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\title{Instrument Modelling in Observational Astronomy} 

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\author{P.\ Ballester}
\affil{European Southern Observatory, 
       Karl-Schwarzschildstr. 2, D-85748 Garching Germany Email: pballest@eso.org}

\author{Michael R.\ Rosa\altaffilmark{1}}
\affil{Space Telescope European Coordinating Facility,
       Karl-Schwarzschildstr. 2, D-85748 Garching Germany Email: mrosa@eso.org}

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\altaffiltext{1}{Affiliated to the Astrophysics Division of the 
Space Science Department of the European Space Agency}

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\contact{Pascal Ballester}
\email{pballest@eso.org}

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\paindex{Ballester, P.}
\aindex{Rosa, M. R.}

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\authormark{Ballester \& Rosa}

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\keywords{instrument modelling, calibration, VLT, HST}

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%			       Abstract 
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By constructing instrument models which incorporate as full as 
possible a knowledge of optical and detector physics, the 
calibration of astronomical data can be placed on a firmer 
footing than is currently the norm. A number of developments make 
it more practical today to efficiently use optical models in the 
whole observational process: At first, the proposer can prepare 
observations using model based exposure time estimators and data 
simulators. Second, the observatory controls the instrumental 
configuration, tests data analysis procedures and provides 
calibration solutions with the help of instrument and environment 
models. We show in particular how such models can be used to 
ease very significantly the calibration and operation of 
complex instruments from the Hubble Space Telescope and 
the Very Large Telescope and provide a high level of 
homogeneity and integrity in the post-operational archives. 
We review the role of instrument models for observatory 
operations, observing, pipeline processing and data 
interpretation and describe the current usage of 
instrument modelling at the ST-ECF and ESO.
\end{abstract}

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\section{Introduction}

With ever more complex instruments entering service at general 
user type observatories an increasingly high importance is assigned 
to calibration tasks. The currently employed concepts for the calibration 
of observational data, evolutionary products of the historical development of 
instrument and detector technology, do not necessarily implement the best 
possible schemes to handle the precious data from large ground and space based 
facilities. Most of our calibration concepts are based on the principle of 
comparison with empirical readings from so called standards. Admittedly, 
computational tools have become much more elaborate, so that the 
standard readings (i.e. the calibration reference data) can 
be filtered and long term trends can be studied with the help of 
archives. However, at the very basis are empirical relations obtained 
from data containing noise. A typical example is the fitting of low order 
polynomials to obtain dispersion relations from wavelength 
calibration lamp spectra. The calibration strategies currently adopted 
in most of optical astronomy are empirical methods aimed at cleaning data from 
instrumental and environmental effects by a sequential application of instrumental 
corrections. Usually the transformation of detector based units into astrophysical 
units is done as a separate last step at the boundary between calibration and 
data analysis. The whole sequence makes it hard to arrive at a 
physical understanding of variations in instrumental characteristics which 
ought to drive maintenance measures or algorithmic solutions. It further requires 
a long learning process to determine the optimum deployment of resources - observing time 
for calibration observations and manpower for subsequent analysis. 

The gap between the calibration of individual observations and the monitoring of instrument 
performance has been closed by new operational concepts such as the one designed for the 
VLT data flow (Haggouchi et al., 2004). But we need in addition to implement
a physical calibration concept. As will be shown in the following, instrument 
models which provide the means to 
simulate the observational data gathering process from first principles form 
the basis for instrument configuration control, 
predictive calibration, and forward analysis. In this paper we review the use 
of instrument models at the Very Large Telescope (VLT) and 
the Hubble Space Telescope (HST) throughout the life-cycle of an observation:  exposure 
time calculators for the VLT, first use of model based calibrations for the HST 
Faint Object Spectrograph, bootstrapping calibration of the VLT UV-Echelle Spectrograph, 
predictive calibration for the HST Space Telecope Imaging Spectrograph and 
finally controlling the performance of the  VLT Interferometer. 

\section{Exposure Time Calculators} 

For the astronomer to be able to plan his observations, and for the 
Observation Programme Committee to be able to evaluate the quality of a proposed 
observation schedule, it is necessary to have good estimates of telescope performance and time 
required to reach the desired science objective. Exposure Time Calculators (ETCs) 
are one class of tools used to support this by modeling astronomical targets, 
telescopes and components in the optical path, and 
producing estimates of signal-to-noise ratios and exposure times. Some 
of the ETCs also provide 
information on the raw data products, such as simulated images and 
spectra or spectral formats. Furthermore, 
instrument scientists also use ETCs to evaluate the efficiency of instruments
during the design phase. Moreover, ETCs may have a number of additional applications, 
for example for the estimation of limiting magnitudes in an archive environment 
(Voisin, 2004).

Exposure time calculators predict the observational performance of an instrument in 
terms of signal-to-noise reached or exposure time needed for a given target and 
under certain conditions. At the VLT we record since 1998 an increasing usage of 
such models from an initial value of 20 simulations per proposal to the present figure of 
40 to 50. Peak usage takes place during the few days 
before the proposal deadline and can reach a total of 3000 a day. For this reason 
the ETC models are primarily designed for speed and easy maintenance, and less 
emphasis is placed on the accuracy of detail.

% We reset the footnote counter for the hyperlink since it does not
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We provide centralised access from a single Web 
\htmladdnormallinkfoot{address}{http://www.eso.org/observing/etc}
and through a system of templates provide a homogeneous interface across the instruments. 
User interface templates, macro language, 
components database and configuration files, as well 
as the design of generic models facilitate the traceability and the 
maintainability of the system. We also cover the complete range of 
instrumentation at the VLT by supporting more than 25 
instruments. These are not independent models, but rather six main programs 
for the optical imaging and spectroscopy, infrared imaging and spectroscopy 
as well as echelle and adaptive optics modes.

The ETCs rely on a variety of components, such as the transmissions 
and detector performance, and many specialised models are involved 
for the performance of adaptive optics systems, fiber optics or atmospheric 
sky emission. The accuracy of these models is about 10\%. 
The engineering database contains laboratory measurements of the 
optical components. The level of accuracy is adequate for the preparation 
of observations given the fact that the actual realisation of observation 
conditions, intensity of atmospheric lines, level of continuums, seeing 
reached will produce a variability of the observing results. Exposure time calculators 
provide a first level of verification of the performance of an instrument
if compared to measured performance. 

\section{Model-based Calibration for the HST Faint Object Spectrograph}

In order to use models for data calibration, we need to reach a better accuracy than
that of observation preparation models. Instrument models which are based on first 
principles clearly have the predictive power required to make substantial progress in the 
areas of calibration and data analysis. Before implementing the concepts of predictive
calibration and model-based analysis, two central questions have to be answered.

\begin{itemize}
\item Can one construct from first principles physical software models of astronomical 
instruments which are able to predict observations of standards, ie, calibration 
relations, to a sufficient degree of accuracy, and work reasonably fast 
so that they can be embedded into routine calibration and data analysis procedures?
\item Are such implementations affordable in comparison with 
the effort needed to maintain 
the classical calibration scheme, and do they bring any additional benefits for the 
operation of common-user instrumentation?
\end{itemize}

\subsection{Scattered Light Model} 

A first application of such modeling has been the FOS Scattered Light correction: 
calibrated UV spectra of a variety of targets show an unphysical 
upturn at the shorter wavelengths. 
The  effect is much more pronounced for 
intrinsically red energy distributions. It is obvious that the intrinsic
UV properties of such targets can only be
recovered via a software model that
simulates to a precision of better than one
part per million the dispersion and image
formation of the aperture-collimator-grating-detector combination. 
A physical model of the FOS, cast into software, was 
finally capable of correctly predicting the scattered light for any kind of intrinsic spectrum 
in all modes of the instrument (Bushouse, Rosa, Mueller, 1995). This model correctly replicates the relevant 
reflection, diffraction, dispersion and detection processes in the optical train of the FOS, and 
shows that the ultimate source of the scattered light are the far wings of the line spread function. 
The shape and level of these predicted far wings are very similar to the LSF measured pre--flight 
in the laboratory, indicating that the effect is not simply due to a deterioration of surfaces or 
adjustments. In fact the LSF is very close to the theoretical case of superb optics (Fig. 1), and only very 
small amounts of surface roughness and dust are required to explain the actual profile. 

\begin{figure}[ht]
\plotone{O5-1_1.ps}
\caption{Line profile with extended wings of the HST FOS high 
resolution mode. The thick line is the actual laboratory pre-flight 
measurement while the undulating thin line is the prediction from the 
FOS optical model with unlimited spatial resolution}
\end{figure}

This example demonstrates that a
software model going beyond a simple
throughput calculation, i.e. correctly
describing all relevant physical effects,
can be very powerful in solving problems
encountered during the scientific analysis
of data. Used in this way, the model appears
simply as an additional data analysis tool
in support of the calibration process.
Ultimately, the goal is to go one step
further and to advance from the calibration
strategy of signature removal currently
in use.

\subsection{Dispersion Relation and Geomagnetic Image Motion} 

Another model could be successfully developed and applied to FOS data. The 
dispersion relation from the standard FOS pipeline was represented by a polynomial 
solution. The usual problem with polynomial solutions is that outliers and 
misidentifications will be included in the polynomial solution and make it 
converge away from the true solution. Instead, using a dispersion model 
based on the grating equation and the 
S-distortion model of the detector yield a substantial 
improvement of the calibration accuracy (Fig. 2).

\begin{figure}[ht]
\plottwo{O5-1_2a.ps}{O5-1_2b.ps}
\caption{Residuals of measured line positions with respect to the predicted location when
using the standard FOS pipeline polynomial dispersion solutions (left panel), and
when using the optical model based solutions of the POA-FOS pipeline (right panel)}
\end{figure}

Since the FOS on HST used detector systems (Digicons) with electron optics, the
observations were affected by the geomagnetic field. Insufficient magnetic 
shielding of these detectors resulted in an imprint of the changing geomagnetic 
environment during HST orbiting the earth onto the raw data - the so called geomagnetically 
induced image motion problem (GIMP). It had been identified early on after launch and an 
on-board correction had been applied to all observations from April 1993 onwards. However
this correction itself was inadequate. As part of the Post-Operational Archive project 
(Rosa, 2000) of the ST-ECF both, the original effect and the inadequate procedure were modelised to yield
an optimal correction for all 24000 measurements in the FOS archive at once.

After correcting for the dispersion relation and the GIMP, a very substantial 
improvement of the radial velocities measurements could be obtained (Fig. 3). What 
was initially believed to be a grating wheel repeatability error turned out to be 
residual from an inadequate procedure that could only be proven and corrected 
through the physical model based approach 
(Rosa, Alexov, Bristow, Kerber, 2001)

\begin{figure}[ht]
\plotone{O5-1_3.ps}
\caption{Physical model based correction to a FOS science exposure: wavelength 
zero-point offsets in the original data (black symbols) and two versions of
the model based correction (red and blue symbols) on a rapid-readout
dataset spanning some 40 minutes of exposure time. More details are given in the 
above reference Rosa et al., 2001}
\end{figure}

\section{Predictive Calibration for the VLT UV-Echelle Spectrograph} 

If the model based approach proves to be useful for a low resolution first order spectrograph, 
we should consider to calibrate much more complex instruments. One of the most 
demanding cases of data calibration and analysis are 2D echelle spectra. Traditionally, 
they require complex data reduction procedures to cope simultaneously with both, the 
geometrical distortion of the raw data introduced by order curvature and line tilt, and 
with the spread of the signal across the tilted lines and between successive orders 
respectively. Unsupervised wavelength calibration for these instruments can only be achieved 
by reducing to a minimum the information needed to start the calibration process. This requires 
the most efficient use of the a priori knowledge from the optical properties of the 
instrument under consideration. 

A generic description of spectrographs based on first optical principles has been 
developed (Ballester \& Rosa, 1997). It incorporates off plane grating equations and 
rotations in three dimensions in order to 
adequately account for line tilt and order curvature. This formalism was validated by 
confronting the models for actual spectrographs (CASPEC and UVES from ESO, and STIS from HST) 
with ray tracing results and arc lamp exposures.

The geometric calibration is a complete definition of the spectral format including the order position and the 
wavelength  associated  to each detector pixel. This step was traditionally carried out via visual identification 
of a few lines and for this reason new methods had to be developed. The precision with which the geometric 
calibration is performed determines the accuracy of all successive steps, in particular the optimal extraction 
and the photometric calibration. In the case of UVES,  the high non-linearity of the  dispersion relation 
made it necessary to develop a physical model of dispersion in order to predict the  dispersion relation and to 
efficiently calibrate the several hundreds of optical configurations offered by the instrument (Fig. 4).

\begin{figure}[ht]
\plotone{O5-1_4.ps}
\caption{Sequence of events during UVES geometric pipeline calibration. Model predicted positions 
of wavelength calibration lines are projected onto a format-check frame. 
The predicted positions are adjusted to the observed positions and an initial dispersion relation 
solution is produced. The order position is automatically defined on a narrow flat-field exposure 
using the physical model. The initial dispersion relation is refined on a Th-Ar frame in order to 
fully take into account the slit curvature}
\end{figure}

To cover the many possible configurations the pipeline uses header information to select the appropriate
optical layout. This layout is based on the specific knowledge of the optical design of the 
instrument, cast into a physical model. The model 
is involved at many stages of the calibration to provide reference values and initial solutions for the 
current configuration making the data reduction completely automatic. In addition to the calibration solutions, 
the pipeline delivers quality control information to assist the user in assessing the proper execution of the data 
reduction process. At the VLT this approach has been extended to other
instruments, FLAMES-UVES which is a fiber adaptation of UVES, and also for the infrared ISAAC 
spectrograph where especially at the longer wavelength range there is a lack of atmospheric 
calibration lines (Yung, 2004).

%\vspace{-0.5cm}

\section{The STIS Calibration Enhancement project} 

Since the benefits of using the 2D generic spectrograph model in the UVES pipeline have 
proved to be so essential for the quality and the repeatability of the science data,
it was obvious to take the model based approach even further. The STIS/CE (Space Telescope 
Imaging Spectrograph Calibration Enhancement) project is a 
full fledged implementation of the physical model based approach for STIS on HST. 
The deliverables (ESA to NASA) involve the construction of a pipeline which is
heavily based on first principles such as physical optics and CCD physics.

A major component of the STIS/CE pipeline is the STIS dispersion model, based on 
the formalism previously developed for UVES.
In the case of STIS a mechanism is needed to optimize the configuration description of the instrument to an 
accuracy better than that provided by the engineering data, which can only be a starting point. There are fixed 
parameters precisely known from engineering such as number of grooves or focal lengths.  
Other parameters are not known to an accuracy sufficient for calibration purpose, for example 
grating angles. Finally some parameters may vary from exposure to exposure (repetition errors). 
Each mode is represented by a master configuration file that has been optimised using dedicated,
long exposure time, calibration frames. From that individualised configuration files can be produced 
with the help of short exposures accompanying the science data. Currently the dispersion model
predicts the location of wavelength catalog lines to a precision better than 0.1 pixel over an 
area of 2000x2000 pixels (Modigliani \& Rosa, 2004). Paramount to this success was not only
the refinement of the optical description, but also the re-measurement of wavelength calibration lamps
at the National Institute of Standards and Technology (NIST) (Kerber, Rosa, Sansonetti, Reader, 2003). 

Another interesting component of the STIS/CE pipeline is the corrective module for Charge Transfer Efficiency
(CTE). The Charge Coupled Devices (CCDs) operating in hostile radiation environments suffer a gradual 
decline in their CTE, or equivalently, an increase in charge transfer inefficiency (CTI). STIS and 
WFPC2 have both had their CTE monitored during their operation in orbit and both indeed show a measurable 
decline in CTE that has reached a level that can significantly affect scientific results. As part of the Instrument 
Physical Modelling Group effort to enhance the calibration of STIS a model was developed of the readout 
process for CCD detectors suffering from degraded charge transfer efficiency. This model is based on
simulating on the microscopic (single electrons) level the transfer of charge across the CCD chip during
readout, and the trapping of electrons in defects of the silicon lattice. The model enables one to 
make predictive corrections to data obtained with such detectors (Bristow, 2004).

\section{Models for the VLT Interferometer} 

With a maximum baseline length of 202 metres, the VLT Interferometer makes it possible to reach 
high angular resolution of the order of a few milliarcseconds. In March 2001, first fringes have been obtained with the VLT Interferometer using 
the test instrument VINCI. Three more instruments will be active at the VLT Interferometer, providing capabilities 
for coherent combination in the mid-infrared wavelength domain with MIDI, up to three near-infrared optical beams 
with AMBER, and simultaneous interferometric observations of two objects with PRIMA. 

Preparing an interferometric observation requires adequate tools that can handle the geometrical configuration of the array 
and target/calibrator positions. Most observations in interferometry will involve measurements for different spatial frequencies 
and are likely to require different configurations and spread over extended periods of time, several weeks or several months. 
The geometrical constraints on the observation of the science and calibrator targets and the limited observability of 
the objects due to both the range of delay lines and shadowing effects will make it necessary to assess the technical 
feasibility of observations at both stages of phase 1 and phase 2 preparation. During phase 1, general tools like the 
WEB-based visibility calculator and exposure time calculators will be provided. In phase 2, the details of the observation 
can be validated more accurately. 

\begin{figure}[ht]
\plotone{O5-1_5.ps}
\caption{The VLTI Visibility Calculator and the Calibrators Selection Tool}
\end{figure}

The VLTI Visibility Calculator (Fig. 5) is the tool used for such calculations. 
The VLTI Visibility Calculator computes the fringe visibility as a function of the 
object diameter, the position of the target in the sky at the time of the observation, 
and the selected configuration. It takes into account the horizon map of the observatory 
and shadowing effects induced by telescope domes and structures on the observatory 
platform. It computes the optical path length, the optical path difference, and takes 
into account the range of the Delay Lines to estimate the period of observation for a 
given target. A second tool, the VLTI Calibrators Selection Tool, allows the user to query 
the list of VLTI calibrators and select the adequate targets for a given science object. 
With the longest VLTI baseline (202 m), angular resolutions can be measured 
at the scale of one milliarcsec (1 mas). Unresolved objects, namely objects much smaller 
than the 1 mas limit, will yield maximal visibility. The fringe visibility decreases 
with the angular size of the observed target.

We need also to understand the performance of the system and developed to this end a unified model 
of performance for the instruments of the VLT Interferometer. The commissioning instrument VINCI 
has been used for measurements for two years and the first science instrument MIDI has been commissioned 
in the course of 2003.  The model predicts the accuracy on the visibility measurements. We are dealing here with 
new properties of the atmosphere, in particular one needs to characterise the atmospheric piston 
noise and its influence on the measurements. The predictions from this model are compared to 
the aggregated two years of VINCI measurements. Further verifications are now performed in the
mid-infrared using MIDI measurements. 

\section{Conclusion} 

In conclusion, our ample experience with the physical model based approach 
allows us to address the questions that we raised at the beginning of
this paper. Firstly, physical models can be developed, and they are accurate enough to 
solve many practical calibration problems in various operational environments (ground- and space-based). 
In particular, in several cases that we have described here, it would not have been 
possible to correct the data without a model. Today models are used in 
operations critical areas. The main additional benefit of models is that they 
provide an independent control on the data gathering process with the assurance 
that they are properly understood.

% Finally, we have a little acknowledgments section.

\acknowledgments

In this review we have tried to give credit to many contributors by citation of primary
articles. It is clear that we could not include all of them and therefore would like to 
acknowledge also those contributions not mentioned explicitly in the text. 

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%			      References
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% publication year, journal name, volume, and page number for
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%      for a full list of journal macros
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%      and publishing information for the ADASS VIII conference
%      proceedings.  Such macros are defined for ADASS conferences I
%      through X.
%   o  When referencing a paper in the current volume, use the
%      \adassviii and \paperref macros.  The argument to \paperref is
%      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 
%      determine the paper ID code for the paper you reference.  
%
\begin{references}

\reference Ballester, P., Rosa, M.R.\  1997, \aaps, 126, 563-571
\reference Ballester, P., et al.\  2000, ESO Messenger, 101, 31-36
\reference Bristow, P. 2004, \adassxiii, \paperref{P10-3}
\reference Bushouse, H., Rosa, M.R., M\"uller, Th. 1995, \adassiv, %Vol. 77,
   p. 345
\reference Haggouchi, K.., et al. 2004, \adassxiii, \paperref{O8-3}
\reference Kerber, F., Rosa, M.R., Sansonetti, 
   C.J., Reader, J.\  2003, ST-ECF Newsletter, 33, 2
\reference Modigliani, A. 2004, \adassxiii, \paperref{P10-10}
\reference Rosa, M.R. 2000, ST-ECF Newsletter, 27, 3
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