$\begin{figure} { \resizebox{8cm}{!}{\includegraphics{fig3a.ps}}\resizebox{8cm}{!}{\includegraphics{fig3b.ps}} } \end{figure}$

Figure 3: Raw data for the Fe I 6301 and 6302 Å lines showing line inclination due to the off-axis spectrograph setup. I+V is shown at left and I-V at right

The analytical form of the curve can be calculated using coefficients that can be fitted from the flat-field data. But we can also rely on statistical methods which provide directly the shearing needed for each line of pixels. In this second approach we consider that in the FFI all the profiles along the slit are of the same shape, but displaced by an indeterminated quantity. We took as a reference the first row of the CCD and applied a Principal Component Analysis (PCA) technique to generate a base of the space of displaced profiles (Rees et al. 1999). Every other profile along the slit was compared against this database and a correction displacement determined within a precision of 0.01 pixels. To assess the limitations of both methods (the statistical and the fitting of a 2nd order function) we compared the two orthogonally-polarised images of the Fe I line at 5576 Å. For this line, insensitive to magnetic fields, both images should be strictly identical except for global displacements of wavelength or flat field errors. Each image was redressed independently and compared to the other one, line by line, to see the difference in the calculated displacement. We did that for all the 314 component images of the FFI. In Fig. 4 we show the statistics of these comparisons. In the standard deviation graph we see a constant value for all the flats of about 0.03 pixels, which we can take as the error in the determination of these displacements.

$\begin{figure} \includegraphics[width=\hsize]{fig4.ps}\hfill \end{figure}$

Figure 4: Statistics of the difference of displacements between the two orthogonally-polarised images for a series of 314 component images of the FFI for the 5576 Å Fe I line. Note the net displacement error of one path relative to the other due to an error in the superposition of the two first rows of each one of the comparison images

The origin of this error is unclear. The PCA and the fitting methods have been tested with analytical profiles with extraordinary results, so we consider that the 0.03 pixels limit is mainly due to differences of the profiles along the slit, differences due probably to instrumental effects. For instance, the same line in both CCD's does not correspond necessarily to strictly the same zone in the sun. Therefore, slightly different radial velocities can affect the two profiles and a different displacement is determined for them.

3.1.2 Dilatations and spatial corrections

Indeed, one of the main problems of this first set of spectropolarimetric data from THEMIS has been the diagnostic of this spatial displacement. For the correct determination of the polarimetric signal, the orthogonally-polarized paths must be compared in exactly the same point of the sun. To ensure this, we have used the solar details as seen in the continuum by the two cameras. A correlation was established between them and a displacement was calculated. This method was sufficient whenever the solar details were highly contrasted, as in the presence of a spot, for instance. But with just granulation some errors appeared here and there which perturbed the measurements. This method must therefore be substituted or, at least improved by adding some other spatial references, such as for instance slit hairs or the limits of the field of view.

Slit hairs would also help in the future for the other severe problem in the determination and correction of the geometry of the two paths: the different optical reduction for the two cameras. One objective is used in front of each CCD to reduce the desired field of view to the size of the CCD chip. These objectives must be identically positioned for the two paths. In case they are not, a dilatation coefficient between both images must be applied. This dilatation is the same in both directions: spatial and spectral. The method used to measure and correct this dilatation has been the comparison of two separate spectral zones, inside the range covered by the camera, to their counterparts from the other camera. The effect of the dilatation was to produce an apparent displacement of one zone relative to the other. This displacement was used to calculate the dilatation coefficient. The results of this method depend on the spectral region. For instance we obtained very good corrections for the 6149 - 6151 Å domain. A small window was defined around each one of the two lines and was compared to equally defined windows in the other camera. The 2 Å which separate the two lines resulted in a high precision in the measurement of the coefficient of dilatation. In the case of the 6301 - 6302 Å domain, the separation of just 1 Å appeared to be in the limit of the method.

3.2 Flat field calibration

The flat field image (FFI) was calculated as described in the previous section. The FFI showed absence of any solar details, however the pseudo-random movement around the disk center (with a radius of about half the solar disk radius) had the undesired effect of widening the profiles. Nevertheless a mean profile was obtained from the FFI, useful to determine the necessary geometrical corrections. Also from the FFI we obtained a correction matrix in which all the defects of the optical path were apparent. Between these effects special attention must be paid to fringes. Two systems of interference fringes appeared in one of the paths. This asymmetry between the two paths served to identify some of the optical pieces that cause these fringes. One of these systems shows a low frequency ( $\sim$ 75 pixels in period) and the correction matrix sufficed to remove it from the observations. The second one, on the other hand, with a much higher frequency ( $\sim$ 3 pixels in period) was not removed by this correction. Probably, small displacements due to real image displacements, to errors in the determination of the superposition of the correction matrix and each one of the observations, or to movements of the system of fringes are the origin of this problem. A small difference in the position of the fringes in the correction matrix from that of the observations, coupled with the high frequency of these fringes, makes full correction impossible. For these first observations and data reduction, we used a simple filtering in the Fourier space, although a better solution must be found for the future.

Wavelength-dependent scattered light was found throughout the observing run. Most of this light came from the F1 focus environment which was not well isolated from the spectrograph, allowing all the light rejected by the slit to enter it as scattered light. This effect was the origin of one of the toughest problems encountered during these observations. Figure 5 shows the equivalent widths of the Fe I line at 5576 Å for the two paths and for all the Stokes signals. There is a clear difference between them of about 10%. The 3 polarisations for each path give similar values and the observed difference is much bigger than the errors in the determination of the equivalent width. This difference between the two paths results in a spurious signal in the subtracted image which strongly limits the polarimetric sensitivity.

$\begin{figure} {\psfig{file=fig5.ps,width=\hsize} } \par\end{figure}$

Figure 5: Measured equivalent width (in mÅ) of the Fe I 5576 Å line for the two paths and all the analyzer positions ( $I\pm Q$ , $I\pm U$ and $I\pm V$ ) in a series of 100 images of solar disk center on August 23rd

3 Reduction techniques

3.1 Geometry corrections

3.1.1 Inclination and curvature corrections

3.1.2 Dilatations and spatial corrections

3.2 Flat field calibration