3 Data reduction

   
3.1 Background subtraction

 

 

Table 2: Exposure times (in seconds) for the three filters (1550 Å, 1600 Å and 1700 Å) are given together with the date of the observations.

date	exp. time	exp. time	exp. time
	1550 Å	1600 Å	1700 Å
	[s]	[s]	[s]
13/09/99	5.6	1.0	2.0
29/09/00	9.7	1.2	3.4

Dark exposures were taken on several days. These images show a diagonal light/dark pattern which changes in time and moves over the CCD. To calculate the dark offset this pattern could be successfully removed with a 2-d Fourier filter from each individual dark exposure. All dark images are then averaged to give the final dark correction applied to the actual images of the time sequence. This readout-pattern (which is electronic in its nature) is also present in the time sequence data. Due to the fact that the pattern moves over the CCD and the solar structures also move over the chip due to solar rotation this pattern introduces an oscillatory variation of intensities. Tests with using different darks (with and without the pattern removed) as well as using the standard dark (nearest temporal dark in the TRACE database provided with the TRACE software) confirm that the lowest temporal frequencies in our resulting intensity power spectra are most strongly influenced by this effect. Note that the pattern has a peak-to-peak value of about 4 DN/s and an rms of about 1.4 DN/s. Thus, areas that have low intensity levels will be influenced most (e.g. the sunspot umbra). In addition, Aschwanden et al. (2000) found a temperature dependence of the dark background and the noise level of the readout, which is an orbital effect. Checking for this effect, I indeed find a sinusoidal variation of the temperature of the CCD with a period of about 50 min. These changes also introduce a regular variation in the background and can create additional power peaks at the lowest frequencies.

Figure 1: Two examples of power spectra from a network (left) and an internetwork region (right). The horizontal line gives the 99% significance level calculated according to Groth. The crosses on top of the figures indicate the significant frequencies found by the randomisation test.

   
3.2 Normalization to exposure time

The data are normalized to exposure time, which gives data numbers per second (DN s^-1). No absolute calibration of the data is applied.

   
3.3 Despiking

In its near-Earth high-inclination orbit TRACE passes regularly through the Earth's radiation belts whose high energetic particles produce high intensity pixels. To remove them I use the same routine as described in Aschwanden et al. (2000), which applies a temporal and spatial mean for comparison and substitutes the hot pixel (the same routine is also used to clean the dark images). The contamination with spikes is worst in the 1550 Å images as they have the longest exposure time. I carried out a second deep cleaning of the 1550 Å data. Still, the 1550 Å data contain some residual noise during the time of the passage which has to be considered with care in the results.

   
3.4 Filling in missing images

In some cases single images were missing in the sequence. Missing images were interpolated by using the average of the two neighbouring images (although in case of the data I selected for this article, no images were missing).

   
3.5 Flatfielding

Due to a long-term burn-in effect there is a large-scale decrease of sensitivity of the CCD from the edge to the center of the chip (see Handy et al. 1999). We take this into account by fitting a paraboloid to some quiet sun calibration data taken in September 1999 and 2000. Since no reliable flatfield images are available no pixel-to-pixel correction is applied. There seems to be no spatially repetitive pattern in the flatfield image shown in the instrument guide. Thus, I do not think the missing flatfield will introduce artificial peaks into the power spectra, although it will increase the overall noise.

   
3.6 Co-alignment

As we are interested in the temporal change of single pixels, we have to properly co-align the sequence of images so as not to mix temporal and spatial changes. This is done in a two step procedure.

   
3.6.1 Correction of differential solar rotation

First, we calculate the shift between the first and the last image of the sequence due to solar differential rotation (according to the equation of Howard et al. 1990). This is calculated for the central column of the image. A shearing of the images due to differential rotation is corrected. This amounts to a difference of 0.5 pixels between the equatorial and the northern-most part of the image over 1 h for the Sep. 2000 data.

   
3.6.2 Correction of image drift

As a second step we calculate the shift of the images by a two-dimensional cross-correlation of the image sequence. The resulting drift amounts to about 3 pixels in E-W and 3 pixels in N-S in 2000 and 1.5 pixels in E-W and 2 pixels in N-S in 1999. This reflects the drift in the pointing of the spacecraft and shows an orbital variation (cf. Handy et al. 1999, Sect. 2.5). Note that there is an additional spatial offset between the filters of about 1 pixel that also displays an orbital variation. This has not been taken into account when displaying the final power maps.

The final time sequences can be viewed as movies, which can be found on my home page (http://aipsoe.aip.de/ $\sim$muglach/). This can be used to check the quality of the co-alignment. Note that the co-alignment is calculated for the complete two-dimensional images. Thus, it represents the global drift of the images. Due to the evolution of the various solar structures one can still see some of them move within the FOV. For example the bright structures in the moat around the sunspot produce the radial streaks seen around the spot in the temporally averaged image of Fig. 2.

Figure 2: Left: image of TRACE in the 1700 Å filter, brighter colour means higher intensity. It is a temporal average of the complete 4 h time sequence. The dark feature in the center of the field of view is the sunspot with umbra and penumbra. The bright patches are plage and network areas, in between is the internetwork. Right: snapshot of MDI magnetogram, the black and white represent the two magnetic polarities. The images show AR 9172, observed on September 29, 2000.

   
3.7 Power spectra

After this procedure one has a time sequence of intensity (DN s^-1) for each pixel of the image. Finally, I calculate power spectra using a Fast Fourier Transform (FFT), which includes a subtraction of the mean intensity and a 10% cosine apodisation. Note that the FFT requires equidistant time series, which is not strictly the case for the data of 1999 due to the interleaved white light images.

   
3.8 Statistical tests

Several components contribute to the power spectra: (i) oscillations, sinusoidal intensity variations which produce peaks in the power spectra and are the main topic of interest in this work. (ii) Non-oscillatory intensity variations that are still solar in their origin, e.g. short-term brightenings that propagate over the FOV along arcs (which are probably low-lying magnetic field structures) and that last for a few minutes (see UV movies on my home page or on the TRACE home page, http://vestige.lmsal.com/TRACE/). In the time domain these brightenings act like step functions and produce considerable power at all frequencies in the Fourier domain. The higher the formation height the more often they occur. They are especially frequent in the data of the 1550 Å filter and are a signature of transition region dynamics. Other possible sources of non-oscillatory power are convection and the evolution of the solar structures. (iii) Non-solar changes in intensity that I will in general call here noise and that might also produce power. I already mentioned one such source in Sect. 3.1. Aschwanden et al. (2000) give a detailed discussion of errors in TRACE data.

The following statistical tests are performed to estimate the reliability of the peaks that one can find in the power spectra.

   
3.8.1 Randomisation test

To get an approximation of the influence of noise in the power spectra and to have an estimate of the significance of their peaks, I performed a randomisation test. The aim of this test is to find how significant the peak amplitude of a periodic signal is within a sequence of data points that also contains noise. Note that it is a purely statistical method so one still has to consider if the periodicities one finds are of solar origin or not (this is also true for the other tests described below). The randomisation test has successfully been applied to various problems in stellar physics: Linnell Nemec & Nemec (1985) used to obtain the statistical significance of the identification of secondary pulsations in Cepheids and RR Lyrae variable stars. It is also widely implemented in the search for periodic signals from other stars, e.g. to look for a planet in the radial velocity data of $\epsilon$ Eri (Hatzes et al. 2000), to study the X-ray variation of AB Dor (Kürster et al. 1997) or to search for brown dwarfs around solar-like stars (Murdoch et al. 1993). O'Shea et al. (2001, 2002a,b) have used it to reliably determine oscillations in sunspot umbrae. This test has the advantage that it is parameter free, which means that it does not rely on any assumptions about the distribution of the noise. It also can be used in connection with various kinds of periodicity finding algorithms (e.g. a wavelet analysis is used in O'Shea et al. 2001, 2002a,b). More detailed descriptions of the method are given in Linnell Nemec & Nemec (1985) and O'Shea et al. (2001).

The method has been applied in the following way: the original measured time sequence S₁ of each single pixel is redistributed using a random generator. Then the power spectrum of this new sequence S₂ is calculated and compared with the power spectrum of S₁. This randomisation is repeated a number of times to get statistically significant results (see below). Then we count the fraction of times that a particular peak in the original power spectrum is higher than any peak in the randomised power spectra. This represents the probability p (in %) that the original peak is due to an actual periodicity in the data. In the following I set a probability limit of $p \geq 95\%$ (which means that power peaks with $p \geq 95\%$ are considered to be statistically significant).

A closer inspection of the power spectra (and other power spectra obtained within JOP097) reveals that the fluctuations seem to be frequency dependent. This means that the noise in the power spectra is also frequency dependent (non-white noise). To take this properly into account one would have to model all contributions to the power spectra (oscillatory and non-oscillatory solar and non solar effects) which is currently not possible. As the randomisation produces white noise out of the measured data it is valid only for white noise in the data.

Nevertheless, we can take this partly into account by applying the randomisation test iteratively: first, take the original sequence and look for significant power peaks as described above. In a second step those frequencies are filtered out (using a Fourier filter). Then the search for significant power peaks is repeated. In case one finds additional frequencies they are filtered out as well. The whole procedure is repeated until no significant frequencies are found anymore. After the final iteration all frequencies below the significance limit are set to zero.

Figure 3: The same as Fig. 2 but showing the TRACE 1700 Å temporally averaged image (left) and a snapshot MDI magnetogram (right) for AR 8693 and 8699 observed on September 13, 1999.

Figure 4: Spatial distribution of intensity oscillatory power for the Sept. 2000 TRACE data (1700 Å channel). The left image shows the integrated power between 2.3-4.3 mHz (5 min range) and the right one between 5.5-7.5 mHz (2-3 min range). Only those power peaks that are statistically significant according to the randomisation test have been included, which means that all power with a probability less than 95% has been set to 0 and is thus not included when calculating the frequency bins (cf. Sect. 3.8.1). Dark areas have less power than bright ones.

Figure 5: Spatial distribution of intensity oscillatory power for the Sept. 2000 TRACE data (1700 Å channel). The left image shows the integrated power between 2.3-4.3 mHz (5 min range) and the right one between 5.5-7.5 mHz (2-3 min range). All power below a 99% significance level according to Groth (1975) has been subtracted before calculating the frequency bins (cf. Sect. 3.8.2). Dark areas have less power than bright ones.

Figure 6: Spatial distribution of intensity oscillatory power for the Sept. 2000 TRACE data (1700 Å channel). The left image shows the integrated power between 2.3-4.3 mHz (5 min range) and the right one between 5.5-7.5 mHz (2-3 min range). To increase the statistics I first sum up 3 $\times$ 3 pixels, then the iterative randomisation test has been applied before calculating the frequency bins (cf. Sect. 3.8.3). Dark areas have less power than bright ones.

The biggest disadvantage of this method is that it is very time-consuming. It has to be done for each time sequence of the complete map separately and to get statistically reliable results one should carry out a sufficient number of permutations m of S₁. To test how large m should be I used the 1700 Å data from 2000 and set m= 100, 200, ..., 1000 and calculated the power maps for each value of m. Due to its statistical nature not all frequencies are found every time, but with increasing m the differences get smaller and smaller. So finally, m = 500 was chosen for the following analysis. To speed up the procedure all frequencies below 2 mHz are filtered out before starting the randomisation. These frequencies are of no interest for the current study and can also be influenced by the observing procedure (cf. Sect. 3.1).

   
3.8.2 Significance level according to Groth

Groth (1975) gives the significance of a power peak depending on the level of the (white) noise in the power spectra. We assume that all power above 7.5 mHz is due to noise and determine the average noise level to calculate the 99% significance level. To produce the final power maps we subtract this constant value from the power spectra. This method has frequently been used in solar oscillation studies (e.g. Balthasar et al. 1987; Solanki et al. 1996; Rüedi et al. 1998; Settele et al. 2002).

Figure 1 shows two examples of power spectra taken at a location in a network region (left) and an internetwork cell (right). The horizontal line gives the 99% significance level calculated according to Groth. The crosses on top of the figures indicate the frequencies where significant peaks were found by the randomisation test as described above (Sect. 3.8.1). The randomisation test is more rigorous in rejecting some peaks. E.g. a number of high frequency peaks (>6 mHz) in the internetwork spectrum lie slightly above the Groth level, but are not significant peaks in the randomisation test. It might well be that the randomisation test rejects peaks that still contain some oscillatory signal, on the other hand there is no doubt that the peaks that it finds significant represent an oscillation.

   
3.8.3 Spatial binning

Hill et al. (2001) have shown that simulating seeing on 2-d images can produce ring-like structures in power maps. Settele et al. (2002) also simulated image motion on their sunspot maps and found the largest artificial fluctuations of all measured parameters are at the location of the largest spatial derivatives. Although our data does not suffer from seeing, all numerical corrections have their limitations and small inaccuracies in the co-registration (as described in Sect. 3.6) may happen. Although the co-alignment of the data is probably better than one pixel I test this by summing up $2 \times 2$ and $3 \times 3$ pixels, thus decreasing the influence of co-alignment errors. Again, power spectra are calculated and the randomisation test as described above is applied to make the power maps.

   
3.9 Frequency binning

The final data set is a 3-dimensional data cube, with x and y, the spatial coordinates on the sun and frequency $\nu$. To display these results as 2-dimensional figures I carry out a frequency binning. Two different frequency domains are of interest, one at 3.3 mHz (5 min) and the other one around 6 mHz (3 min periods). Thus, I integrate all power in the range of 2.3-4.3 mHz and 5.5-7.5 mHz. These maps are shown in Figs. 4-9.