6 Discussion

The precise reduction applied on the raw spectrum P significantly decreased the noisy signal coming from the flat-field. The relative standard deviation of the signal of the raw spectrum P was 5.61% (see Table 2) and the resulting spectrum R₃ had a relative standard deviation of 4.72%. Although the last number seems to be rather big the variations of the signal in the resulting spectrum R₃ are caused by real physical variations of the solar granulation. The other numbers given in Table 2 documented that both the simple reduction (spectrum R₁) and the extended reduction did not help to minimize the noise in the raw spectrum P. The simple spectrum R₁ shows a smaller standard deviation of the signal than the spectrum R₂ but the shapes of the spectral lines in the spectrum R₁ were affected. The signal of the reduced spectra is higher comparing to the raw observed spectrum P due to correction of the spectrum by segment-flat M₆.

We have also analyzed the degradation of the signal in the spectra, caused by different types of flat-fields and by the background. Table 3 clearly shows that an averaging of only 7 background images G suppressed the noise in the mean background  $G_{\rm mean}$ significantly. On the other hand no significant difference of the standard deviation is seen in all flat field frames but they all have smaller noise than the raw spectrum P due to their "flat'' character.

The contribution of the different components of the flat-field matrix M to the noise of the original raw spectrum P is documented by the last four columns of the Table 3. The "hard'' part M_1,4,5 is more significant in this way ( $STD=2.59\%$) than the "soft'' part M_2,3 with a standard deviation of 1.44$\%$. More, the main contributor to the noise of the flat-field matrix M_1,4,5 is the slit-flat M₅ as can be seen in the two last columns of the Table 3. The standard deviation of M₅ is roughly two times higher compared to the pixel and camera flats M_1,4. This clearly documents that the slit-flat M₅ plays an important role in the flat-fielding of the solar spectra.

b) spectral characteristics
Spectral characteristics determined from spectra reduced with different accuracy exhibit significant differences. The widely used simple approach of reduction, produced fully insufficient accuracy of the results. The most influenced were the central parts of the spectral lines as a consequence of applying of an incorrect flat-field matrix M_2,3 i.e. the illumination-flat and shutter-flat (see Fig. 4a). Significant errors for line centre intensities $I_{\rm o}$ and for full width at half of line maxima FWHM are documented in this case. Only a small fraction of the data has accuracy better than 3% (see the part of 400-600 pixel in Figs. 10a,b). The relative errors of the rest of the data fluctuate by up to 10%. The humps of data in these two panels are caused by insufficient correction of the curvature of the spectral lines in the simple reduction. Only the central parts of the image are sufficiently corrected using the flat-field matrix M_2,3 (cf. Fig. 4). Then the difference between the correct data derived from precise reduction and the data resulting from the simple reduction shows humps.

Figure 10: The relative differences $RD_{\rm 3,1}$ of ( precise - simple) results a-c) and relative differences $RD_{\rm 3,2}$ of ( precise - extended) results  d-f) calculated for continuum intensity ($I_{\rm c}$), line centre intensity ($I_{\rm o}$) and full width at half maximum ( FWHM) according to Eq. (14). Notice the scaling of factor 10 between the graphs of FWHM.

Figure 11: Absolute differences of line centre shifts  $L_{\rm sh}$ expressed in m s-1, calculated according to Eq. (15). a) Difference between precise and simple flat-fielding; b) difference between precise and extended flat-fielding.

Figure 12: Absolute differences of bisectors Bi expressed in m s-1, calculated using the Eq. (15). a) Difference between precise and simple flat-fielding; b) difference between precise and extended flat-fielding. The panels from the top represent calculations for bisectors determined at 0.2, 0.4 and 0.6 of the line intensity respectively.

In the simple case the continuum intensities $I_{\rm c}$ were less influenced: they differ from the precise values by 1%-2%. Several values exceeded 3%. These excesses appeared at those positions where the intensity of the original raw spectrum P was strongly reduced by dust particles on the slit.

The differences of line centre Doppler shifts  $L_{\rm sh}$ are also significant. They exceeded 200 m s^-1 (cf. Fig. 11a). The $L_{\rm sh}$ measured in the precisely reduced spectrum reached $\sim$1 km s^-1. Thus a velocity of 200 m s^-1 represents an error of $\sim$20%. Similar results were found for bisectors (cf. Figs. 12a-c).

The extended reduction improved the results slightly but still not sufficiently. The relative differences of $I_{\rm c}$ as well as of $I_{\rm o}$ (cf. Figs. 11d,e) remain under 1% except those cases when the intensity of the original raw spectrum was reduced by dust particles on the slit. In this case the relative differences of $I_{\rm c}$ as well as of $I_{\rm o}$ exceeded 2% and some "spikes'' reached 5% and more.

Almost identical values of FWHM resulting from the extended and from the precise reductions were found. The differences of the FWHM were less than 0.2%. Only a few values exceeded 0.5%, again at those positions where the wires imposed by dust on the slit appeared in the original raw spectrum.

Similarly to the FWHM case, the line centre Doppler shifts  $L_{\rm sh}$ were also not very sensitive to the temporal changes of the flat-field conditions. The differences of  $L_{\rm sh}$ were smaller than 3 m s^-1 and only a few values reached 10 m s^-1 (cf. Figs. 12d-f). The relative errors of this data are of 0.3%-1.0%.

The small sensitivity of the FWHM and  $L_{\rm sh}$ to the change of flat-field conditions can be explained by the fact that the shift of the slit-flat M₅ did not significantly change the position and the shape of the particular line profile but it mostly changed the intensity of the line and of the continuum. This fact is also documented by the differences of bisectors which gradually increased at higher intensities of the spectral line (cf. Figs. 12d-f). The differences of bisectors at a line intensity 0.2 were smaller than 2 m s^-1 and gradually increased up to 10 m s^-1 for bisectors calculated at line intensity of 0.8 (not shown in figure). The relative errors in all these cases fluctuated between 0.2%-1.0%.

In the case of the simple reduction, the accuracy of the determination of the bisectors increased gradually when the bisectors were calculated for higher positions of line intensity. An opposite trend in the case of the extended reduction was found, but the accuracy of this approach was still one and half times better than in the simple reduction.