Use of the Wigner-Ville distribution in interpreting and

Use of the Wigner-Ville distribution in

interpreting and identifying ULF waves in

triaxial magnetic records

P. J. Chi1 and C. T. Russell1

Received 13 April 2007; revised 2 August 2007; accepted 8 October 2007; published 31 January 2008.

[1] Magnetospheric ultra-low-frequency (ULF) waves (f = 1 mHz to 1 Hz) exhibit highlytime-dependent characteristics due to the dynamic properties of these waves and, forobservations in space, the spacecraft motion. These time-dependent featuresmay not be properly resolved by conventional Fourier techniques. In this study weexamine how the Wigner-Ville distribution (WVD) can be used to analyze ULF waves.We find that this approach has unique advantages over the conventional Fourierspectrograms and wavelet scalograms. In particular, for Pc1 wave packets, field line/cavitymode resonances in the Pc 3–4 band, and Pi2 pulsations, the start and end timesof each wave packet can be well identified and the frequency better defined. In addition,we demonstrate that the Wigner-Ville distribution can be used to calculate thepolarization of wave signals in triaxial magnetic field data in a way analogous to Fourieranalysis. Motivated by the large amount of ULF wave observations, we have alsodeveloped a WVD-based algorithm to identify ULF waves as a way to facilitate the rapidprocessing of the data collected by satellite missions and the vast network ofground magnetometers.

Citation: Chi, P. J., and C. T. Russell (2008), Use of the Wigner-Ville distribution in interpreting and identifying ULF waves in

triaxial magnetic records, J. Geophys. Res., 113, A01218, doi:10.1029/2007JA012469.

1. Introduction

[2] The ultra-low-frequency (ULF) waves (f = 1 mHz to1 Hz) in the magnetosphere are intimately related to manymagnetospheric processes, ranging from the interaction ofparticles, the normal-mode resonances of magnetosphericfield lines, to the onset of substorms [see, e.g., Takahashi etal., 2006]. One of the fundamental characteristics of theULF waves, as clearly indicated by the classification intothe subgroups of continuous pulsations (Pc) and irregularpulsations (Pi) [Jacobs et al., 1964], is that the wavefrequency and phase can be quite changeable, creatingcomplications and challenges in wave analysis. For satelliteobservations the motion of the spacecraft can also introducean artificial phase drift that produces frequency shifts.[3] Fourier analysis has been the dominant spectral tech-

nique in ULF research, as its formulation closely matchesthose used in wave physics. A few other techniques havealso been used in ULF analysis. For example, complexdemodulation was occasionally used due to its power indistinguishing time-dependent changes in wave phase[Beamish et al., 1979]. Since the 1990s, as the wavelettechnique became popular, there has gradually been more

use of wavelet in ULF research [e.g., Rees et al., 2003]. Apertinent question then is, Are the existing techniquessufficient for a thorough investigation of ULF waves orcan any other techniques not presently in our arsenalprovide advantages in analyzing ULF waves?[4] In this study we demonstrate that the use of the

Wigner-Ville distribution (WVD) in time-frequency analy-sis can better characterize the time-frequency properties ofsome types of ULF waves than Fourier spectrograms andwavelet scalograms. In addition, the WVD can be used tocalculate the polarization of wave signals in triaxial mag-netic field data in a way analogous to Fourier analysis. Alsomotivated by the formidable task of investigating theenormous amount of ULF wave events collected by satel-lites, we have developed an algorithm based on the WVDtogether with statistical tests to automatically identify ULFwaves. This procedure can significantly facilitate statisticalstudies of ULF waves as many future multisatellite missionswill dramatically increase the amount of data that can beanalyzed.[5] In the following we start with a brief description of

the WVD and its use in time-frequency analysis, followedby the examples of several often-studied classes of ULFwaves, including Pc 1 wave packets, Pi 2 pulsations, and thefield line/cavity mode resonances in the Pc 3–4 band.Section 4 lays out the procedure of the automatic identifi-cation of wave events based on WVD, and section 5describes how to use WVD in polarization analysis. Dis-

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1Institute of Geophysics and Planetary Physics, University ofCalifornia, Los Angeles, Los Angeles, California, USA.

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cussed at the end are some considerations of the choice oftime-frequency methods.

2. Wigner-Ville Distribution

[6] The Wigner-Ville distribution Ws of a time seriessignal s(t) is defined as

Ws t;wð Þ ¼Z 1

�1s t þ t

2

� �s* t � t

2

� �e�iwtdt: ð1Þ

[7] This distribution was first introduced by EugeneWigner in his calculation of the quantum corrections ofclassical statistical mechanics [Wigner, 1932]. It was inde-pendently derived again by J. Ville in 1948 as a quadraticrepresentation of the local time-frequency energy of asignal. The time series function s(t) in equation (1) can beeither real or complex. The only type of complex signalconsidered in this paper is the analytic signal, which isdefined as (s(t) + iH[s(t)]) where H denotes the Hilberttransform [e.g., Cohen, 1994]. The analytic signal has thesame spectrum (or, to be precise, the spectrum multiplied bytwo) as that of the original real signal at positive frequenciesbut keeps the spectrum at negative frequencies zero.[8] A significant characteristic of WVD, which essentially

correlates the signal with a time- and frequency-translatedversion of itself, is that it does not contain a windowingfunction as those in the Fourier and wavelet frameworks.This unique feature frees WVD from the smearing effectdue to the windowing function, and, as a result, the WVDprovides the representation that has the highest possibleresolution in the time-frequency plane. For example, if awave packet is nonzero only for t 2 (t1, t2), it can be shownthat the corresponding Ws has to be 0 for t outside this exacttime interval.[9] There is, however, a well-known disadvantage of

using WVD. When there are multiple components in asignal, the WVD can become difficult to interpret. Supposethat the signal s(t) is composed by the sum of two signals,s1(t) and s2(t), the WVD becomes

Ws t;wð Þ ¼ Ws1 t;wð Þ þWs2 t;wð Þ þ 2<Ws1;s2 t;wð Þ; ð2Þ

in which Ws1,s2(t, w)

R1�1 s1(t +

t2) s*2(t � t

2) e�iwt dt is

called the ‘‘cross’’ Wigner-Ville function. Representing the‘‘interference’’ between the two signals, the term 2<Ws1,s2

(t,w)has significant nonzero values located between the auto termsin the time-frequency plane. This cross-term interferencemakes it difficult at times to interpret signal properties fromWVD.[10] One of the special properties of the interference term

is that it is highly oscillatory in the time-frequency planecompared to the auto terms that represent the true signals[e.g., Cohen, 1994]. This property inspired the idea that asmoothing function, or a ‘‘kernel,’’ can be used to suppressthe interference pattern without much of an effect on thedesired signal. In this approach, the kernel is placed in theWigner-Ville function as

fWs t;wð Þ ¼Z 1

�1

Z 1

�1f t0;w0ð ÞWs t � t0;w� w0ð Þdt0dw0; ð3Þ

in which f(t, w) is a two-dimensional (2-D) low-pass filterthat can suppress the cross-term interference in the time-frequency plane. In the last 2 decades, significant progresshas been made in understanding the properties of thesmoothed version of the WVD. There are also a wide rangeof kernels constructed, each carrying its special advantagesin suppressing the interference term. In this study we use thepopular kernel, ‘‘Choi-William function,’’ that has provencapable of suppressing the interference effectively [Choiand Williams, 1989].[11] In this study we also converted real-valued signals to

analytic signals before calculating the WVD. The reason isthat real-valued signals have spectral power at negativefrequencies, which can create significant cross-term inter-ference in the vicinity of w = 0. Because the analytic signalis a halfband function, the resulting WVD effectively avoidsall cross-term interference associated with the negativefrequency components [Qian, 2002].[12] An example of the comparison among the Wigner-

Ville distribution, the short-time Fourier transform (STFT),and the wavelet scalogram is demonstrated in Figure 1. Thesample time series consists of a wave packet with a constantfrequency, a sharp pulse, and a wave packet with decreasingfrequency (Figure 1a). Figures 1b, 1c, and 1d show thecorresponding STFT, wavelet scalogram (where the Morletwavelet is used as the basis), and the magnitude of WVD.For the STFT, there is a well-known tradeoff between timeand frequency resolution. A Fourier window that canproperly address low-frequency waves will smear the rep-resentation for sharp pulses. The wavelet scalogram hassmarter architecture in which the window size is inverselyproportional to the wave frequency of interest. Nevertheless,the wavelet representation still smears the sharp pulse in thetime domain due to the wavelet basis involved. In contrast,the WVD has a better defined representation of both wavepackets and a very narrow line for the sharp pulse. Thetradeoff is the interference that cannot be removed com-pletely even when the smoothing function is used.

3. Time-Frequency Analysis of ULF Waves

[13] In the following, we examine the Wigner-Villerepresentations for three classes of ULF waves and comparethem with the conventional Fourier spectrograms and wave-let scalograms. Each class of these ULF waves has been awell-recognized research subject because of its specificimplications in magnetospheric physics. The detailedtime-dependent characteristics of these waves, however,may not have been studied extensively or still awaitdefinitive physical explanation.

3.1. Pc 1 Wave Packets

[14] The Pc 1 waves (f = 0.5–2 Hz) are essentiallygenerated by the ion cyclotron instability in the magneto-sphere. One major objective of Pc 1 research is the wave’sinteraction with ring current particles and its impact on ringcurrent decay. These waves may occur in the form ofseparate bursts, gradually developing into a series of pulsa-tions that appear as ‘‘pearls’’ in time series plots. They mayalso occur in the form of consecutive groups of pulsationswith varying frequency. A review of Pc 1 waves can befound, for example, in the paper by Fraser et al. [2006].

A01218 CHI AND RUSSELL: WIGNER-VILLE DISTRIBUTION

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[15] Figure 2a shows an example of Pc 1 waves observedby the Polar satellite. Three wave packets, each containingfine structure, can be seen in the 3-min interval presented.The magnetosphere was in the recovery phase of a magneticstorm that started 3 d earlier, and Polar was located at L ’3.5 and ’�15� magnetic latitude, traveling approximatelyalong the field line and toward the equator. Thecorresponding STFT, wavelet scalogram, and WVD(Figures 2b–2d) all identify the three wave packets. A hintof frequency splitting in the third wave packet is seen in thewavelet scalogram but is more clearly evident in the STFTand WVD plots. We have experimented with several dif-ferent Fourier windows for STFT and with the numbers ofanalyzed frequencies for wavelets, but, as demonstrated bycomparing Figures 2b–2d, WVD provides the best defini-tion of the detailed time-frequency structures of these wavepackets.[16] The basic properties of these waves are consistent

with the generation via the ion cyclotron instability in theequatorial region. The carrier frequency of all the wavepackets is 1.8 Hz, which was below the equatorial value offHe+, 3.9 Hz, and implied the possibility of O+ participatingin the process. The relatively steady wave frequency is aclear indication that the wave source was far from theobservation location because the ambient field measuredby the satellite fell off by 12% during the 3 min ofobservation. Instead, the steady frequency is consistent withthe field mapped to the equator that was almost unchanged.Additional interesting features of this Pc 1 event reside inthe periodicity of wave packets, the drift of the carrierfrequency, and frequency splitting. The repetitive wave

packets have a periodicity of approximately one minute,but Polar did not observe any other concurrent perturbationsat a similar periodicity, which is a possible mechanismof wave modulation [e.g., Coroniti and Kennel, 1970;Rasinkangas and Mursula, 1998]. Also mysterious is theslight frequency drift, increasing from 1.62 Hz to 1.68 Hz,in the second wave packet and the frequency splitting in thethird wave packet. Because the Polar satellite essentiallystayed on the same field line during the Pc 1 event, thesedetailed characteristics are more likely associated with thewave source at the equator. In summary, this Pc 1 examplesuggests that the source of EMIC waves in the equatorialregion can be bursty, and the WVD is a powerful tool topinpoint the detailed evolution of time-frequency features ofPc 1 wave packets.

3.2. Irregular Pulsation Pi 2

[17] Pi 2 pulsations are associated with the onset ofmagnetospheric substorms or the bursty bulk flows in themagnetotail (see an overview by Olson [1999] and thereferences therein). They are excited by the movement ofplasma due to reconnection, and the signal can propagatevia the MHD waves. The signatures of Pi 2 pulsations aremostly seen in the records of ground magnetometers locatedin the nightside, but occasionally they can be found in thedayside as well [e.g., Sutcliffe and Yumoto, 1991]. Althoughthe detailed dynamics of these waves are being activelystudied, the frequency of Pi 2 could be the consequence ofseveral different processes, such as surface waves at theplasmaspheric boundary [Lester and Orr, 1983], field lineresonance [Takahashi et al., 1988], cavity mode resonance

Figure 1. (a) A signal containing a wave packet with constant carrier frequency, a sharp pulse, and achirp; (b), (c), and (d) the short-time Fourier transform (STFT), the wavelet spectrogram, and the Wigner-Ville distribution (WVD) of the signal.


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[e.g., Sutcliffe and Yumoto, 1991], and the periodicity ofbursty bulk flows [Kepko et al., 2001]. These waves attractsubstantial interest because they are associated with theexplosive phenomena in the nightside magnetosphere, andtheir timing and other properties may reveal the propertiesof substorms and the magnetotail.[18] The Pi 2, the second type of impulsive pulsations,

was so designated because of its broadband and short-duration nature. Its small number of cycles imposes diffi-culty on traditional Fourier analysis as it is not capable ofidentifying the time-dependent features within a shortperiod of time. Beamish et al. [1979] have attempted tocircumvent this problem by performing complex demodu-lation analysis on the instantaneous values of amplitude andphase of Pi 2. This technique, however, assumes that thesignal is confined in a narrow frequency band and can focuson only the amplitude modulation but not the possiblefrequency drift.[19] To assess if WVD is useful for analyzing the time-

frequency relation in Pi 2 pulsations, we compare the threetechniques using the Pi 2 event observed by several groundmagnetometer stations in North America at 0850 UT on28 January 2003. This Pi 2 event occurred at the onset of asubstorm, after which the AL index dropped to a minimumof �300 nT at approximately 0920 UT. The magneticfield in the H-component observed at Cambridge, MN(CAM, L = 3.20), and Island Lake (ISLL, L = 5.23), areshown in Figures 3a and 3e. The trend in BH for Pi 2pulsations is often in a form that is more complicated thanthe first-order and second-order polynomial fits, causing afurther complication in spectral analysis. To alleviate this

problem, we use the first difference of BH, which preservesthe frequency content of the signal as d

dt(A exp�iwt) = �iw A

exp�iwt, for suppressing the low-frequency modulationswithout affecting the time-dependent frequency of Pi 2.Both STFT spectrogram (Figures 3b and 3f) and waveletscalogram (Figures 3c and 3g) show that a dominant Pi 2frequency at ’14.5 mHz was seen at both stations. Thewavelet scalogram shows a lower frequency at 5.7 mHz,which is not seen in other time-frequency representationsand could be related to the duration of the wave packet.Both STFT and WVD plots show the slightly dropping tonein the Pi 2 wave packet, and the WVD provides the bestfrequency determination.[20] The WVD results of this Pi 2 event demonstrate the

similarities and differences in frequency observed by thetwo stations. Both stations saw an identical frequency at14.5 mHz initially, and then the higher-latitude station ISLLexperienced a stronger decline in frequency to 9.7 mHz,compared to the end frequency 11.9 mHz at CAM. It shouldbe noted that the two stations were located on different sidesof the plasmapause, evidenced by the opposite polarizationpresented in the time series plots (Figures 3a and 3e). Theinitial frequency at 15 mHz could be dictated by the surfacewaves on the plasmaspheric boundary, but the wave periodwas then modified by the local field line resonance after 2–3 cycles when the wave energy had sufficient time tobounce between the two ionospheres. At the lower-latitudestation CAM, on the other hand, a cavity resonance could beexcited and result in the weak oscillations seen after t =600 s. It is clear that, to study the detailed evolution of Pi 2frequency, one needs to address the change in wave period

Figure 2. (a) Structured Pc 1 waves observed by the Polar satellite on 7 May 1998. The start time is2121 UT. (b), (c), and (d) the STFT spectrogram, the wavelet spectrogram, and the WVD for the timeseries in Figure 2a.


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from one cycle to the next, and the WVD is a powerful toolfor such time-frequency analysis.

3.3. Pc 3–5 Waves: Field Line Resonance andPhase Skipping

[21] Evidenced by the success of the field line resonanceobservations in the last 3 decades, the traditional Fouriermethod is sufficient in handling most analysis need. Thereare additional benefits, however, that WVD could bring inthrough its superior resolution in both time and frequency.Figure 4 presents the first derivative of the azimuthalmagnetic field (Bf), the corresponding STFT, waveletscalogram, and WVD plot for a Pc 3–4 event observedby the ISEE-1 satellite on 24 November 1977. This event isidentical to one of the examples shown by Kivelson et al.

[1997] for demonstrating the observational evidence of theputative cavity mode resonance. The ISEE-1 satellite waslocated at ’5� magnetic latitude in the morning sector,traveling inbound from 10.5 RE to 8.6 RE in the distance tothe Earth. Shown in all the three time-frequency plots(Figures 4b–4d) are the intermittent wave activities atfrequencies between 7 and 41 mHz recorded during the80-min interval. The wave power was confined in severalfield line resonance harmonics, exhibiting a trend of in-creasing frequency as the satellite was moving closer to theEarth.[22] Although the STFT, wavelet scalogram, and WVD

show consistent results with regard to the frequencies offield-line and cavity-mode resonances, it is evident that the

Figure 3. Pi 2 waves observed at (left) Cambridge, Minnesota (CAM, L = 3.20) and (right) Island Lake(ISLL, L = 5.23) on 28 January 2003, and their STFT, wavelet, and Wigner-Ville representations. Thestart time is 0850 UT.


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WVD best preserves the narrow peaks of the wave spec-trum. Not only can this improved precision provided byWVD help compare with the modeling of cavity moderesonance, it also means a better determination of theplasma mass density inferred from the satellite and groundobservations of field line resonance harmonics [Denton etal., 2004; Chi et al., 2000, 2005]. The WVD plot alsoreveals many broadband (i.e., vertical) power enhancementsthat are not found in either STFT or wavelet scalogram.These broadband structures are associated with large valuesin dBf/dt or, equivalently, sudden changes in wave phases.A consequence of impulsive energy input, phase skipping isa common feature of dayside Pc 3–4 waves [Chi andRussell, 1998] but not suited for Fourier or wavelet analysis.

Figure 4 demonstrates that without the need of any window-ing function, the WVD has a unique advantage to properlyexhibit phase skips rather than broaden the spectral peaksfor neighboring wave cycles.

4. Wave Identification

[23] After decades of operation, satellites and groundmagnetometers have generated a vast amount of data.However, these data have not been fully explored forULF analysis. Identifying the ULF events in a few yearsof data recorded by a single satellite can already be aformidable task if it is done without a computer algorithmfor wave detection. This issue will likely be more serious

Figure 4. Field line resonance in the Pc 3–5 band observed by the ISEE-1 satellite on 24 November1977, showing (a) time series; (b), (c), and (d) STFT, wavelet, and WVD representations. The start time is2101:19 UT.


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when more multisatellite missions are in operation and theamount of data increase dramatically. The need to investi-gate a greater amount of data more efficiently motivated usto develop a WVD-based algorithm to identify wave eventsautomatically.[24] We propose that the selection of wave events in the

WVD can be performed in a very similar way to theselection in Fourier and wavelet spectrum. For Fourier-based analysis, one popular selection approach is based onthe comparison with background noise, which can bemodeled as either white noise or red noise. Because theFourier spectrum follows the chi-square distribution, we cancalculate the level of background noise for any assignedconfidence interval (such as 99%). Any spectral peak abovesuch confidence interval can be considered as a wave eventsince the probability that it is generated by noise is ex-tremely small. Torrence and Compo [1998] show thatwavelet-based analysis can follow the same procedure, butno study to our knowledge has demonstrated whether thesame approach is still valid for WVD-based analysis. Infact, the WVD has an important frequency marginal condi-tion that Z 1

�1Ws t;wð Þdt ¼ jS wð Þj2 ð4Þ

where S(w) denotes the Fourier transform of the signal s(t).Further integration with respect to the frequency, 1

2p

R1�1

jS(w)j2dw =R1�1 js(t)j2dt, yields the total energy of the

signal. In other words, the WVD can be considered as thesignal’s time frequency density function. When the intervalof time integration is long enough to cover the wave packetsof interest, we can use the finite integral

RDT

0Ws(t, w)dt to

approximate the spectrum as shown in equation (4). Wehave also confirmed, through Monte Carlo analysis, that thetime-integrated WVD follows the chi-square distribution,and the details are presented in Appendix A.[25] To illustrate an example of wave detection, we

applied the above method to the Pc 1 wave packets in thesame event presented in section 3.1. For ion cyclotronwaves in triaxial records, the wave power of interest is thepolarized power that is defined as the total power multipliedby the degree of polarization [e.g., Means, 1972]. Moredetails on the wave polarization expressed by WVD aredescribed in the next section and Appendix B. Figure 5ashows the WVD representation of polarized power and theselected wave events, based on a confidence interval at99%, that are denoted by black contour lines. All three wavepackets visualized in Figure 2, including the frequencysplitting in the third wave packet, are successfully selected.

5. Polarization Analysis

[26] For any spectral analysis method, it is desirable toestablish a formulation for polarization analysis to examinethewave properties frommultichannel data, such as the triaxialmagnetometer data recorded by satellites. The Fourier-basedalgorithm for polarization analysis has been widely used for4 decades [e.g., Fowler et al., 1967; Means, 1972], and it isespecially useful for studying the polarization properties of thePc 1 waves. The polarization analysis can also be based on

wavelets, and such approach has been applied to the studies ofseismic waves [e.g., Lilly and Park, 1995].[27] Here we demonstrate for the first time that the

Wigner-Ville distribution can be used to calculate thepolarization properties of waves. The polarization is oftenexpressed by the coherence matrix:

J ¼sx tð Þsx* tð Þh i sx tð Þsy* tð Þ

� �sx tð Þsz* tð Þh i

sy tð Þsx* tð Þ� �

sy tð Þsy* tð Þ� �

sy tð Þsz* tð Þ� �

sz tð Þsx* tð Þh i sz tð Þsy* tð Þ� �

sz tð Þsz* tð Þh i

24 35 ð5Þ

in which the brackets indicate average values within thecoherence time [see, e.g., Means, 1972]. If we define amatrix JW based on the cross WVD as

JW t;wð Þ ¼Wxx t;wð Þ Wxy t;wð Þ Wxz t;wð ÞWyx t;wð Þ Wyy t;wð Þ Wyz t;wð ÞWzx t;wð Þ Wzy t;wð Þ Wzz t;wð Þ

24 35; ð6Þ

it can be shown that, for a monochromatic wave, JW can bean alternative coherence matrix to calculate polarizationparameters. A simple proof for the two-dimensional case isprovided in Appendix B.[28] Using the same Pc 1 example, Figures 5b and 5c

show the details of the propagation angle (to the back-ground B) and the ellipticity for the same event. It is shownthat all three Pc 1 wave packets propagated in directionsroughly parallel or antiparallel to the background magneticfield. The ellipticity for the first two wave packets isbasically linear. The third wave packet, however, is moreleft-handed polarized. Although it is difficult to provide adefinitive explanation from single-spacecraft measurements,the difference in ellipticity for the third wavelet is consistentwith the conventional model of ion cyclotron waves in aplasma with multiple ion species [Perraut et al., 1984]:When a wave generated near the equator propagates tohigher latitudes, all the critical frequencies increase due tothe increasing magnetic field, and the originally left-handedpolarized wave becomes linearly polarized when the wavefrequency matches the crossover frequency. In this Pc 1event, the position of the Polar satellite could be in theregion where the wave frequency was close to the crossoverfrequency, and the satellite moved closer to the equatorwhere the wave was more left-handed.

6. Concluding Remarks

[29] In this study we present how the Wigner-Villedistribution can be used to study ULF waves. We havedemonstrated that the method can either better determine thetime dependence of wave frequency or reveal previouslylittle explored properties of ULF signals. Compared toSTFT and wavelets, the WVD is most powerful when it isapplied to the types of pulsations that contain rapid varia-tions in signal amplitude or phase, such as Pi 2, Pc 1packets, and the phase skips in Pc 3–4 waves. The WVDstill satisfies the Heisenberg-Gabor uncertainty principle,but it is free from any reduction in resolution due to the useof a windowing function. For completeness, we also pro-vided the formulation for wave detection and polarizationanalysis based on the WVD.


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[30] One caution in using the WVD is that the WVD ismore susceptible to trends than STFT or wavelets. TheWVD results can be seriously affected if the trend is notproperly removed beforehand. In most cases, the often-usedapproaches in trend estimation such as a linear fit or a low-order polynomial fit are sufficient. For some types of ULFwaves, such as Pi 2 pulsations, the trend can be the fastvarying magnetic field related to substorm onsets. In orderto maintain the consistency with the Wigner-Ville method, itis preferable not to remove the trend by Fourier-basedfiltering. One possibility is to decompose the time seriesinto multiple subsets, each at a different timescale, throughthe empirical mode decomposition (EMD) [e.g., Huang etal., 1998], and then calculate the WVD for the subsetsassociated with Pi 2 timescales. Although EMD is anappealing method and does not rely on any functional basis,there has been no theoretical foundation for it to interpretthe meaning of each subset. In this study we have demon-

strated that differencing the time series is a useful techniquein preparing Pi 2 events for WVD analysis, and it alsoprovides an additional benefit to better reveal the phaseskipping in the Pc 3–4 waves.[31] Another technical issue with using the WVD tech-

nique is its higher computation costs. Our tests show that,for a 512-point time series, the STFT and WVD calculationstake 0.075 s and 1.5 s on a PC with a 2.0 GHz CPU.Although WVD needs a computation time 20 times longer,it is still an affordable method for wave detection even onpersonal computers.[32] Last, we would like to address an often asked ques-

tion: ‘‘What is the best time frequency/timescale method touse?’’ As seen in our presentations, each of STFT, wavelet,and WVD has its own strengths and weaknesses, and oneshould not expect that any of these methods can clearlyexpress all the features in a wave signal. For analyzing waveswith known properties, the question is often reduced to

Figure 5. WVD representation of (a) polarized wave power, (b) angle of propagation, and (c) ellipticity.Contour lines in black or white denote the wave events selected for a confidence interval at 99%.


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identifying the simplest but sufficient method. For timeseries with unknown characteristics, we argue that oneshould generate all the three, and possibly other, timefrequency/timescale representations to make the best possi-ble interpretation of wave signals. This is a feasible approachnow considering that the needed computer algorithms andfast CPUs are readily available. We hope that this study canbe a start of such practice for the analysis of ULF waves andother types of wave signals.

Appendix A: Relationship Between WVD andChi-Square Distribution

[33] A commonly used method to identify wave events isthrough the comparison between the wave spectrum and thebackground noise. If a spectral peak is much stronger than

the background noise, it can be considered as a true waveevent because the probability for such hypothesis to be falseis minuscule. One necessary piece of knowledge to makesuch a comparison is the probability distribution of spec-trum. It is known that the Fourier spectrum of a time seriesfollows the chi-square distribution [Jenkins and Watts,1968], and the wavelet spectrum also has the same property[Torrence and Compo, 1998]. This Section demonstratesthat the time-integrated Wigner-Ville distribution also fol-lows the chi-square distribution.[34] The background noise can be modeled as white noise

xn ¼ zn; ðA1Þ

where zn follows the Gaussian distribution N (0, s2), or asred noise that follows the first-order autoregressive process,or AR(1),

xn ¼ axn�1 þ zn: ðA2Þ

[35] When a = 0, the equation is reduced to white noise.The spectral density function for the AR(1) process isknown to be

P fð Þ ¼ s2

1þ a2 � 2a cos 2pfð Þ : ðA3Þ

[36] Because of the frequency marginal condition of theWigner-Ville distribution (see equation (4)), the integrationof W(t, w) with respect to time is expected to follow thesame spectrum and the associated significance levels. Toverify this argument, we performed a Monte Carlo analysison 10,000 Wigner-Ville distributions derived from timeseries that follow equations (7) and (8). Owing to the useof analytic signals, the WVD values were divided by two forcomparison. The mean spectrum and the confidence inter-vals at 80%, 90%, and 95%, for both the time-integratedWVD the chi-square distribution, are shown in Figure A1.The comparison confirms that the Wigner-Ville spectrumclosely follow the chi-square distribution.

Appendix B: Polarization Parameters Expressedby Cross Wigner-Ville Functions

[37] It is shown here that the coherence matrix as shownin equation (5) can be expressed by cross Wigner-Villefunctions. If we consider two orthogonal wave fields, Hx

and Hy, of frequency w,

Hx ¼ a1 tð Þei wtþf1ð Þ

Hy ¼ a2 tð Þei wtþf2ð Þ

the coherence matrix J can be expressed as

J ¼ Hx tð ÞHx* tð Þh i Hx tð ÞHy* tð Þ� �

Hy tð ÞHx* tð Þ� �

Hy tð ÞHy* tð Þ� ��

¼ a21 tð Þ ei f1�f2ð Þa1 tð Þa2 tð Þe�i f1�f2ð Þa2 tð Þa1 tð Þ a22 tð Þ

" #: ðB1Þ

Figure A1. (top) Spectral density function for white noiseat the mean value and three different confidence intervals at80%, 90%, and 95%. Lines and symbol represent thetheoretical values calculated from the chi-square distribu-tion and the Monte Carlo results from the Wigner-Villedistributions of 104 time series samples. (bottom) The sameas the upper plot except for the red noise following anAR(1) process with a = 0.7.


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The quantity ai tð Þaj tð Þ is the average value over thecoherence time. The corresponding expression of J usingthe cross Wigner-Ville distribution can be written as

JW t;wð Þ ¼ Wxx t;wð Þ Wxy t;wð ÞWyx t;wð Þ Wyy t;wð Þ

� ; ðB2Þ

where

Wxx ¼Z

a1 t þ t2

� �a1 t � t

2

� �dt;

Wxy ¼ ei f1�f2ð ÞZ

a1 t þ t2

� �a2 t � t

2

� �dt;

Wyx ¼ e�i f1�f2ð ÞZ

a2 t þ t2

� �a1 t � t

2

� �dt;

Wyy ¼Z

a2 t þ t2

� �a2 t � t

2

� �dt:

[38] To compare the values of J and JW, we can consider asimple case where the signal has constant amplitude over afinite time such that ai(t) = ai for �T

2� t � T

2and ai(t) = 0

otherwise. The elements in J are simply ai tð Þaj tð Þ = aiaj.

One can also show that, for the integrals in JW,R1�1 ai(t +

t2)

aj(t � t2) dt = aiaj (2T � 4jtj) for �T

2� t � T

2. For each t 2

[�T2, T

2], JW is equal to J multiplied by a constant, and

therefore polarization parameters derived from JW remainthe same. This result also indicates that the matrix expressedby the cross Wigner-Ville functions (as equation (11)) canbe used to calculate the properties of wave polarization.Several commonly used parameters are as follows:

B1. Propagation Angle

[39] Following Means [1972], we can use the Wigner-Ville coherence matrix (in three dimensions) to obtain thecomponents of the wave normal vector k = [kx, ky, kz]

kx ¼JW ;yz

K; ky ¼

�JW ;xz

K; kz ¼

JW ;xy

K; ðB3Þ

in which K = (JW,yz2 + JW,xz

2 + JW,yz2 )1/2. The direction of the

wave propagation with respect to B is

qkB ¼ cos�1 k � B� �

: ðB4Þ

B2. Polarized Power

[40] Most polarization parameters are more convenientlycalculated in the coordinate system where z0 = k, y0 = z0 � B,and x0 = y0 � z0. The coherence matrix in this coordinatesystem can be obtained through the similarity transformation

J0W ¼ R�1JWR; ðB5Þ

which R is the rotation matrix that connects the originalcoordinates (x, y, z) and the new coordinates (x0, y0, z0). If

the 2 � 2 submatrix of J0W that contains elements for x0 andy0 is denoted as J0W[x

0, y0; x0, y0], the degree of polarization his

h ¼ 1� 4jJ0W x0; y0; x0; y0½ �j

J 0W ;x0x0 þ J 0W ;y0y0

� �2

2643751=2

ðB6Þ

and the polarized power is the total power multiplied by thedegree of polarization

hTr JWð Þ: ðB7Þ

B3. Ellipticity

[41] The auxiliary angle b = tan�1(ay0/ax 0) can beexpressed as

sin 2b ¼2Im J 0W ;x0y0

� �J 0W ;x0x0 þ J 0W ;y0y0

� �2

�4jJ0W x0; y0; x0; y0½ �j� �1=2

; ðB8Þ

and the ellipticity is

� ¼ tan b: ðB9Þ

For right-handed waves, k � B > 0 and this quantity remainsthe same. For left-handed waves, however, k � B < 0 and theellipticity is multiplied by �1.

[42] Acknowledgments. This work was supported in part by NASANAG5-12953 and the Polar project. The magnetometer from the IslandLake station is provided by the Canadian Space Science Data Portal. Wethank the authors of the tftb toolbox for developing the computer programsfor time-frequency calculations.[43] Amitava Bhattacharjee thanks Lars Blomberg and Fouad Sahraoui

for their assistance in evaluating this paper.

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Use of the Wigner-Ville distribution in interpreting and

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