Timescale Analysis of Spectral Lags

Chin. J. Astron. Astrophys. Vol. 4 (2004), No. 6, 583- -598Chinese Journal of( http:// www.chjaa.org or htt:/ chjaa. bao. ac.cn )Astronomy andAstrophysicsTimescale Analysis of Spectral Lags *Ti-Pei Li1,2,3, Jin-Lu Qu2, Hua Feng', Li-Ming Song", Guo-Qiang Ding2 andLi Chen41 Department of Physics & Center for Astrophysics, Tsinghua University, Beijing 100084;litp@mail.tsinghua.edu.cn2 Particle Astrophysics Lab., Institute of High Energy Physics, Chinese Academy of Sciences3 Department of Engineering Physics & Center for Astrophysics, Tsinghua UniversityA Department of Astronomy, Beijing Normal UniversityReceived 2004 May 17; accepted 2004 June 28.Abstract A technique for timescale analysis of spectral lags performed directlyin the time domain is developed. Simulation studies are made to compare the timedomain technique with the Fourier frequency analysis for spectral time lags. Thetime domain technique is applied to studying rapid variabilities of X-ray binariesand rγ-ray bursts. The results indicate that in comparison with the Fourier analysisthe timescale analysis technique is more powerful for the study of spectral lags inrapid variabilities on short time scales and short duration faring phenomena.Key words:methods: data analysis一binaries: general 一X-rays: stars - 一gamma rays: bursts 一X-rays: bursts1 INTRODUCTIONThe analysis of spectral lag between variation signals in different energy bands is an im-portant approach to obtain useful information on their producing and propagation processesin celestial objects. Observed intensity variations are usually produced by various processeswith different time scales and different spectral lags. A lag spectrumn, a distribution of timelags over Fourier frequencies, can be derived from two related time series with the aid of theFourier transformation. Let x(t;) and y(ti) be two light curves observed simultaneously in twoenergy bands at times ti, their Fourier transforms are X(fj) and Y(fj;) respectively, and thecross spectrum C(f;) = X*(fj)Y (fj). The argument of the cross spectrum C(fj) is the phasediference between the two processes at frequency f;, or the time lag of photons in band 2relative to that in band 1中国煤化工r(f;) = arg[C(f;)]MHCNMHG(1)The Fourier analysis technique has been most widely used in studying spectral lags.* Supported by the National Natural Science Foundation of China.584T.P. Li, J. L. Qu, H. Feng et al.The time domain method for studying spectral lags can be based on the correlation analysis.For two counting series x(ti), y(ti) (or x(i), y(i) ), the observed counts in the correspondingenergy band in the time interval (ti,ti+1) with ti = (i- 1)△t, the cross-correlation function(CCF) of the zero-mean time series at lag kOt is usually defined asCCF(k)= E u(i)o(i + k)/o(u)o() (k= 1,土1,.)(2)with u(i)= x(i) - i,v(i)= y(i)- y, σ2(u) = E;[u(i)]2 and σ2(v) = E;[0(i)]2. With CCF thetime lag can be defined as A = km△t where CCF(k)/CCF(0) has maximum at k: = km. Insteadof a lag spectrum provided by Fourier analysis, the correlation technique gives only a singlevalue A of time lag. To an understanding of a physical process occurring in the time domain,we need to know spectral lags at different timescales, i.e. a timescale spectrum A(Ot).We cannot simply equate a Fourier period with the timescale and interpret a Fourier spec-trum in the time domain as the timescale spectrum. For example, a Fourier power spectrumcan not be interpreted as the distribution of variability amplitude vs. timescale. A sinusoidalprocess with frequency f has no Fourier power at any frequency except f, but it does not meanthat no variation exists at timescales shorter than 1/f. One can make light curves with timesteps smaller than 1/f and find that non-Poissonian variations of intensities do exist in suchlight curves. In fact, a frequency analysis is based on a certain kind of time-frequency transfor-mation. Different mathematically equivalent representations with diferent bases or functionalcoordinates in the frequency domain exist for a certain time series, a Fourier spectrum with thetrigonometric basis does not necessarily represent the true distribution of a physical process inthe time domain. It has to be kept in mind that a mathematical transform may distort physicalinformation contained in the observational data. For correctly understanding the real process,one has to invert results obtained through a time-frequency transform into the real physicalspace. It is usually not easy to complete such an inversion. A sinusoidal process is the simplestsignal in the frequency domain, but a complex one in the time domain. The correct procedureto invert a Fourier power spectrum p=(f) into the timescale spectrum p(Ot) in the physicalspace (the time domain) isp(Ot)= | Pp(f) p(Ot|f)△t，where p(△t|f) is the timescale spectrum of a sinusoidal process with frequency f and unitamplitude, which is not a simple value or function and can not be derived from the Fourieranalysis.To correctly understanding a time process, we have to make timescale analysis directly inthe time domain and need to develop spectral analysis technique in the time domain withoutusing the Fourier transform or other time-frequency transformation. A preliminary algorithmto modify the conventional cross-correlation technique was proposed by Li, Feng & Chen (1999).After then the algorithms to evaluate timescale spectra of power density, coherence, spectralhardness, variability duration, and correlation coefficient between two characteristic quantitieswere worked out (Li 2001), the modifed cross-correlation techniaue is a Dart of the timescaleanalysis method in the time domain. Recently we ha中国煤化工ted the modifedcross- correlation technique, improved its sensitivity al|YHC N M H Gntly. This paperpresents the timescale analysis technique of spectral lags and its application to analyzing spacehard x-ray and γ-ray data. The general procedure of timescale analyzing and the modifiedcross-correlation function for spectral lag analysis in the time domain are introduced in SectionTimescale Analysis of Spectral Lags5852. The technique has been applied to studying spectral lags of hard X-rays from X-ray binaries,y-ray bursts and terrestrial y-ray fashes, some examples are shown in Section 3. Relevantdiscussions are made in Section 4.2 METHOD2.1 Timescale AnalysisTemporal analysis is an important approach to study dynamics of physical processes inobjects. Usually we take some quantities, e.g., power density (variation amplitude), spectral lag,and coherence, etc, to characterize temporal property of observed light curves. The complexvariability of high- energy emission shown in diferent time scales is a common character forX-ray binaries, super massive black holes and r-ray bursts. The variability is caused by variousphysical processes at different timescales. It is not easy to study the variation phenomena ona given time scale. Large time bin used in calculation will erase the information on shortertime scales. Moreover, the analysis result with a short time bin reflects not only the variationproperty on the short time scale, but may also be affected by that on longer ones up to thetotal time period used in the calculation. The fact that a light curve with time step Ot doesnot include any information of variabilities at timescales shorter than△t can be used as afoundation of timescale analysis. A set of light curves with different time steps Ot producedby rebinning the same originally observed data with a time resolution δt is the basic materialin timescale analysis.Usually the originally observed data for temporal analysis is a counting series x(j;8t) (j =1,..) with a time resolution δt. To study variability on a timescaleOt=Msδt，(3)we need to construct a new light curve with the time step Ot from the native series by combiningits Mst successive bins byiMotx(i;△t) =x(j;8t) .(4)j=(i- 1)Ms++1As the light curve {x( Ot)} does not include any information about the variation on any timescaleshorter than△t, it is suitable for studying variability over the region of timescale≥Ot.Let A denote the quantity under study. The value A(△t) of the quantity at the timescaleOt can be seen as a function of the light curve x(i; Ot)A(Ot)= fA[{x(Ot)}] .(5)The key point in timescale analysis is to find a proper algorithm fs to calculate the value ofthe studied quantity A at a certain timescale△t.The procedure (4) of binning the native series {x(8t)} to get {x(Ot)} with a larger timestep Ot = Mstδt is started from the first bin of ax(j = 1;δt). From the native light curve wecan obtain Mst different light curves with the same time step Ot中国煤化工xm(i;Ot)=(6)j=(i-1)Ms:YHCNMHGwhere the combination starts from the mth bin of the native series, the phase factor m =1,.., MAt (see the diagrammatic sketch Fig. 1). .586T. P. Li,J. L. Qu, H. Feng et al.m=1|m=2m=3m=5Fig.1 From an originally observed time series with time resolution δt (schematically shownat the top where each small square represents a time bin with width 8t) five diferent lightcurves with time step△t = 5δt can be constructed with different phase parameter m.For suficiently using the information about variation on the timescale Ot included in theoriginally observed light curve, we can calculate the studied quantity A(Ot) for each {xm(△t)}and take their average as the resultant valueMatA(Ot)=Msr 2 f(m(0)}].(7)m=1In timescale analysis the observed light curve {x(8t)} is usually divided into L segments,each segment includes nearly equal number of successive bins. For each segment i, we canacquire a value A:(Ot) by Eq. (7), the average N(Ot) and its standard deviation σ(A(Ot) canbe derivedA(Ot) =, A:(Ot)/L，o(N(t)) = 1(A:(Ot) - I(O)2/L(L-1).(8)Usually we can use some convenient statistical methods based on the normal distributionto make statistical inference, e.g. significance test, on A(Ot). For the case of short time scale△t, although the number of counts per bin may be too small for it to be assumed as a normalvariable, it is easy from a certain observation period to obtain the total number L of segmentslarge enough to satisfy the condition for applying the central limit theorem in statistics andusing the normal statistics for the mean A(Ot).2.2Modified Cross Correlation FunctionIn timescale analysis for spectral lags, the observed data are two related counting series{x(i;8t)} and {y(i;8t)} in two energy bands with time resolution St. If the timescale Ot understudy is larger than the time resolution, we need to construct two new light curves, {x(i; △t)}nd {y(i; Ot)}, by re-binning the originally observed series with the time step Ot. With thetraditional CCF defined by Eq. (2), we can calculate the time lag A only if A : >△t. However,for many physical processes the real time lag is short中国煤化工than the processtimescale. For the purpose of applying correlation alTHCN M H Ge of lag alysis,a modified CCF at lag kδt has been proposed (Li, Feg w Jxl 1.05)MCCFo(k;Ot)= 2 u(; Ot)vk+1(i; Ot)/o(u)o(v)，(9)Timescale Analysis of Spectral Lags587where the time step Ot = Mstit, {ue(Ot}} and {vx(Ot)} are the zero-mean series of {xn(Ot)}and {yr(△t)}, respectively, andxm(i;Ot) =2x(j;8t)，j=(i-1)Mat+miMst+m-1Ym(i;Ot) =y(j;8t) .(10)j=(i-1)Mst +mThe lag resolution of CCF defined by Eq.(2) is Ot, and that of MCCFo defined above is theoriginal time resolution δt.For sufficiently using the information contained in the observed lightcurve, we propose toimprove the definition of MCCFo further by following the procedure described by Eq. (7). Thenew and complete definition of MCCF at lag k8t isMCCF(k;Ot) =Mat_Mo.:EE, Um(i; St)0m+e(i; Ot)/o(u)σ(v),(11)m=1iwhere um(i;Ot) = xm(i; Ot) - zm(Ot), vm(i;Ot)= ym(i; Ot) - ym(Ot), im(Ot) and ym(Ot)are the current averages for the used segments of lightcurve {x} and {y}, respectively. Theprocedure of calculating a modifed cross-correlation cofficient is schematically shown by Fig. 2.We can find a value km of k to satisfy the conditionMCCF(k = km; Ot)/MCCF(0;Ot) = max ,(12)then the lag of band 2 relative to band 1 on timescale△tA(Ot) = km8t .(13){x(6t)}{,(O)}[F Eix,(i; Ot)y(;Ot){u。(Ot)} [{x,()}t Six.(; 0t)y,(;Ot){y,(Ot)}Fig.2 MCCF of two native time series {x(δt)} and {y(5t)} at a time lag T = δt (r =k8t, k= 1) and on a timescale Ot = 28t. MCCF(k= 1;Ot) =结汇; xr(i;△t)y2(i;Ot) +E: x2(i; Ot)y,(i; ot)].For an observed time series x(j; δt) with time resolution δt, we can similarly define a mod-ified auto-correlation function at lag k8t on timescal中国煤化工MACF(k;Ot) =“TYHCNMHG1: EEr, um(i; Ot)um+k(i; Ot)/o2(u), .(14)Mot.m=1 i588T. P. Li,J. L. Qu, H. Feng et al.where Um(i;Ot) = xm(i; St)-元, xm(i; Ot)= Ej=t-1)Mas+m x(j;8t). The FWHM of MACF(T; Otcan be taken as a measure of the duration of variation on time scale Ot. With MACF we canstudy the energy dependence of average shot width in a random shot process on different timescales.2.3 Simulations2.3.1 Poissonian signalsTo compare the above MCCF technique of estimating time lags with the traditional CCFtechnique and Fourier analysis, we produce two photon event series of length 1000 s with aknown time lag between them. The series 1 is a white noise series with average rate 200 cts s-1and series 2 consists of the same events in series 1 but each event time is delayed 13 ms.Besides the signal photons mentioned above, the two series are given independent additionalnoise events at average rate 300 cts s-1. By binning the two event series, two light curves withtime resolutionot= 1 ms are produced. We make time lag analysis at timescales Ot from 1 msto 2 s for the two light curves by CCF, MCCF and Fourier analysis techniques separately (inFourier analysis we use Fourier cross spectrum with 1 ms light curves and 4096-point FFT andtake Fourier frequency f = 1/Ot), the results are shown in Fig.3. In the left panel of Fig. 3,the cross signs show lags derived by CCF and plus signs by MCCFo defined by Eq. (9). For thetimescale region of Ot shorter or approximately equal to the magnitude of the true lag 0.013 swhere CCF works, MCCFo can provide more reliable results with better accuracy. The circlesin the right panel of Fig. 3 indicate the Fourier lags, for the short timescale region of Ot≤0.3 s(or high frequency region of f 2 30 Hz) the Fourier analysis can not give any meaningful result.In the timescale region of Ot > 0.1 s, the estimnates of time lag by MCCFo defined by Eq. (9)(plus signs in the left panel of Fig. 3) show significant fuctuation about the expectation. Wuse the improved MCCF defined by Eq. (11) to calculate the lag spectrum again, the plus signsin the right panel of Fig. 3 indicate the result. Comparing the lag spectra with MCCFo [Eq. (9)]and MCCF [Eq. (11)], plus signs in the left panel and right panel of Fig. 3, we can see that usingthe improved MCCF can improve the lag spectrum significantly in the large timescale region.8t田000-包。q0.010.1中国煤化工Ot (s)YHCNMHGFig. 3Time lag vs. time scale of two white noise series with 13 ms time lag shown by thedotted horizontal line. Left panel: Cross - CCF lag; Plus - lag evaluated by MCCFo [Eq. (9)].Right panel: Circle - lag from Fourier analysis; Plus - lag by MCCF [Eq. (11)].Timescale Analysis of Spectral Lags5892.3.2 Transient signalsTo study the relative timing of transient emission at different energies is a difficult task inastrophysics. Neither the Fourier analysis nor the traditional crOss correlation technique canobtain meaningful result of spectral lags from the observed data of short y-ray bursts. Weshow here the ability of MCCF to study spectral lags in prompt emission of short y-ray burstsby simulation. Figure 4 shows light curves with 5 ms time bin observed by BATSE on CGROmission for a y-ray burst, GRB 911025B (BATSE trigger number 936), in channel 25- -55 keV,55- -110keV, 110- -320keV, and > 320 keV, respectively. The time-tagged event (TTE) data ofBATSE contain the arrival time (2μs resolution) of each photon for the short duration burstGRB 911025B. In our simulation, we use the 25- 55 keV photons between 1.35 s and 1.55 s inthe TTE data as the burst events in channel 1. The burst events in channel 2 are the samein channel 1 but each time is delayed 0.01 s. By binning the two event series with time bin5 ms, expected signal series in channels 1 and 2 are produced. Two simulated light curves inchannels 1 and 2 are generated by taking random samples from the expected signal series andadding independent background noise at the average of 15 counts each bin, shown in Fig. 5.&B[GRB 911025BGRB 91 1025825-55 k@V55-110 keV导81.52.1.5"ime (s)Time (s)品p110- 320 keV吕E>320 keV只tC中国煤化工Fig. 4 Light curves of a y-ray burst GRB 91 1025B cMHCNMHGels 25 55keV,55- 110keV, 110- 320keV, and > 320 keV, separately.T. P. Li,J. L. Qu, H. Feng et al.品[品Light curve8fChannel 1Channel 22?呆个导RIol1.5O1Time (s)lime (s)Fig. 5 Simulated light curves for two channels. The burst process for channel 2 is delayed 0.01sto channel 1.From the two simulated lightcurves we use MCCF to calculate the time lags at timescale△t = 0.005 s, 0.015 s, 0.045 s, 0.14 s, and 0.43 s, separately, the results are shown in Fig. 6. Theerror bar of time lag at each timescale is estimated from 200 bootstrap samples. The simulationresult indicates that MCCF is a useful tool in relative timing of transient processes.8CH:...+0.010.1Time scale (s)Fig.6 Time lag vs timescale of two simulated light curves in Fig. 5.Dotted line一expected time lag. Cross - measured by MCCF.中国煤化工2.3.3 Timescale dependent processMHCNMHGSpectral time lags observed for X-ray binaries and AGNs are usually timescale dependent.For example, the Fourier lag between 14.1- 45 keV and 0- 3.9 keV of X-ray emission from Cyg X-1 in the low state continuously varies with Fourier frequency, from ~ 30 ms at 0.1 Hz decreasingTimescale Analysis of Spectral Lags591down to ~ 2 ms at 10 Hz (Nowak et al. 1999). Real data in timing can be seen as a complextime series with multiple timescale components.The timescale dependence of spectral lagscan be the intrinsic property of the emission process and/or comes from different processesdominating at different timescales. Correctly detecting the timescale dependence of spectrallags is important to studying the undergoing physical processes. Now we compare the abilitiesof MCCF and Fourier technique to study spectral lags of time series with multiple timescalecomponents. Two light curves of a complex process consisting of five independent random shotcomponents are produced by Monte Carlo simulation. Each signal component i (i = 1 - 5)consists of random shots with shape of profile a . exp[(t - to)/r;)]2, where the peak height a israndomly taken from the uniform distribution between zero and the maximum. The separationbetween two successive shots is exponentially distributed with average separation Ti. For eachcomponent i, we produce a 3000 s counting series of band 1 with a step of δt = 1 ms andaverage rate 200 cts s~ 1. The corresponding series in band 2 consists of the same events inseries 1 but each event time is delayed Ai s. The characteristic time constants T; of the fivesignal components are 0.005, 0.01, 0.02, 0.04, and 0.08 s, and their time lags are 0.004, 0.008,0.012, 0.016, and 0.02s, respectively. Summing up the five series for each band, we producetwo expected signal light curves. Two synthetic light curves with time step 1 ms are made byrandom sampling the expected light curves with Poisson fuctuation plus a independent whitenoise at mean rate of 100 ctss-1. For the two light curves, we use their Fourier cross spectrumand 4096-point FFT and MCCF in the time domain to calculate the time lags on different timescales△t and show the results in Fig. 7.To derive the expected lag spectrum of the above synthetic light curves, we calculate thetimescale distribution of power density for each shot component using the algorithm of esti-mating power density spectrum p(Ot) in the time domain (Li 2001; Li & Muraki 2002). As anexample, Fig. 8 shows the distribution of variation power p(Ot)△t (rms2) vs. time scale△t forthe expected light curve of the shot process with T = 0.01 s. As shown in Fig. 8, an individualshot component with characteristic time Ti has its variation power distributed over a certaintime scale region. The time lag between two bands should appear in the whole timescale regionwhere the signal variation power exists. The expected lag of the two synthetic light curves withmultiple components on timescale△t should be estimated as a weighted average of five lags Aiof individual component with corresponding weight factor pi(Ot)△tA(Ot)= > p:(Ot)Ot .A.(15)The solid line in Fig.7 is the expected lag distribution calculated by Eq. (15). We can seefrom Fig. 7 that the Fourier cross spectrum fails to detect lags in the short timescale region,but MCCF works well. More simulations with diferent signal to noise ratios show that theMCCF technique is capable of correctly detecting time lags from severely noisy data and theinfficiency of detecting time lag in short time scale region is an intrinsic weakness of the Fouriertechnique, even for data with much higher signal to noise ratio the Fourier analysis still can notdetect lags in the high frequency region.中国煤化工MYHCNMHG592T.P.Li,J.L.Qu,H.Fengetal.品|Channel 1只&210Time (a)吊Channel 2B。6Time (3)00O1￥↓+十8中包c0号C79-L⊥⊥wL.0.010.1Time Scale Ot (s)Fig.7 Top and Middle: Synthetic light curves中国煤化工complexprocess consisting of different characteristic t:tral lags.Bottom: Timescale spectra of time lag; Solid lirYHCN M H Grum, Plus- MCCF lag, Circle一lag from Fourier analysis.Timescale Analysis of Spectral Lags5931.5x 1041050000.010.1At (s)Fig. 8Power distribution p(△t)△t vs. time scale Ot of a randomshot process with a characteristic timescale T = 0.01 s, where p(Ot)(rms2 s" 1) is the power density on the timescale△t of the process.3 APPLICATIONSThe timescale spectral method for time lag analysis is a powerful tool in revealing thecharacteristic of emission process in objects. With the help of MCCF technique, as a example,ve can judge between diferent production models of x-rays from accreting black holes. Theenergy spectra of hard X rays from black hole binaries can be fitted well by Comptonization ofsoft photons by hot electrons in the vicinity of the compact sources. To explain the observedenergy spectra the uniform corona model was suggested initially (Payne 1980)， in which thesoft photons from the central region of the system are Comptonized by the hot electrons ofcorona. The Comptonization process makes the observed hard photons undergo more scatteringthan the low energy photons and therefore the hard photons are naturally delayed from softphotons. Hard lags are not correlated with the variability timescale (or variability frequency)in the uniform corona model, but the later study showed strong timescale- dependence of timelags (Miyamoto & Kitamoto 1988). For overcoming the contradiction between prediction bythe uniform corona model and observed results, other models, such as the non-uniform coronamodel (Kazanas et al. 1997), the magnetic fAare model (Poutanen & Fabian 1999) and thedrifting-blob model (Bottcher & Liang 1999) are proposed. The hard X-ray lags are studiedby observations with PCA detector on board RXTE mission in using the Fourier technique tothe black hole candidate Cyg X-1 in the low state中国煤化工h state (Cui et al.1997a), and during spectra transitions (Cui et al.Fourier spectra oftime lag from PCA/ RXTE data are all limited inYHCNM H Goency≤30 Hz (ortimescale 2 0.03 s) and, except that with an uniform corona, all models mentioned above canfit the observed lag spectra of Cyg X-1. To test these models, we need to compare the expected594T. P. Li,J. L. Qu, H. Feng et al.and observed lags in the higher frequency range or on the shorter time scales.Kazanas et al. (1997) presented that for the PCA/RXTE observation of June 16 1996(ObsID P10512), hard X-ray time lags of Cyg X-1 in the soft state as a function of Fourierfrequency over the region of 8 ~ 30Hz can be well ftted by the non-uniform corona model.From the same data, we measure the time lags between 13-60 keV and 2-5 keV on short timescales down to Ot ~ 1 ms with MCCF and the results, shown in the top panel of Fig. 9 cannot be ftted by this model (the solid line in the figure). The drifting-blob model (Bottcher &Liang 1999) can also explain the observed time lags in the Fourier period region above 0.1s. Onshort time scales, the Comptonization process of the drifting-blob model is similar with that ofthe non-uniform corona model. Thus, the time lag will have similar time scale dependence, i.e.,time lag will decrease with time scale as quickly as the behavior of non-uniform corona modelon short time scales. The drifting blob model also can not explain the observed time lags ofCyg X-1 on the short time scales.可?号重2 10-50.010.Tme Scale (s)5f田'eT中↓! 100.1Time Sccle (sFig.9 Hard X-ray time lag vs. time scale of Cyg中国煤化工;- ime lag between13- 60keV and 2-5 keV in the soft state of Cyg X-1 onObsID P10512)measured by MCCF; Solid line - expected by the non-uMHCNMHGmpanel:Circle-lags between 13- -60 keV and 2- -5 keV of Cyg X-1 in the hard state on 1996 October 23 (PCA/RXTEObsID P10241) measured by MCCF; Solid line - lags between 27 keV and 3 keV predicted by themagnetic fare model.Timescale Analysis of Spectral Lags595Poutanen & Fabian (1999) proposed the magnetic fare avalanche model to explain theobserved time lags of Cyg X-1, and parameterize the spectral evolution of a magnetic flareand the avalanche process based on the PCA/ RXTE observation on October 23 1996 (ObsIDP10241). In the bottom panel of Fig. 9, we show the measured time lags with the observationdata and MCCF technique (the circles) and the predicted by the model (the solid line). Thepredicted time lags can fit the measured time lags well in the time scales range between ~ 4 msand 1s, much better than other models, although there seemns to be overestimated on theshortest timescales. This example shows that the capability of MCCF for detecting time lagson short time scales can help us to reveal the underlying physics in high energy process inobjects. .GRB 91 1C25BD-GRB 9110258| 110- 320 keVvs 25-55 keV可+置Ez vs (25- 55)keVBf?f0.010.150100200500Time scale (s)Fz (keV)Fig.10”Hard lags of GRB 911025B measured by MCCF. Left panel: Timescale spectra oftime lag of (110- 320) keV vs. (25- 55) keV. Right panel: Energy dependence of time lag of hardphotons vs. (25-55) keV, averaged for timescales 0.005s, 0.01s, 0.25s, 0.06s, and 0.14s.The MCCF is particularly useful in studying transient processes. The BATSE detectorhas discovered an unexplained phenomenon: a dozen intense flashes of hard X-ray and y-rayphotons of atmospheric origin (TGFs) (Fishman et al. 1994). As all the observed TGFs wereof short duration (just a few milliseconds), it is dificult to study their temporal property byconventional techniques. With the aid of the preliminary MCCF (MCCFo, Eq. (9) in thispaper), Feng et al. (2002) revealed that for all the Alashes with high signal to noise ratio γ-rayvariations in the low energy band of 25- 110 keV relative to the high energy band of > 110 keVare always late in the order of ~ 100 us in the timescale region of6x 10-6-2x 10-4 s andpulses are usually wide. The above features of energy dependence of time profiles observed inTGFs support models that TGFs are produced by upward explosive electrical discharges athigh altitude.Efforts have been made to measure the temporal correlation of two GRB energy bandsby the CCF technique (e.g. Link, Epstein & Prid中国煤化工al. 1995; Wu &Fenimore 2000; Norris 2002). The CCF technique|_ty to make timinganalysis for weak events. For strong bursts the DIS:0YHC NMH Gtabase 4-hanelight curves with 64 ms time resolution, are usually analyzed, but it fails with the traditionalACF and CCF in the case that the existed spectral lags are comparable or smaller than 64 ms596T.P. Li, J. L. Qu, H. Feng et al.whatever how strong the burst is. For short bursts of duration < 2 s, we can use TTE data toconstruct high resolution light curves, as the four light curves of time resolution δt = 5 ms forGRB 911025B shown in Fig. 4. We failed to obtain statistically meaningful results in spectrallag analysis for short GRBs by using CCF and MCCFo. The MCCF technique can help usto reveal spectral lags in strong short bursts. As an example, with the light curves of GRB9110258B and MCCF, we calculate the time lag between two channels at timescale△t = 0.005s, 0.01 s, 0.025 s, 0.06 s, and 0.14 s, respectively. In our calculation only partial data havinghigher signal to noise ratio recorded during 1.29 s and 1.62 s are used. The obtained timescalespectrum and energy dependence of time lags are shown in Fig. 10. For long r-ray bursts, theBATSE Time-to-Spill (TTS) data record the time intervals to accumulate 64 counts in eachof four energy channels. The TTS data have fine time resolution than 64 ms of DISCSC datawhen the count rate is above 1000 cts s-1. The TTS data can be binned into equal time binswith a resolution of δt ~ 10 ms and our simulations show that from the derived light curvesthe temporal and spectral properties with the time resolution δt can be reliably studied withMCCF for typical GRBs recorded by BATSE. As an example, the left panel of Fig. 11 showsthe lag spectrum of GRB 910503 detected by BATSE with a duration ~ 50 s. From the MCCFlag spectra, we can further derive the energy dependence of lag at different timescales, shownin the right panel of Fig 11. The results shown in Figs. 10 and 11 indicate that MCCF can beused to explore temporal and spectral properties for both long and short γ-ray bursts.Symbol Timescole (s)0.6◎。①C0.30.0φE口GRB 910503iE Ezvs 20-60 keV>325 keV vs 20-60 keV一+110-325 keV Vs 20-60 keV0.010.11100200Timescale (s)E2 (keV)Fig. 11Hard lags of GRB 910503 measured by MCCF. Left panel: Timescale spectra of timelag. Circle- (20- 60) keV vs. (60-100) keV; Plus- (20- 60) keV vs. (110- -325) keV; Diamond - (20-60)keV vs. > 325 keV. Right panel: Time lag of 20- 60 keV photons vs. energy of hard photons.Circle - timescale 0.01s; Plus - timescale 0.3 s; Diamond - timescale 0.6s.4DISCUSSION中国煤化工The technique for timescale analysis with MCCF.0HC N M H Gandard cross cor-relation function CCF. For determining CCF from unevenly sampiea ata which are commonin astronomical contexts, several different methods have been introduced, i.e., interpolatingthe data between observed points to form a continuous function (Gaskell & Sparke 1986), theTimescale Analysis of Spectral Lags597discrete correlation function (DCF, Edelson & Krolik 1988), evaluating CCF with the dis-crete Fourier transform (Scargle 1989), and z- transformed discrete correlation function (ZDCF,Alexander 1997). Introducing MCCF was motivated by the need of improving the resolutionand sensitivity of the standard correlation analysis. Comparing the two definitions, Eq. (11)for MCCF and Eq.(2) for CCF, we can see that MCCF includes more information from theobserved data than CCF does. That the lag T = kδt in MCCF has the same resolution withthe originally observed data but the resolution of CCF lag is Ot and that MCCF is calculatedby summing over m = 1,2,.. , Mst (using all possible light curves that can be derived fromthe native data with timescale Ot) but CCF only uses one light curve for a given time bin△t. make MCCF has better resolution and sensitivity in measuring spectral lags.Three diferent temporal quantities exist in the timescale analysis: time t, time resolution δt,and time scale△t. The originally observed data, based on which both the frequency analysisand timescale analysis are performed, are time series x(t) in the time domain with a timeresolution δt. With frequency analysis, we can derive a frequency spectrum A(f) for a studiedcharacteristic quantity A (in this paper A is time lag, in other spectral analysis it may be powerdensity, coherence, or other quantity) from the observed time series. In frequency analysis,the observed time series has to be transformed into the frequency domain first with the aidof time-frequency transformation, e.g., the Fourier transform, and the maximum frequency isdetermined by the time resolution 8t. The timescale analysis is performed directly in the timedomain without any time-frequency transformation. With timescale analysis we can derive atimescale spectrum A(Ot) for the studied quantity A where the argument△t, i.e., the variationtimescale, is a variable similar to the frequency f in frequency analysis. The minimum timescalein timescale analysis is the time resolution δt of the originally observed data. To study temporalproperty on a certain timescale Ot, the used light curves should have a time step equal to thetimescale under study. With MCCF we can measure any lag greater than the time resolutionof observation at any given timescale and produce a timescale spectrum of time lags.There exist two kinds of spectral analysis: frequency analysis and timescale analysis. Al-though the Fourier method is a common technique to make spectral analysis, it can not replacethe timescale analysis in the time domain. As any observable physical process always occurs inthe time domain, a frequency spectrum obtained by frequency analysis needs to be interpretedin the time domain. However, a frequency analysis is dependent on a certain time frequencytransformation. A Fourier spectrum by using Fourier transform with the trigonometric basisdoes not necessarily represent the true distribution of a physical process in the time domain. .The rms variation vs. timescale of a time-varying process may differ substantially from itsFourier spectrum, as an example, the Fourier spectrum of a random shot series significantlyunderestimates the power densities at shorter timescales (Li & Muraki 2002). The present workshows that, like Fourier power spectra, Fourier lag spectra also always significantly underes-timate time lags at short timescales. The timescale analysis performed directly in the timedomain can derive a real timescale distribution for quantities characterizing temporal property.In comparison with the Fourier technique, timescale spectra of power density and time lag fromthe timescale analysis can more sensitively reveal temporal characteristics at short timescalesfor a complex process.Welsh (1999) pointed out that the lag determi中国煤化工ld be consideredonly a characteristic time scale. Care has to beYHC N M H Gasured lags witha particular physical model. Most techniques in timing can only treat timescale just in asynthetic meaning. For example, a correlation function lag of two shot series may caused not598T.P.Li,J.L.Qu,H.Fengetal.only by a time displacement between the two series, but also by changes in shot shape andintervals between two successive shots. Distinguishing diferent kinds of timescale and physicalprocess is obviously helpful to study physics, that should be a goal in future development oftime domain technique. Using a single analysis technique alone is often difficult to definitelydistinguish the possible processes and compiling different results of analysis from different viewangles will be helpful. In comparison with the Fourier technique, the time domain techniquehas the freedom of choosing a proper statistic for a particular purpose. Recently Feng, Li &Zhang (2004) introduced a statistic w( Ot) to study widths of random shots and diagnosed blackhole and neutron star X- ray binaries by timing with the new designed statistic.Acknowledgements This work is supported by the Special Funds for Major State BasicResearch Projects and the National Natural Science Foundation of China. 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