Forum Springer Astron. Astrophys.
Forum Whats New Search Orders

Astron. Astrophys. 353, 389-395 (2000)

Previous Section Next Section Title Page Table of Contents

2. Data and analysis procedures

2.1. Observational data

The present analysis is based on a sequence of white light granulation images ([FORMULA] nm) that was obtained on June 5, 1993 with the Swedish Vacuum Solar Telescope on La Palma (SVST, aperture 48 cm, see Scharmer et al. 1985 and Scharmer 1989). The observing and inititial reduction procedures have been described in detail by Simon et al. (1994). Frames were recorded by a Kodak Megaplus Model 1.4 CCD camera at a rate of 3.7 Hz. Each frame consisted of 1310 by 970 pixels with 8-bit digitization. A frame selection sytem (Scharmer & Lofdahl 1991) determined the rms contrast over a sub-field in real time. The two best images out of the 55 frames sampled per 15 s interval were recorded. The subsequent storage to magnetic tape took about 6 s; thus the total cycle time was about 21 s. For final analysis the better frame of each pair was selected. The image scale was 0.125 arcsec/pixel. Typical exposure times were 10-14 ms. The field of view was centered slightly south-east of the sunspot group NOAA 7519 which was situated at N05/E15 in the morning of June 5, 1993.

Thanks to excellent seeing conditions and the real-time frame selection, an image sequence of outstanding quality and length was obtained. From 8:07 to 19:07 UT the rms intensity contrast of most frames ranged between 6% and 10.6% with a median value of 8.1%. Fig. 1 of Paper II shows the temporal variation of the measured rms intensity contrast for the sequence. After corrections for dark current, flat field, rotation, transparency, and exposure variations the field of view was reduced to [FORMULA] pixels (41.7 Mm [FORMULA] 39.3 Mm); this field displayed quiet photosphere without magnetic activity. Mean photospheric intensity was normalized to 1.0. The frames were registered, corrected for instrumental profile, destretched to minimize seeing distortions, and interpolated to 21.05 s cadence, but not filtered subsonically, since this would spoil information on solar oscillations.

2.2. Fourier mapping

In order to reduce the frame-to-frame contrast changes caused by variable seeing, all granulation images were histogram equalized over 200 bins; i.e., the pixel brightnesses were rescaled such that there were equal numbers of pixels in each brightness bin. For example, the bin of brightness 1.00-1.01 contains 0.5% of the pixels, as does the maximum brightness bin 1.99-2.00.

The Fourier maps were constructed from three partly overlapping 89.5 min segments of 256 images each, centered at 12:45, 13:15, and 14:13 UT (see Fig. 3 for a schematic overview of the data sets); they represent a total of 513 images observed between 12:00 and 14:58 UT. The segments were apodized with a cosine bell to an effective duration of roughly 60 min to produce pixel-by-pixel Fourier amplitude maps for frequencies between 0 and 8.37 mHz (periods longer than 2 min) with a frequency resolution of 0.186 mHz. This duration is short enough to resolve meso-scale pattern evolution, while long enough to resolve wave modes with periods as long as 20 to 30 minutes. Since the three 89.5 min segments partly overlap, the results from the sequences are not completely independent. However, the overlap is not too large and the effect of overlap is further reduced by the apodizing which enhances the importance of images near the center of the segments and reduces it for images near the begin and end.

Fig. 1a (left column) shows an example of an 8 min amplitude map (middle panel) together with the intensity averaged over 1 h (top panel) as well as the divergence of the granular flow pattern (bottom panel). The right column, i.e., Fig. 1b, displays enlarged sections of each of the lower left sub-fields of Fig. 1a. A visual inspection of this figure puts forward the question which features in the amplitude map are co-spatial with positive or negative divergence. This is quantified by means of the correspondence factor C in Fig. 4. In the analysis below we make a distinction between amplitude maps of frequencies above 0.3 mHz and amplitude maps of lower frequency (i.e. the two lowest frequency channels) because the latter more or less display an intensity average of the granulation during the sequence. For frequencies above 0.3 mHz we investigate regions with amplitudes higher than twice the map average. They cover about 5% of the field.

[FIGURE] Fig. 1. a : granulation pattern averaged over 1 h (top); amplitude of the oscillation at periods around 8 minutes (middle); divergence of the granular proper motion determined by local correlation tracking (LCT) with positive divergence brighter than average (bottom). b : same quantities as in a shown for the lower left sub-field of each panel in a .

For the lowest frequencies we took the maps at 0 mHz representing the average granulation and discarded those at 0.186 mHz because the latter were corrupted by the apodizing function that has precisely this frequency. The results of the co-location tests between averaged granulation and the divergence pattern are given in Fig. 4 at 0 mHz and at 0.186 mHz. However, instead of working with pixels of twice the average amplitude (perhaps better called high brightness here), here we set the cutoff to larger than average and to larger than 1.10 the average. The former fill approximately 50% of the field, the latter about 45%. Note that we employ histogram-equalized images with a brightness distribution which is very different from the original images.

2.3. Divergence maps

Local correlation tracking with a 1.36 arcsec FWHM window was used to derive granular proper motion flow fields from pairs of images 21.05 sec apart. From sets of 86 frames thus 85 flow fields were computed covering a time span of [FORMULA] s, or 29.8 min each. After averaging sets of 85 flow fields a total of 10 averaged maps for the period 10:30 to 15:28 UT were obtained and the divergence of the averaged flows was computed; in most cases it ranged between [FORMULA] s-1. Fig. 2 shows the histograms of the divergence values of the 10 flow fields plotted on top of each other. The averaged distribution is rather smooth and very nearly symmetric, with a FWHM of [FORMULA] s-1. From each half-hour divergence map 10 segmented sub-maps were constructed, such that each sub-map contained 10% of the pixels, as shown schematically in Fig. 2. Thus sub-map 1 contains 10% of the pixels with strongest negative divergence, sub-map 2 the next 10%, and so on up to sub-map 10, which contains the 10% most positive divergence pixels. The 0.5 h average sub-maps were then submitted to a test for co-location with the corresponding Fourier amplitude maps as described in Sects. 2.4 and 2.5. The results for six of the sub-maps are presented in Fig. 4.

[FIGURE] Fig. 2. Histograms of ten half-hour divergence maps, each of area 41.7 Mm [FORMULA] 39.3 Mm, plotted on top of each other; bin width is [FORMULA]. For further analysis each divergence map was segmented into ten sub-maps 1...10 each containing 10% of the number of pixels, as shown schematically.

2.4. Correspondence factor

Spatial alignment is measured here with the correspondence factor C, a statistical tool which quantifies the cospatiality between two sub-sets of images or maps. This tool is comparable to the correlation factor but differs from it in that correlations must be calculated from full images, whereas correspondences may employ sub-fields. It was introduced in Sect. 3 of Paper I and is defined as [FORMULA], where [FORMULA] is the fraction of pixels of type A in one map that also belong to type B in another map and [FORMULA] is the filling factor of pixels of type B. C quantifies cospatiality as a random-draw likelihood. [FORMULA] suggests that A and B are independent phenomena while values [FORMULA] imply preferential co-location and [FORMULA] implies spatial avoidance. This statistical probability depends on spatial distribution only and may be estimated reliably from large data sets.

For example: the upper left panel of Fig. 1 displays the average of a sequence of 256 granulation images. Almost 50% of the pixels display a brightness above the average of the image. However, such pixels of high brightness constitute as much as 68% of the white regions in the lower left panel of Fig. 1 which represent the 10% of the field with the highest divergence. Therefore, the correspondence C between the regions with the highest divergence (sub-map 10 in Fig. 2) and pixels that are brighter than the field average is [FORMULA]. This tells us that if one chooses a pixel from the regions with highest divergence one has a 36% larger than random chance to pick one that is brighter than average.

2.5. Data structure and time delay charts

The temporal arrangement of the data sets and their use is demonstrated schematically in Fig. 3. As an example, the center line at [FORMULA] = 0 in the top panel of Fig. 4 is constructed from the computation of the correspondence between 3 simultaneous maps: sub-map 1 (10% strongest negative divergence pixels) of d5 vs. the amplitude maps of A1, sub-map 1 of d6 vs. A2, and sub-map 1 of d8 vs. A3. Values for [FORMULA] = 0.5 h represent C computed from sub-map 1 of d6 vs. A1, sub-map 1 of d7 vs. A2, and sub-map 1 of d9 vs. A3. Values for the other time lags in the top panel of Fig. 4 are computed accordingly - all for the 10% most negative divergence pixels of the 10 divergence maps. Each location in this frequency vs. time lag plot thus represents the mean of 3 correspondence values, except the lines at [FORMULA] = 1.5 h and 2.0 h for which A3 has no divergence maps taken 1.5 and 2 h later. From the variance of these averages one can estimate the [FORMULA] and thus obtain a measure of the reliability. We find that the contour levels are spaced at roughly 1.5-2[FORMULA]; in other words, the rms variations of C are about 0.05 to 0.07. It must be pointed out, however, that for [FORMULA] h the derived [FORMULA] is not a very reliable estimate for the error, because it is derived from only 2 amplitude maps that are not fully independent.

[FIGURE] Fig. 3. Schematic presentation of the data sets used for analysis. The abscissa represents the time axis in UT of the observing day 5 June 1993. d1... d10 are 10 divergence maps of 29.8 min duration; the divergence maps were derived from averaging sets of 85 granule flow fields computed from neighbouring frames. A1... A3 are 3 sets of Fourier amplitude maps derived from partly overlapping 89.5 min sections of the image time series; time apodization is indicated by the bell shaped curves.

The second panel from the top of Fig. 4 is created accordingly from the 10 divergence sub-maps 2, i.e., 10% of the pixels of divergence around [FORMULA] (see Fig. 2). The middle two panels of Fig. 4 represent C for very small divergence values just below and above zero, i.e. sub-maps 5 and 6, while the two bottom panels refer to the two most positive divergence bins (sub-maps 9 and 10). In the following we occasionally use the term "convergent" for regions of negative divergence.

[FIGURE] Fig. 4. Contour plot of the spatial correspondence C between high oscillatory amplitudes (exceeding twice the average) and sub-maps 1 ... 10 of Fig. 2. Abscissa is oscillation frequency, ordinate is time lag. Values of [FORMULA] are brighter than the grey background, those [FORMULA] are darker. Time lag is positive if amplitude maps are sampled before the divergence maps; temporal resolution is about 0.5 h. From top to bottom: sub-maps 1, 2, 5, 6, 9, 10 of Fig. 2.

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 2000

Online publication: December 8, 1999