2 Data preparation

The dataset used was taken from the BATSE current catalogue. The "discsc'' files are available at http://www.batse.msfc.nasa.gov/batse/grb/catalog /4b/ (Paciesas et al. 1999). The files contain the data from the four energy channels, which were combined into a single channel to maximise the signal to noise ratio. The shapes of GRB pulses vary little with energy and pulses in different energy channels can be added together and nearly retain their initial shape. A subset of the BATSE catalogue was selected based on the criteria (Norris et al. 1996) that the GRB duration was greater than two seconds (T₉₀>2 s) and the peak flux $P_{\rm 256 ms} > 3.28$ photons cm^-2 s^-1. In this way a sample of 324 bursts with good signal to noise and clearly resolved features was obtained. Five of these bursts could not be analysed properly due to data gaps, and so our final sample consisted of 319 GRBs. All 319 GRBs were used for the timing analyses. The 250 GRBs that were summed over only two Large Area Detectors (LADs) were used for all analyses involving pulse amplitude and area.

2.1 Background subtraction

The first step in the data preparation involved selecting the appropriate background for subtraction from the GRB. The start and end times for each burst were identified. A further margin of 10 s was added to both the beginning and end of this chosen section. Two background sections of duration 30 s were then selected, one finishing 20 s before the start of the section containing the burst and the other starting 20 s after the end of the burst (Fig. 1). These two regions were used to fit a linear background that was subtracted from the burst section.

2.2 Denoising technique

One of the difficulties in analysing the time profiles of GRBs is in overcoming the limitations imposed by the presence of noise in the signal and the overlap of the individual pulses. The transient nature of GRBs also means that the usual assumptions for Fourier transform techniques do not hold (Chang & Yi 2000; Suzuki et al. 2001). An alternative method of filtering the signal is with wavelets. Wavelet analysis was pioneered by Daubechies (1992) and others during the 1980's (Meyer 1993).

Wavelets are specific functions that, when convolved with the signal under investigation, produce a transformed signal that represents the location and strength of variations within the original data. The convolution is applied repeatedly to the data over a range of scales with the convolution function gradually stretched to coarser and coarser scales, revealing variations at corresponding scales in the original signal. This iteration with scaling of the convolved function allows the identification of structure with a variety of extents both in the spatial and frequency domains.

If the wavelet function is written as $\psi(x)$ then, more formally, the transform of f(x) at a particular scale s, can be written as

$$\begin{eqnarray*}W_{s}f(s,x)=f*\psi_s(x) \end{eqnarray*}$$

where $\psi_s(x) \equiv (1/s)\psi(x/s)$ represents the wavelet dilated by a scale factor s. Let discrete signals be denoted with the superscript d. At each scale 2^j, a discrete wavelet transform that we denote by W^d_2^j can be computed. For the particular choice of scale, $s=2^j, j=1 \ldots J$ the sequence

$$\begin{eqnarray*}\left\{ S^d_{2^J}f,\left( W^d_{2^j} f\right)_{1\leq j\leq J }\right\} \end{eqnarray*}$$

is called the discrete dyadic wavelet transform of the input discrete signal $D=(S_1 f(n))_{n\in Z}$. The W^d_2^j fcomponents provide the detail at each scale, meaning the response of the (scaled) wavelet function to the detailed variation of the signal. The coarse signal, S^d_2^J, provides the low frequency (slowly varying) component of the signal remaining at scales larger than 2^J. The higher frequency components can all be recovered from the dyadic wavelet transform $\left(W^d_{2^j}f\right)_{1\leq j\leq J}$ between scales 2¹and 2^J. A fast algorithm for calculating the wavelet transform of a signal was developed by Mallat & Zhong (1992) and implementation of this algorithm is at the core of the denoising procedure used. In Fig. 2, GRB 920513 is shown together with its dyadic wavelet transform for scales $s=2^{2\ldots4}$ (the lowest scale, s=2¹, is not shown because it is dominated by noise). The transforms at each of the scales are shown (Figs. 2c, d and e), along with the low frequency signal S^d_2⁴ containing the remaining information for scales i>4 (Fig. 2f).

Figure 2: The wavelet reconstruction of GRB 920513. a) The GRB profile and wavelet fit with maxima (identified by the + symbol) and minima (identified by the x symbol). A total number of 33 pulses were identified above a threshold of 5$\sigma$. b) The residuals between the wavelet fit and the actual GRB profile. c)-e) The decomposition of GRB 920513 (BATSE trigger 1606) into its component wavelet scales. The wavelet transform for scales $s=2^{2\ldots4}$ (the finest scale, s=21, is not shown because it was dominated by noise). f) The low frequency signal Sd24 containing the remaining information for scales i>4.

Figure 3: A diagramatic representation of the algorithm for identifying pulses with appropriate significance. The error bars were calculated using the count rate in the particular bin, multiplied by the threshold significance level, ${\rm\tau _\sigma }$. The dashed circles highlight maxima and minima where the error bars overlap. Larger values of ${\rm\tau _\sigma }$ eliminate overlapping regions that were more widely separated. With a slightly smaller choice of ${\rm\tau _\sigma }$ in this example, pulses 6 and 14 would have been accepted.

The function $\psi(x)$ used in the algorithm was chosen so that the wavelet acts like an edge detector with a delta response function to a step edge. In fact W^d_2^jf is proportional to the derivative of the original signal smoothed at the scale 2^j. Thus calculating the positions of the modulus maxima of the transform, |W^d_2^J f|, is analogous to locating the sharp variations in the original signal. Mallat & Zhong (1992) also developed an algorithm for allowing the reconstruction of a signal given just the modulus maxima of the wavelet transforms across a set of scales along with the low frequency signal remaining at the coarsest scale. This reduced representation can reconstruct an accurate copy of the original, using an iterative algorithm that converges quite quickly to acceptable levels.

A technique for identifying white noise and removing it without losing any other information was introduced by Mallat & Hwang (1992). This technique relies on characterising the behaviour of noise across the various scales in the wavelet transform using just the information present in the extrema wavelet representation described in the previous paragraph. The change in the amplitude of the extrema between scales allows the number called the Lipschitz exponent to be calculated. More precisely, each extremum describes a particular curve in $\left(\log (s), \log \vert Wf(s,x) \vert\right)$ space representing its increase or decay on all the scales for which Wf(s,x) has been computed. Then the Lipschitz exponent, $\alpha$, is just the maximum slope of a straight line that remains above this curve (Young et al. 1995). Using results from an analysis of white noise ( $\alpha<-\frac{1}{2}$, Mallat & Hwang 1992), as well as the investigation described below, the characteristic distribution of Lipschitz exponents for the noise present in BATSE GRB profiles was determined. In general, noise is expected to have a negative Lipschitz exponent indicating a decreasing amplitude with increasing scale. Extrema in the wavelet transform which fall in the range expected for noise can be removed using an algorithm developed for this purpose.

As an example consider the extremum at around 110 s on the top (finest) scale of GRB 920513 (Fig. 2c). On the next scale, the amplitude of this extremum is considerably smaller and by the following scale it is hardly detectable. It is clear that this extremum corresponds to a noise feature in the original signal and visual examination of the transform indicates that the amplitude of the wavelet transform decays quickly when moving to progressively coarser scales, as expected for noise. The extremum at around 95 s corresponds to a pulse with intensity over 25 000 counts per 64 ms. The amplitude of the wavelet maximum increases on coarser scales, contrary to the behaviour expected for noise.

Figure 4: The number of pulses selected from the 319 GRBs as a function of the isolation level ${\rm\tau _I}$ and the threshold ${\rm\tau _\sigma }$. The plot also shows a projection of the data on to the ${\rm\tau _I,\tau _\sigma }$ plane and the contour levels are as given in the legend.

The extrema removal algorithm was combined with a simple thresholding procedure based on the analysis of a section of the burst background. For each burst being prepared, a section of the same burst well separated from the signal was used to generate the wavelet transform of the background. The standard deviation of the transform at the scale 2² (the finest scale not dominated by noise) was combined with a significance level ($\sigma_T$) to calculate an amplitude threshold for the wavelet coefficients of the signal at the same scale. Any extrema in the signal with amplitudes less than this threshold on scale 2² were assumed to be due to noise and were removed.

A denoised signal was then reconstructed (e.g. Fig. 2a) using the algorithm described by Mallat & Zhong (1992). The reconstructed functions have no spurious oscillations or sharp variations, and are very similar to the original signal. The residuals between the background subtracted burst and the reconstructed burst are shown in Fig. 2b. The residuals were calculated assuming that the errors in the raw data were Poisson distributed. The variation of the residuals in the section containing signal do not differ significantly from a section due to background noise (Fig. 2b).

2.3 Pulse selection

Each pulse was examined to find minima on each side which were separated in amplitude from the maximum by more than a chosen significance level. If the search for minima failed on a pulse it was rejected and the search for pulses continued. The algorithm was designed so that the maximum and minima finally selected for a given pulse were the extreme values in that region of the signal. The significance level of the maxima and minima was calculated by multiplying the error on the counts by a scale factor, $\tau_\sigma$. Given two data points n₁ and n₂, where n₁>n₂, the points overlap if

$$\begin{eqnarray*}(n_1-\tau_{\sigma} \Delta n_1)<(n_2+\tau_{\sigma} \Delta n_2). \end{eqnarray*}$$

The errors on the counts in the cleaned signal were assumed to be Poissonian.

The pulse selection process is illustrated in Fig. 3. Starting with pulse 2, minima 1 and 3 were easily identified. Next, pulse 4 was considered. Pulse 4 overlaps the adjacent minimum 3 (based on appropriately sized error bars) and both turning points were rejected and minimum 5 was considered as the true minimum associated with pulse 2. A minimum, maximum, minimum triplet consisting of points 1, 2 and 5 was found. At the conclusion of the analysis the overlapping maxima/minima pairs consisting of points 6 and 7, 9 and 10 and 13 and 14 were rejected. The algorithm selected points 2, 8 and 12 as the maxima and identified associated minima 1, 5, 11 and 15.

The analysis was then extended to allow the identification of pulses that were well separated from their neighbours such that the overlap from the surrounding signal did not significantly affect the profile of the pulse. The fraction of the total height, $H_{\rm b}$, (from pulse to background) which was above the higher minimum was determined and a threshold was applied to this fraction, above which pulses were considered isolated. In Fig. 3 the pulses 8 and 12 are not very well separated from their neighbours whereas pulse 2 is effectively isolated and not strongly influenced by surrounding signal. The fractional isolated height of pulse 8 (to which the threshold is applied) was obtained using (Fig. 3):

$$\begin{eqnarray*}f_{\rm I}=\min (H_{\rm l},H_{\rm r})/H_{\rm b} = H_{\rm r}/H_{\rm b} = 8/21 \approx 0.38, \end{eqnarray*}$$

where $H_{\rm l}$ is the height difference between the pulse 8 and the point 5 on the left and $H_{\rm r}$ is the height difference between the pulse 8 and the point 11 on the right.

For pulse 2 (with point 1 being the higher minimum the smaller height difference is on the left) the estimate is:

$$\begin{eqnarray*}f_{\rm I}=\min(H_{\rm l},H_{\rm r})= H_{\rm l}/H_{\rm b} = 15/17 \approx 0.88. \end{eqnarray*}$$

Thus, if the threshold, $\tau_{\rm I}$, were set above $\sim$40% then pulse 2 would be accepted as isolated and pulse 8 would be rejected. This method provided an objective way to identify and quantify pulses that were influenced by neighbouring signals.

2.4 Properties of the pulses

The next task was to characterise the GRB profiles based on the properties of the constituent pulses. The following characteristics which had been studied previously (McBreen et al. 1994; Li & Fenimore 1996; Hurley et al. 1998) were investigated: the number of pulses per burst,  N; the time intervals between pulses,  $\Delta T$; the pulse amplitudes,  C; the pulse area,  ${A_{\rm p}}$; the rise and fall times,  ${t_{\rm r}}$ and   ${t_{\rm f}}$; and the pulse durations or full width at half maximum, FWHM.

The total number of pulses in the sample of 319 GRBs was determined for a range of thresholds ${\rm\tau _\sigma }$ and varying isolation levels, $\tau_{\rm I}$ (Fig. 4). The variation in isolation level has a much larger effect on the sample than the threshold, and caused a reduction in the number of pulses from over 3000 at the isolation level of 20% to under 800 at the 80% level. Figure 4 also shows that the number of pulses falls quite quickly as ${\tau}_{\rm\sigma}$ increases from 3 to 5. The initial rapid reduction in the number of pulses selected may be an indication of the removal of the small population of noise pulses remaining after the denoising process. The total number of pulses is not very sensitive to the threshold level in the region of 5 ${\rm\sigma}$. The 319 GRBs are listed in Table 1, along with the total number of pulses above 5 ${\rm\sigma}$ for each burst. Also included in the table are the number of isolated pulses at and above the 50% and 75% levels.

 

 

Table 1: The BATSE trigger number of the GRBs used in the analysis. N is the total number of pulses in the GRB, N(50/75) are the numbers of pulses that are isolated at and above the 50% and 75% levels respectively.

GRB	N	N(50/75)	GRB	N	N(50/75)	GRB	N	N(50/75)	GRB	N	N(50/75)	GRB	N	N(50/75)	GRB	N	N(50/75)
105	4	3/2	1974	2	0/0	2994	36	20/6	3929	3	2/1	6113	5	3/1	7318	12	5/3
109	19	10/2	1997	13	8/2	3001	3	3/2	3930	19	6/1	6124	30	14/5	7329	3	0/0
130	11	5/1	2037	6	5/3	3035	21	4/1	3936	5	1/1	6168	2	1/1	7343	11	2/1
143	13	6/3	2053	1	1/1	3039	5	3/1	3954	1	1/1	6198	10	1/1	7360	5	4/1
179	4	3/2	2067	8	1/1	3042	19	12/6	4039	33	21/7	6235	6	3/1	7374	1	1/1
219	13	4/3	2080	26	12/2	3057	52	3/1	4048	9	3/3	6242	3	3/3	7429	3	1/1
222	6	0/0	2083	2	2/1	3067	8	4/3	4312	4	2/1	6251	3	3/2	7446	1	1/1
249	21	4/1	2090	15	7/2	3105	30	19/9	4368	12	2/2	6266	9	2/1	7464	3	2/1
394	25	13/3	2110	13	3/1	3110	13	8/3	4556	3	1/1	6274	18	16/6	7475	2	2/2
451	2	2/2	2138	6	2/2	3115	9	6/3	4701	20	8/4	6303	1	1/1	7477	9	3/2
467	3	1/1	2151	3	3/3	3128	36	13/1	4814	2	1/1	6321	3	3/3	7491	40	11/1
469	1	1/1	2156	39	8/3	3138	4	2/1	5080	7	7/3	6329	9	1/1	7503	4	1/1
503	1	1/1	2213	5	5/2	3178	6	3/1	5304	14	0/0	6335	2	2/2	7527	2	2/2
543	3	2/1	2228	28	12/5	3227	17	5/3	5389	13	7/3	6336	3	0/0	7529	1	1/1
647	3	1/1	2232	14	8/6	3241	15	10/3	5417	2	0/0	6397	1	1/1	7530	2	2/2
660	2	2/1	2316	1	1/1	3245	71	24/6	5419	2	1/1	6400	3	3/3	7549	33	6/2
676	4	2/1	2321	11	4/1	3255	12	6/3	5439	4	4/3	6404	13	7/2	7560	12	8/6
678	52	21/6	2329	20	3/1	3269	15	13/7	5447	2	1/1	6413	8	7/3	7575	30	10/4
829	3	0/0	2362	2	2/2	3287	10	6/1	5450	13	5/4	6422	2	0/0	7605	9	9/3
841	6	6/3	2367	3	3/2	3290	6	4/3	5451	2	0/0	6436	3	3/3	7607	8	8/3
869	10	4/1	2371	2	2/1	3306	16	13/4	5470	7	4/4	6451	3	2/1	7678	37	5/2
907	4	3/2	2387	1	1/1	3330	13	5/3	5473	39	27/19	6453	25	5/1	7688	14	9/3
973	3	1/1	2393	2	1/1	3345	4	1/1	5477	12	9/5	6472	47	18/5	7695	27	16/5
999	2	2/2	2431	1	1/1	3351	10	7/5	5486	12	1/1	6525	8	6/5	7711	1	1/1
1025	4	2/1	2436	11	8/2	3408	44	17/5	5489	16	8/4	6528	8	3/1	7766	22	15/6
1085	2	0/0	2446	4	3/2	3415	14	8/4	5512	5	1/1	6560	20	16/9	7770	3	2/1
1122	10	3/1	2450	22	7/4	3436	5	4/1	5523	2	2/2	6576	18	11/5	7775	1	1/1
1141	9	0/0	2533	49	9/1	3458	8	2/2	5526	34	34/14	6587	28	9/4	7781	4	2/2
1157	10	8/2	2537	5	2/1	3480	1	1/1	5530	7	5/1	6593	20	13/2	7788	21	14/6
1159	1	1/1	2586	11	7/7	3481	11	1/1	5548	9	8/4	6621	2	2/2	7845	17	15/11
1190	4	2/2	2611	3	2/2	3488	6	6/5	5563	1	1/1	6629	6	4/3	7858	5	0/0
1204	3	3/3	2628	7	5/2	3489	12	2/2	5567	5	1/1	6630	2	1/1	7884	21	15/6
1385	23	4/1	2700	4	3/1	3491	3	1/1	5568	3	0/0	6665	8	2/1	7906	13	7/5
1419	3	1/1	2736	1	1/1	3512	1	1/1	5572	3	3/3	6672	4	3/1	7929	6	4/1
1425	7	5/1	2790	20	10/1	3516	10	7/2	5575	6	2/1	6683	10	10/3	7954	16	14/13
1440	15	9/2	2793	7	4/3	3523	16	0/0	5591	10	7/4	6694	10	6/1	7969	2	1/1
1443	2	2/2	2797	3	1/1	3569	4	2/2	5593	1	1/1	6764	8	3/1	7987	6	2/1
1468	19	17/6	2798	11	1/1	3593	13	6/1	5601	1	1/1	6814	1	1/1	7994	13	1/1
1533	32	20/12	2799	8	6/6	3598	3	3/2	5614	2	2/2	6816	8	7/3	7998	4	3/2
1541	33	11/1	2812	15	12/9	3634	7	7/3	5621	6	3/1	6824	2	2/2	8008	6	3/3
1578	3	0/0	2831	58	12/2	3648	3	3/1	5628	9	4/1	6904	1	1/1	8019	2	2/1
1606	33	15/4	2852	23	5/1	3649	2	2/2	5644	4	1/1	6930	2	2/2	8022	5	5/4
1609	8	2/1	2855	13	2/1	3658	4	2/1	5654	5	3/2	6963	17	10/2	8030	7	6/3
1625	15	2/1	2856	86	19/3	3663	9	7/3	5704	9	8/4	7028	5	3/2	8050	1	1/1
1652	6	1/1	2889	26	12/4	3765	4	2/2	5711	2	2/1	7113	52	8/1	8063	4	4/3
1663	16	0/0	2891	18	15/4	3776	7	1/1	5725	13	12/6	7170	16	6/2	8087	8	7/3
1664	5	1/1	2894	5	1/1	3788	4	3/1	5726	3	3/1	7172	1	1/1	8098	4	2/2
1676	35	21/7	2913	5	2/2	3860	13	9/3	5773	4	1/1	7185	19	16/7	8099	1	1/1
1683	6	3/1	2919	3	3/2	3866	1	1/1	5867	7	6/3	7240	4	4/4	8111	2	2/1
1709	2	0/0	2929	44	16/5	3870	1	1/1	5955	4	2/2	7247	2	1/1
1711	8	2/1	2953	6	2/2	3891	3	2/1	5989	8	7/5	7255	5	1/1
1815	10	5/2	2958	2	2/2	3893	3	1/1	5995	7	1/1	7290	1	1/1
1883	1	1/1	2984	22	16/6	3905	4	4/1	6090	3	1/1	7301	51	13/7
1886	5	0/0	2988	2	2/2	3912	1	1/1	6100	3	0/0	7305	3	3/3

Figure 5: The distribution of the number of pulses (N) per GRB. The shaded regions highlight the division into four categories, namely M, N, O and P with $1 \leq N \leq 2$, $3 \leq N \leq 12$, $13 \leq N \leq 24$ and $N \geq 25$ respectively.

In the analysis of pulse shapes, non-parametric methods were used to estimate the various characteristics of the pulse profiles. This approach was chosen to make the conclusions more robust since no assumptions were made about the pulse shapes. Also since the measurements are made on the isolated pulses selected by the algorithm, the degree of isolation can be varied arbitrarily. If a particular measurement was sensitive to influence from surrounding pulses then the threshold ${\rm\tau _I}$ was increased until the influence was reduced sufficiently, with the proviso that the number of pulses in the sample remains statistically useful.

The classification of pulses into isolated and non-isolated categories based on the algorithm allowed the measurement of characteristics of the temporal profile which are affected by neighbouring signals. The level of interference between pulses and the surrounding signal is dependent on the threshold at which the selection of these pulses is made. In fact a broad range of threshold levels were used to examine the time profiles and ${\rm\tau _I}$ was typically varied from 20% to 80%. It was decided based on these results to adopt pulses with ${\rm \tau_\sigma}\geq 5 {\rm\sigma}$ and ${\rm\tau _I} \geq 50\%$ for the main analysis of the pulse properties.

The pulse amplitude was measured as the maximum count in a 64 ms time interval after background subtraction. The pulse area was measured using the sum of the background subtracted count rates starting at 5% of the height of the pulse above the left minimum to 5% of the pulse height above the right minimum on the falling edge of the profile. The starting point at 5% above the minimum was chosen to eliminate contributions from background noise for pulses with minima widely separated from the maximum.

For similar reasons the rise time was measured from 5% of the height of the pulse above the left minimum to 95% of that height. The upper value of 95% ensures that the finishing point is robust against flat topped pulses and noise in the profile near the maximum. The fall time was measured in a similar way to the rise times i.e. from 95% to 5% of the pulse height above the right minimum.

The duration of the individual pulses was measured using the FWHM of the pulse. This approximation is valid only for well isolated pulses and tended to give poorer estimates for the true pulse width as the effect of neighbouring signals increased and the left and right minima of the pulse rose out of the background.

2.5 Number of pulses per GRB

The frequency distribution of the number of pulses (N) per GRB is given in Fig. 5. N has a range from 1 to 86 with a peak at a value of 3, a median of 6 and only 10% of GRBs have $N \geq 25$. For convenience N is divided into four categories (Fig. 5). There are 34 GRBs in category P and only 7 have N > 50. Many of the timing studies on GRBs have concentrated on the categories with large N (e.g. Norris et al. 1996; Li & Fenimore 1996), which means that these analyses have focussed solely on the tail of the distribution shown in Fig. 5.