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\markboth{Vlasyuk}{Reduction of Bidimensional Spectral Data}
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\title{Reduction of Bidimensional Spectral Data
    Obtained with the Integral Field Spectrographs
    of the 6-m Telescope}  

\author{Valery V.\ Vlasyuk}
\affil{Special Astrophysical Observatory of Rus.\ Ac.\ Sci., 
       Nyzhnij Arkhyz, RUSSIA 357147} 
%Email : vvlas@sao.stavropol.su}

\begin {abstract}
	This paper presents an outline of the software, created by the author 
for reducing of bidimensional spectral data, obtained
with the Russian 6-meter telescope by means of the Multi-Pupil Field 
Spectrograph (MPFS) in its modifications : with different detectors--- an IPCS 
and CCDs, in combination with scanning Fabry-Perot interferometer, etc.
	The principal algorithms, concerning to the basic reducing steps,
as flatfield correction, spectra extraction
and separation, wavelength calibration are briefly described.
	Procedures are designed to be run on IBM-compatible computer under
MS-DOS operating system and may be transferred into any platform, being
written on C language.
\end{abstract}

\keywords{data analysis, image processing, spectroscopy, spectral reductions}

\section {MPFS Data Reduction}

\subsection{Method of Bidimensional Spectroscopy}

Method of bidimensional spectroscopy, proposed by Prof.\ G.\ Courtes 
(Courtes et al.\ 1987), allows to get independent spectra of each contiguous
spatial element of the field, using a bidimensional array of lenses in order
to form a grid of micro-pupils instead of slit in a classical spectrograph, where
these images are dispersing into spectra.

Number of spectra, registered simultaneously, as well as a spectral
range of data are limited by size of detector and entrance field of spectrograph.
Using of square lenses permits to realize correct data sampling,and such 
scheme is ideal for spectrophotometry tasks, allowing complete integration of 
energy.

     Main advantages of integral field spectroscopy are its ability to cover extended sources
with necessary spatial sampling, matched with seeing conditions, independence
of a spectral resolution from seeing, continuous spatial sampling.

\subsection{Integral Field Spectrograph of 6-m Telescope}

The Multi-Pupil Field Spectrograph (MPFS) have been developed in the Special Astrophysical Observatory
of Russian Academy of Sciences in 1990 and now is extensively using for 
spectral observations with the 6-m telescope (Afanasiev et al.\ 1995). 
It is providing simultaneous detection of about 100
individual spectra, produced by raster from 9 by 11 (or 8 by 16) lenses. 

The spatial sampling of data is choosing by observer accordingly to seeing 
quality and may be varied from 0.3\arcsec\ per lens to 1.6\arcsec\ per lens.

There are two different type of detectors may be used for spectra detection, being
attached to spectrograph's camera : 

a) bidimensional photon counting system on the base of 4-stage EMI image 
intensifier and TV head. Image processing complex of the 6-m telescope 
performs frame from 512$\times$512 pixels of 30$\mu$m size each.

b) CCD detector with 530$\times$580 pixels (pixel size - 18$\times$24$\mu$m) with
15 e$^{-}$ readout noise, developed in the Observatory.

The reduction of the MPFS data consists of the following steps :

-- bias subtraction;

-- cosmic particles hits removal;

-- data flat-fielding ( fulfilled if nonuniformities exceeds 2-3\%);

-- recognizing of the individual spectra on frame and its extraction;

-- transformation of spectra into uniform wavelength scale;

-- correction of spectra for variations in lenses transparency and vignetting;

-- spectral analysis, decomposing of complex profiles, measurements of astrophysical
valuable parameters, image reconstruction, etc.

The software package for data reduction, described here, is written on C language
and operating on personal computers AT/386 and higher under MS-DOS operating system 
control.
No any particular hardware using is necessary. All procedures may be easily
transferred into another operaing system and platform.


\subsection{Spectral Data Flat-Fielding}
Fullfilment of this step is rather clear then using for CCD data correction,
but it became a complex problem when observed data are acquired with the photon
counting system. It is widely known that all types of photon detectors on the
base of image intensifiers suffer from instabilities, depending on time, orientation
of system in space, etc. The instabilities, 
providing frame shifts, make useless the ordinary flat-fielding
procedure. In order to solve this problem the method of preliminary matching
of spectral and flatfield accumulations was developed.

As a reper points, indicating differential shifts of image, the most prominent
nonuniformities are using. To match it with appropriate accuracy 
we decided to use the bidimensional cross-correlations, which
are calculating in corresponding fragments of spectral data and flatfield of
64$\times$64 pixels. The position of the cross-correlation maximum yields value
of relative shift of one fragments with respect to another. The total
field of the relative shifts is approximating by bidimensional second-order
polynoms.

The regular form of differential shift suggests the presence of real correlation
between images. The total flatfileld image with high signal-to-noise ratio needs
long exposures also is preparing in such manner : individual sub-exposures
are correlating one with another and adding after corresponding shift.

\subsection{Spectra Extraction}
This procedure became to be very delicate, as spectra are packing on detector frame
very close one to another : mean distance between adjacent spectra didn't exceed
5 pixels usually. Moreover, some optics' drawbacks in combination with CCD's
charge transfer non-efficiency are slightly blurring spectra.

Preliminary step in extraction was defining of approximate spectra' location
on the frame. It was doing by means of a description of spectral trajectories 
by a bidimensional low-order polynom, approximating positions 
of a set of user-predefined repers, which were images of spectral lines on
arc lamp acquisitions. 

Following procedure are 
doing in two stages : first, accurate position 
and current profile for each
spectrum are defining by minimizing of observed summary profile, obtained
by stacking of some detector columns, around
each spectrum.
Second stage consists of fitting the amplitudes of
cross-dispersion profiles for each column, taking into account neighboring spectra,
in order to reach minimum of residuals between real and modelled profiles.
The observed cross-dispersion profiles are usually fairly presenting by a combination of 2 Gaussians,
so all fitting procedures are solving as multi-gaussian decomposing tasks.


\section{PYTHEAS-6 Data Processing}

\subsection{Brief Description of the Instrument}
PYTHEAS-6 is, in essence, an integral field spectrograph,
including scanning Fabry-Perot interferometer, which is illuminating by the
sharp angular beam, being mounted on the optical axis between enlarger and a
multi-lens array. The basic concept of this instrument is described in (leCoarer et al.\ 1995)

This instrument allows to observe extended sources with a spectral resolution,
defined by a used scanning Fabry-Perot etalon, which is providing spectral 
channeling and subsequent filling by executing tuning of etalon.

It is evident that choice of available etalon depends on the used linear dispersion
of spectrograph : successive etalon's spectral orders are to be separated enough
for accurate measurements.

Observations with PYTHEAS-6 consist of sequence of spectral integrations for
each spectral channel of etalon, obtained with changing distance between its
plates. Input spectra are modulating by the etalon transparency function, where
wavelength of maximum $\lambda$ for a particular order {\it p} is
defined by law :

\centerline{${\it p}\times\lambda=2\times n\times e\times cos(i).$}

Here n is the refractive index of internal substance of etalon,
e is a distance between plates,
i is an angle between axes of etalon and beam ($\approx$ 0).

Therefore, individual spectra are representing on spectral images as set of the 
spots with a various brightness, corresponding to the spectral intensity :
for continual source these spots will occupy all spectral range, for the linear
spectrum such spots will be seen only at lines location on the frame.

As following from the main law of etalon, the complete covering of {\it free spectral range} by a 
individual interference order depends on wavelength. In order
to acquire all available information necessary to choose total scanning distance,
corresponding to reddest wavelength. Then some information will be redundant for
blue wavelengths, as part of spectra will be scanned twice.


\subsection{Integration of Spectra}

For accurate integration of some overlapped spectra 
the method, similar to one of Gaussian extraction, briefly explained above, is applying
for each individual spectra over all successive spectral channels. Because during
etalon scanning the wavelengths, which is corresponding to interference order 
peaks, are increasing, spots will be shifted from one spectral channel to another.
Then, the integrated data from successive etalon channel for a particular spectrum,
being placed one under another, will show inclined tracks.

Restoration of result spectra is providing by integration of signal over this
image along tracks in dispersion direction. In order to separate accurately
individual interference orders, Gaussian extraction is applying also.

\subsection{Wavelength Calibration}

For sampling of spectra with a uniform step on wavelength necessary to reduce
data for comparison spectrum acquisition in such style as object's data. 
One can assign for these lines corresponding number of interference order, using
information, derived from continual source data and define number of spectral
channel, corresponding to each line peak. 

Let first spectral channel corresponds to integer order's number. Then to 
obtain real (non-integer) order value $FloOrder_{i}$ for particular {\it i}-th line with wavelength
$W_{i}$, found in channel $Chan_{i}$ and assigned to $IntOrder_{i}$, 
the following expression is using :

$
FloOrder_{i}\;=\;IntOrder_{i}\;+\;\frac{WaveScan}{W_{i}}\times\frac{Chan_{i}}{NumChan}
$

where NumChan - total amount of scanning channels used.

Using the derived values of $FloOrder_{i}$ one can to get ``empirical phase
curve'', representing a polynom, approximating set of products $FloOrder_{i}\times W_{i}$ as wavelength
function.
Errors of this approximation define resulted error 
of transformation into wavelength scale.  

As followed from the main etalon law, zeroth order coefficients is so-called 
``constant of etalon'' (right side of etalon's law) for a particular wavelength;
first order coefficient is offset (difference between true and assigned values)
for interference orders;
coefficients of higher orders represent phase shift, i.e., variations of etalon's
constant with wavelength.

Finally,  ``empirical phase curve'' allows to compute for each specified
wavelength corresponding value of interference order and to built result spectrum.

\section{Acknowledgements}

The author wishes to thank V.\ L.\ Afanasiev for useful discussions.
The creation of the described package was supported by an American Astronomical
Society grant.


%\begin{thebibliography}{}
\begin{references}
\reference Afanasiev, V.\ L., Shapovalova, A.\ I., Burenkov, A.\ N., Dodonov, S.\ N., \&
Vlasiouk, V.\ V.\ 1995, in Tridimensional Optical Spectroscopic Methods in Astrophysics,
eds.\ G.\ Comte \& M.\ Marcelin, 261
\reference le Coarer, E., Bensammar, S., Comte, G., Gach, J.\ L., \& Georgelin, Y.\  1995, \aaps, 111, 359
\reference Courtes, G., Georgelin, Y., Bacon, R., Monnet, G., \& Boulesteix, J.\
1987, in Instrumentation for Ground-Based Optical Astronomy : 
Present and Future, ed.\ L.\ Robinson, 266
%\end{thebibliography}{}
\end{references}
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