J. Geophys. Res., 103, 6503-6511, 1998

Shock profile analysis using wavelet transform

M. Gedalin

Department of Physics, Ben-Gurion University, Beer-Sheva, Israel

J. A. Newbury and C. T. Russell

Institute of Geophysics and Planetary Physics, Universityof California, Los Angeles, USA

Abstract. We study the fine structure of the collisionless shock front,payingparticular attention to the large gradients and internal structure within the rampand to quasiperiodic structures in the ramp vicinity. In order to separate randomsteep gradients with low amplitudes from steep large-amplitude formations, weapply Gaussian wavelet transforms suitable for this task. We analyze several highMach number shocks and show that ramp substructure is not uncommon and thatthe typical scales of the steepest gradient features are significantly smaller thanthe shock transition width (provided they are stationary inthe shock frame). Weapply the Morlet wavelet transform to the magnetic field datain order to identifyquasiperiodic patterns which may last only several periods. It is likely that shortquasiperiodic wave trains dominate in the wave activity. Such wave trains are notalways associated with the ramp itself.

1. Introduction

Shock transition layers are very peculiar in the sense thatthe typical scale of the quasi-stationary magnetic field pro-file varies in them by two orders of magnitude. In particular,scales in supercritical, high Mach number shocks vary fromthe largeVu=u scale size of the foot [Woods, 1969;Sck-opke et al., 1983] (whereVu is the upstream plasma veloc-ity andu is the upstream ion gyrofrequency) to the small� =!pe scale found in the ramp [Newbury and Russell,1996] (where!2pe = 4�ne2=me is the electron plasma fre-quency). The magnetic field profile of the low Mach num-ber shock is generally laminar, containing one large, essen-tially monotonic increase in magnetic field at the ramp witha width� =!pi and a whistler precursor with the typicalwavelength2� os �=(M2� 1)1=2!pi [Russell et al., 1982;Mellott and Greendstadt, 1984;Mellott and Livesey, 1987;Farris et al., 1993] (hereM is the Alfvenic Mach number ofthe upstream plasma,� is the angle between the shock nor-mal and upstream magnetic field, and!pi is the ion plasmafrequency).

The high Mach number shock, however, has a much nar-rower ramp [Scudder et al., 1986; Newbury and Russell,1996], and, as a rule, strong wave activity is superimposed

on what is usually believed to be its stationary structure. Thiswave activity consists of large-amplitude, large-scale mag-netic field spikes in the ramp itself. It is not known whetherthese waves are, in fact, quasi-stationary (phase standing),are transient, or are representative of the intrinsic shockfrontdynamics [Krasnosel’skikh, 1985;Galeev et al., 1988]. Up-stream whistlers are believed to be associated with upstreamion beams [Hoppe et al., 1982] or generated in the ramp it-self [Krasnosel’skikh et al., 1991;Orlowski et al., 1995;Du-dok de Wit et al., 1995]. Finally, there is a large amountof activity in the magnetic field downstream of the shock,including a series of large-amplitude overshoots and under-shoots and a number of wave modes, convected across theramp or generated locally [Lacombe et al., 1990]. A part ofthis “wave activity” may reveal an ordered pattern and be apart of the fine structure of the shock front (scales less thanVu=u). Other activity may be random turbulence causedby microinstabilities [Wu et al., 1984]. The separation of thequasi-stationary structure of the shock from the time-varyingstructure and the determination of the scales in the shockfront are among important and most difficult problems ofshock physics. The second problem, the conversion of theobserved time series into a spatial profile, also requires re-liable knowledge of the shock normal and shock velocity,

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GEDALIN ET AL.: SHOCK ANALYSIS USING WAVELETS 2

uncertainties in both of which can produce additional errorin scale measurements. However, even the determination oftypical temporal scales in the measured time series encoun-ters serious problems, related to the rapid variation of thelarge-scale magnetic field, strong inhomogeneity along theshock normal, and the absence of a clear distinction accord-ing to frequency (because of Doppler shifts, different mech-anisms working at the same frequencies, etc.).

The technique commonly used to determine ramp posi-tion and width is to visually mark the ramp beginning andend. This procedure becomes very subjective, particularlywhen the shock transition is not a sharp monotonic mag-netic field increase but shows several successive jumps withcomparable amplitude or is “spoiled” with a large-amplitudewave superimposed on it. Similarly, the Fourier spectrum,which is quite successful far upstream, is not appropriate inthe shock front itself, since some quasiperiodic patterns mayonly last for several periods, while higher-frequency wavesare definitely influenced by varying ambient magnetic fieldand plasma (for example, typical gyrofrequencies are spa-tially dependent). Moreover, studying waves with the typi-cal wavelengths of several =!pe in the ramp vicinity usingFourier analysis would require averaging over a much largerspatial length, with the ramp itself contributing substantiallyin the same frequency range.

Techniques involving wavelets have been used exten-sively for turbulence analysis in other fields for some time(seeArgoul et al. [1989]; Everson et al. [1990]; Meneveau[1991]; Farge [1992]; Liu [1994], andChakraborty andOkaya [1995] for a rather random selection of references),but they are relatively new to space physics. Wavelet analy-sis is capable of providing information about the scale of anevent and its position altogether, within the limits of the un-certainty principle. Wavelet analysis is also capable of trac-ing localized structures, whereas the Fourier transform ofwhich has a broadband spectrum. Application of the discreteDaubechies wavelet transform to the magnetosonic wavesteepening byMuret and Omidi [1995] is an excellent exam-ple. Recently, wavelet transforms were successfully appliedto the analysis of upstream waves whereDudok de Wit etal. [1995] applied the Morlet wavelet transform. The Mor-let transform is especially useful for the analysis of wavesbut not very useful for the analysis of localized structuressuch as the shock ramp or magnetic spikes. Other wavelets[Lewalle, 1994] are applicable to this task.

In the present paper we apply wavelet transforms to thelocalized structures and large-amplitude quasiperiodic wavetrains found in the collisionless shock front. In particu-lar, we focus on the vicinity of the ramp, where the inho-mogeneity of the magnetic field is the strongest. We usewavelets formed from Gaussian derivatives to determine the

position and spatial duration of the ramp and any meaning-ful large-amplitude structures within the ramp. We also usethe Morlet wavelet to measure the period of any quasiperi-odic structures. The main objective of the present study is todetermine whether the shock front (ramp, in particular) con-tains any fine-scale substructure and to determine physicallymeaningful scales of these formations. We are also going todemonstrate the capabilities of the wavelet transform and tointroduce this powerful tool by applying it to observationsof bow shocks made by the ISEE 1 and 2 spacecraft and ex-tracting information which would be otherwise difficult toobtain. The paper is organized as follows: In section 2 wediscuss our selection of wavelets and calibrate the wavelettransforms based on a model magnetic field profile. In sec-tion 3 we apply wavelet transforms to several observed bowshock profiles. The results are discussed in section 4.

2. Wavelet Choice and Calibration

In contrast with the Fourier transform, which providesthe frequency spectrum of a time series by comparing itwith a monochromatic wave of infinite length, the wavelettransform compares a time series with compact forms of es-sentially finite length and thus provides information aboutboth scale (inverse frequency) and position (time) where thisscale is encountered in the data. Wavelet analysis is basedon the convolution of the time series with a family of well-localized functions of the same shape but different width[Farge, 1992]. The mother wavelet (t) (the function whichdetermines the basic shape and from which the others arederived) should satisfy the following admissibility criteria:Z 1�1 (t)dt = 0; Z 1�1 j (t)j2dt <1: (1)

After a mother wavelet is chosen, the whole wavelet familyis obtained using translations and dilations as follows: (a; l; t) = 1pl ( t � al ); (2)

wherea is the position (or time, which shows the position ofsome reference point at the wavelet shape relative to the timeseries) whilel is the scale (or duration, which is the measureof the wavelet width).

The wavelet transform of a functionf(t) gives a two-dimensional function in the time-duration space, accordingto the following prescription:Wg[f ℄(a; l) = Z 1�1 f �(a; l; t)dt: (3)

The choice of the proper mother wavelet (t) is dictatedby our needs. The identification of the ramp is equivalent to

GEDALIN ET AL.: SHOCK ANALYSIS USING WAVELETS 3

the search for the steepest gradients with the largest mag-netic field jump. The first and second derivatives of theGaussian (the latter also known as the Mexican Hat wavelet)g1(t) = t exp(�t2=2); (4)g2(t) = (t2 � 1) exp(�t2=2); (5)

are the appropriate mother wavelets for this task [Farge,1992;Lewalle, 1994]. The first is especially sensitive to gra-dients,df(t)=dt, while the latter is sensitive to inflections,d2f=dt2. The corresponding wavelet families, according to(2), look as follows:g1(a; l; t) = t� apl3 exp[� (t� a)22l2 ℄; (6)g2(a; l; t) = 1pl [( t� al )2 � 1℄ exp[� (t � a)22l2 ℄: (7)

For the analysis of periodic structures or quasi-monochrom-atic waves we choose the Morlet wavelet:m(t) = exp[(2�it� t2=2)℄; (8)m(a; l; t) = 1pl exp[2�i(t� a)l � (t� a)22l2 ℄; (9)

which is simply a monochromatic wave with the frequency!0 = 2� and an amplitude modulated with a Gaussian en-velope. The two real waveletsg1(t) andg2(t) and the realpart of the Morlet waveletRem(t) are shown in Figure 1. In

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what follows, we shall plotPg(a; l) = jReWg [a; l℄(f)j2=l.For g1 andg2 wavelets, which are real, this quantity is thewavelet energy density, since the total energy isE = Z jW [a; l℄j2dadll : (10)

For the Morlet wavelet transform the interpretation is notso straightforward, although it is clear that by normalizingon l we preserve the proper energy density scaling. Takingonly the real part ofWm allows us to also retain phase in-formation (correct sequence of maxima and zeroes), whichis desirable when periodic structures are analyzed.

Analytical expressions describing the above transformscan be obtained only for sufficiently simple functions,whereas we are planning to apply the technique to rathercomplicated bow shock observations. To validate use ofwavelet transforms for quantitative analysis of shock pro-files, we shall calibrate this technique using a model profile.With the following “building blocks”:B0 = B1 +B2 tanh[1:5(t� t0)T0 ℄; (11)A0 = a1 sin[2�(t� t1)T1 ℄ + a2 sin[2�(t� t2)T2 ℄; (12)C0 = exp[(t3 � t)(t� t4)T 23 ℄; (13)

it is possible to construct profiles that reasonably resemblethe measured shocks, with the proper choice of parameters.HereB0 describes atanh profile of the width2T0, andA0adds two superimposed waves with periodsT1 andT2 andrelative amplitudesa1 anda2 and phasest1 and t2, whileC0 provides damping into upstream and downstream. Threesuch profiles with the corresponding wavelet transforms areshown in Figures 2-4. The model profile in Figure 2 is the

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Figure 2. (top to bottom) Model shock profile (consisting oftanh profile and superimposed monochromatic wave of con-stant amplitude),g1 transform,g2 transform, andm trans-form.

simplest. It is constructed asB = B0+A0, with the follow-ing choice of parameters:B1 = 2,B2 = 1, t0 = 32, T0 = 1,

GEDALIN ET AL.: SHOCK ANALYSIS USING WAVELETS 4

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Figure 3. (top to bottom) Model shock profile (consisting oftanh profile and superimposed monochromatic wave withthe peak amplitude proportional to thetanh field and damp-ing into upstream and downstream),g1 transform,g2 trans-form, andm transform.a1 = 0:35, t1 = 0:3, T1 = 1, anda2 = 0. It consists of thetanh-like profile with a 2 s width (we measure “time” in sec-onds for convenience), centered att = 32 s, and one mono-chromatic wave with constant amplitude and period 1 s. Forthe wavelet transform representation the filled contour plot(30 contours per whole range) ofPg(a; l) is chosen, whereais time andl is duration. The magnitude ofPg(a; l) is shownby color. Since the exact value of this magnitude is not ofdirect significance here and the significant part of the infor-mation is contained in the pattern of the wavelet transform,the gray scale representation is chosen. As shown below, theposition and shape of dark regions allows us to draw con-clusions about the features of the profile. The so called in-fluence cone (dark conical pattern which is wide for largedurations and becomes progressively narrower for smallerdurations) corresponds to the ramp transition. This cone uni-formly converges to the width 2 s (betweent = 31 s andt = 33 s) for small durations. Thus the ramp width is givenby the width to which the influence cone tends in the limitof small durations. Finer features (like two small “tongues”growing from the cone near the horizontal axis) correspondto the distortions of the profile.

The same width can be found from theg2 transform dia-

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,sFigure 4. (top to bottom) Model shock profile (consisting oftanh profile and two superimposed monochromatic wave,each having the peak amplitude proportional to thetanhfield and damping into upstream and downstream),g1 trans-form, g2 transform, andm transform.

gram, where two influence cones converge to approximatelythe same 2 s distance between them. Both transforms ac-knowledge the presence of the large amplitude monochro-matic wave as a sequence of gradients (g1 transform) or ex-trema (g2 transform). The last diagram shows them trans-form, which is not especially sensitive to thetanh profile (itis seen as the structures converge towardt = 32), but clearlyidentifies the monochromatic wave, even within the ramp, asa horizontal stripe of “ellipses” (two per each wave period)with the same duration 1s, exactly equal to the wave periodin the model.

The model profile in Figure 3 is obtained asB = B0(1+A0C0), with the following choice of parameters:B1 = 2,B2 = 1, t0 = 32, T0 = 1, a1 = 0:35, t1 = 0:7, T1 = 2:4,a2 = 0, t3 = t4 = 32, andT 23 = 100. It shows the sametanh profile upon which a wave is superimposed with theperiod 2.4 s and peak amplitude proportional to the undis-turbed field magnitude. The wave damps into upstream anddownstream as well. Theg1 transform now shows a slightlysmaller width, which is the result of the accidental interfer-ence of the ramp and wave, as seen from the top diagram.The same changes are in theg2 diagram. Them diagramagain clearly shows the wave period, although now the inter-

GEDALIN ET AL.: SHOCK ANALYSIS USING WAVELETS 5

ference with the ramp is strong. The transforms also showthe wave damping.

In Figure 4 a rather pathological case is presented wherethere are two waves with the periods 1 s and 2.4 s and compa-rable amplitudes, which are proportional to the undisturbedtanh field, and definite phase difference. The waves dampinto upstream and downstream, but the damping starts at theovershoot behind the ramp. This model is obtained as abovewith the sameB0 and other parameters chosen as follows:a1 = 0:55, t1 = 0:7, T1 = 2:4, a2 = 0:35, t2 = 0:3,T2 = 1, t3 = 40, t4 = 36, andT 23 = 150. Despite the verystrong interference between the waves andtanh field, theg1 transform is quite successful in the identification of theramp width, showing the large scale� 3 s and small scale� 1 s. One may be tempted to state that the true width isbetween the two scales. However, with such large wave am-plitude it is difficult to speak about stationary structure onwhich wave activity is superimposed. Rather, if the wave ismoving relative to thetanh profile, the shock itself shouldbe considered as nonstationary with the typical scale varyingbetween the determined large and small scales. Theg2 trans-form diagram supports the above scale estimates, althoughin this case it is less clear and less useful. Them transformidentifies both waves, despite the very strong interferencebetween them and thetanh field (it should be understoodthat the coupling is nonlinear as follows from the construc-tionB = B0(1 + A0C0)).3. Shock Analysis

The above model examples show that the selected wavelettools are, indeed, appropriate for shock profile analysis. As afurther demonstration we now apply the technique to the lowMach number, November 26, 1977, 0610 UT shock whichhas been comprehensively analyzed earlier by conventionalmethods. The shock profile and the corresponding wavelettransforms are shown in Figure 5. The top diagram shows 2min measurements of the total magnetic field, measured here(and hereafter) in nanoteslas. Time is measured in seconds,with t = 0 corresponding to 0610:00 UT. This shock is a typ-ical laminar, quasi-perpendicular, low Mach number shockwith a Mach numberM = 2:7, kinetic to magnetic pressureratios�i = 0:16 and�e = 0:36, and the angle between theshock normal and upstream magnetic field�Bn = 67Æ. Theion inertial length is calculated to be 58.2 km. The shocknormal was determined from coplanarity. The separation ofthe ISEE 1 and 2 spacecraft along the shock normal is deter-mined to be19:37� 0:50 km, and the velocity of the shockfront along the normal direction is5:5 � 0:8 km/s. The di-mensionless spatial separation along the shock normal corre-sponds toLN � 0:3( =!pi), while the separation along the

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,sFigure 5. (top to bottom) Total magnetic field profile for lowMach number shock of November 26, 1977,g1 transform,g2transform, and Morlet transform.

shock front isLT � 0:8( =!pi). The ramp duration for thisshock, determined directly from the shock profile, is about10.8 s which corresponds to1:02� 0:15 =!pi [Newbury etal., 1997]. No substantial wave activity is observed to inter-fere with the ramp, except for the small bump nearly in themiddle of it. No prominent features are seen in the turbu-lence except a visually periodic structure betweent = �59andt = �30 s.

The g1 transform diagram clearly shows the ramp posi-tion and width. Owing to the low level of the wave ac-tivity the pattern is almost as clean as the model profile.The wavelet determined width is slightly less than the abovementioned 10.8 s and is rather the scale of the steepest partof the shock front than the width of the transition from up-stream to downstream magnetic field. It is interesting to notethat the transform provides the second scale correspondingto the bump inside the ramp, despite its rather low amplitude.This sensitivity of the tool happens because of the absenceof large-amplitude fluctuations.

The third diagram shows theg2 transform, which is ex-pected to be sensitive tod2B=dt2. Indeed, the contoursclearly converge toward the upstream edge of the ramp andto the bump in the middle of the ramp, where the gradientof the gradient is expected to be the largest. Usually theg2transform contains few new details compared to theg1 trans-

GEDALIN ET AL.: SHOCK ANALYSIS USING WAVELETS 6

form and is used here as a double check.

The final Morlet transform diagram is the most informa-tive. It shows the features of the wave activity. The upstreamwave train is clearly identified as a 5 s periodic structure be-tween t = �50 and t = �30 s. The period of the wavesuddenly increases almost by a factor of 2 neart = �50 s.For the parameters of this shock the wavelength of the phasestanding whistler�W = 2� os �Bn=(M2 � 1)1=2!pi cor-responds to the observed period of about 9.8 s, which is sub-stantially larger than the period of the wave train in the partwhich is closer to the ramp. Moreover, the wave train is notconnected to the ramp at all. Some quasiperiodic activity isgenerated at the ramp, including the wave train with the tem-poral length of 20 s and gradually decreasing period from theramp into downstream. Another wave train with the periodnear 6 s starts about 40 s beyond the ramp. In general, thewave activity down to periods 1 s (frequencies below 1 Hz)seems to consist mainly of distinct short wave trains whichlast from 4 to 10 wave periods.

It is interesting to compare the Morlet transform with thedynamic Fourier spectrum shown in Plate 1. In Plate 1 on thelog(f[Hz℄)

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log(P[nT2 =Hz℄)Plate 1.Dynamic spectrum for 2 min of magnetic field mea-surements of November 26, 1977, 0610 UT shock profile.

vertical axis is the logarithm of the frequencyf , which is re-lated to the duration asf = 1=duration, so thatlogf = 0corresponds to 1 s duration (period) in the lowest diagramof Figure 5. Fluctuations at 10 s period that would lie atthe top of the Morlet diagram (Figure 5) lie at the bottomof Plate 1. The upstream power spectrum shows a kind ofa plateau which extends up to the frequency 0.4 Hz, whichcorresponds to the period 2.5 s. The visual sharp increase ofthe wave activity near 0610 UT corresponds to the broad-band contribution of the ramp into the Fourier transform.The downstream power spectrum decreases monotonically

toward higher frequencies. The dynamic spectrum analysisdoes not provide the information about the periodic wavetrains in either shock parts (neither upstream nor down-stream) and therefore is not quite appropriate for the descrip-tion of the wave activity in the shock front.

Besides the physical picture of the low Mach numbershock structure, the above application of the wavelet trans-form has provided us with additional calibration of the toolwhich can now be applied to high Mach number shocks,where other methods are not quite successful. We start withthe nearly perpendicular (�Bn = 86Æ), high Mach number(M = 4:0), moderate beta (� = 0:91) shock, observed onAugust 2, 1980, at 0858 UT. For this shock the ramp positionand width are determined quite satisfactorily using conven-tional methods (within the uncertainty of the determinationof the normal vector). The magnetic field profile for thisshock and the corresponding wavelet transforms are shownin Figure 6. The magnetic field diagram covers 1 min of

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Figure 6. (top to bottom) ISEE 1 measured total magneticfield profile for high Mach number August 2, 1980, 0858 UTshock,g1 transform,g2 transform, and Morlet transform.

ISEE 1 measurements centered at the ramp crossing, witht = 0 corresponding to 0858:50 UT. The overall transitionfrom the relatively quiet upstream state to the downstreamstate with a low level of turbulence occurs within 20 s (fromt = �5 s tot = 15 s). This transition includes several dis-tinguishable features, such as the steep upstream precursorneart = 14 s and a series of steep elevations of the mag-

GEDALIN ET AL.: SHOCK ANALYSIS USING WAVELETS 7

netic field, the steepest of which is the jump by a factor of> 2 neart = 0 s, which occurs during less than a second.The second diagram shows theg1 transform of this profile.It is clearly seen that two scales are present in the transform:the big scale� 8 s, associated with the transition as a whole,and a small scale� 2 s, which identifies the two successivejumps neart = 0 s, a single steep transition. The precursorat t = 14 s stands alone. The same picture is seen from theg2 transform, which very clearly indicates the small scalewithin the ramp and standing alone precursor. Them trans-form is a little messy, showing that a number of frequenciesare present in the local spectrum and none of them domi-nates, except probably the� 3 s periodicity, which could berelated in part to the spacecraft 3 s rotation. It is worthwhileto note that the precursor att = 14 s is clearly seen in allthree diagrams, including the Morlet diagram, where darkshapes converge to this position starting at large durations.

In the previous case of a low Mach number shock we re-stricted ourselves only to the ISEE 1 measurements, sincethere was no internal structure of the ramp, which would re-quire additional stationarity analysis. In the present case theexistence of two scales requires analysis as to whether thisis an occasional transient feature or should be consideredas a part of the stationary shock structure. Figure 7 showsthe ISEE 2 measured profile and the corresponding wavelet

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Figure 7. (top to bottom) ISEE 2 measured magnetic fieldof the high Mach number August 2, 1980, 0858 UT shock,g1 transform,g2 transform, and Morlet transform.

transforms. The time separation for the two spacecraft wasfound earlier to be 7.5 s, which is less than, but of the or-der of, the upstream ion gyroperiodTi = 2�=u � 12 s.The spatial separation along the shock normal isLN �0:3( =!pi) and along the shock front isLT = 0:9( =!pi).Visually, the profiles are slightly different, although thesharp magnetic field increase in two successive jumps looksalmost identical to that measured by ISEE 1. Indeed, theg1transform shows that this small-scale transition is the moststable feature in the ramp, and its width,� 2 s, does notchange noticeably at the timescale of the order of the ion gy-roperiod thus corresponding to� 0:2( =!pi) in the shockframe. Comparison of the position of this transition at theISEE 1 and ISEE 2g1 transforms confirms well the aboveestimate of the time separation. The larger scale is less pro-nounced because of the appearance of substantial turbulencein the upstream part of the transition betweent = 10 sandt = 17 s. The precursor is still clearly seen on theg1transform diagram and on other diagrams too, although lessclearly, apparently because of the interaction with waves.Itis seen that the position of the precursor relative to the rampdoes not change, which means that it is standing in the shockframe. The most substantial difference from the ISEE 1 pro-file is in them diagram, which shows much less energy in 3 spulsations and much less turbulent ramp. However, it shouldbe understood that the plotted wavelet transforms show rel-ative energy content, which simply means that ISEE 1 mea-sured field contains more energy in the low-frequency partof the spectrum than the ISEE 2 profile. It is worth men-tioning also that ISEE 1m transform shows patterns similarto damping wave trains in Figures 3 and 4, suggesting thatthe 3 s pulsations originate at the ramp and damp into down-stream and upstream. An interesting feature is observed inthe Figure 6m transform diagram betweent = �18 s andt = �5 s, where the wave train with the period of� 2 sseems to be excited (its amplitude increases fromt = �5 stowardt = �10 s) and damped subsequently, with the ac-companying frequency increase.

This shock does not exhibit a classical foot. Associat-ing the upstream 15 s with the reflected ion caused magneticfield perturbations we find its length as� 1:5( =!pi) �0:4(Vu=u) in agreement with what could be expected.

Finally, we apply the wavelet transform to the shockcrossing observed on August 4, 1978. This shock also hasa high Mach number (M = 4:9), is a quasi-perpendicular(�Bn = 62:9Æ) shock, and has a moderate� of 0.87. Thespacecraft separation along the shock normal isLN = 202�25 km � 2( =!pi), the separation along the shock front isLT � 2( =!pi), and the shock velocity is11:9� 1:6 km/s.The ion inertial length is determined to be91:2 km. Theprofile measured by ISEE 1 (1 min of measurements cen-

GEDALIN ET AL.: SHOCK ANALYSIS USING WAVELETS 8

tered at 1809:30 UT) is presented in Figure 8 along with

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Figure 8. (top to bottom) ISEE 1 measured total mag-netic field profile for the high Mach number August 4, 1978shock,g1 transform,g2 transform, and Morlet transform.

the corresponding wavelet transforms. The complexity ofthe profile (three successive magnetic field jumps, each onewith Bmax=Bmin � 3) requires comparison with the ISEE2 profile, shown in Figure 9. The two profiles do not seemidentical, especially since the ISEE 2 profile shows only twomagnetic field jumps, and the shock is clearly not station-ary (at least not as stationary as the previous two). How-ever, the presence of some distinct downstream features inboth profiles suggests that a common quasi-stationary pat-tern exists. Indeed, comparison of the twog1 transformsshows a remarkable stability of the pattern, consisting of theramp and two downstream features. The relative temporalseparations between two successive features changes by asmuch as 2 s during the� 18 s lasting between the ISEE1 and ISEE 2 measurements (about two upstream ion gy-roperiods), suggesting that this pattern is stationary within20% error. It seems that the second jump in the ISEE 1 pro-file, which itself lasts less than 2 s, is slowly overtaken bythe magnetic field behind it and merges with it to result inthe more smooth ISEE 2 profile. In any case the velocity ofthese steepest transitions in the spacecraft frame is very closeto the shock velocity, so that the sharpest gradient scale canbe estimated as� 0:1( =!pi) or� 4( =!pe) thus approach-ing the typical electron width. These shock “fingerprints”

0 5 10 15 20 25 30 35 40 45 50 55 60

2

4

6

2

4

6

2

4

6

0 5 10 15 20 25 30 35 40 45 50 55 60

10

20

30

40

Time, s

jBjD

ura

tion

,sD

ura

tion

,sD

ura

tion

,sFigure 9. (top to bottom) ISEE 2 measured total mag-netic field profile for the high Mach number August 4, 1978shock,g1 transform,g2 transform, and Morlet transform.

are also seen quite clearly in theg2 diagrams. Comparisonof them transforms suggests that large-amplitude perturba-tions start att = 20 s (ISEE 1) and close up to the ramp(ISEE 2) while steepening. The steepening is seen in theISEE 1 pattern as the period decrease and afterward in theISEE 2 pattern as a much stronger converging pattern in theramp vicinity. We also speculate that the following strongfeature, observed in the downstream att = 35 s (ISEE 1),will also steepen towards the ramp and merge with it. It ispossible that this behavior results in slow partial recyclingof the ramp structure on a timescale larger than the ion gy-roperiod. The corresponding foot duration for this shock canbe estimated as about 8 s, which gives the length of about =!pi � 0:2(Vu=u), which is less than typically observed.It may be due to the slow motion of the substructure. Sinceion reflection occurs in the tail of the ion distribution, it maybe sensitive to the fine structure of the shock profile.

4. Conclusions

We have analyzed the fine structure of three shocks, com-bining conventional direct analysis of the measured mag-netic field profile with the wavelet transform of this profile.The proper choice of the transforming wavelet (in particular,the first and second derivatives of the Gaussian) allows forthe determination of ramp position and scale. Application

GEDALIN ET AL.: SHOCK ANALYSIS USING WAVELETS 9

of the method to three shock observations of different types(a quasi-laminar, low Mach number shock, a high Machnumber shock with a clearly identified ramp, and a highMach number shock with a structured ramp) has success-fully and unambiguously determined both the position andthe physically meaningful scales of the shock front, whichprovides a substantial level of independence from the an-alyst’s eye. Encouragingly, the wavelet transform ignoreslarge-amplitude, large-gradient alternating fluctuations andfocuses on the structures which exhibit the whole complexof features: large-amplitude and large-gradient and clearlynonoscillating character. This allows one to interpret thesestructures as parts of the shock profile itself (stationary ornonstationary) and not as superimposed wave activity, gen-erated by some instabilities. This conclusion seems to befirmly supported even in the case of the highly structuredshock, which has several magnetic field jumps instead of aclear visually identified monotonic ramp. It is impossibleto conclude, in general, only from the wavelet transform,whether these structures are standing or moving in the shockframe. In the present case this task has been successfully ac-complished, since the wavelet transform provided us withthe shock “fingerprints” almost identical for both ISEE 1and 2 measurements. The substructure being almost sta-tionary, the wavelet-transform-determined scale of the maxi-mum gradient accompanied by the maximum magnetic fieldjump (actually the transform performs optimal search onthese two parameters) appears to be� 0:1 � 0:3( =!pi),based on the two cases of high Mach number shocks.

It is appropriate here to briefly discuss possible implica-tions of the observed short duration, large-amplitude mag-netic field jumps to the general interpretation of shock mea-surements, even when it is impossible to conclude abouttheir stationarity in the shock frame. Let us for simplicityassume that the shock profile is one-dimensional. Then, themagnetic field in the shock frame isB = B(x; t), where thedependence on time is weak if the shock is quasi-stationary.In the spacecraft frame one would measure the magneticfield Bs = B(x + Vsht; t), whereVsh is the shock veloc-ity in the spacecraft frame (for conveniencex is directedfrom upstream into downstream, and the shock velocity inthe spacecraft frame is in fact�Vsh, Vsh > 0). The timevariation of the magnetic field in the spacecraft frame isdBsdt = Vsh �B�x + �B�t ; (14)

where the first term on the right-hand side describes thestructure crossing by the spacecraft, while the second oneis due to the temporal variations of the nonstationary fieldin the shock frame. We shall now approximate each term in(14) by simply dividing the magnetic field change over the

corresponding time (or length) at which this change occurs:�Bs�s = Vsh�BL + �Bt�t ; (15)

where subscriptt means that the variation is caused by in-trinsic variability. Let us now assume that the spacecraft-observed magnetic field variation�Bs is of the order of thetotal magnetic field jump (which is what we have lookedfor in the above analysis). In this case,�B + �Bt ��Bs. The interpretation of the data depends on the rel-ative contribution of the two terms on the right-hand sideof (15). If the first term dominates, that is,�B � �Bsand�Bt=�Bs � Vsh�t=L, then the spacecraft-observedmagnetic field change is due to the quasi-stationary struc-ture crossing, andL � Vsh�s, as usual. However, if thesecond term is at least comparable with the first one, it ap-pears that�t �< �s, in which case the ramp is nonstationarywith the variation timescale not larger than the observed du-ration of the considered large-scale structure. In the lattercase the observed durations of� 1 s (as for example, about2 s for the gradient scale for both high Mach number shocksconsidered here) mean that the ramp would be nonstation-ary at the time scale of about0:1 of the ion gyroperiod. Thephysically meaningful definition of shock stationarity wouldrefer to particle motion within the transition layer. Ions andelectrons traverse the ramp of the widthL � ( =!pi) inaboutL=Vu � 1=Mu, so that the fields in the ramp arestationary or marginally nonstationary for them. However,ittakes about half a gyroperiod for a reflected ion to come backto the ramp, so in the nonstationary shock case they shouldbe considered as scattered by essentially a randomly movingramp transition, which may alter the reflection conditionsand the distribution of reflected ions. Finally, strong nonsta-tionarity would require the reconsideration of the fields in-side the shock transition, including the relation between thenormal incidence frame and de Hoffman-Teller frame cross-shock potentials.

The above analysis shows that the shock front is non-stationary, although in this particular case nonstationarity isweak, and the measured duration can be converted into spa-tial scales, which appear to be quite small. It, however, doesnot exclude completely the possibility of fast reforming inother cases, taking into account the variety of shock parame-ters and profiles.

In addition to the scale analysis we have applied the Mor-let wavelet transform for a preliminary study of quasiperi-odic patterns near the ramp. In all studied cases the waveactivity seems to be dominated by short wave trains withclearly identified periodicity. Some of these wave trainsoriginate at the ramp, and their periods correspond to theshortest scale within the ramp. There are indications of wave

GEDALIN ET AL.: SHOCK ANALYSIS USING WAVELETS 10

excitation and damping downstream. In the case of a non-stationary shock them transform pattern suggests that large-amplitude downstream formations approach the ramp, pro-gressively steepening and overtaking it eventually. Thesequestions require more thorough investigation and are be-yond the scope of the present paper. Finally, the above analy-sis inevitably brings us to the conclusion that whether lowMach number shocks are more or less alike, there is no suchthing as a “typical” high Mach number shock.

Acknowledgments. The research was supported in part bythe Binational Science Foundation under grant 94-00047, and theNational Science Foundation under grant ATM 94-13081.

The Editor thanks G. Paschmann and another referee for theirassistance in evaluating this paper.

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M. Gedalin, Department of Physics, Ben-Gurion Uni-versity, P.O. Box 653, Beer-Sheva 84105, Israel. (e-mail:gedalin@bgumail.bgu.ac.il)

J.A. Newbury and C.T. Russell, IGPP/UCLA, 405 Hil-gard Ave., Los Angeles, CA 90095-1567, USA (new-bury@igpp.ucla.edu, ctrussel@igpp.ucla.edu)September 18, 1997; revised December 1, 1997; accepted December 2,1997.

This preprint was prepared with AGU’s LATEX macros v4, with the ex-

tension package ‘AGU++ ’ by P. W. Daly, version 1.6b from 1999/08/19.