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\title{Spectral Data Models for the Virtual Observatory}
%\titlemark{ }

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\author{Jonathan C.\ McDowell, Stephen Lowe, Mark Cresitello-Dittmar, Janet
DePonte Evans, Ian Evans, Arnold Rots, Michael Harris}
\affil{Smithsonian Astrophysical Observatory, 60 Garden St, Cambridge, MA 02138}

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\contact{Jonathan McDowell}
\email{jcm@cfa.harvard.edu}

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\paindex{McDowell, J. C.}
\aindex{Lowe, S.}     % Remove this line if there is only one author
\aindex{Cresitello-Dittmar, M.}
\aindex{DePonte Evans, J.}
\aindex{Evans, I.}
\aindex{Rots, A. H.}
\aindex{Harris, M.}
%-----------------------------------------------------------------------
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\authormark{McDowell et al.}

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\keywords{data: structures, software: data model, Virtual Observatory, spectroscopy}

%-----------------------------------------------------------------------
%			       Abstract
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\begin{abstract}          % Leave intact
There is no standard way in astronomy to represent digital spectroscopic
data.
We present requirements for a standardized 1-dimensional spectral data
model for use in the Virtual Observatory. We discuss the   
different kinds of spectra and the different observables used, as well
as the appropriate instrumental calibrations. Our model is intended
as a special case of an n-dimensional model for image and spectral
data, and would incorporate the FITS spectral WCS proposal for
coordinate descriptions.
The problem of describing spectral data is closely related to the
problems of defining bandpasses and photometric calibrations, as well as
of abstract instrument descriptions such as spectral responses and
efficiencies.

\end{abstract}

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\section{Introduction}

A significant fraction of the public data  available to the astronomical
community is in the form of spectra.  Although most 
archives store image metadata in fairly standard ways the history of
archiving spectral data is much less
successful. A recent survey of spectral
archives (see Tody on www.ivoa.net/forum/dal) which revealed a
heterogeneous collection of formats, many in ASCII tables, FITS tables,
or FITS images. This is in contrast to simple sky
images which, despite problems with how to represent mosaics, are mostly
in some variation of FITS image extensions. The current FITS WCS
proposal (Greisen et al.) for wavelength transformations is only
one of the steps needed to use archived spectra interoperably. The
VO will need to specify a uniform way to describe spectra.


This study attempts to isolate the metadata needed for representing
spectra to the VO, and proposes ways to structure this
metadata. 
Our model separates metadata needed by applications using the idealized,
generalized spectrum (pixel values, coordinates, errors, units,
resolution)  from metadata describing the idealized observation (sky
region, observation date) and from metadata which are needed by
specialized applications which deal with particular observational
strategies (e.g. spectral extraction details).


\section{\bf What is a spectrum?}

A spectrum is the value of an observable 
as a function of a spectral
coordinate, corrected or not for
various instrumental effects. The spectral survey confirms that existing
public data use the full range of possible parameters for the
electromagnetic spectrum; ( see Greisen et al. 2004 for spectral
coordinates in FITS).

We distinguish between a theoretical spectrum
sense, the energy output versus e.g. frequency, and
a spectral dataset in the observer's sense,
which maps such a spectrum onto an instrument.
Spectral datasets often have degeneracy
when two celestial axes and one spectral coordinate are 
projected onto two instrument coordinates.
Here we describe spectra (the
idealized $F(\nu)$) rather than spectral datasets.

The 1-D spectrum is clearly a special case of
a 1-D histogram, and our final VO scheme should unify
common metadata with other 1-D histograms (e.g. lightcurves)
and with n-dimensional generalizations such as the 2-D image.
This case study will be used to ensure that the n-D observation model
can encompass everything we need to represent a spectrum.

We can identify several other kinds of `spectrum':
\par -
Other observables as a function of wavelength:
percentage polarization, extinction coefficient. These can use
the present model.
\par - Arrays of spectra such as spectral-spatial data cubes.
They are a simple extension
if we model spatial images compatibly.
\par - Spectral coordinates for particles other than
photons: massless (gravitational
waves) or massive (electron energies in a jet, cosmic ray spectrum).
\par - Spectral coordinates not a particle property: power spectra
of source variability or CMB anisotropies, Fourier transforms generally.
Needs a slightly different model.

%\begin{table}[ht]
\begin{center}
{\small
\begin{tabular}{|ll|}
\hline
\hline
Observable    & Typical unit \\
\hline
Energy flux Density vs $\lambda$ &  erg cm$^{-2}$ s$^{-1}$~\AA$^{-1}$ \\
Energy flux Density vs $\nu$     &  Jy\\
Energy flux Density vs $\log\nu$ (for SED)    &  Jy Hz\\
Photon flux density vs Energy  &  photon cm$^{-2}$ s$^{-1}$ keV$^{-1}$ \\
Luminosity (at source) & erg s$^{-1}$ \AA$^{-1}$\\
Luminosity per decade & $L_\odot$ \\
Radiation energy density &  erg cm$^{-3}$ Hz$^{-1}$\\
Flux per solid angle (e.g. at source surface) &  erg cm$^{-2}$ s$^{-1}$ \AA$^{-1}$ sr$^{-1}$ \\
Antenna temperature & K \\
Brightness temperature & K\\
Magnitude in given band & mag\\
AB magnitude    & mag\\
Surface brightness flux density &  Jy / arcsec$^2$\\
Flux per resolution element  & Jy / beam \\
Surface brightness mag. &  mag / arcsec$^2$ \\
Instrumental reading  & ADU, count\\
Ratio of two spectra   & Dimensionless\\
\hline
\hline
\end{tabular}
\\
Table 1. An incomplete list of spectral observables
}
%\caption{An incomplete list of spectral observables}
\end{center}
%\end{table}


\section{Observables}

A crucial task for the VO is to standardize how data providers describe the
observable. What do the pixel values represent? At the moment, if  you are lucky
there is a BUNIT keyword in a FITS image to at least tell you the unit,
but that is not really sufficient. The VO will use tags such as Uniform Content
Descriptors (UCD2, Derriere et al. 2004) to unambiguously
characterize the physical concept being measured. Our spectral data
model must define a standard place to store this metadata. Examples
of spectral observables are listed in table 1.

\section{A Partial Model}

There are three main parts of our model: the dataset description,
(Fig. 1), the data container description (Fig. 2), and the observation coverage
description (not presented here). 
The complete Dataset object,  a simplified version of the one
presented in Cresitello-Dittmar et al. (2004), 
contains curation and coverage objects as well as several Data Container
objects. The dataset will have at least one Data Container for the
main data, and may have additional ones for a background spectrum,
an exposure array, and a sensitivity array.

\begin{figure}[ht]
\epsscale{0.70}\plotone{P3-4_f1.eps}
\caption[Dataset]{UML class diagram for the Dataset model.}
\end{figure}

The Data Container has a Data Storage
object containing Value, Error, Quality and Resolution sub-objects.
Our abstraction is that the data consists of an ordered array
of values (accessed by the Index object) which may be coupled
to one or more PixelMap objects locating each value in a coordinate
system (Cresitello-Dittmar et al. 2004). 
In the spectral case, the PixelMap would provide a bijection between
pixel number and the spectral coordinate. 

\begin{figure}[ht]
\epsscale{0.62}\plotone{P3-4_f2.eps}

\caption[aa]{UML class diagram for the Data Container model.}
\end{figure}

A simple case of such a map
is a set of regularly spaced, contiguous wavelength bins. However,
our abstraction also supports irregular or sparse arrays. 

One may in general obtain value, error, quality and resolution numbers
for each pixel, although in many cases things like the resolution may be
constant for all pixels; the four separate objects, accessed using the
Index, hide this implementation detail.


\section{Remaining Design Issues}

The observable is declared with a UCD, this needs to be 
eloborated to fully model a Photometric System object.

The resolution is grouped within the Data Container together
with values and errors, but the resolution object should 
ideally be  a line spread function at each pixel.
In contrast, the sensitivity, exposure
and background are treated as separate data containers for two
reasons: firstly, their effects are considered to be calibrated
out, and accounted for in the error object; and secondly, they
often have their own error, quality and resolution information
different from the main data - although we should require them
to have compatible pixel maps in some sense.
Alternative choices would be to include all these arrays in
a single Data Storage object, or at the other extreme to consider
them as separate but associated Dataset objects and replicate 
all the observation information.

The sensitivity and exposure require care when we extend the
model to a 3D energy-position cube, where practical implementations are
likely to express things separably as, e.g., an on-axis energy
sensitivity and a spatial sensitivity map.

We need to add  appropriate UCDs in the observation description for
specifying that a spectrum is in the rest frame and corrected for Milky
Way but not intergalactic absorption, or corrected for detector QE but
not vignetting. 

% You can also add an acknowledgments section as indicated below.

\acknowledgments

We acknowledge support from NSF grant no. AST 0121296 and Cooperative
Agreement AST 0122449, as well as the Chandra X-ray Center under NASA
contract NAS8-39073.








%-----------------------------------------------------------------------
%			      References
%-----------------------------------------------------------------------
% List your references below within the reference environment
% (i.e. between the \begin{references} and \end{references} tags).
% Each new reference should begin with a \reference command which sets
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% the User Guide listing "macro-ized" journals.   
%
% EXAMPLE:  \reference Hagiwara, K., \& Zeppenfeld, D.\  1986, 
%                Nucl.Phys., 274, 1
%           \reference H\'enon, M.\  1961, Ann.d'Ap., 24, 369
%           \reference King, I.\ R.\  1966, \aj, 71, 276
%           \reference King, I.\ R.\  1975, in Dynamics of Stellar 
%                Systems, ed.\ A.\ Hayli (Dordrecht: Reidel), 99
%           \reference Tody, D.\  1998, \adassvii, 146
%           \reference Zacharias, N.\ \& Zacharias, M.\ 2003,
%                \adassxii, \paperref{P7.6}
% 
% Note the following tricks used in the example above:
%
%   o  \& is used to format an ampersand symbol (&).
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%   o  \adassvii is a macro that expands to the full title, editor,
%      and publishing information for the ADASS VII conference
%      proceedings.  Such macros are defined for ADASS conferences I
%      through XI.
%   o  When referencing a paper in the current volume, use the
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%      the paper ID code for the paper you are referencing.  See the 
%      note in the "Paper ID Code" section above for details on how to 
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%

\begin{references}
\reference Cresitello-Dittmar, M. et al.\ 2004, \adassxiii,
           \paperref{P3-6}
\reference Derriere et al.\ 2004, \adassxiii,
           \paperref{P3-17}
\reference Greisen et al., 2004, in prep.
\end{references}

% Do not place any material after the references section

\end{document}  % Leave intact
