4 Reduction of spectra

Figure 2: The mean flat-field frame $F_{\rm mean}$ as constructed using Eq. (9) with K =40. WL = pictures of the artificial wires put on the spectrograph slit; SL = spectral lines; TL = terrestrial spectral line; DL = "dust'' lines caused by dust particles lying on the slit; IP = imperfections coming mainly from invalid pixels of the CCD chip. Inclination and stretching of the spectral line(s) and inclination of the horizontal wires are pointed out by the white vertical and horizontal lines, respectively. Different sensitivities of the four segments of the chip are obvious. The grey-scale bar gives relative intensities expressed in ADU.

We describe in detail all the steps (sometimes rather trivial) of the reduction process to keep an instructive scheme of the whole one. Spectral characteristics at three different steps of the reduction process were retrieved to demonstrate an increase in the precision of the results by using more precise steps of reduction. The precise reduction process consists of:

• 1. subtraction of the background G from the particular raw spectrum P;
• 2. construction of the mean flat-field frame $F_{\rm mean}$;
• 3. extraction of the segment-flat M₆ from the mean flat-field frame $F_{\rm mean}$;
• 4. retrieving of the components M_1,2,3,4,5 from the mean flat-field frame  $F_{\rm mean}$;
• 4.1 precise restoration of the X inclination of the image;
• 4.2 de-stretching - the restoration of the curvature and Y inclination of the spectral line;
• 4.3 construction of the flat-field matrix M;
• 4.4 extracting the slit-flat M₅ from the flat-field matrix M;
• 4.5 shift of the slit-flat M₅;
• 5. flat-fielding.

In the simple approach of the flat-fielding we omitted the steps 4.1, 4.2, 4.4, and 4.5. This means that the flat-fielding was done with the flat-field matrix M which was constructed directly from the raw mean flat-field frame  $F_{\rm mean}$ (cf. Fig. 2) corrected for the segment-flat M₆. In the extended approach of the flat-fielding we omitted step 4.5. The consequence is that the flat-field matrix M is shifted compared to the spectrum S.

The reduction was performed with programs written in the Interactive Data Language (IDL) using also the IDL KIS package at the Kiepenheuer-Institut für Sonnenphysik, Freiburg.

Figure 3: Average background  $G_{\rm mean}$  belonging to the spectra in Figs. 1 and 2. The grey-scale bar gives relative intensities expressed in ADU. Segment numbers are denoted inside the figure.

4.1 Subtraction of the background G from the particular raw spectrum P

In real spectra, the background G given in the Eqs. (1), (2), (4) and (5) consists of the noise of the CCD camera (dark current D), of the bias B, introduced by the readout electronics of the CCD and of the scattered light L coming from the spectrograph with a closed shutter. As was mentioned in Sect. 2, the background G can vary during the observation. The bias B varies with temperature, so the best way to keep it constant is to use the cooling control of the CCD camera. If this is not the case, one must estimate the sensitivity of the bias B to the temperature changes experimentally and then use an appropriate background G image according to the actual temperature during the acquisition of the spectra. Measurement of the background G should also be repeated if any change of the setup of the instrument was done concerning the change of the scattered light in the instrument.

To keep the S/N ratio as high as possible it is useful to average several background G frames (see Newberry 1991, for a more detailed description of signal-to noise considerations).

Thus, the Eq. (4) takes the following form:

$$S = P-G_{\rm mean} ~,$$ (6)

where

$$G_{\rm mean} = \frac{\sum_{j=1}^{J} G_j}{J}~$$ (7)

is the average of several particular background images G including the dark current D, bias B and scattered light L. The background G was stable during the whole observation run because the CCD camera was equipped with a cooling control. Therefore we used the average of the first 7 background images G for corrections. An example of  $G_{\rm mean}$ is demonstrated in Fig. 3. The mean value of the intensity of the $G_{\rm mean}$ was 44.96 ADU. 100 ADU was automatically added to every background frame (as well as to all other frames) to avoid negative values. The segments I, II, III and IV, denoted in Fig. 3, had relative intensities to the mean 0.636, 1.227, 0.978 and 1.150, respectively. Different intensities of the segments are due to different bias, i.e. due to nonuniform readout of the electronics of the four segments. Each particular raw spectrum P as well as flat-field source frame F was corrected for  $G_{\rm mean}$.

4.2 Construction of the mean flat-field frame F $\mathsfsl{_{mean}}$

In this section we will deal with the difficulties of taking flat field source frames F and mean flat-field frame  $F_{\rm mean}$ that serve as the basis for the computation of the flat-field matrix M. We will focus on the special case of a solar spectra taken with a large CCD array.

As we stated in Sect. 2, the flat-field is used to correct both the non-uniform response across the surface of the chip and non-uniform illumination of the chip. Therefore, it is not enough to illuminate the chip with uniform light and use the resultant exposure for correction. The uniform light must pass through the complete optical system with an identical focus, exposure time and other conditions as those valid for every particular spectrum P. Moreover, the spectral range of the flat-field source frame F should match that of the object being imaged. Therefore, for solar CCD spectroscopy, the ideal source for flat-fielding should be at infinity, should be a source similar to sunlight and the light must pass through the telescope and spectrograph. From these last requirements it is clear that it is impossible to get ideal artificial uniform light from infinity for flat-fielding in this case and the Sun itself is the only source intensive enough to mimic observational conditions. Thus, the flat-field source frames F must be derived from images that are almost identical to the particular raw spectrum P. There are several ways to create the flat-field source frames Fin this case:
a) the simplest way is to sandwich all particular raw P images we made and get an average "soft'' spectrum with smoothed spectral line(s);
b) the second possibility is to de-focus the telescope and to make enough un-sharp images. Then we construct the average of them to obtain a similar result to the previous case;
c) The third method of creation of the flat-field source frame F is to move the image of the Sun across the slit in the focal plane of the telescope and take a lot of images. The average of such images will give a similar result to the previous two cases, because the spectral lines will be smeared due to the fast movement of the solar image across the spectrograph slit. It is also possible to use a combination of the second and third methods (e.g. Johannesson et al. 1992).

It is useful to take the flat-field source frames F very often, to minimize the influence of the temporal changes of the flat-field conditions. Many cases of observations do not allow us to make flat-field source frames F very often. For example, the simultaneous observations with VTT and space instruments or investigation of temporal changes in the solar atmosphere require uninterrupted long series of observations. Then, one can take the flat-field source frames F only at the beginning and at the end of the observing run. So the temporal changes of the flat-field conditions introduce rather serious problems for the reduction of these spectra.

It is clearly seen, from the methods mentioned above, that all non-uniform illuminations of the chip in the case of large solar spectra are grouped together into one mean flat-field frame  $F_{\rm mean}$. Namely, the $F_{\rm mean}$ for the first (sandwiching) method will be defined as:

$$F_{\rm mean} = \frac{\sum_{k=1}^{K} [P_k-G_{\rm mean}]}{K},$$ (8)

where K is the number of all particular raw spectra P and  $G_{\rm mean}$ is the mean background. For the second and third methods the  $F_{\rm mean}$ is given by:

$$F_{\rm mean} = \frac{\sum_{k=1}^{K} [F_k-G_{\rm mean}]}{K},$$ (9)

where K is the number of the flat-field source frames F taken with de-focused telescope and/or with moving solar image across the focal plane of the telescope.

However, all three methods for the construction of  $F_{\rm mean}$ have some limitations. Usually, one tries to observe the long series during the best seeing conditions, keeping the spectrograph slit in the same position of the solar disk or scanning within small areas. Thus, the average of many raw spectra P still exhibits shifted patterns which remain from the real Doppler velocity shifts of a particular part of every particular raw spectrum P. This results in non-uniform  $F_{\rm mean}$ in the case of the first method. The second method with the de-focused telescope tends to optically disturb the spectrum giving rise to lateral displacement of the location of the spectral line(s) in the image plane in the camera and changes the fixed pattern, particularly the variations due to fringing and illumination. A similar displacement of the spectral lines could appear when one moves the telescope, if the scanned area at the solar disc is very large. This smears the spectral line due to averaging of shifted lines caused by real differential rotation of the Sun.

We used the third method for construction of the mean flat-field frame  $F_{\rm mean}$. The quick movement of the telescope was kept very near to the disc centre in a quiet region to minimize the mentioned above displacement effects in every particular flat-field source frame F. The final  $F_{\rm mean}$ was an average of 40 particular flat-field source frames Ffrom which  $G_{\rm mean}$ had previously been subtracted. The mean flat-field frame  $F_{\rm mean}$ is given in Fig. 2 with a description of the most important features in the spectrum. Identical features also appear in the raw spectrum P shown in Fig. 1. The $F_{\rm mean}$ contains all six flat-fields M_i mentioned in Sect. 2, and also the spectral lines as a consequence of the facts discussed in the beginning of this section. In the following sections we show how to retrieve the particular M_i from the mean flat-field frame  $F_{\rm mean}$.

Figure 4: a) An incorrect flat-field matrix M2,3 composed of the illumination-flat and the shutter-flat as a resultant from the mean flat-field frame  $F_{\rm mean}$ which was not corrected for stretching and inclination of the spectral lines. The grey-scale bar gives relative intensities normalized to 1; b) The stretching and inclination of the spectral lines determined in Sect. 4.4.2. Fluctuations of the centres of the spectral profiles along the spectrograph slit are shown by the thin line and the polynomial approximation is represented by the thick line.

4.3 Extraction of the segment-flat M $\mathsfsl{_6}$ from the mean flat-field frame F $\mathsfsl{_{mean}}$

The chip is composed of four independent parts which differ in sensitivity. Therefore, the segment-flat M₆ that bears only this differences is not affected by the presence of the spectral lines in the mean flat-field frame  $F_{\rm mean}$. So, it is possible and useful to extract the M₆ from the mean flat-field frame  $F_{\rm mean}$ before further image manipulation. In addition, the segment-flat M₆ has fixed geometrical, rectangular stability and further manipulations of the mean flat-field frame  $F_{\rm mean}$, which are described in the following subsections, would change the geometry of the segments. On the other hand, if the mean flat-field frame  $F_{\rm mean}$ were not corrected for the segment-flat M₆ before applying of the smoothing procedures required for rubbing out the spectral lines, the different intensities of the segments of the flat-field frame  $F_{\rm mean}$ would affect the smoothing procedure. In fact the four segments of the CCD chip also differ in the bias B, but this was already compensated for in Sect. 4.1.

Because the flat-field M₆ is uniform within every sub-segment, it is enough to estimate the ratios of sensitivities between them and simply construct the segment-flat matrix M₆ as a matrix of four subsegments with appropriate ratios of sensitivities between them. We experimentally determined the sensitivities of the segments which are equal to 1.004, 1.029, 0.930 and 1.037 for segment I, II, III and IV, respectively, comparing to the mean intensity of the whole chip, normalized to unity. The designation of the segments is identical as in Fig. 3. We have found the stability of the sensitivity of the individual segments to be within the range of $\pm$0.1% for the intensity range of 400-3200 ADU. Obtaining the segment-flat M₆, it was possible to correct the mean flat-field frame  $F_{\rm mean}$ by dividing it by M₆. Logically, the spectrum S was also corrected in the same way at this stage of the reduction process. We will name this improved spectrum S₁. It will be used in the simple reduction approach.

4.4 Retrieving of the flat-field matrix (M $\mathsfsl{_{1,2,3,4,5}}$) from the F $\mathsfsl{_{mean}}$

First, several manipulations with the mean flat-field frame  $F_{\rm mean}$ will be done in this section, to prepare it for a correct derivation of the flat-field matrices. Then, we will define the flat-field matrix M which is composed of five ( M_1,2,3,4,5) components. The elimination of the spectral lines from the flat-field matrix M will be described and finally each particular flat-field matrix will be extracted from the mean flat-field frame  $F_{\rm mean}$.

It was possible to extract the segment-flat M₆ from the mean flat-field frame  $F_{\rm mean}$ even when the spectral lines were presented. This is not the case in the other flat-field matrices. The spectral lines constrain the flat-field matrices and we must erase them from the $F_{\rm mean}$. The smoothing processes used for retrieving the flat-field matrix M are applied to spectra in both the X and Y directions in the image. To run them correctly it is necessary to have the mean flat-field frame  $F_{\rm mean}$(as well as the spectrum S₁) parallel to both edges of the CCD camera.

4.4.1 Precise restoration of the $\mathsfsl{X}$ inclination of the image

The horizontal sharp lines in the particular raw spectrum P as well as in the mean flat-field frame  $F_{\rm mean}$ (see the inclined WL and DL in Figs. 1 and 2) are artificial and/or natural. The artificial horizontal lines (WL) are images of wires placed on the spectrograph slit to adjust spectra taken by different CCD cameras in different spectral regions. The natural horizontal lines in the spectrum come from defects of the spectrograph slit. These defects are caused by dust particles on the slit of the spectrograph (DL - dark lines - less intensive) and by holes in the sharp edges of the slit (bright lines - more intensive). One should try to adjust the CCD camera in the focal plane of the spectrograph to align the edge of the image (the X axis) parallel to the horizontal sharp lines appearing in the spectrum. In any case, some inaccuracy (even a fraction of pixel) remains. This introduces problems in the next steps of reduction and also if one tries to calculate continua on the left and right side of a spectral profile using one row of the spectrum. To make the horizontal lines parallel to the edge of the image we used our procedure which allows us to shift the columns of the CCD image in the Y direction gradually with an arbitrary predefined accuracy without shifting of any part of the spectrum in the X direction. Re-binning and shifting of every particular column of the image in the Y direction with an accuracy of 1/100 of a pixel was performed in this particular reduction. As a consequence of our approach several incomplete rows at the top or at the bottom of the image had to be cut away. This didn't affect the next steps of the reduction, because only pure spectra between the two WL wires were finally used for calculations of spectral characteristics. The shifts were applied to both the mean flat-field frame  $F_{\rm mean}$ and the spectrum S₁.

4.4.2 De-stretching - restoration of the curvature and $\mathsfsl{Y}$ inclination of the spectral line

The spectral lines in both the spectrum S and the mean flat-field frame  $F_{\rm mean}$ exhibit curvature and inclination along the slit (along Y). The curvature is caused by the spectrograph design and the line inclination appears due to an inaccuracy in the adjustment of the parallel between the vertical edges of the CCD camera and the spectrum.

The correction of spectral line stretching and inclination is essential for the next steps of flat-fielding. To erase the spectral line from the  $F_{\rm mean}$ the smoothing also runs in the Y direction in the image. If the spectral line in the  $F_{\rm mean}$ is not strictly vertical, i.e. parallel with the Y edge of the image, the process will not work correctly, as we will see later. The best procedure to estimate line "stretching'' is to utilize a terrestrial line if it is available in the spectrum. Such a line has a well defined and stable centre. The positions of the centres determined for every scan directly reflect the curvatures and inclination of the spectral lines in the particular spectral region. But often there are no terrestrial lines available in the spectrum or the terrestrial lines are very weak, e.g. as in our case. Here we used the smoothed observed line itself derived from the mean flat-field frame  $F_{\rm mean}$ as the source of the required information. Again, the vector of the positions of the centres of the particular line profile (row) gave us information about the curvature and inclination of the spectral lines along the slit. Such a vector is given in Fig. 4b. The vector of the positions of the centres was approximated by a polynomial approximation of the second order in this case (thick line in Fig. 4b). The resulting polynomial approximation was finally used for shifts of every particular scan (row) in the X direction to make the spectral lines parallel to the Y edges of the image. An identical polynomial approximation for shift was applied to both the mean flat-field frame  $F_{\rm mean}$ and the spectrum S₁. This again, similar to the previous step, required several incomplete columns on the left and right of the image to be cut away. The prepared spectrum S₂ and the mean flat-field frame  $F_{\rm mean}$ after all steps of the reduction mentioned above are shown in Fig. 5. The spectrum S₂ will be used in the extended and precise reduction.

Figure 5: The prepared spectrum S2 a) and the prepared mean flat-field frame  $F_{\rm mean}$ b) after corrections for segment-flat M6, for X inclination and for stretching and Y inclination of the spectral lines. The grey-scale bars give relative intensities expressed in ADU. The rectangular box marked the area where the standard deviations were calculated (see Sect. 5).

Figure 6: a) The "hard'' flat-field matrix M1,4,5 composed of the pixel-flat, the slit-flat and the camera-flat; b) The "soft'' flat-field matrix M2,3 composed of the illumination-flat and shutter-flat; c) The slit-flat matrix M5; d) The flat-field matrix M1,4 composed of the pixel-flat and camera-flat. The grey-scale bars give relative intensities normalized to 1.0.

4.4.3 Construction of the flat-field matrix $\mathsfsl{M}$

The flat-field matrix M_1,2,3,4,5 will be defined in our case formally with an equation similar to the Eq. (5) which defined the ith particular flat-field matrix M_i but our resulting matrix M will be composed of five flat-fields and it will still contain the spectral lines. These lines must be erased from the matrix M to get the flat-field frame(s) commonly used for flat-fielding. The flat-field matrix equation reads:

$$M = \prod_{i} M_i = \frac{F_{\rm mean}}{A}~, ~~~i=1,2,..5$$ (10)

where M is the flat-field matrix composed of the five individual flat-field matrices and of spectral lines, A is the average pixel value used for normalization and  $F_{\rm mean}$represents the mean-flat field frame defined by Eq. (9). Different  $F_{\rm mean}$ were used for different approaches of the reduction. The $F_{\rm mean}$ created from the raw mean flat-field (Fig. 2) and corrected for segment flat M₆ was used in the simple reduction. The manipulated  $F_{\rm mean}$ shown in Fig. 5b was used in the extended and precise reduction.

Equation (10) is calculated in two steps:

• The flat-field matrix composed of the illumination-flat M₂, of the shutter-flat M₃ and of spectral lines is derived from  $F_{\rm mean}$. This we call a "soft'', part of the flat-field matrix M due to its smooth character;
• The flat-field matrix composed of the pixel-flat M₁, the camera-flat M₄, and the slit-flat M₅ is retrieved from  $F_{\rm mean}$. This we call a "hard'' part of the flat-field matrix M due to its sharp "spiky'' character.

The procedure smooths the image with a rectangular box (window) in both directions. Then additional smoothing of scans (rows) in the X direction by a least squares function spline is done and finally, for each column of the image the linear interpolation along the Y direction between the scans of the predefined box (window) is carried out. We used a window of 9 pixels width.

Dividing the mean-flat field frame  $F_{\rm mean}$ by the "soft'' part of the flat-field matrix M we derived the "hard'' part of the flat-field matrix M. This is shown in Fig. 6a. Then the spectral lines are eliminated from the "soft'' part of the flat-field matrix M dividing this part by "mean scan''. It is constructed as an average of all rows of the image. From this last step it is clear why the spectral lines must be parallel to the Y edge of the image. If it is not the case, some residual intensities in every row of the image will remain. This incorrect resulting "soft'' flat-field matrix M_2,3 is shown in Fig. 4a. The correct "soft'' M_2,3 is shown in Fig. 6b.

4.4.4 Extracting the slit-flat ${\mathsfsl{M}}_{\mathsf{5}}$ from the "hard'' part of the flat-field matrix $\mathsfsl{M}$

In this section, we will first analyze the main reason for the temporal changes of the flat-field conditions, the spectrograph drift. Then we will show that the time dependent parts of the flat-field matrix M are the slit-flat M₅, and illumination-flat M₂. Finally we will demonstrate how to solve post factum the problem of the temporal changes of the flat-field conditions. We split the "hard'' part of the flat-field matrix M into two components, the slit-flat M₅ and the flat-field matrix M_1,4 composed of pixel-flat and camera-flat.

Drift of the spectrograph

Spectrographs with high resolution and dispersion exhibit drift of the spectral lines and changes of the dispersion, e.g. due to thermal bending of mechanical components, like support structures of the slit basement, the collimating and focusing mirrors and the grating itself. It has to be taken into account that asymmetric heating of the solar towers occurs in the daytime with sunshine coming in the morning from the east and in the afternoon from the west. This can be measured with terrestrial spectral lines (e.g. of oxygen), laboratory molecular lines (e.g. of iodine) or laser reference lines. A detailed investigation made by Thiele (1982) showed for the solar spectrograph at Locarno drifts up to 0.5 pm per hour. Such drifts can still be determined using terrestrial oxygen lines. There was a high correlation of the drifts and changes of their sign with the recorded temperature in the environment in the data of Thiele. Very stable lasers used to control a Fourier Spectrometer (Brault 1979) allowed the determination of even shifts of the oxygen lines of the order of 0.01 pm within a day due to wind in the Earth's atmosphere (Balthasar et al. 1982).

For the VTT no published data about the drift of its spectrograph exist. Within the investigation performed in 1988 one of the authors (H.W.) determined drifts using terrestrial oxygen lines 6302.005 Å and 6302.771 Å. He found drifts that ranged from 0.015 pm within a minute up to 0.3 pm within an hour in August 1988.

Figure 7: The "slit-vector'' which represents the average of all columns of the flat-field matrix M1,4,5 shown in Fig. 6a.

We also found the drift of the spectrum in the focal plane of the spectrograph of VTT in both directions in a data set observed in 1996 using different CCD cameras. Again the terrestrial lines of the oxygen molecule 6302.005 Å and 6302.771 Å were used for determination of the drift in wavelength direction. The wires (WL) in the images were used as tracers for determination of the drift in spatial direction. Values of 1.06 pm (1.3 pixel) per hour and 0.26 arcsec (1.5 pixel) per hour were found for the X and Y directions, respectively. In addition we have found that the drift was not linear. It was steepest in the morning and gradually decreased in the middle of the day.

The analysis of the drift made with the data used in this work shows that the drift in the spatial direction (the WL wires were used for tracking) was 1.7 pixel per hour that corresponds to 0.21 arcsec. It is in agreement with the previous results.

Time dependent parts of the flat-field matrix M

As we have documented above the spectrograph drift moves the image (spectrum) in the focal plane of the spectrograph in both the X and Y directions. This will affect only two components of the whole flat-field matrix M, namely the illumination-flat M₂ and the slit-flat M₅, because the other three M_1,3,4, i.e. the pixel-flat, the shutter-flat and the camera-flat are closely connected to the CCD camera body.

The intensity errors caused by the movement of the illumination-flat M₂ are negligible because its gradient is very small across the camera field (see Fig. 6b). Thus a shift of even several pixels does not match any measurable effect in intensity changes. We separated the M₂ (together with the shutter-flat M₃) directly in the flat-fielding procedure as the "soft'' part of the flat-field matrix M.

The movement of the slit-flat M₅ is much more important. This flat-field is caused by dust and defects of the spectrograph slit. These show rapid changes of the intensity across the camera field in the Y direction and they are seen as the horizontal white and black lines and stripes described in Fig. 2 and mentioned above many times. The movement of the slit-flat M₅ acts in both the X and Y direction in the spectrum.

The movement of the slit-flat M₅ in the X direction is not important. The horizontal lines do not significantly change their intensities in the X direction because the inclination of the image was corrected in the previous step of the reduction (cf. Sect. 4.4.1).

But movement of the slit-flat M₅ in the Y direction by even a fraction of a pixel changes dramatically the whole "hard'' flat-field matrix M_1,4,5 which then does not match the conditions valid for the observed spectrum S₂. To solve this problem, one has to separate the slit-flat M₅ from the "hard'' part of the flat-field matrix M_1,4,5 (Fig. 6a), to be able to shift only the slit-flat M₅ in the Y direction, to fit the particular spectrum S₂ well.

Separation of the slit-flat M₅

We benefited from the fact that only the slit-flat M₅ is uniform in the X direction. This is a direct consequence of the imaging of the slit on the focal plane of the spectrograph. Thus, the vector made as an average of all columns of the "hard'' flat-field M_1,4,5 will mainly represent the fluctuations of the slit-flat M₅in the Y direction because the other two components, the pixel-flat M₁ and the camera-flat M₄, are randomly distributed across the CCD image and they will not contribute systematically to the averaged vector. We denoted this vector the "slit-vector'' (see Fig. 7). We created a new array of the same dimensions as the spectrum S₂ and the "slit-vector'' was replicated in all columns of the created array. Thus, this new array represents the pure slit-flat M₅ (see Fig. 6c). Obtaining the pure slit-flat M₅ makes it possible to extract the pixel-flat and the camera-flat M_1,4 dividing the "hard'' flat-field matrix M_1,4,5 by the pure slit-flat M₅. The resulting M_1,4 is shown in Fig. 6d.

Figure 8: a) The spectrum R2 as the resultant of the extended reduction made according to Eq. (12). The symbols V and H denote the sharp intensity spikes resulting from incorrect fitting of the strips and lines caused by dust and defects on the spectrograph slit and by the artificial wires seen in the prepared spectrum S2; b) The spectrum R3 as the resultant of the precise reduction made according to the Eq. (13). The grey-scale bars give relative intensities expressed in ADU.

4.5 Flat-fielding

At the beginning of Sect. 4 we defined the simple, extended and precise reductions. According to those definitions, we can rewrite Eq. (3) valid for flat-fielding in the three following forms:

Simple flat-fielding

This flat-fielding was done with the flat-field matrix  M_1,2,3,4,5 which was constructed according to Eq. (10) directly from the raw mean flat-field frame  $F_{\rm mean}$ (cf. Fig. 2) corrected only for the segment-flat M₆. So the simply reduced spectrum R₁ is defined by the equation:

$$R_1 = {S_1}~ \frac{1}{M_6}\ \prod_{i=1}^5 \frac{1}{M_i} ~,$$ (11)

where the spectrum S₁ defined by Eq. (6) was corrected for background and different sensitivities of the segments of the CCD chip only. The simple spectral characteristics were calculated using this reduced spectrum R₁.

Extended flat-fielding

In the extended approach of the flat-fielding we omitted only step 4.5 of the reduction, i.e. the appropriate shift of the slit-flat matrix M₅. The consequence is that the flat-field matrix M exhibits a shift comparing to the spectrum S₂. Thus, the equation used for extended flat-fielding is formally identical to the previous one:

$$R_2 = {S_2}~ \frac{1}{M_6}\ \prod_{i=1}^5 \frac{1}{M_i} ~,$$ (12)

but the spectrum S₂ represents here the manipulated prepared spectrum S₂ shown in Fig. 5a, and the flat-field matrix M_1,2,3,4,5was retrieved from the prepared mean flat-field frame $F_{\rm mean}$ shown in Fig. 5b. It means that S₂ as well as $F_{\rm mean}$ were corrected not only for background and different sensitivities of the segments of the CCD chip but also for the X inclination of the image and for stretching and the Y inclination of the spectral lines. The resulting R₂ is shown in Fig. 8a. The extended spectral characteristics were calculated using this reduced spectrum R₂. There are still sharp horizontal lines and stripes in R₂. This is due to temporal changes of the flat-field conditions mainly caused by the drift of the spectrograph. Thus, the mean flat-field frame  $F_{\rm mean}$ constructed from the flat-field source frames F taken at the beginning of the observing run does not perfectly match the flat-field conditions valid for the moment of the particular spectrum P registration. The resulting flat-field matrix  M_1,2,3,4,5 is slightly shifted in both directions comparing to S₂. So, some defects are enhanced instead of suppressed in R₂.

   

Table 2: Mean signal $\overline {Si}$ (ADU) and standard deviations "STD'' of the original and reduced arrays.

	P	$P-G_{\rm mean}$	R1	R2	R3
	Fig. 1			Fig. 8a	Fig. 8b
$\overline {Si}$	2192	2149	2308	2291	2282
STD [$\%$]	5.61	5.72	5.67	6.09	4.72

• $P={\rm raw}$ spectrum, $G_{\rm mean}={\rm mean}$ background, R₁= "simple'' spectrum, R₂= "extended'' spectrum, R₃= "precise'' spectrum.

Precise flat-fielding

To avoid the consequences mentioned above the precise flat-fielding differs in the following way: we compared the "Y'' coordinates of the WL wires in the slit-flat M₅(Fig. 6c) and in the prepared spectrum S₂ (Fig. 5a) looking for the highest correlation. The difference and direction in which the slit-flat M₅ must be shifted in the Y direction to fit the prepared spectrum S₂ was found. For the shift of the slit-flat M₅ we used our procedure in which the image is re-binned 100 times up to get a shift accuracy of 1/100 of pixel. This was done step by step, along the X direction to avoid the production of an extremely large array. It means that every particular column of the image was re-binned 100 times, shifted appropriately and re-binned back. The Eq. (12) will be modified in this case to equation:

$$R_3 = {S_2}~ \frac{1}{M_6}\ \prod_{i=1}^4 \frac{1}{M_i}\ \frac{1}{M_5} ~,$$ (13)

where the S₂, M₆ and M_i are identical to those which take place in Eq. (12) and M₅ denotes the new shifted slit-flat M₅. The precise spectrum R₃ reduced according to Eq. (13) is represented in Fig. 8b.

We need to keep the positions of the artificial wires WL for adjustment between the two CCD images taken with different CCD cameras in different spectral regions. Therefore, just before dividing the prepared spectrum S₂ by the flat-field matrices, the central rows of the artificial wires WL in the flat-field matrices were replaced by rows of mean intensity. After dividing the spectrum by the flat-field matrices the two horizontal central WL wires appeared back in identical positions in the reduced spectrum R₃ because they had not been removed from the prepared spectrum S₂.

   

Table 3: Mean signal $\overline {Si}$ (ADU) and standard deviations "STD'' of the measured arrays.

	G	$G_{\rm mean}$	F	F-G	$F-G_{\rm mean}$	$F_{\rm mean}$ (1)	$F_{\rm mean}$ (2)	M1,4,5	M2,3	M5	M1,4
		Fig. 3				Fig. 2	Fig. 5b	Fig. 6a	Fig. 6b	Fig. 6c	Fig. 6d
$\overline {Si}$	44	44	1248	1204	1204	1205	1303	1	1	1	1
STD [$\%$]	2.87	0.80	4.96	5.06	5.04	5.09	4.84	2.59	1.44	2.27	1.23

• $G={\rm background}$, $G_{\rm mean}={\rm mean background}$, $F={\rm flat frame}$, $F_{\rm mean}(1)={\rm raw mean flat}$, $F_{\rm mean}(2)={\rm prepared mean flat}$, M_1,4,5= "hard'' part of M, M_2,3= "soft'' part of M, $M_5={\rm slit}$-flat, $M_{1,4}={\rm pixel}$ and camera flat.