 J δδjt Wsnj( ) i. Reconstruction σ 2 = ∑∑ , (14) CN s Since the wavelet transform is a bandpass filter with δ n=0 j=0 j a known response function (the wavelet function), it is possible to reconstruct the original time series us- where σ2 is the variance and a δ function has been as- ing either or the inverse filter. This is sumed for reconstruction. Both (11) and (14) should straightforward for the orthogonal wavelet transform be used to check wavelet routines for accuracy and to δ (which has an orthogonal basis), but for the continu- ensure that sufficiently small values of s0 and j have ous wavelet transform it is complicated by the redun- been chosen. dancy in time and scale. However, this redundancy For the Niño3 SST, the reconstruction of the time also makes it possible to reconstruct the time series series from the wavelet transform has a mean square using a completely different wavelet function, the easi- error of 1.4% or 0.087°C. est of which is a delta (δ) function (Farge 1992). In this case, the reconstructed time series is just the sum of the real part of the wavelet transform over all scales: 4. Theoretical spectrum and significance levels

12 J ℜ ( ) δδjt {}Wsnj To determine significance levels for either Fourier x = ∑ n ψ ( ) 12 . (11) or wavelet spectra, one first needs to choose an appro- Cδ 0 0 j=0 s j priate background spectrum. It is then assumed that different realizations of the geophysical process will ψ The factor 0(0) removes the energy scaling, while the be randomly distributed about this mean or expected 1/2 sj converts the wavelet transform to an energy den- background, and the actual spectrum can be compared sity. The factor Cδ comes from the reconstruction of a against this random distribution. For many geophysi- δ function from its wavelet transform using the func- cal phenomena, an appropriate background spectrum ψ η tion 0( ). This Cδ is a constant for each wavelet func- is either white noise (with a flat Fourier spectrum) or tion and is given in Table 2. Note that if the original red noise (increasing power with decreasing frequency). time series were complex, then the sum of the com- A previous study by Qiu and Er (1995) derived the plex Wn(s) would be used instead. mean and variance of the local wavelet power spec- To derive Cδ for a new wavelet function, first as- trum. In this section, the theoretical white- and red- sume a time series with a δ function at time n = 0, given noise wavelet power spectra are derived and compared δ by xn = n0. This time series has a Fourier transform to Monte Carlo results. These spectra are used to es- $ −1 $ xk = N , constant for all k. Substituting xk into (4), at tablish a null hypothesis for the significance of a peak time n = 0 (the peak), the wavelet transform becomes in the wavelet power spectrum.

68 Vol. 79, No. 1, January 1998 a. α=0.00 a. Fourier red noise spectrum 1 Many geophysical time series can be modeled as t) either white noise or red noise. A simple model for red δ 10 noise is the univariate lag-1 autoregressive [AR(1), or 95% Period ( Markov] process: 100 95%

0 100 200 300 400 500 =+α Time (δt) xxznnn−1 , (15) b. α=0.70 1 α where is the assumed lag-1 autocorrelation, x0 = 0, t) δ 10 and zn is taken from Gaussian white noise. Following Gilman et al. (1963), the discrete Fourier power spec- Period ( 100 trum of (15), after normalizing, is 0 100 200 300 400 500 Time (δt) 1− α 2 = FIG. 4. (a) The local wavelet power spectrum for a Gaussian Pk 2 , (16) 122+−ααcos( πkN) white noise process of 512 points, one of the 100 000 used for the Monte Carlo . The power is normalized by 1/σ2, and where k = 0 … N/2 is the frequency index. Thus, by contours are at 1, 2, and 3. The thick contour is the 95% confidence level for white noise. (b) Same as (a) but for a red-noise AR(1) choosing an appropriate lag-1 autocorrelation, one can process with lag-1 of 0.70. The contours are at 1, 5, and 10. The use (16) to model a red-noise spectrum. Note that α = 0 thick contour is the 95% confidence level for the corresponding in (16) gives a white-noise spectrum. red-noise spectrum. The Fourier power spectrum for the Niño3 SST is shown by the thin line in Fig. 3. The spectrum has been normalized by N/2σ2, where N is the number of points, Therefore, the lower dashed curve in Fig. 3 also and σ2 is the variance of the time series. Using this corresponds to the red-noise local wavelet spectrum. normalization, white noise would have an expectation A random vertical slice in Fig. 1b would be of 1 at all frequencies. The red-noise background to have a spectrum given by (16). As will be shown in spectrum for α = 0.72 is shown by the lower dashed section 5a, the average of all the local wavelet spectra curve in Fig. 3. This red-noise was estimated from (α tends to approach the (smoothed) Fourier spectrum of ⎯ 1 α / α α the time series. + √ 2) 2, where 1 and 2 are the lag-1 and lag-2 of the Niño3 SST. One can see the broad set of ENSO peaks between 2 and 8 yr, well c. Significance levels above the background spectrum. The null hypothesis is defined for the wavelet power spectrum as follows: It is assumed that the time series b. Wavelet red noise spectrum has a mean power spectrum, possibly given by (16); The wavelet transform in (4) is a series of bandpass if a peak in the wavelet power spectrum is significantly filters of the time series. If this time series can be above this background spectrum, then it can be as- modeled as a lag-1 AR process, then it seems reason- sumed to be a true feature with a certain percent con- able that the local wavelet power spectrum, defined fidence. For definitions, “significant at the 5% level” as a vertical slice through Fig. 1b, is given by (16). To is equivalent to “the 95% confidence level,” and im- test this hypothesis, 100 000 Gaussian white-noise plies a test against a certain background level, while time series and 100 000 AR(1) time series were con- the “95% ” refers to the range of structed, along with their corresponding wavelet power confidence about a given value. spectra. Examples of these white- and red-noise wave- The normalized Fourier power spectrum in Fig. 3 |$ |2/2σ2 let spectra are shown in Fig. 4. The local wavelet spec- is given by N xk , where N is the number of points, $ σ 2 tra were constructed by taking vertical slices at time xk is from (3), and is the variance of the time series. n = 256. The lower smooth curves in Figs. 5a and 5b If xn is a normally distributed , then $ show the theoretical spectra from (16). The dots show both the real and imaginary parts of xk are normally the results from the Monte Carlo simulation. On av- distributed (Chatfield 1989). Since the square of a erage, the local wavelet power spectrum is identical normally distributed variable is chi-square distributed |$ |2 to the Fourier power spectrum given by (16). with one degree of freedom (DOF), then xk is chi-

Bulletin of the American Meteorological Society 69 χ2 square distributed with two DOFs, denoted by 2 a. α=0.00 (Jenkins and Watts 1968). To determine the 95% con- 4 fidence level (significant at 5%), one multiplies the background spectrum (16) by the 95th value 95% χ2 for 2 (Gilman et al. 1963). The 95% Fourier confi- 3

dence spectrum for the Niño3 SST is the upper dashed ) 2 curve in Fig. 3. Note that only a few frequencies now σ have power above the 95% line. 2 In the previous section, it was shown that the local wavelet spectrum follows the mean Fourier spectrum. If the original Fourier components are normally dis- Variance ( Mean tributed, then the wavelet coefficients (the bandpassed 1 inverse Fourier components) should also be normally distributed. If this is true, then the wavelet power spec- | |2 χ2 trum, Wn(s) , should be 2 distributed. The upper 0 curves in Figs. 5a and 5b show the 95% Fourier red- 1000 100 10 1 noise confidence level versus the 95% level from the Period (δt) Monte Carlo results of the previous section. Thus, at α each point (n, s) in Fig. 1b, assuming a red-noise pro- b. =0.70 χ2 20 cess, the distribution is 2. Note that for a wavelet transform using a real-valued function, such as the 95% Mexican hat shown in Fig. 1c, there is only one de- gree of freedom at each point, and the distribution is 15

2 )

χ 2 1. In summary, assuming a mean background spec- σ trum, possibly red noise [(16)], the distribution for the 10 Fourier power spectrum is

Variance ( Mean 2 Nxˆ 1 5 k ⇒ P χ 2 (17) \\\2σ 2 2 k 2 at each frequency index k, and “⇒” indicates “is dis- 0 tributed as.” The corresponding distribution for the 1000 100 10 1 δ local wavelet power spectrum is Period ( t) FIG. 5. (a) Monte Carlo results for local wavelet spectra of white noise (α = 0.0). The lower thin line is the theoretical mean white- 2 Ws( ) 1 noise spectrum, while the black dots are the mean at each scale n ⇒ P χ 2 (18) σ 2 2 k 2 of 100 000 local wavelet spectra. The local wavelet spectra were slices taken at time n = 256 out of N = 512 points. The top thin χ2 line is the 95% confidence level, equal to 2 (95%) times the mean at each time n and scale s. The 1/2 removes the DOF spectrum. The black dots are the 95% level from the Monte Carlo α factor from the χ2 distribution. (For a real wavelet the runs. (b) Same as (a) but for red noise of = 0.70. χ2 distribution on the right-hand side would be Pk 1.) The value of Pk in (18) is the mean spectrum at the Fourier frequency k that corresponds to the wavelet scale s (see As with , smoothing the wavelet section 3h). Aside from the relation between k and s, power spectrum can be used to increase the DOF and (18) is independent of the wavelet function. After find- enhance confidence in regions of significant power. ing an appropriate background spectrum and choos- Unlike Fourier, smoothing can be performed in either ing a particular confidence for χ2 such as 95%, one can the time or scale domain. Significance levels and then calculate (18) at each scale and construct 95% DOFs for smoothing in time or scale are discussed in confidence contour lines. section 5.

70 Vol. 79, No. 1, January 1998 Inside the COI, the distribution is still χ2, but if the (20) one can then find confidence intervals for the time series has been padded with zeroes, then the mean peaks in a wavelet power spectrum to compare against − /τ spectrum is reduced by a factor of (1 − ½e 2t s), where either the mean background or against other peaks. τ s is from Table 1, and t is the distance (in time) from either the beginning or end of the wavelet power spec- e. Stationarity trum. It has been argued that wavelet analysis requires the The 95% confidence level for the Niño3 SST is use of nonstationary significance tests (Lau and Weng shown by the thick contours on Figs. 1b and 1c. Dur- 1995). In defense of the use of stationary tests such as ing 1875–1910 and 1960–90, the variance in the 2–8- those given above, the following points are noted. yr band is significantly above the 95% confidence for red noise. During 1920–60, there are a few isolated 1) A nonarbitrary test is needed. The assumption of significant regions, primarily around 2 yr, and at the stationary statistics provides a standard by which edge of the usual 2–8 yr ENSO band. The 95% confi- any nonstationarity can be detected. dence implies that 5% of the wavelet power should be 2) The test should be robust. It should not depend above this level. In Fig. 4b, approximately 5% of the upon the wavelet function or upon the actual dis- points are contained within the 95% contours. For the tribution of the time series, other than the assump- Niño3 wavelet spectrum, 4.9% of the points are above tion of a background spectrum. 95%, implying that for the Niño3 time series a test of 3) A non– is preferred. In addi- enclosed area cannot distinguish between noise and tion to the savings in computation, the chi-square signal. However, the spatial distribution of variance test simplifies comparing one wavelet transform can also be examined for . In Fig. 4b, the with another. variance shows a gradual increase with period, with 4) Many wavelet transforms of real data appear simi- random distributions of high and low variance about lar to transforms of red-noise processes (compare this mean spectrum. In Figs. 1b and 1c, the significant Figs. 1b and 4b). It is therefore difficult to argue regions are clustered together in both period and time, that large variations in wavelet power imply indicating less randomness of the underlying process. nonstationarity. 5) One needs to ask what is being tested. Is it d. Confidence interval nonstationarity? Or low-variance versus high-vari- The confidence interval is defined as the probabil- ance periods? Or changes in amplitude of Fourier ity that the true wavelet power at a certain time and modes? The chi-square test gives a standard mea- scale lies within a certain interval about the estimated sure for any of these possibilities. wavelet power. Rewriting (18) as In short, it appears wiser to assume stationarity and the statistical tests accordingly. If the tests show Ws( ) 2 χ 2 n ⇒ 2 large deviations, such as the changes in ENSO vari- σ 2 , (19) Pk 2 ance seen in Figs. 1b and 1c, then further tests can be devised for the particular time series. σ2 one can then replace the theoretical wavelet power Pk W2 with the true wavelet power, defined as n(s). The W2 5. Smoothing in time and scale confidence interval for n(s) is then a. Averaging in time (global wavelet spectrum) 2 2 2 2 Ws( ) ≤ W2( s) ≤ Ws( ) If a vertical slice through a wavelet plot is a mea- χχ2 (p 2) nn2 (12− p ) n, 2 2 sure of the local spectrum, then the time-averaged (20) wavelet spectrum over a certain period is

where p is the desired significance (p = 0.05 for the n2 1 2 95% confidence interval) and χ2(p/2) represents the 2 ( ) = ( ) 2 Wsn ∑ Wsn , (21) χ2 n = value of at p/2. Note that for real-valued wavelet a nn1 χ2 functions, the right-hand side of (19) becomes 1, and the factor of 2 is removed from the top of (20). Using where the new index n is arbitrarily assigned to the

Bulletin of the American Meteorological Society 71 sure of the background spectrum, against which peaks 14 in the local wavelet spectra could be tested (Kestin et al. 1998). 12 By smoothing the wavelet spectrum using (21), one can increase the degrees of freedom of each point and ) 10 2

σ increase the significance of peaks in wavelet power. 8 To determine the DOFs, one needs the number of in- dependent points. For the Fourier spectrum (Fig. 3), 6 the power at each frequency is independent of the oth-

Variance ( ers, and the average of the power at M frequencies, 4 each with two DOF, is χ2 distributed with 2M degrees of freedom (Spiegel 1975). For the time-averaged 2 wavelet spectrum, one is also averaging points that are χ2 0 2 distributed, yet Figs. 1b and 4 suggest that these points are no longer independent but are correlated in 64 32 16 8 4 2 1 0.5 Period (years) both time and scale. Furthermore, the correlation in time appears to lengthen as scale increases and the FIG. 6. Fourier power spectrum from Fig. 3, smoothed with a wavelet function broadens. Designating ν as the DOFs, five-point running average (thin solid line). The thick solid line is ν ∝ ν ∝ −1 the global wavelet spectrum for the Niño3 SST. The lower dashed one expects na and s . The simplest formula line is the mean red-noise spectrum, while the upper dashed line to consider is to define a decorrelation length τ = γs, is the 95% confidence level for the global wavelet spectrum, ν δ /τ such that = 2na t . However, Monte Carlo results assuming α = 0.72. τ show that this is too abrupt at small na or large scales; even though one is averaging points that are highly correlated, some additional information is gained. − + midpoint of n1 and n2, and na = n2 n1 1 is the num- The Monte Carlo results are given in Fig. 7, which ber of points averaged over. By repeating (21) at each shows the mean and 95% levels for various na. These time step, one creates a wavelet plot smoothed by a certain window. The extreme case of (21) is when the average is over 4 all the local wavelet spectra, which gives the global wavelet spectrum 95% 3

− ) 5

N 1 2 2 2 ( ) = 1 ( ) σ Ws ∑ Wsn . (22)

N n=0 15 2 25

In Fig. 6, the thick solid line shows the normalized 51

⎯ Variance ( 101 2 /σ2 201 global wavelet spectrum, W (s) , for the Niño3 SST. Mean 512 The thin solid line in Fig. 6 shows the same spectrum 1 as in Fig. 3, but smoothed with a five-point running average. Note that as the Fourier spectrum is smoothed, it approaches the global wavelet spectrum more and 0 more closely, with the amount of necessary smooth- 1000 100 10 1 ing decreasing with increasing scale. A comparison of Period (δt) Fourier spectra and wavelet spectra can be found in FIG. 7. Monte Carlo results for the time-averaged wavelet Hudgins et al. (1993), while a theoretical discussion spectra (21) of white noise using the Morlet wavelet. The to the right of each curve indicate n , the number of times that were is given in Perrier et al. (1995). Percival (1995) shows a that the global wavelet spectrum provides an unbiased averaged, while the black dots are the 95% level for the Monte Carlo runs. The top thin lines are the 95% confidence from (23). and consistent estimation of the true power spectrum The lower thin line is the mean white-noise spectrum, while the of a time series. Finally, it has been suggested that the black dots are the means of the Monte Carlo runs (all of the means global wavelet spectrum could provide a useful mea- are identical).

72 Vol. 79, No. 1, January 1998 χ2/ν curves are best described by the distribution Pk ν , Comparing (24) and (14), the scale-averaged wavelet where Pk is the original assumed background spectrum power is a time series of the average variance in a cer- χ2 ν and ν is the chi-square distribution with degrees of tain band. Thus, the scale-averaged wavelet power can freedom, where be used to examine modulation of one time series by another, or modulation of one frequency by another within the same time series. 2 ⎛ ntδ ⎞ As an example of averaging over scale, Fig. 8 shows ν =+21 ⎜ a ⎟ . (23) ⎝ γ s ⎠ the average of Fig. 1b over all scales between 2 and 8 yr (actually 2–7.34 yr), which gives a measure of the average ENSO variance versus time. The variance plot Note that for a real-valued function such as the Mexi- shows a distinct period between 1920 and 1960 when can hat, each point only has one DOF, and the factor ENSO variance was low. Also shown in Fig. 8 is the of 2 in (23) is removed. The decorrelation factor γ is variance in the Southern Oscillation index (SOI), determined empirically by an iterative fit of absolute which correlates well with the changes in Niño3 SST error to the 95% Monte Carlo level and is given in Table 2 for the four wavelet NINO3 0.8 SOI 2.5

functions. The relative error (or percent ) 2

0.6 )

2.0 2 C o difference) between the Monte Carlo and 1.5 χ2/ν 0.4 the ν distribution was everywhere less 1.0 SOI (mb

NINO3 ( 0.2 0.5 than 7% for all scales and na values. The 0.0 0.0 thin lines in Fig. 7 show the results of 1880 1900 1920 1940 1960 1980 2000 (23) using the Morlet wavelet. Note that Year even the white noise process has more FIG. 8. Scale-averaged wavelet power (24) over the 2–8-yr band for the Niño3 stringent 95% confidence levels at large SST (solid) and the SOI (dashed). The thin solid line is the 95% confidence level scales compared to small. As a final note, from (26) for Niño3 SST (assuming red noise α = 0.72), while the thin dashed α if the points going into the average are line is the 95% level for the SOI (red noise = 0.61). within the cone of influence, then na is re- duced by approximately one-half of the number within variance (0.72 correlation). Both time series show the COI to reflect the decreased amplitude (and infor- consistent interdecadal changes, including a possible mation) within that region. modulation in ENSO variance with a 15-yr period. To A different definition of the global wavelet spec- examine more closely the relation between Niño3 SST trum, involving the discrete wavelet transform and and the SOI, one could use the cross-wavelet spectrum including a discussion of confidence intervals, is given (see section 6c). by Percival (1995). An example using Percival’s defi- As with time-averaged wavelet spectrum, the DOFs nition can be found in Lindsay et al. (1996). are increased by smoothing in scale, and an analytical The 95% confidence line for the Niño3 global relationship for significance levels for the scale-aver- wavelet spectrum is the upper dashed line in Fig. 6. aged wavelet power is desirable. Again, it is conve- Only the broad ENSO peak remains significant, al- nient to normalize the wavelet power by the expecta- though note that power at other periods can be less than tion value for a white-noise time series. From (24), this δ δ σ2 σ2 significant globally but still show significant peaks in expectation value is ( j t )/(CδSavg), where is the local wavelet power. time-series variance and Savg is defined as b. Averaging in scale −1 ⎛ j2 To examine fluctuations in power over a range of 1 ⎞ S = scales (a band), one can define the scale-averaged avg ⎜ ∑ ⎟ . (25) ⎝ = s j ⎠ wavelet power as the weighted sum of the wavelet jj1 power spectrum over scales s1 to s2: The black dots in Fig. 9 show the Monte Carlo re- 2 δδ j2 Ws( ) sults for both the mean and the 95% level of scale- 2 = jt nj Wn ∑ . (24) averaged wavelet power as a function of various na, Cδ jj= s j − + 1 where na = j2 j1 1 is the number of scales averaged.

Bulletin of the American Meteorological Society 73 δ 0.5(j1+j2) j / 4 where Smid = s02 . The factor Savg Smid corrects for 95% the loss of DOF that arises from dividing the wavelet power spectrum by scale in (24) and is observed in the Monte Carlo results. Note that for a real-valued func- 3 DOG2 DOG6 tion such as the Mexican hat, each point only has one Paul4 DOF, and the factor of 2 in (28) is removed. The δ decorrelation distance j0 is determined empirically by 2 Morlet an iterative fit of absolute error between (28) and the 95% level of the Monte Carlo results and is given in Table 2. The thin lines in Fig. 9 show the results of Mean (28) for the Morlet, Paul (m = 4), DOG2, and DOG6

Normalized Variance 1 wavelet functions. For these wavelets, the relative er- χ2 ror between the ν distribution using (28) and the Monte Carlo results is less than 1.5%. It should be 0 noted that (28) is valid only for confidences of 95% 0 2 4 6 8 10 or less. At higher confidence levels, the distribution δ na j (# scales in avg, compressed) begins to deviate significantly from χ2, and (28) is no FIG. 9. Monte Carlo results for the wavelet spectra averaged longer valid. n over a scales from (24), using white noise. The average from (24) In Fig. 8, the thin solid and dashed lines show the δ is centered on scale s = 16 t for convenience, but the results are 95% confidence levels for the Niño3 SST and the SOI independent of the center scale. To make the graph independent δ of the choice for δj, the x axis has been compressed by the Monte using (25)–(28). In this case, j = 0.125, the sum was Carlo δj of 0.25. The top black dots are the 95% level for the between Fourier periods 2 and 8 yr (actually 2.1–7.6 δ Monte Carlo runs, while the lower black dots are the means. The yr), na = 16, Savg = 0.221 yr, Smid = 3.83 yr, j0 = 0.60, means for all four wavelet bases are all the same, while the 95% and ν = 6.44. Since the two time series do not have level depends on the width of the basis in Fourier space, with the the same variance or the same red-noise background, Morlet being the most narrow. The top thin lines are the 95% the 95% lines are not equal. confidence from (28). The lower thin line is the mean white-noise spectrum.

6. Extensions to wavelet analysis Using the normalization factor for white noise, the distribution can be modeled as a. Filtering As discussed in section 3i, the wavelet transform (4) is essentially a bandpass filter of uniform shape and CS χ 2 δ avg 2 ⇒ ν varying location and width. By summing over a sub- WPn , (26) δδσjt 2 ν set of the scales in (11), one can construct a wavelet- filtered time series: where the scale-averaged theoretical spectrum is now given by δδ 12 j2 ℜ{}Ws( ) ′ = jt nj xn ∑ 12 . (29) Cδψ (0) = s 0 jj1 j j2 P = j PSavg ∑ . (27) = s This filter has a response function given by the sum jj1 j of the wavelet functions between scales j1 and j2. This filtering can also be done on both the scale and Note that for white noise this spectrum is still unity time simultaneously by defining a threshold of wave- (due to the normalization). The degrees of freedom ν let power. This “denoising” removes any low-ampli- in (26) is modeled as tude regions of the wavelet transform, which are pre- sumably due to noise. This technique has the advan- 2 tage over traditional filtering in that it removes noise 2nS ⎛ njδ ⎞ ν =+a avg 1 ⎜ a ⎟ at all frequencies and can be used to isolate single ⎝ δ ⎠ , (28) Smid j0 events that have a broad power spectrum or multiple

74 Vol. 79, No. 1, January 1998 o o events that have varying frequency. A more complete (a) Power 2-8 yrs slp 5.0 S to 15.0 S (b) Zonal Avg description including examples is given in Donoho 1990 and Johnstone (1994). 1980

Another filtering technique involves the use of the 1970 two-dimensional wavelet transform. An example can be found in Farge et al. (1992), where two-dimensional 1960 turbulent flows are “compressed” using an orthonor- 1950 mal wavelet packet. This compression removes the 1940 low-amplitude “passive” components of the flow, while retaining the high-amplitude “dynamically ac- 1930 tive” components. 1920

1910 b. Power Hovmöller By scale-averaging the wavelet power spectra at 1900 multiple locations, one can assess the spatial and tem- 1890 poral variability of a field of data. Figure 10a shows a 1880 power Hovmöller (time–longitude diagram) of the 1870 o o o o o o o ) wavelet variance for sea level pressure (SLP) anoma- 2 0 60 E 120 E 180 120 W 60 W 0 0.30 0.15 0.00 Av. Var. (mb2) 0.40 lies in the 2–8-yr band at each longitude. The original 0.30 0.20 time series at each longitude is the average SLP be- 0.10 0.00 tween 5° and 15°S. At each longitude, the wavelet (c) Av.Var (mb 0o 60oE 120oE 180o 120oW 60oW 0o power spectrum is computed using the Morlet wave- FIG. 10. (a) Power Hovmöller of 2–8-yr averaged wavelet let, and the scale-averaged wavelet power over the 2– power in SLP. The original time series at each longitude is the 8-yr band is calculated from (24). The average wave- average SLP between 5° and 15°S. The contour interval is 0.1 mb2. let-power time series are combined into a two-dimen- The thick contour is the 95% confidence level, using the sional contour plot as shown in Fig. 10a. The 95% corresponding red-noise spectrum at each longitude; (b) the confidence level is computed using the lag-1 auto- average of (a) over all longitudes; (c) the average of (a) over all times. correlation at each longitude and (26). Several features of Fig. 10 demonstrate the useful- ness of wavelet analysis. Fig. 10c shows the time-av- eraged 2–8-yr power as a function of longitude. Broad power in the 1915–30 period. The generally low power local maxima at 130°E and 130°W reflect the power observed in Figs. 1 and 8 between 1930 and 1950 associated with the Southern Oscillation. This longi- mainly reflects a lack of power in the Australian re- tudinal distribution of power is also observed in the gion, with the eastern Pacific having some significant 2–8-yr band for Fourier spectra at each longitude (not fluctuations in the 1940s. The large zonal-scale fluc- shown). The zonal average of the power Hovmöller tuations in both regions return in the 1950s, with the (Fig. 10b) gives a measure of global 2–8-yr variance strongest amplitudes after 1970. The diminished in this latitude band. Comparing this to Fig. 8, one can power after 1990 is within the COI, yet may reflect see that the peaks in zonal-average power are associ- the changes in ENSO structure and evolution seen in ated with the peaks in Niño3 SST variance, and, hence, recent years (Wang 1995). the 2–8-yr power is dominated in this latitude band by ENSO. c. Cross-wavelet spectrum With the power Hovmöller in Fig. 10a, the tempo- Given two time series X and Y, with wavelet trans- X Y ral variations in ENSO-associated SLP fluctuations forms Wn (s) and Wn(s), one can define the cross-wave- XY X Y* Y* can be seen. While the low power near the date line let spectrum as Wn (s) = Wn (s)Wn (s), where Wn (s) is Y region is apparent throughout the record, the high the complex conjugate of Wn(s). The cross-wavelet power regions fluctuate on interdecadal timescales. spectrum is complex, and hence one can define the | XY | From the 1870s to the 1920s, strong decadal fluctua- cross-wavelet power as Wn (s) . tions in the 2–8-yr power are observed in the Austra- Confidence levels for the cross-wavelet power can lian region. In contrast, the eastern Pacific fluctuations be derived from the square root of the product of two are strong only through 1910 and appear to have little chi-square distributions (Jenkins and Watts 1968).

Bulletin of the American Meteorological Society 75 Assuming both wavelet spectra are χ2 distributed with give a useful measure of coherence. This smoothing ν DOFs, the distribution is given by would also seem to defeat the purpose of wavelet analysis by decreasing the localization in time. Liu 2−ν (1994) suggests plotting the real and imaginary parts 2 ν − fz( ) = z1Κ ( z) ν ⎛ ν ⎞ 0 , (30) (the co- and quadrature-wavelet spectra) separately, Γ 2 ⎝ ⎠ and also plotting the coherence phase, defined as 2 −1 [ℑ XY /ℜ XY ] tan {Wn (s)} {Wn (s)} . The co- and quadrature-wavelet spectra for the where z is the random variable, Γ is the Gamma func- Niño3 SST and the SOI (not shown) do not appear to tion, and K0(z) is the modified Bessel function of or- give any additional information, especially in conjunc- der zero. The cumulative distribution function is given tion with the coherence phase shown in Fig. 11d. The Zν(p) by the p = ∫0 fν(z) dz, where Zν(p) is the con- shaded region in Fig. 11d shows where the phase dif- fidence level associated with probability p. Given a ference between Niño3 SST and the SOI is between probability p, this integral can be inverted to find the 160° and 200°. It is well known that the Niño3 SST confidence level Zν(p). and the SOI are out of phase, yet this shows that the If the two time series have theoretical Fourier spec- time series are within ±20° of being 180° out of phase X Y tra Pk and Pk , say from (16), then the cross-wavelet over all periods between 2 and 8 yr. Furthermore, this distribution is out-of-phase behavior is consistent with changes in the cross-wavelet power, with periods of low variance, say X Y* between 1920 and 1960, associated with more random WsWs( ) ( ) Zp( ) n n ⇒ ν PPX Y phase differences. σσ ν k k , (31) XY

σ σ where X and Y are the respective standard deviations. 7. Summary ν For = 1 (real wavelets), Z1 (95%) = 2.182, while for ν = 2 (complex wavelets), Z2 (95%) = 3.999. Wavelet analysis is a useful tool for analyzing time Figure 11a shows the wavelet power spectrum of series with many different timescales or changes in Niño3 SST using the Paul (m = 4) wavelet, while Fig. variance. The steps involved in using wavelet analy- 11b shows the wavelet power for the SOI. Note that sis are as follows:1 the narrow width in time of the Paul gives better time localization than the Morlet but poorer frequency lo- 1) Find the Fourier transform of the (possibly padded) calization. Finally, Fig. 11c shows the cross-wavelet time series. power for the Niño3 SST and the SOI and indicates 2) Choose a wavelet function and a set of scales to large between the time series at all scales analyze. between 2 and 8 yr. The 95% confidence level was 3) For each scale, construct the normalized wavelet derived using (31) and assuming red-noise spectra function using (6). (16) with α = 0.72 for Niño3 SST and α = 0.61 for 4) Find the wavelet transform at that scale using (4); the SOI. 5) Determine the cone of influence and the Fourier wavelength at that scale. d. Wavelet coherence and phase 6) After repeating steps 3–5 for all scales, remove any Another useful quantity from Fourier analysis is the padding and contour plot the wavelet power coherence, defined as the square of the cross-spectrum spectrum. normalized by the individual power spectra. This 7) Assume a background Fourier power spectrum gives a quantity between 0 and 1, and measures the (e.g., white or red noise) at each scale, then use the cross-correlation between two time series as a func- chi-squared distribution to find the 95% confidence tion of frequency. Unfortunately, as noted by Liu (5% significance) contour. (1994), this coherence is identically one at all times and scales. In Fourier analysis, this problem is circum- For other methods of wavelet analysis, such as the vented by smoothing the cross-spectrum before nor- malizing. For wavelet analysis, it is unclear what sort 1Software and examples are available from the authors at URL: of smoothing (presumably in time) should be done to http://paos.colorado.edu/research/wavelets/.

76 Vol. 79, No. 1, January 1998 orthogonal wavelet transform, see Farge a. NINO3 SST (1992). The results presented in section 1 2 4 on statistical significance testing are 4 presumably valid for higher-dimensional 8 16 wavelet analysis (assuming an appropri- 32 ate background spectrum can be chosen), Period (years) 64 1880 1900 1920 1940 1960 1980 2000 but this has not been tested and is left to Time (year) future research. More research is also b. GMSLP SOI needed on the properties and usefulness 1 of the cross-wavelet, wavelet coherence, 2 4 and co- and quadrature spectra. 8 In the wavelet analysis of Niño3 sea 16 32 surface temperature, the Southern Oscil- Period (years) 64 lation index, and the sea level pressure, 1880 1900 1920 1940 1960 1980 2000 it was found that the variance of the El Time (year) ✕ Niño–Southern Oscillation changed on c. NINO3 SST GMSLP SOI 1 interdecadal timescales, with a period of 2 low variance from 1920 to 1960. Using 4 8 both the filtered 2–8-yr variance and the 16 32

cross-wavelet power, the changes in Period (years) 64 Niño3 SST variance appear to be well 1880 1900 1920 1940 1960 1980 2000 correlated with changes in the SOI. The Time (year) SLP power Hovmöller suggests that d. Phase difference (160o−200o) these changes are planetary in scale, 1 2 while Torrence and Webster (1997) use 4 wavelet analysis to show that inter- 8 16 decadal changes in ENSO are also related 32 to changes in Indian monsoon variance. Period (years) 64 1880 1900 1920 1940 1960 1980 2000 Further studies are necessary to deter- Time (year) mine the extent and possible causes of these interdecadal changes. FIG. 11. (a) The wavelet spectrum for the Niño3 SST using the Paul (m = 4) wavelet. The contours are at normalized variances of 2, 5, and 10, while the thick It is hoped that the analysis presented contour is the 95% confidence level (red noise α = 0.72). (b) same as (a) but for here will prove useful in studies of the GMSLP SOI (red noise α = 0.61); (c) the cross-wavelet power for the Niño3 nonstationarity in time series, and the SST and the SOI. Contours are at 2, 5, and 10, while the thick contour is the 95% addition of statistical significance tests confidence from (31), with the red noise given in (a) and (b); (d) the phase will improve the quantitative nature of difference between Niño3 SST and the SOI, with the filled contour enclosing wavelet analysis. Future studies using regions between 160° and 200°. wavelet analysis can then concentrate on the results rather than simply the method. References

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