A Practical Guide to Wavelet Analysis

Transcription

1 A Practical Guide to Wavelet Analyi Chritopher Torrence and Gilbert P. Compo Program in Atmopheric and Oceanic Science, Univerity of Colorado, Boulder, Colorado ABSTRACT A practical tep-by-tep guide to wavelet analyi i given, with example taken from time erie of the El Niño Southern Ocillation (ENSO). The guide include a comparion to the windowed Fourier tranform, the choice of an appropriate wavelet bai function, edge effect due to finite-length time erie, and the relationhip between wavelet cale and Fourier frequency. New tatitical ignificance tet for wavelet power pectra are developed by deriving theoretical wavelet pectra for white and red noie procee and uing thee to etablih ignificance level and confidence interval. It i hown that moothing in time or cale can be ued to increae the confidence of the wavelet pectrum. Empirical formula are given for the effect of moothing on ignificance level and confidence interval. Extenion to wavelet analyi uch a filtering, the power Hovmöller, cro-wavelet pectra, and coherence are decribed. The tatitical ignificance tet are ued to give a quantitative meaure of change in ENSO variance on interdecadal timecale. Uing new dataet that extend back to 87, the Niño3 ea urface temperature and the Southern Ocillation index how ignificantly higher power during 88 9 and 96 9, and lower power during 9 6, a well a a poible 5-yr modulation of variance. The power Hovmöller of ea level preure how ignificant variation in 8-yr wavelet power in both longitude and time.. Introduction Wavelet analyi i becoming a common tool for analyzing localized variation of power within a time erie. By decompoing a time erie into time frequency pace, one i able to determine both the dominant mode of variability and how thoe mode vary in time. The wavelet tranform ha been ued for numerou tudie in geophyic, including tropical convection (Weng and Lau 99), the El Niño Southern Ocillation (ENSO; Gu and Philander 995; Wang and Wang 996), atmopheric cold front (Gamage and Blumen 993), central England temperature (Baliuna et al. 997), the diperion of ocean wave (Meyer et al. 993), wave growth and breaking (Liu 99), and coherent tructure in turbulent flow (Farge 99). A Correponding author addre: Dr. Chritopher Torrence, Advanced Study Program, National Center for Atmopheric Reearch, P.O. Box 3, Boulder, CO torrence@ucar.edu In final form October American Meteorological Society complete decription of geophyical application can be found in Foufoula-Georgiou and Kumar (995), while a theoretical treatment of wavelet analyi i given in Daubechie (99). Unfortunately, many tudie uing wavelet analyi have uffered from an apparent lack of quantitative reult. The wavelet tranform ha been regarded by many a an intereting diverion that produce colorful picture, yet purely qualitative reult. Thi miconception i in ome ene the fault of wavelet analyi itelf, a it involve a tranform from a one-dimenional time erie (or frequency pectrum) to a diffue two-dimenional time frequency image. Thi diffuene ha been exacerbated by the ue of arbitrary normalization and the lack of tatitical ignificance tet. In Lau and Weng (995), an excellent introduction to wavelet analyi i provided. Their paper, however, did not provide all of the eential detail neceary for wavelet analyi and avoided the iue of tatitical ignificance. The purpoe of thi paper i to provide an eay-toue wavelet analyi toolkit, including tatitical ignificance teting. The conitent ue of example of Bulletin of the American Meteorological Society 6

2 ENSO provide a ubtantive addition to the ENSO literature. In particular, the tatitical ignificance teting allow greater confidence in the previou wavelet-baed ENSO reult of Wang and Wang (996). The ue of new dataet with longer time erie permit a more robut claification of interdecadal change in ENSO variance. The firt ection decribe the dataet ued for the example. Section 3 decribe the method of wavelet analyi uing dicrete notation. Thi include a dicuion of the inherent limitation of the windowed Fourier tranform (WFT), the definition of the wavelet tranform, the choice of a wavelet bai function, edge effect due to finite-length time erie, the relationhip between wavelet cale and Fourier period, and time erie recontruction. Section preent the theoretical wavelet pectra for both white-noie and red-noie procee. Thee theoretical pectra are compared to Monte Carlo reult and are ued to etablih ignificance level and confidence interval for the wavelet power pectrum. Section 5 decribe time or cale averaging to increae ignificance level and confidence interval. Section 6 decribe other wavelet application uch a filtering, the power Hovmöller, cro-wavelet pectra, and wavelet coherence. The ummary contain a tepby-tep guide to wavelet analyi.. Data Several time erie will be ued for example of wavelet analyi. Thee include the Niño3 ea urface temperature (SST) ued a a meaure of the amplitude of the El Niño Southern Ocillation (ENSO). The Niño3 SST index i defined a the eaonal SST averaged over the central Pacific (5 S 5 N, 9 5 W). Data for are from an area average of the U.K. Meteorological Office GISST.3 (Rayner et al. 996), while data for January June 997 are from the Climate Prediction Center (CPC) optimally interpolated Niño3 SST index (courtey of D. Garrett at CPC, NOAA). The eaonal mean for the a. NINO3 SST ( o C) Period (year) Period (year) b. Morlet c. DOG Time (year) Time (year) entire record have been removed to define an anomaly time erie. The Niño3 SST i hown in the top plot of Fig. a. Gridded ea level preure (SLP) data i from the UKMO/CSIRO hitorical GMSLP.f (courtey of D. Parker and T. Banett, Hadley Centre for Climate Prediction and Reearch, UKMO). The data i on a 5 global grid, with monthly reolution from January 87 to December 99. Anomaly time erie have been contructed by removing the firt three harmonic of the annual cycle (period of 365.5, 8.65, and.75 day) uing a leat-quare fit. The Southern Ocillation index i derived from the GMSLP.f and i defined a the eaonally averaged preure difference between the eatern Pacific ( S, 5 W) and the wetern Pacific ( S, 3 E) Scale (year) FIG.. (a) The Niño3 SST time erie ued for the wavelet analyi. (b) The local wavelet power pectrum of (a) uing the Morlet wavelet, normalized by / σ (σ =.5 C ). The left axi i the Fourier period (in yr) correponding to the wavelet cale on the right axi. The bottom axi i time (yr). The haded contour are at normalized variance of,, 5, and. The thick contour encloe region of greater than 95% confidence for a red-noie proce with a lag- coefficient of.7. Cro-hatched region on either end indicate the cone of influence, where edge effect become important. (c) Same a (b) but uing the real-valued Mexican hat wavelet (derivative of a Gauian; DOG m = ). The haded contour i at normalized variance of.. Scale (year) 6 Vol. 79, No., January 998

3 3. Wavelet analyi Thi ection decribe the method of wavelet analyi, include a dicuion of different wavelet function, and give detail for the analyi of the wavelet power pectrum. Reult in thi ection are adapted to dicrete notation from the continuou formula given in Daubechie (99). Practical detail in applying wavelet analyi are taken from Farge (99), Weng and Lau (99), and Meyer et al. (993). Each ection i illutrated with example uing the Niño3 SST. a. Morlet.3. ψ (t / ) b. Paul (m=) c. DOG (m=) d. DOG (m=6) t / 6 ψ^ ( ω) ω / (π) FIG.. Four different wavelet bae, from Table. The plot on the left give the real part (olid) and imaginary part (dahed) for the wavelet in the time domain. The plot on the right give the correponding wavelet in the frequency domain. For plotting purpoe, the cale wa choen to be = δt. (a) Morlet, (b) Paul (m = ), (c) Mexican hat (DOG m = ), and (d) DOG (m = 6). 6 a. Windowed Fourier tranform The WFT repreent one analyi tool for extracting local-frequency information from a ignal. The Fourier tranform i performed on a liding egment of length T from a time erie of time tep δt and total length Nδt, thu returning frequencie from T to (δt) at each time tep. The egment can be windowed with an arbitrary function uch a a boxcar (no moothing) or a Gauian window (Kaier 99). A dicued by Kaier (99), the WFT repreent an inaccurate and inefficient method of time frequency localization, a it impoe a cale or repone interval T into the analyi. The inaccuracy arie from the aliaing of high- and low-frequency component that do not fall within the frequency range of the window. The inefficiency come from the T/(δt) frequencie, which mut be analyzed at each time tep, regardle of the window ize or the dominant frequencie preent. In addition, everal window length mut uually be analyzed to determine the mot appropriate choice. For analye where a predetermined caling may not be appropriate becaue of a wide range of dominant frequencie, a method of time frequency localization that i cale independent, uch a wavelet analyi, hould be employed. b. Wavelet tranform The wavelet tranform can be ued to analyze time erie that contain nontationary power at many different frequencie (Daubechie 99). Aume that one ha a time erie, x n, with equal time pacing δt and n = N. Alo aume that one ha a wavelet function, ψ (η), that depend on a nondimenional time parameter η. To be admiible a a wavelet, thi function mut have zero mean and be localized in both time and frequency pace (Farge 99). An example i the Morlet wavelet, coniting of a plane wave modulated by a Gauian: ωη η ψ ( η)= π i e e, () where ω i the nondimenional frequency, here taken to be 6 to atify the admiibility condition (Farge 99). Thi wavelet i hown in Fig. a. The term wavelet function i ued generically to refer to either orthogonal or nonorthogonal wavelet. The term wavelet bai refer only to an orthogonal et of function. The ue of an orthogonal bai implie the ue of the dicrete wavelet tranform, while a nonorthogonal wavelet function can be ued Bulletin of the American Meteorological Society 63

4 with either the dicrete or the continuou wavelet tranform (Farge 99). In thi paper, only the continuou tranform i ued, although all of the reult for ignificance teting, moothing in time and cale, and cro wavelet are applicable to the dicrete wavelet tranform. The continuou wavelet tranform of a dicrete equence x n i defined a the convolution of x n with a caled and tranlated verion of ψ (η): N n n t Wn( )= x ψ δ n, () n = ( ) where the (*) indicate the complex conugate. By varying the wavelet cale and tranlating along the localized time index n, one can contruct a picture howing both the amplitude of any feature veru the cale and how thi amplitude varie with time. The ubcript on ψ ha been dropped to indicate that thi ψ ha alo been normalized (ee next ection). Although it i poible to calculate the wavelet tranform uing (), it i coniderably fater to do the calculation in Fourier pace. To approximate the continuou wavelet tranform, the convolution () hould be done N time for each cale, where N i the number of point in the time erie (Kaier 99). (The choice of doing all N convolution i arbitrary, and one could chooe a maller number, ay by kipping every other point in n.) By chooing N point, the convolution theorem allow u do all N convolution imultaneouly in Fourier pace uing a dicrete Fourier tranform (DFT). The DFT of x n i ˆx k = N N πikn N xe n n=, (3) where k = N i the frequency index. In the continuou limit, the Fourier tranform of a function ψ(t/) i given by ψ$ (ω). By the convolution theorem, the wavelet tranform i the invere Fourier tranform of the product: N Wn( )= x ˆ ˆψ k ( ωk) e k= iω nδt where the angular frequency i defined a k, () ω k πk N : k Nδt = πk N. (5) : k > Nδt Uing () and a tandard Fourier tranform routine, one can calculate the continuou wavelet tranform (for a given ) at all n imultaneouly and efficiently. c. Normalization To enure that the wavelet tranform () at each cale are directly comparable to each other and to the tranform of other time erie, the wavelet function at each cale i normalized to have unit energy: π ψˆ( ω ψˆ k)= ωk ( ). (6) δt Example of different wavelet function are given in Table and illutrated in Fig.. Each of the uncaled ψ$ are defined in Table to have + ( ) = ψˆ ω dω ; that i, they have been normalized to have unit energy. Uing thee normalization, at each cale one ha N k= ( ) = ˆψ ω k N, (7) where N i the number of point. Thu, the wavelet tranform i weighted only by the amplitude of the Fourier coefficient x$ k and not by the wavelet function. If one i uing the convolution formula (), the normalization i ψ ( ) n n δt δ = t ( ) n n δt ψ, (8) where ψ (η) i normalized to have unit energy. d. Wavelet power pectrum Becaue the wavelet function ψ(η) i in general complex, the wavelet tranform W n () i alo complex. The tranform can then be divided into the real part, 6 Vol. 79, No., January 998

5 TABLE. Three wavelet bai function and their propertie. Contant factor for ψ and ψ $ enure a total energy of unity. e-folding Fourier Name ψ (η) ψ (ω) time τ wavelength λ Morlet (ω = frequency) π i ( ω ω) e e π H( ω) e ωη η ω π + + ω Paul (m = order) m m i m! iη π( m)! ( ) + ( m ) m m H( ω)( ω) e m( m )! ω π m+ DOG (m = derivative) ( ) m+ Γ m + m d dη m η ( e ) m i m ( ω) e Γ m + ( ω) π m+ H(ω) = Heaviide tep function, H(ω) = if ω >, H(ω) = otherwie. DOG = derivative of a Gauian; m = i the Marr or Mexican hat wavelet. R{W n ()}, and imaginary part, I{W n ()}, or amplitude, W n (), and phae, tan [I{W n ()}/R{W n ()}]. Finally, one can define the wavelet power pectrum a W n (). For real-valued wavelet function uch a the DOG (derivative of a Gauian) the imaginary part i zero and the phae i undefined. To make it eaier to compare different wavelet power pectra, it i deirable to find a common normalization for the wavelet pectrum. Uing the normalization in (6), and referring to (), the expectation value for W n () i equal to N time the expectation value for x$ k. For a white-noie time erie, thi expectation value i σ /N, where σ i the variance. Thu, for a white-noie proce, the expectation value for the wavelet tranform i W n () = σ at all n and. Figure b how the normalized wavelet power pectrum, W n () /σ, for the Niño3 SST time erie. The normalization by /σ give a meaure of the power relative to white noie. In Fig. b, mot of the power i concentrated within the ENSO band of 8 yr, although there i appreciable power at longer period. The 8-yr band for ENSO agree with other tudie (Trenberth 976) and i alo een in the Fourier pectrum in Fig. 3. With wavelet analyi, one can ee variation in the frequency of occurrence and amplitude of El Niño (warm) and La Niña (cold) event. During and 96 9 there were many warm and cold event of large amplitude, while during 9 6 there were few event (Torrence and Webter 997). From 875 9, there wa a light hift from a period near yr to a period cloer to yr, while from 96 9 the hift i from horter to longer period. Thee reult are imilar to thoe of Wang and Wang (996), who ued both wavelet and waveform analyi on ENSO indice derived from the Comprehenive Ocean Atmophere Data Set (COADS) dataet. Wang and Wang analyi howed reduced wavelet power before 95, epecially The reduced power i poibly due to the parene and decreaed reliability of the pre-95 COADS data (Folland et al. 98). With the GISST.3 data, the wavelet tranform of Niño3 SST in Fig. b how that the pre-9 period ha equal power to the pot-96 period. Bulletin of the American Meteorological Society 65

6 Variance (σ ) Period (year).5 FIG. 3. Fourier power pectrum of Niño3 SST (olid), normalized by N/(σ ). The lower dahed line i the mean rednoie pectrum from (6) auming a lag- of α =.7. The upper dahed line i the 95% confidence pectrum. e. Wavelet function One criticim of wavelet analyi i the arbitrary choice of the wavelet function, ψ (η). (It hould be noted that the ame arbitrary choice i made in uing one of the more traditional tranform uch a the Fourier, Beel, Legendre, etc.) In chooing the wavelet function, there are everal factor which hould be conidered (for more dicuion ee Farge 99). ) Orthogonal or nonorthogonal. In orthogonal wavelet analyi, the number of convolution at each cale i proportional to the width of the wavelet bai at that cale. Thi produce a wavelet pectrum that contain dicrete block of wavelet power and i ueful for ignal proceing a it give the mot compact repreentation of the ignal. Unfortunately for time erie analyi, an aperiodic hift in the time erie produce a different wavelet pectrum. Converely, a nonorthogonal analyi (uch a ued in thi tudy) i highly redundant at large cale, where the wavelet pectrum at adacent time i highly correlated. The nonorthogonal tranform i ueful for time erie analyi, where mooth, continuou variation in wavelet amplitude are expected. ) Complex or real. A complex wavelet function will return information about both amplitude and phae and i better adapted for capturing ocillatory behavior. A real wavelet function return only a ingle component and can be ued to iolate peak or dicontinuitie. 3) Width. For concretene, the width of a wavelet function i defined here a the e-folding time of the wavelet amplitude. The reolution of a wavelet function i determined by the balance between the width in real pace and the width in Fourier pace. A narrow (in time) function will have good time reolution but poor frequency reolution, while a broad function will have poor time reolution, yet good frequency reolution. ) Shape. The wavelet function hould reflect the type of feature preent in the time erie. For time erie with harp ump or tep, one would chooe a boxcar-like function uch a the Harr, while for moothly varying time erie one would chooe a mooth function uch a a damped coine. If one i primarily intereted in wavelet power pectra, then the choice of wavelet function i not critical, and one function will give the ame qualitative reult a another (ee dicuion of Fig. below). Four common nonorthogonal wavelet function are given in Table. The Morlet and Paul wavelet are both complex, while the DOG are real valued. Picture of thee wavelet in both the time and frequency domain are hown in Fig.. Many other type of wavelet exit, uch a the Haar and Daubechie, mot of which are ued for orthogonal wavelet analyi (e.g., Weng and Lau 99; Mak 995; Linday et al. 996). For more example of wavelet bae and function, ee Kaier (99). For comparion, Fig. c how the ame analyi a in b but uing the Mexican hat wavelet (DOG, m = ) rather than the Morlet. The mot noticeable difference i the fine cale tructure uing the Mexican hat. Thi i becaue the Mexican hat i real valued and capture both the poitive and negative ocillation of the time erie a eparate peak in wavelet power. The Morlet wavelet i both complex and contain more ocillation than the Mexican hat, and hence the wavelet power combine both poitive and negative peak into a ingle broad peak. A plot of the real or imaginary part of W n () uing the Morlet would produce a plot imilar to Fig. c. Overall, the ame feature appear in both plot, approximately at the ame location, and with the ame power. Comparing Fig. a and c, the Mexican hat i narrower in time-pace, yet broader in pectral-pace than the Morlet. Thu, in Fig. c, the 66 Vol. 79, No., January 998

7 peak appear very harp in the time direction, yet are more elongated in the cale direction. Finally, the relationhip between wavelet cale and Fourier period i very different for the two function (ee ection 3h). f. Choice of cale Once a wavelet function i choen, it i neceary to chooe a et of cale to ue in the wavelet tranform (). For an orthogonal wavelet, one i limited to a dicrete et of cale a given by Farge (99). For nonorthogonal wavelet analyi, one can ue an arbitrary et of cale to build up a more complete picture. It i convenient to write the cale a fractional power of two: = δ, =,, K, J (9) J δ log Nδt, () = ( ) where i the mallet reolvable cale and J determine the larget cale. The hould be choen o that the equivalent Fourier period (ee ection 3h) i approximately δt. The choice of a ufficiently mall δ depend on the width in pectral-pace of the wavelet function. For the Morlet wavelet, a δ of about.5 i the larget value that till give adequate ampling in cale, while for the other wavelet function, a larger value can be ued. Smaller value of δ give finer reolution. In Fig. b, N = 56, δt = / yr, = δt, δ =.5, and J = 56, giving a total of 57 cale ranging from.5 yr up to 6 yr. Thi value of δ appear adequate to provide a mooth picture of wavelet power. g. Cone of influence Becaue one i dealing with finite-length time erie, error will occur at the beginning and end of the wavelet power pectrum, a the Fourier tranform in () aume the data i cyclic. One olution i to pad the end of the time erie with zeroe before doing the wavelet tranform and then remove them afterward [for other poibilitie uch a coine damping, ee Meyer et al. (993)]. In thi tudy, the time erie i padded with ufficient zeroe to bring the total length N up to the next-higher power of two, thu limiting the edge effect and peeding up the Fourier tranform. Padding with zeroe introduce dicontinuitie at the endpoint and, a one goe to larger cale, decreae the amplitude near the edge a more zeroe enter the analyi. The cone of influence (COI) i the TABLE. Empirically derived factor for four wavelet bae. Name C δ γ δ ψ () Morlet (ω = 6) π / Paul (m = ) Marr (DOG m = ) DOG (m = 6) C δ = recontruction factor. γ = decorrelation factor for time averaging. δ = factor for cale averaging. region of the wavelet pectrum in which edge effect become important and i defined here a the e-folding time for the autocorrelation of wavelet power at each cale (ee Table ). Thi e-folding time i choen o that the wavelet power for a dicontinuity at the edge drop by a factor e and enure that the edge effect are negligible beyond thi point. For cyclic erie (uch a a longitudinal trip at a fixed latitude), there i no need to pad with zeroe, and there i no COI. The ize of the COI at each cale alo give a meaure of the decorrelation time for a ingle pike in the time erie. By comparing the width of a peak in the wavelet power pectrum with thi decorrelation time, one can ditinguih between a pike in the data (poibly due to random noie) and a harmonic component at the equivalent Fourier frequency. The COI i indicated in Fig. b and c by the crohatched region. The peak within thee region have preumably been reduced in magnitude due to the zero padding. Thu, it i unclear whether the decreae in 8-yr power after 99 i a true decreae in variance or an artifact of the padding. Note that the much narrower Mexican hat wavelet in Fig. c ha a much maller COI and i thu le affected by edge effect. h. Wavelet cale and Fourier frequency An examination of the wavelet in Fig. how that the peak in ψ$ (ω) doe not necearily occur at a frequency of. Following the method of Meyer et al. (993), the relationhip between the equivalent Fourier period and the wavelet cale can be derived analytically for a particular wavelet function by ubtituting a coine wave of a known frequency into () and computing the cale at which the wavelet power pec- Bulletin of the American Meteorological Society 67

8 trum reache it maximum. For the Morlet wavelet with ω = 6, thi give a value of λ =.3, where λ i the Fourier period, indicating that for the Morlet wavelet the wavelet cale i almot equal to the Fourier period. Formula for other wavelet function are given in Table, while Fig. give a graphical repreentation. In Fig. b,c, the ratio of Fourier period to wavelet cale can be een by a comparion of the left and right axe. For the Morlet, the two are nearly identical, while for the Mexican hat, the Fourier period i four time larger than the cale. Thi ratio ha no pecial ignificance and i due olely to the functional form of each wavelet function. However, one hould certainly convert from cale to Fourier period before plotting, a preumably one i intereted in equating wavelet power at a certain time and cale with a (poibly hortlived) Fourier mode at the equivalent Fourier period. i. Recontruction Since the wavelet tranform i a bandpa filter with a known repone function (the wavelet function), it i poible to recontruct the original time erie uing either deconvolution or the invere filter. Thi i traightforward for the orthogonal wavelet tranform (which ha an orthogonal bai), but for the continuou wavelet tranform it i complicated by the redundancy in time and cale. However, thi redundancy alo make it poible to recontruct the time erie uing a completely different wavelet function, the eaiet of which i a delta (δ) function (Farge 99). In thi cae, the recontructed time erie i ut the um of the real part of the wavelet tranform over all cale: x n δδ t = C ψ δ ( ) J = { } R ( ) Wn. () The factor ψ () remove the energy caling, while the / convert the wavelet tranform to an energy denity. The factor C δ come from the recontruction of a δ function from it wavelet tranform uing the function ψ (η). Thi C δ i a contant for each wavelet function and i given in Table. Note that if the original time erie were complex, then the um of the complex W n () would be ued intead. To derive C δ for a new wavelet function, firt aume a time erie with a δ function at time n =, given by x n = δ n. Thi time erie ha a Fourier tranform x$ k = N, contant for all k. Subtituting x$ k into (), at time n = (the peak), the wavelet tranform become N Wδ( )= ψ ˆ ( ωk). () N k= The recontruction () then give C δ δδ t = ψ () J = { } R ( ) Wδ. (3) The C δ i cale independent and i a contant for each wavelet function. The total energy i conerved under the wavelet tranform, and the equivalent of Pareval theorem for wavelet analyi i σ δδ t = CN δ N J n= = ( ) Wn, () where σ i the variance and a δ function ha been aumed for recontruction. Both () and () hould be ued to check wavelet routine for accuracy and to enure that ufficiently mall value of and δ have been choen. For the Niño3 SST, the recontruction of the time erie from the wavelet tranform ha a mean quare error of.% or.87 C.. Theoretical pectrum and ignificance level To determine ignificance level for either Fourier or wavelet pectra, one firt need to chooe an appropriate background pectrum. It i then aumed that different realization of the geophyical proce will be randomly ditributed about thi mean or expected background, and the actual pectrum can be compared againt thi random ditribution. For many geophyical phenomena, an appropriate background pectrum i either white noie (with a flat Fourier pectrum) or red noie (increaing power with decreaing frequency). A previou tudy by Qiu and Er (995) derived the mean and variance of the local wavelet power pectrum. In thi ection, the theoretical white- and rednoie wavelet power pectra are derived and compared to Monte Carlo reult. Thee pectra are ued to etablih a null hypothei for the ignificance of a peak in the wavelet power pectrum. 68 Vol. 79, No., January 998

9 a. Fourier red noie pectrum Many geophyical time erie can be modeled a either white noie or red noie. A imple model for red noie i the univariate lag- autoregreive [AR(), or Markov] proce: Period (δt) a. α=. 95% 95% x = αx + z, (5) n n n where α i the aumed lag- autocorrelation, x =, and z n i taken from Gauian white noie. Following Gilman et al. (963), the dicrete Fourier power pectrum of (5), after normalizing, i P k = α, (6) + α αco( πk N) where k = N/ i the frequency index. Thu, by chooing an appropriate lag- autocorrelation, one can ue (6) to model a red-noie pectrum. Note that α = in (6) give a white-noie pectrum. The Fourier power pectrum for the Niño3 SST i hown by the thin line in Fig. 3. The pectrum ha been normalized by N/σ, where N i the number of point, and σ i the variance of the time erie. Uing thi normalization, white noie would have an expectation value of at all frequencie. The red-noie background pectrum for α =.7 i hown by the lower dahed curve in Fig. 3. Thi red-noie wa etimated from (α + α )/, where α and α are the lag- and lag- autocorrelation of the Niño3 SST. One can ee the broad et of ENSO peak between and 8 yr, well above the background pectrum. b. Wavelet red noie pectrum The wavelet tranform in () i a erie of bandpa filter of the time erie. If thi time erie can be modeled a a lag- AR proce, then it eem reaonable that the local wavelet power pectrum, defined a a vertical lice through Fig. b, i given by (6). To tet thi hypothei, Gauian white-noie time erie and AR() time erie were contructed, along with their correponding wavelet power pectra. Example of thee white- and red-noie wavelet pectra are hown in Fig.. The local wavelet pectra were contructed by taking vertical lice at time n = 56. The lower mooth curve in Fig. 5a and 5b how the theoretical pectra from (6). The dot how the reult from the Monte Carlo imulation. On average, the local wavelet power pectrum i identical to the Fourier power pectrum given by (6). Period (δt) 3 5 Time (δt) b. α= Time (δt) FIG.. (a) The local wavelet power pectrum for a Gauian white noie proce of 5 point, one of the ued for the Monte Carlo imulation. The power i normalized by /σ, and contour are at,, and 3. The thick contour i the 95% confidence level for white noie. (b) Same a (a) but for a red-noie AR() proce with lag- of.7. The contour are at, 5, and. The thick contour i the 95% confidence level for the correponding red-noie pectrum. Therefore, the lower dahed curve in Fig. 3 alo correpond to the red-noie local wavelet pectrum. A random vertical lice in Fig. b would be expected to have a pectrum given by (6). A will be hown in ection 5a, the average of all the local wavelet pectra tend to approach the (moothed) Fourier pectrum of the time erie. c. Significance level The null hypothei i defined for the wavelet power pectrum a follow: It i aumed that the time erie ha a mean power pectrum, poibly given by (6); if a peak in the wavelet power pectrum i ignificantly above thi background pectrum, then it can be aumed to be a true feature with a certain percent confidence. For definition, ignificant at the 5% level i equivalent to the 95% confidence level, and implie a tet againt a certain background level, while the 95% confidence interval refer to the range of confidence about a given value. The normalized Fourier power pectrum in Fig. 3 i given by N x$ k /σ, where N i the number of point, x$ k i from (3), and σ i the variance of the time erie. If x n i a normally ditributed random variable, then both the real and imaginary part of x$ k are normally ditributed (Chatfield 989). Since the quare of a normally ditributed variable i chi-quare ditributed with one degree of freedom (DOF), then x$ k i chi- Bulletin of the American Meteorological Society 69

10 quare ditributed with two DOF, denoted by χ (Jenkin and Watt 968). To determine the 95% confidence level (ignificant at 5%), one multiplie the background pectrum (6) by the 95th percentile value for χ (Gilman et al. 963). The 95% Fourier confidence pectrum for the Niño3 SST i the upper dahed curve in Fig. 3. Note that only a few frequencie now have power above the 95% line. In the previou ection, it wa hown that the local wavelet pectrum follow the mean Fourier pectrum. If the original Fourier component are normally ditributed, then the wavelet coefficient (the bandpaed invere Fourier component) hould alo be normally ditributed. If thi i true, then the wavelet power pectrum, W n (), hould be χ ditributed. The upper curve in Fig. 5a and 5b how the 95% Fourier rednoie confidence level veru the 95% level from the Monte Carlo reult of the previou ection. Thu, at each point (n, ) in Fig. b, auming a red-noie proce, the ditribution i χ. Note that for a wavelet tranform uing a real-valued function, uch a the Mexican hat hown in Fig. c, there i only one degree of freedom at each point, and the ditribution i χ. In ummary, auming a mean background pectrum, poibly red noie [(6)], the ditribution for the Fourier power pectrum i Nxˆ k σ Pkχ (7) at each frequency index k, and indicate i ditributed a. The correponding ditribution for the local wavelet power pectrum i ( ) Wn σ Pkχ (8) at each time n and cale. The / remove the DOF factor from the χ ditribution. (For a real wavelet the ditribution on the right-hand ide would be P k χ.) The value of P k in (8) i the mean pectrum at the Fourier frequency k that correpond to the wavelet cale (ee ection 3h). Aide from the relation between k and, (8) i independent of the wavelet function. After finding an appropriate background pectrum and chooing a particular confidence for χ uch a 95%, one can then calculate (8) at each cale and contruct 95% confidence contour line. Variance (σ ) Variance (σ ) 3 Period (δt) 5 5 a. α=. 95% Mean b. α=.7 95% Mean Period (δt) FIG. 5. (a) Monte Carlo reult for local wavelet pectra of white noie (α =.). The lower thin line i the theoretical mean whitenoie pectrum, while the black dot are the mean at each cale of local wavelet pectra. The local wavelet pectra were lice taken at time n = 56 out of N = 5 point. The top thin line i the 95% confidence level, equal to χ (95%) time the mean pectrum. The black dot are the 95% level from the Monte Carlo run. (b) Same a (a) but for red noie of α =.7. A with Fourier analyi, moothing the wavelet power pectrum can be ued to increae the DOF and enhance confidence in region of ignificant power. Unlike Fourier, moothing can be performed in either the time or cale domain. Significance level and DOF for moothing in time or cale are dicued in ection 5. 7 Vol. 79, No., January 998

11 Inide the COI, the ditribution i till χ, but if the time erie ha been padded with zeroe, then the mean pectrum i reduced by a factor of ( ½e t/τ ), where τ i from Table, and t i the ditance (in time) from either the beginning or end of the wavelet power pectrum. The 95% confidence level for the Niño3 SST i hown by the thick contour on Fig. b and c. During and 96 9, the variance in the 8- yr band i ignificantly above the 95% confidence for red noie. During 9 6, there are a few iolated ignificant region, primarily around yr, and at the edge of the uual 8 yr ENSO band. The 95% confidence implie that 5% of the wavelet power hould be above thi level. In Fig. b, approximately 5% of the point are contained within the 95% contour. For the Niño3 wavelet pectrum,.9% of the point are above 95%, implying that for the Niño3 time erie a tet of encloed area cannot ditinguih between noie and ignal. However, the patial ditribution of variance can alo be examined for randomne. In Fig. b, the variance how a gradual increae with period, with random ditribution of high and low variance about thi mean pectrum. In Fig. b and c, the ignificant region are clutered together in both period and time, indicating le randomne of the underlying proce. d. Confidence interval The confidence interval i defined a the probability that the true wavelet power at a certain time and cale lie within a certain interval about the etimated wavelet power. Rewriting (8) a ( ) Wn σ P k χ, (9) one can then replace the theoretical wavelet power σ P k with the true wavelet power, defined a W n (). The confidence interval for W n () i then W χ p ( ) p n( ) Wn( ) W n( ) χ( ), () where p i the deired ignificance (p =.5 for the 95% confidence interval) and χ (p/) repreent the value of χ at p/. Note that for real-valued wavelet function, the right-hand ide of (9) become χ, and the factor of i removed from the top of (). Uing () one can then find confidence interval for the peak in a wavelet power pectrum to compare againt either the mean background or againt other peak. e. Stationarity It ha been argued that wavelet analyi require the ue of nontationary ignificance tet (Lau and Weng 995). In defene of the ue of tationary tet uch a thoe given above, the following point are noted. ) A nonarbitrary tet i needed. The aumption of tationary tatitic provide a tandard by which any nontationarity can be detected. ) The tet hould be robut. It hould not depend upon the wavelet function or upon the actual ditribution of the time erie, other than the aumption of a background pectrum. 3) A non Monte Carlo method i preferred. In addition to the aving in computation, the chi-quare tet implifie comparing one wavelet tranform with another. ) Many wavelet tranform of real data appear imilar to tranform of red-noie procee (compare Fig. b and b). It i therefore difficult to argue that large variation in wavelet power imply nontationarity. 5) One need to ak what i being teted. I it nontationarity? Or low-variance veru high-variance period? Or change in amplitude of Fourier mode? The chi-quare tet give a tandard meaure for any of thee poibilitie. In hort, it appear wier to aume tationarity and deign the tatitical tet accordingly. If the tet how large deviation, uch a the change in ENSO variance een in Fig. b and c, then further tet can be devied for the particular time erie. 5. Smoothing in time and cale a. Averaging in time (global wavelet pectrum) If a vertical lice through a wavelet plot i a meaure of the local pectrum, then the time-averaged wavelet pectrum over a certain period i W n n a n= n ( )= Wn( ) n, () where the new index n i arbitrarily aigned to the Bulletin of the American Meteorological Society 7

12 Variance (σ ) midpoint of n and n, and n a = n n + i the number of point averaged over. By repeating () at each time tep, one create a wavelet plot moothed by a certain window. The extreme cae of () i when the average i over all the local wavelet pectra, which give the global wavelet pectrum W 6 8 Period (year) N ( )= Wn( ). () N n=.5 FIG. 6. Fourier power pectrum from Fig. 3, moothed with a five-point running average (thin olid line). The thick olid line i the global wavelet pectrum for the Niño3 SST. The lower dahed line i the mean red-noie pectrum, while the upper dahed line i the 95% confidence level for the global wavelet pectrum, auming α =.7. In Fig. 6, the thick olid line how the normalized global wavelet pectrum, W ()/σ, for the Niño3 SST. The thin olid line in Fig. 6 how the ame pectrum a in Fig. 3, but moothed with a five-point running average. Note that a the Fourier pectrum i moothed, it approache the global wavelet pectrum more and more cloely, with the amount of neceary moothing decreaing with increaing cale. A comparion of Fourier pectra and wavelet pectra can be found in Hudgin et al. (993), while a theoretical dicuion i given in Perrier et al. (995). Percival (995) how that the global wavelet pectrum provide an unbiaed and conitent etimation of the true power pectrum of a time erie. Finally, it ha been uggeted that the global wavelet pectrum could provide a ueful meaure of the background pectrum, againt which peak in the local wavelet pectra could be teted (Ketin et al. 998). By moothing the wavelet pectrum uing (), one can increae the degree of freedom of each point and increae the ignificance of peak in wavelet power. To determine the DOF, one need the number of independent point. For the Fourier pectrum (Fig. 3), the power at each frequency i independent of the other, and the average of the power at M frequencie, each with two DOF, i χ ditributed with M degree of freedom (Spiegel 975). For the time-averaged wavelet pectrum, one i alo averaging point that are χ ditributed, yet Fig. b and ugget that thee point are no longer independent but are correlated in both time and cale. Furthermore, the correlation in time appear to lengthen a cale increae and the wavelet function broaden. Deignating ν a the DOF, one expect ν n a and ν. The implet formula to conider i to define a decorrelation length τ = γ, uch that ν = n a δt/τ. However, Monte Carlo reult how that thi τ i too abrupt at mall n a or large cale; even though one i averaging point that are highly correlated, ome additional information i gained. The Monte Carlo reult are given in Fig. 7, which how the mean and 95% level for variou n a. Thee Variance (σ ) 3 95% Mean 5 Period (δt) FIG. 7. Monte Carlo reult for the time-averaged wavelet pectra () of white noie uing the Morlet wavelet. The number to the right of each curve indicate n a, the number of time that were averaged, while the black dot are the 95% level for the Monte Carlo run. The top thin line are the 95% confidence from (3). The lower thin line i the mean white-noie pectrum, while the black dot are the mean of the Monte Carlo run (all of the mean are identical) Vol. 79, No., January 998

13 curve are bet decribed by the ditribution P k χ ν /ν, where P k i the original aumed background pectrum and χ ν i the chi-quare ditribution with ν degree of freedom, where δ ν = + na t γ. (3) Note that for a real-valued function uch a the Mexican hat, each point only ha one DOF, and the factor of in (3) i removed. The decorrelation factor γ i determined empirically by an iterative fit of abolute error to the 95% Monte Carlo level and i given in Table for the four wavelet function. The relative error (or percent difference) between the Monte Carlo and the χ ν /ν ditribution wa everywhere le than 7% for all cale and n a value. The thin line in Fig. 7 how the reult of (3) uing the Morlet wavelet. Note that even the white noie proce ha more tringent 95% confidence level at large cale compared to mall. A a final note, if the point going into the average are within the cone of influence, then n a i reduced by approximately one-half of the number within the COI to reflect the decreaed amplitude (and information) within that region. A different definition of the global wavelet pectrum, involving the dicrete wavelet tranform and including a dicuion of confidence interval, i given by Percival (995). An example uing Percival definition can be found in Linday et al. (996). The 95% confidence line for the Niño3 global wavelet pectrum i the upper dahed line in Fig. 6. Only the broad ENSO peak remain ignificant, although note that power at other period can be le than ignificant globally but till how ignificant peak in local wavelet power. NINO3 ( o C ) NINO3 SOI Comparing () and (), the cale-averaged wavelet power i a time erie of the average variance in a certain band. Thu, the cale-averaged wavelet power can be ued to examine modulation of one time erie by another, or modulation of one frequency by another within the ame time erie. A an example of averaging over cale, Fig. 8 how the average of Fig. b over all cale between and 8 yr (actually 7.3 yr), which give a meaure of the average ENSO variance veru time. The variance plot how a ditinct period between 9 and 96 when ENSO variance wa low. Alo hown in Fig. 8 i the variance in the Southern Ocillation index (SOI), which correlate well with the change in Niño3 SST Year FIG. 8. Scale-averaged wavelet power () over the 8-yr band for the Niño3 SST (olid) and the SOI (dahed). The thin olid line i the 95% confidence level from (6) for Niño3 SST (auming red noie α =.7), while the thin dahed line i the 95% level for the SOI (red noie α =.6). variance (.7 correlation). Both time erie how conitent interdecadal change, including a poible modulation in ENSO variance with a 5-yr period. To examine more cloely the relation between Niño3 SST and the SOI, one could ue the cro-wavelet pectrum (ee ection 6c). A with time-averaged wavelet pectrum, the DOF are increaed by moothing in cale, and an analytical relationhip for ignificance level for the cale-averaged wavelet power i deirable. Again, it i convenient to normalize the wavelet power by the expectation value for a white-noie time erie. From (), thi expectation value i (δ δt σ )/(C δ S avg ), where σ i the time-erie variance and S avg i defined a SOI (mb ) b. Averaging in cale To examine fluctuation in power over a range of cale (a band), one can define the cale-averaged wavelet power a the weighted um of the wavelet power pectrum over cale to : W n δδ t = C δ = Wn ( ). () Savg = =. (5) The black dot in Fig. 9 how the Monte Carlo reult for both the mean and the 95% level of caleaveraged wavelet power a a function of variou n a, where n a = + i the number of cale averaged. Bulletin of the American Meteorological Society 73

14 Normalized Variance 3 95% Mean Uing the normalization factor for white noie, the ditribution can be modeled a CS δ avg W δδσ t n χν P, (6) ν where the cale-averaged theoretical pectrum i now given by P = S avg = P. (7) Note that for white noie thi pectrum i till unity (due to the normalization). The degree of freedom ν in (6) i modeled a δ ν = ns a avg + n a S δ mid DOG DOG6 Paul Morlet 6 8 n a δ (# cale in avg, compreed) FIG. 9. Monte Carlo reult for the wavelet pectra averaged over n a cale from (), uing white noie. The average from () i centered on cale = 6δt for convenience, but the reult are independent of the center cale. To make the graph independent of the choice for δ, the x axi ha been compreed by the Monte Carlo δ of.5. The top black dot are the 95% level for the Monte Carlo run, while the lower black dot are the mean. The mean for all four wavelet bae are all the ame, while the 95% level depend on the width of the bai in Fourier pace, with the Morlet being the mot narrow. The top thin line are the 95% confidence from (8). The lower thin line i the mean white-noie pectrum., (8) where S mid =.5( + )δ. The factor S avg /S mid correct for the lo of DOF that arie from dividing the wavelet power pectrum by cale in () and i oberved in the Monte Carlo reult. Note that for a real-valued function uch a the Mexican hat, each point only ha one DOF, and the factor of in (8) i removed. The decorrelation ditance δ i determined empirically by an iterative fit of abolute error between (8) and the 95% level of the Monte Carlo reult and i given in Table. The thin line in Fig. 9 how the reult of (8) for the Morlet, Paul (m = ), DOG, and DOG6 wavelet function. For thee wavelet, the relative error between the χ ν ditribution uing (8) and the Monte Carlo reult i le than.5%. It hould be noted that (8) i valid only for confidence of 95% or le. At higher confidence level, the ditribution begin to deviate ignificantly from χ, and (8) i no longer valid. In Fig. 8, the thin olid and dahed line how the 95% confidence level for the Niño3 SST and the SOI uing (5) (8). In thi cae, δ =.5, the um wa between Fourier period and 8 yr (actually. 7.6 yr), n a = 6, S avg =. yr, S mid = 3.83 yr, δ =.6, and ν = 6.. Since the two time erie do not have the ame variance or the ame red-noie background, the 95% line are not equal. 6. Extenion to wavelet analyi a. Filtering A dicued in ection 3i, the wavelet tranform () i eentially a bandpa filter of uniform hape and varying location and width. By umming over a ubet of the cale in (), one can contruct a waveletfiltered time erie: δδ t xn = C ψ δ ( ) { } R Wn( ) =. (9) Thi filter ha a repone function given by the um of the wavelet function between cale and. Thi filtering can alo be done on both the cale and time imultaneouly by defining a threhold of wavelet power. Thi denoiing remove any low-amplitude region of the wavelet tranform, which are preumably due to noie. Thi technique ha the advantage over traditional filtering in that it remove noie at all frequencie and can be ued to iolate ingle event that have a broad power pectrum or multiple 7 Vol. 79, No., January 998

15 event that have varying frequency. A more complete decription including example i given in Donoho and Johntone (99). Another filtering technique involve the ue of the two-dimenional wavelet tranform. An example can be found in Farge et al. (99), where two-dimenional turbulent flow are compreed uing an orthonormal wavelet packet. Thi compreion remove the low-amplitude paive component of the flow, while retaining the high-amplitude dynamically active component (a) Power -8 yr lp 5. o S to 5. o S (b) Zonal Avg b. Power Hovmöller By cale-averaging the wavelet power pectra at multiple location, one can ae the patial and temporal variability of a field of data. Figure a how a power Hovmöller (time longitude diagram) of the wavelet variance for ea level preure (SLP) anomalie in the 8-yr band at each longitude. The original time erie at each longitude i the average SLP between 5 and 5 S. At each longitude, the wavelet power pectrum i computed uing the Morlet wavelet, and the cale-averaged wavelet power over the 8-yr band i calculated from (). The average wavelet-power time erie are combined into a two-dimenional contour plot a hown in Fig. a. The 95% confidence level i computed uing the lag- autocorrelation at each longitude and (6). Several feature of Fig. demontrate the uefulne of wavelet analyi. Fig. c how the time-averaged 8-yr power a a function of longitude. Broad local maxima at 3 E and 3 W reflect the power aociated with the Southern Ocillation. Thi longitudinal ditribution of power i alo oberved in the 8-yr band for Fourier pectra at each longitude (not hown). The zonal average of the power Hovmöller (Fig. b) give a meaure of global 8-yr variance in thi latitude band. Comparing thi to Fig. 8, one can ee that the peak in zonal-average power are aociated with the peak in Niño3 SST variance, and, hence, the 8-yr power i dominated in thi latitude band by ENSO. With the power Hovmöller in Fig. a, the temporal variation in ENSO-aociated SLP fluctuation can be een. While the low power near the date line region i apparent throughout the record, the high power region fluctuate on interdecadal timecale. From the 87 to the 9, trong decadal fluctuation in the 8-yr power are oberved in the Autralian region. In contrat, the eatern Pacific fluctuation are trong only through 9 and appear to have little (c) Av.Var (mb ) o 6 o E o E 8 o o W 6 o W o o 6 o E o E 8 o o W 6 o W o.3.5. Av. Var. (mb ) FIG.. (a) Power Hovmöller of 8-yr averaged wavelet power in SLP. The original time erie at each longitude i the average SLP between 5 and 5 S. The contour interval i. mb. The thick contour i the 95% confidence level, uing the correponding red-noie pectrum at each longitude; (b) the average of (a) over all longitude; (c) the average of (a) over all time. power in the 95 3 period. The generally low power oberved in Fig. and 8 between 93 and 95 mainly reflect a lack of power in the Autralian region, with the eatern Pacific having ome ignificant fluctuation in the 9. The large zonal-cale fluctuation in both region return in the 95, with the tronget amplitude after 97. The diminihed power after 99 i within the COI, yet may reflect the change in ENSO tructure and evolution een in recent year (Wang 995). c. Cro-wavelet pectrum Given two time erie X and Y, with wavelet tranform W nx () and W ny (), one can define the cro-wavelet pectrum a W n XY () = W nx ()W n Y* (), where W n Y* () i the complex conugate of W ny (). The cro-wavelet pectrum i complex, and hence one can define the cro-wavelet power a W n XY (). Confidence level for the cro-wavelet power can be derived from the quare root of the product of two chi-quare ditribution (Jenkin and Watt 968). Bulletin of the American Meteorological Society 75