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Astron. Astrophys. 336, 743-752 (1998)

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3. Data reduction

The essential steps to be performed in the data reduction have minutely been described in a thesis (Al 1997), and many of them also in Bendlin & Volkmer (1995), but there is no comprehensive overview readily available. The unusual and complex problems arising from the data obtained with the two-dimensional spectrometer therefore suggest a somewhat abridged, yet sufficiently detailed description here.

3.1. Preparatory steps

Among the narrow-band images which differ considerably within a scan, neither a correlation nor a "destretching" algorithm will work satisfactorily, therefore the results of both have to be derived from the white-light images first and must afterwards be applied to the narrow-band images taken strictly simultaneously. As a necessary prerequisite for this procedure, the common field of view must have been determined from the images of CCD 1 and CCD 2. After proper preparation, the grid images taken for this purpose revealed slightly different imaging ratios which entailed a reduction of the field of view for all white-light images. Finally, the distortions induced by the optical components characteristic of the light path leading to CCD 1 (as compared to the CCD 2-images) were removed using a destretching programme. With the information gained by the reduction steps performed with the grid images, everything was prepared for the subsequent alignment of the white-light with the narrow-band images.

3.2. White-light images

After subtraction of an averaged dark image from every raw white-light image, the solar images were divided by an averaged flat-field image. (Here, "averaged" always refers to the mean values for the individual pixels.) The field of view of the solar images was slightly reduced to match the imaging ratio of the narrow-band filtergrams, and the frames were all "cut" to their common image size. Then, the destretching coefficients obtained from the destretching algorithm used to compare the grid images of CCD 1 and CCD 2 were applied to all white-light images. After that, the rms value of the intensity was calculated for every image, and the "best" (or "reference") white-light image defined by the highest rms value within each scan was determined and stored separately. Despite the short exposure time and the fairly high image acquisition rate it proved necessary to correlate the images within each scan with their corresponding reference image to correct for image motion. The resulting shifts were first applied to the individual white-light images. Afterwards, the same procedure could be repeated with the corresponding narrow-band filtergrams taken simultaneously (see Sect. 3.3). Apart from image motion, seeing-induced distortions had to be compensated for by cross-correlation as well. Again, the destretching coefficients determined (and used) here were also valid for the narrow-band filtergrams and were thus kept for subsequent reduction steps. Noise was reduced in both spatial dimensions of the white-light data via an optimum filter working in the Fourier domain.

After the correlation of all images within each of the 128 scans making up the time series, the 128 reference images still had to be correlated with each other because of the "residual" image motion shifting "whole scans" against each other. By this procedure, the possible occurrence of guiding errors of the telescope was also taken into account. Next, the same correlation as for their corresponding reference image was performed for the other images in each scan as well, after which the common image size in all (128 [FORMULA] 30) frames was found to be 357 [FORMULA] 262 pixels or 71:004 [FORMULA] 52:004, respectively. Finally, the intensities of the reference images were normalized to make up for the increasing intensity the telescope received during the observation due to the rising Sun. Besides, the divison of each reference image by the mean intensity value of its corresponding scan ensured that in case of varying transparency of the Earth's atmosphere any resulting intensity fluctuations had also been compensated for.

3.3. Narrow-band filtergrams

Analogous to the reduction of the white-light images, first, the corresponding averaged dark image was subtracted from each image taken with the same exposure time. The flat-field correction, however, is quite different: In principle, every narrow-band image of each solar scan has to be divided by a flat-field image taken at exactly the same wavelength position, yet, the latter must not contain the Na D2 line. The scan obtained with a continuum source fulfils this requirement, but these flat-field images represent a different spatial intensity pattern than a uniform solar surface would have produced. Obviously, the opposite is true of the solar flat-field scan. It therefore suggested itself to combine the advantages of both scans and eliminate their shortcomings in creating a so-called artificial flat-field scan (see Bendlin & Volkmer (1995) for details) which is "ideal" in every respect. The intensity pattern of the new scan looks as if it were obtained with uniform solar light, which means no spatial intensity variation on the Sun and constant intensity in the spectral range of the Na D2 line. The transmission curves for each pixel on the CCD-chip thus also show the same blueshift according to their relative position to the optical axis as the line profiles contained in the narrow-band scans (apart from Doppler shifts) do. The fact that the automatic stabilization programme for the FPI allows the interferometer's transmission region to move a little around the wavelength of Na D2 between two successive scans was taken into account (1) by using a flat-field scan which covers a significantly larger spectral range than any single solar scan and (2) by determining meticulously for each individual scan of the time series which part of the flat-field scan matches it in wavelength. This was done by first averaging intensities over a suitable small area around the optical axis in the images of each narrow-band scan. A curve is thus obtained which contains the averaged line profile of Na D2. This curve was then divided by the averaged transmission curve of the artificial flat-field scan. The procedure usually had to be repeated several times by "shifting" these curves, the averaged curve and the transmission curve, against each other in wavelength until the resulting line profile closely resembled the exemplary profile from a standard atlas of the solar spectrum. Usually, the best results could only be achieved after some interpolation between the flat-field images, thus allowing finer wavelength steps. With the results from the averaged curves, the individual images of each solar scan were correspondingly divided by the matching images of the flat-field scan.

The necessary corrections for image motion and distortions were then applied to the narrow-band filtergrams by perfect analogy with the results obtained from the correlation of the corresponding white-light images taken simultaneously.

Compensation for the blueshift of the profiles (see Sect. 2.2) was achieved by dividing the flat-field scan obtained with solar light by the artificial flat-field scan, where care was taken again that the individual images used for division corresponded to the same wavelength. Thus, for every pixel in the field of view, a line profile resulted. The position of every line (intensity) minimum was determined by a fourth-order polynomial fit. With these values, an array was created giving the amount of wavelength shifts according to the pixels' position in the field of view. This array was employed to correct the individual line profiles of every pixel in every scan of the time series for the blueshift due to the FPI's position at the pupil's image.

Finally, noise in the data cube (one spectral and two spatial dimensions) was reduced by using a three-dimensional optimum filter in the Fourier domain.

3.4. New images

The proper ingredients of diagnostic diagrams are "minimum (intensity) images" and velocity maps. For the first type of images, the minimum intensity of each profile in the field of view was calculated using fourth-order polynomial fits to the immediate neighbourhood of the line core (i.e. [FORMULA]120 mÅ around the line minimum) to avoid "photospheric contamination" by contributions from the line wings. The velocity fields were derived from the Doppler shifts of the profiles which had been determined from the positions of the line minima represented by the polynomial fits as well. The "image numbers" of a scan also give a wavelength scale as the spectral difference between neighbouring images was chosen to be 30 mÅ .

The minimum images had to be correlated with each other and needed to be intensity-scaled in the same way as the reference white-light images whereas for the velocity maps, evidently only the correlation was necessary.

An example of small subfields of minimum images is given in Fig. 3.

Finally, the criterion used to discriminate between the intra-network and the network regions (the comparison between which will become of some importance in the following section) should be mentioned in this context: The averaged minimum intensity image which was obtained from all 128 minimum images shows bright parts, presumably bordering a supergranule (with a typical diameter of about [FORMULA]), and darker ones which were considered to belong to the intra-network. Whenever no clear distinction between the two seemed possible (here, the averaged minimum image prompted an intensity range from about 2.7% to 3.5% above the mean intensity [FORMULA]), the corresponding regions were totally neglected. At first sight it might be confusing that the intensity limit for the intra-network ([FORMULA] 2.7% above [FORMULA]) also lies above the mean intensity, but this is simply a consequence of the intensities in the intra-network regions fluctuating around some lower value than [FORMULA] (as to be expected), and thus surpassing it in some areas. Most intensity values in the network exceeded [FORMULA] considerably, especially for those areas persisting throughout the whole observing period.

According to the criterion given here, the intra-network covered about four times the size of the network areas.

3.5. Diagnostic tools

In the next step, one- and two-dimensional power, phase, and coherence spectra were computed. Although under certain conditions, it is possible to obtain V-V and I-I phase spectra from only one spectral line (using "fixed" positions in the line wings as well, see e.g. Deubner et al. (1996)), the observed set of data was best suited to yield [FORMULA] phase spectra, i.e. the phase differences between the Doppler shifts of the line cores and the line-centre intensities as a function of frequency and, in the two-dimensional case, of wavenumber which was taken as an average over suitable intervals in the [FORMULA]-[FORMULA] plane. Because of the apodization necessary in the Fourier transform which had been performed before, 5 pixels from each border of the images were lost, so that the resolution in wavenumber was slightly deteriorated. As an area of 69:004 [FORMULA] 50:004 corresponds to 50.315 Mm [FORMULA] 36.54 Mm on the solar surface, the resolution was limited in one spatial dimension by [FORMULA] Mm-1 and by [FORMULA] Mm-1 in the other.

The temporal Nyquist frequency of (only) about 8.9 mHz naturally suggests a practical improvement if the two-dimensional spectrometer were to be used regularly for observing solar oscillations. With faster PCs and enhanced disk storage it would be possible to take successive scans in shorter intervals and get even longer time series to improve the resolution in frequency (which is 0.14 mHz here). Nevertheless, as the following section shall show, the concept devised for this observation has proved a fairly good compromise.

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© European Southern Observatory (ESO) 1998

Online publication: July 20, 1998