is defined asGEOPHYSICAL RESEARCH LETTERS, VOL. 29, NO. 3, 10.1029/2001GL013992, 2002
[4] To study the energy dependence of temporal profiles, which is important to diagnose the emission mechanism, we construct two counting series f1(t) and f2(t) for each burst for the energy band 25–110 keV and >110 keV, respectively. The average pulse widths in the low and high energy bands and the relative time delays between the two bands can be calculated by a modified correlation analysis technique used in studying X-ray rapid variability of the black hole binary Cyg X-1 [Li et al., 1999; Li, 2001].
[5] The cross-correlation function of two time series f1 and f2 at time lag is defined as
where 
, f(t) is the number of photons in the time interval (t, t + 
t), t is the time step. If the function CCF()/CCF(0) has maximum at = 
, the time lag of the energy band 1 relative to the band 2 at the time scale Deltat is then defined as . Monte Carlo simulations have been done and the results show that with this technique we can measure the relative time delay between two bands over a wide range of time scale t with high time resolution. At large scales one usually can get enough signal photons in a time bin and desirable correlation values from finite time bins. And at small scales the effect of serious Poisson fluctuation of signal counts in a time bin can be compensated by the large amounts of time bin and accurate correlation values can also be derived with Eq. (1). A distribution of time lag vs. time scale can reflect the character of the physical process to produce the delay better than a single value of lag at only one time scale. A physical process usually occurs in a range of time scale, the spectral delay caused by the process should appear at different time scales, smoothly distributed in the range. On the other hand apparent delays from statistical fluctuation will fluctuate between positive and negative values.
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[6] For each studied burst and m = 25 different values of time step t which are logarithmically uniformly placed in the region of 10-5 -10-3 s, we calculate the time lags of f1 relative to f2. All obtained time lags i (i = 1,…, m) of each burst with high signal to noise ratio are always positive. We average each 5 successive i and show the distribution of average time lag vs. time scale in Figure 1. The global average 
and the standard deviation 
for each selected TGF is listed in Table 1. The total counting profile and profiles in 4 energy bands of a TGF with BATSE trigger number 2955 are plotted in Figure 2, where the delay of lower energy photons relative to higher energy ones is apparent.
[7] The width Wl of a temporal profile in a band l can be defined as the FWHM of the autocorrelation function
The widths W1 and W2 of studied TGFs in the low and high energy bands are calculated and their ratios W1/W2 are presented in Table 1, from which we can see that the lower energy pulses are wider than higher energy ones for most studied bursts.
Citation: Temporal and spectral properties of gamma-ray flashes, Geophys. Res. Lett., 29(3), 10.1029/2001GL013992, 2002.
-ray lag vs. time scale of TGFs. The quantity beside an arrow is the BATSE trigger number of the indicated burst.