source: trunk/MagicSoft/Simulation/Detector/ReflectorII/doc/Tdas0211.tex@ 18342

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%%%  Kopyleft (K) 2000 J C Gonzalez
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\documentclass[12pt]{article}

\usepackage{magic-tdas}
\usepackage{amssymb}

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\begin{document}

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\title{ The Reflector simulation program v.0.6 }
\author{A.Moralejo\\ 
        \texttt{<moralejo@pd.infn.it>}} 
\date{January 20, 2003\\}
\TDAScode{MAGIC-TDAS 02-11\\ 030120/AMoralejo}
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\maketitle

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\begin{abstract}
In this document we provide a brief description of Reflector program
(version 0.6) and a guide to install and run it. Some of the
information contained in this document is also present in MAGIC-TDAS
02-05 dealing with the previous version of the program, but it has also
been included here for clarity. Two important bugs regarding the
ray-tracing routine have been corrected in this new version.
\par
NOTE: In December 2002, a first release of Reflector 0.6 was made,
together with a first version of the present TDAS note. Immediately
after that (too short a time to justify a new version number), some
other changes were implemented in the program to improve the
atmospheric absorption routines, and the dependence of mirror
reflectivity with wavelength was introduced. This document is the
manual of the final 0.6 version of the reflector, including
explanations of these last modifications.

\end{abstract}

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%------------------------------------------------------------
\section{Introduction}

The Reflector program was originally written by Jose Carlos Gonz\'alez 
and then improved by Harald Kornmayer. The Reflector program reads in
MMCS output files (cerxxxxxx files from corsika) and writes an output
file with the information about all the photons which reach the
telescope focal plane (taking into account atmospheric and mirror
absorption) and which are within the camera radius defined in the file 
{\bf magic.def}. In September 2001 D. Bastieri and C. Bigongiari
released a new version (v.0.4) to adapt the program to a format change
in the MMCS output. In June 2002 the version 0.5 was released, with
small changes in the ouput file format and some bug fixing (see TDAS
02-05).
\par
The aim of releasing the present version 0.6 was in principle only to
add some missing information in the output. However, the ray-tracing
routine was checked in detail and some inconsistencies were
found. As a result of this, an important bug was detected which
produced a systematic blurring of all the images. Later another bug
was found in the calculation of the photon arrival time on the
camera.These bugs, fixed in this version, are discussed in detail in
section \ref{notes}. 

%------------------------------------------------------------
\section{Description of simulation \label{descrip}}

The main steps of the simulation are:

\begin{enumerate}

\item Atmospheric absorption. 

\item Checking if the photon hits the dish.

\item Aluminum  absorption.

\item Determination of the mirror hitted.

\item Mirror reflection.

\item Checking if the photon is inside the camera borders. 

\item Calculation of photon arrival time on camera.

\end{enumerate}

The reflection of each mirror element is simulated in a realistic way
by introducing a gaussian spread of the reflected photons positions on
the camera plane. The sigma of this PSF is defined via the
{\bf point\_spread} parameter in the telescope description file {\bf
magic.def}, and has a value of 0.5 cm, which corresponds approximately
to the quality of the MAGIC mirrors produced up to date. Also the
possible misalignment of mirror elements is simulated (see section
\ref{neededfiles}).

\section{Notes on version 0.6 \label{notes}}

\subsection {First bug fixed in the ray tracing}

The previous versions of the reflector were extensively checked and a
strange behaviour was found in them. It is a well known fact (see for
instance the discussions in the MAGIC design report) that, to focus a
telescope on an object placed at a finite distance, one has to shift
the camera plane {\it away} from the mirror dish, with respect to the
position in which an object at infinity (a star) would be
focused (see fig. \ref{colimation}). For instance, with a paraboloid
of focal distance f = 1697 cm, an object placed 10 km above the
telescope would be focused on a plane at $\simeq 1700$ cm from the
dish (a distance measured along the telescope axis).
%
\begin{figure}[h!]
  \begin{center}
    \epsfig{file=eps/colimation.eps,width=0.7\textwidth}
    \caption{To focus an object at a finite distance h, the camera
plane must be moved away from the mirror a distance d, given
(approximately) by the formula on the right. For MAGIC, the shift is
of around 3 cm for a source located 10 km above the
telescope.\label{colimation}}
  \end{center} 
\end{figure}
%
\par
With reflector program up to version 0.5, and using the mirror
parameters in the standard magic.def file, we found out just the
opposite behaviour. We wrote a program to produce files with the same
structure as the Corsika Cherenkov output, but containing photons from
a point source. If we set the source at infinity, we found the best
spot placing the camera at a distance of 1697 cm (this can be changed in
the magic.def file via the {\it focal\_distance} parameter). We can
see the resulting spot in fig. \ref{spot_inf_f1697}. A completely
independent ray-tracing program was used to verify that this is the
spot that a f/1 16.97 m $\varnothing$ tesellated paraboloid should produce.
%
\begin{figure}[h]
  \begin{center}
    \epsfig{file=eps/spot_inf_f1697.eps,width=0.85\textwidth}
    \caption{Image of a point like source placed at infinity with
Reflector 0.5 (units of x and y are cm). Camera plane placed at 1697 cm
from the mirror. The circle on the left indicates (small) pixel
size. On the right side, projections on x and y of the
spot. \label{spot_inf_f1697}}
  \end{center} 
\end{figure}
%
If we now
place the source 10 km above the telescope, the best spot is achieved
at 1694 cm from the dish, instead of the expected 1700 cm
(figs. \ref{spot10kmf1700} and \ref{spot10kmf1694}). It must be
noted that for these and the following checks we removed all the
gaussian smearing which the program introduces to simulate the
possible mirror misalignments and surface irregularities (by the way,
a feature which somehow optimistically was not included in the
simulations shown in the MAGIC proposal). In this way we can check
just the ray tracing.
%
\begin{figure}[h!]
  \begin{center}
    \epsfig{file=eps/spot10kmf1700.eps,width=0.85\textwidth}
    \caption{Image of a point like source placed 10 km above the
telescope with Reflector 0.5 (units of x and y are cm). Camera plane
placed at f = 1700 cm from the mirror. The fact that the spot is so large
indicates a problem in the simulation, since with this camera position
the telescope should be focused at 10 km (see
fig. \ref{spot_inf_f1697}) and hence produce a spot similar to the one
shown in fig. \ref{spot_inf_f1697}. The hole in the middle of the spot
corresponds to the hole in the mirror dish, and indicates by itself a
focusing problem.
\label{spot10kmf1700}}
  \end{center} 
\end{figure}
%
\begin{figure}[h!]
  \begin{center}
    \epsfig{file=eps/spot10kmf1694.eps,width=0.85\textwidth}
    \caption{Image of a point like source placed 10 km above the
telescope with Reflector 0.5 (units of x and y are cm). Camera plane
placed at f = 1694 cm from the mirror. Now the spot is small, but the
camera plane is not at the expected position (see text).
\label{spot10kmf1694}}
  \end{center} 
\end{figure}
%
\par
In order to rule out a possible mistake in the generation of the
``false'' Cherenkov files, we repeated the check using real Corsika
files, and directly looking at shower images ($\Theta = 10^\circ$
incident gammas with a Crab-like spectrum). We must bear in mind that
the shower maximum for gammas of 100 GeV lies around 10 km above the MAGIC
level. From the result we can clearly see that the images are best
focused at 1694 cm (fig. \ref{evcompare}), therefore confirming the
existence of a real problem in the reflector simulation.
%
\begin{figure}[p]
  \begin{center}
    \epsfig{file=eps/evcompare.eps,width=0.85\textwidth}
    \caption{Gamma shower images obtained with Reflector 0.5 placing
the camera at 1700 cm (left) and 1694 cm (right) from the mirror
dish. Primary gammas at $\theta = 10^\circ$, E = 16, 46 and 232 GeV.
\label{evcompare}}
  \end{center} 
\end{figure}
%
\paragraph {Check of the mirror parameters in the magic.def file\\}
In the standard magic.def file we have been using up to now, the
spherical mirrors centers were found to be distributed on the surface
of a f = 1700 cm paraboloid. Their curvature radii, though discretized
in 8 ``zones'' as explained in the design report, corresponded in
average to the local mean curvature radius of the same parabolic
surface. However, their orientations were those corresponding to a f =
1697 cm paraboloid. Personally I do not think this is something
intended, and looks more like an error, but anyway we checked that this
was not the reason for the problems in the ray tracing: a test
magic.def file was created with {\it all} the mirror parameters
(positions, orientations and radii of curvature)
calculated as for a f = 1697 cm paraboloid, and no significant
difference could be seen. That 
is: i) the individual mirror orientations are the dominant factor, and
the overall dish in the old reflector behaved indeed like a f = 1697
cm parabolic 
mirror, and ii) the bug must be somewhere else. Even though the error
was not there, a new magic.def file has been produced containing the
parameters for the 956 mirrors of the final MAGIC design (instead of
the 920 mirrors foreseen in the beginning).
%
\paragraph {Solution of the problem\\}
Finally, a check of the routines in {\it ph2cph.c} revealed that the
reflector program was actually misinterpreting the information read
from the Cherenkov file written by Corsika. In figure \ref{coorsystems}
we can see the definition of Corsika's coordinate system (see also the
Corsika manual). In it, the x axis points north and the z axis points
up. The zenith angle $\Theta$ of a particle trajectory is measured
between the particle momentum and the negative z-axis, and the
azimuthal angle $\Phi$ between the positive x-axis and the x-y
component of the particle momentum, counterclockwise (0 -
2$\pi$). When the direction cosines of a particle's trajectory are
given, they refer also to its momentum vector (a downgoing
vector)\footnote{In Corsika the third component of the momentum is
measured along the negative z-axis and this is probably the origin of
the confusion which resulted in the bug}. The same is true for the
Cherenkov photons.
%
\begin{figure}[h]
  \begin{center}
    \epsfig{file=eps/coorsystems.eps,width=\textwidth}
    \caption{Coordinate systems of Corsika and MAGIC.\label{coorsystems}}
  \end{center} 
\end{figure}
%
\begin{figure}[h!]
  \begin{center}
    \epsfig{file=eps/parabola.eps,width=0.6\textwidth}
    \caption{Misunderstanding of photon direction cosines from Corsika
by Reflector v 0.5 and older versions.\label{parabola}}
  \end{center} 
\end{figure}
%
Unfortunately, in the output of Corsika, only the direction cosines
(u, v) with respect to the x and y axis are given. These completely
determine the particle's trajectory as long as one knows that they
refer to a downgoing versor, in which case one gets the third
direction cosine as $w = -\sqrt{1-u^2-v^2}$, with a minus sign. However,
in {\it ph2cph.c} we found exactly the opposite (lines 116 to 118 in
v.05):

\begin{verbatim}
  r[0] = ph->u; 
  r[1] = ph->v; 
  r[2] = (float) sqrt(1.0 - r[0]*r[0] - r[1]*r[1]);
\end{verbatim}

This means that the program was interpreting u and v as the direction
cosines of the {\it upgoing} versor towards which one should ``look''
to see the incoming photon. The situation is depicted in figure
\ref{parabola}. The two vectors at the bottom left of the plot are the
two possible interpretations of the direction cosines u and v. These
vectors have the same u and v, and differ only in the sign of their
third components w. As the figure illustrates, the reflector
simulation, by taking the wrong (positive) sign of w, was transforming
the light coming from a point source 10 km away, into a convergent
light beam which was then focused at a distance (1694 cm),
shorter than the focal length of the paraboloid, and produced a 
blurred image at the theoretically optimal distance (1700 cm). Of
course a paralel beam of light was focused at the right distance (1697
cm) since in that case u = v = 0, and both the upgoing and downgoing
versors have the same direction. 
\par
The example in figure \ref{parabola} shows the case in which the
telescope is pointing at zenith. It is easy to see that if the
telescope was pointing in an arbitrary direction a few degrees away
from zenith, the fact of taking the wrong sign in the third direction
cosine would spoil the reflection completely, to the point that the
spot would lie outside the camera limits. However, this is not what we
observed: when using the older reflector versions for different zenith
angles the images were still contained in the camera. The explanation
is that the transformation between Corsika's coordinates system and
the telescope system was also wrong, since the angles $\Phi$ and
$\Theta$ which indicate the telescope orientation (see fixed\_target
option in section \ref{commands}) follow the same convention taken
from Corsika: for instance, for pointing the telescope towards North
we should set $\Phi = 180^\circ$, because that would be the $\Phi$
value for a particle or photon coming from North (see again
fig. \ref{coorsystems}, left). This wrong transformation of coordinates
oriented the telescope in a way that the situation was always like the
one shown in fig. \ref{parabola}, and the image was formed in the
camera also for $\Theta > 0$, though also defocused.
\par
The telescope coordinate system shown in figure \ref{coorsystems}
(right) has its z-axis along the telescope axis, and the origin in the
center of the mirror dish. This system is used in the ray tracing
routine of the reflector simulation. When the telescope points up 
($\Theta = \Phi = 0$) this system matches exactly the one in Corsika. 
The general transformation between both is a simple rotation,
since for the sake of simplicity we assume in the simulation that the
origins always coincide. As we have said, in Reflector v.05 or older the rotation
matrix was wrong: it had been written assuming that ($\Phi$, $\Theta$)
indicated the direction towards which the telescope pointed. Actually,
for the reasons already exposed, the telescope must point to 
($\Phi +\pi$, $\Theta$). The function in charge of building the rotation
matrix is {\it makeOmega} (a part of {\it ph2cph.c}), which is called
from {\it reflector.c}. For the present version we have simply
replaced $\Phi$ by $\Phi + \pi$ in the function call. The transformation of
coordinates is shown in figure \ref{telecoor}, and can be seen as a
rotation of angle $\Phi +\pi$ around the z axis of Corsika plus a
rotation of angle $\Theta$ around the y'' axis of the telescope (the
same way in which the real MAGIC points).
%
\paragraph {New coordinate system of the camera \label{newcoordi}}
We have introduced another change in Reflector 0.6 regarding the
coordinates. In versions up to 0.5 the coordinates ($x_{camera}$,
$y_{camera}$) of the photon impact point on the camera plane were
given in the telescope system (x'', y'', z'') described in
fig. \ref{coorsystems}. This was a bit confusing (a rotation of the
telescope in the zenith axis resulted in a displacement in x'' of the
images), in particular it would have been messy when working in wobble
mode. We have now adopted the camera coordinate system proposed in
TDAS 01-05: when the observer is looking from the center of the
reflector in the direction of the telescope axis (towards the camera)
the $x_{camera}$ axis points horizontally to the right, and the
$y_{camera}$ axis points upwards. It is trivial from figure
\ref{telecoor} to see that the transformation needed to obtain these
coordinates is: $x_{camera} = -y''$, $\:y_{camera} = -x''$. This has been
added at the end of {\it ph2cph.c}.
%
\begin{figure}[h]
  \begin{center}
    \epsfig{file=eps/telecoor.eps,width=\textwidth}
    \caption{Transformation of coordinates between the Corsika and
MAGIC coordinate systems.\label{telecoor}}
  \end{center} 
\end{figure}
%
\paragraph {How old was the ray-tracing bug?\\}
%
The bug was certainly present in versions 0.4 and 0.5, but may be even
older. Nevertheless, there is no doubt that the reflector program used
for the simulation shown in the MAGIC design report was working
fine. Extensive proof 
of this is provided in an appendix of the design report. A plausible
explanation could be that, up to some date, the data being read in by
the reflector program (the Corsika output) contained direction cosines
which really referred to the upgoing versors of the photon directions,
and until then the program worked well. Then may be the output of
Corsika was changed to its present form, and the change went unnoticed.
The Corsika version used then was 4.52, whereas all further work has
been done with Corsika 5.20 or later versions. The Corsika history
file shows no record of any change in this respect, but given that we
have always used a slightly modified Corsika, it would not be
surprising if the Cherenkov output was modified at some point in the
MAGIC version of Corsika (MMCS). There is no documentation on this, so
if anyone has any relevant information, please make it public.
%
\paragraph {Influence of the bug on image analysis\\}
The fixing of the bug (resulting in sharper images) will for sure
improve the results of the image analysis, in particular with regard
to gamma / hadron separation. This could in part explain the
differences we have been observing in the expected performance of
MAGIC with respect to what was foreseen in the design report. However,
we note here again that in the simulation used there, not only was the
reflector working well, but also no noise was introduced in the
reflecting process. This was surely too optimistic, and it implies
that it will be hard to reproduce those results using a more realistic
approach which accounts for mirror imperfections. However, the
introduction of the noise (fig. \ref{refl06images}) has a less
dramatic effect than the defocusing which the bug was producing.
%
\subsection {Performance of ray-tracing in the new version}
In figure \ref{coma} we show the images of a point-like source at 10
km from the telescope, produced with the Reflector version 0.6,
and using the new version of the magic.def file (see see next
section). No noise has been introduced in the reflection, the observed
spots are just the result of optical aberrations. The light source has
been put at slightly different viewing angles
from the telescope. The results are comparable to those in the design
report, actually these are a bit better, the difference probably being
that the focal lengths of the mirror tiles in the older magic.def file
were discretized in only eight values, while now they change rather
continuously. Some images of a point source at infinity (a star) can be
seen in fig. \ref{coma_star}. We can see that for any incidence angle,
the area within which 50$\%$ of the light is concentrated is smaller
than that of a small pixel.
\par
In figure \ref{refl06images} the images of three gamma events ($\theta
= 10^\circ$, E = 16, 46, 232 GeV), the same of fig. \ref{evcompare}
are shown. They have been produced with Reflector 0.6 assuming perfect
spherical mirrors (left) and realistic ones (right). The images look
reasonable, much sharper than with the older versions, even when the
mirror imperfections are taken into account.
%
\begin{figure}[h!]
  \begin{center}
    \epsfig{file=eps/timing.eps,width=\textwidth}
    \caption{Test of reflector isochrony. The arrival time
distributions of photons in the camera are shown for (buggy) Reflector
0.5 and for Reflector 0.6. The sketch in the center shows the test for
the case in which the light beam is paralel to the telescope axis
(left plot). On the right, the same test has been made with light
arriving 1 degree off axis.
 \label{timing}}
  \end{center} 
\vspace*{-1cm}
\end{figure}
%
\subsection {Second bug: photon timing}
%
After a first release of Reflector 0.6 had been made public, another
important bug was found. We tried to check whether the simulated
reflecting dish was really ``isochronous''. A paralel beam of photons,
all sharing the same arrival time on the ground, were processed by the
simulation program and their arrival times on the camera plane were
histogrammed. The result can be seen in fig. \ref{timing} (dashed
histograms). This time the bug was quite evident: since Corsika gives
us the arrival time of photons on ground, the path from the point
where the photon hits the dish to the ground has to be subtracted (or
added, because since the center of the dish in the MC is at $z = 0$,
the mirror reflecting the photon may have $z < 0$ when the dish is
inclined). The sign in this subtraction (in {\it ph2cph.c}) was
wrong. This bug was present in both versions 0.5 and 0.4, and might
be related to the other one (a change of orientation of the z axis
at some point may have produced it). Since in the simulation made for
the design report the timing played little or no role (the camera
simulation did not consider the arrival times of photons) it is not
possible to know whether the bug was already in the code by then.
\par
This bug means that all the studies made up to now regarding photon
arrival times on the camera are completely useless (for instance, the
optimization of the time parameters in the L1 trigger has to be redone
from scratch!).
%
\subsection{The new magic.def file}
A new magic.def file (see sect. \ref{neededfiles}) has been created
and included in the Reflector 0.6 package. Now the number of
individual mirror tiles is 956, matching 
the number and distribution of the final MAGIC design. The mirror
centers and orientations are those corresponding to a paraboloid of
1697 cm focal (hence the camera plane is placed at 1700 cm from the
dish). The focal lengths have been calculated by R. Mirzoyan taking
into account the so called ``shortening effect'' (see design report).
A new axisdev.dat file (se again \ref{neededfiles}) with data for the
956 mirrors is also included.
%
\subsection{The new reflectivity.dat file}
Up to version 0.5 of the program, the reflectivity of the mirrors
(which the program reads in from the file reflectivity.dat) was
considered to be equal to $90\%$ for all wavelengths. Following
measurements performed in Padua of a mirror sample, the reflectivity
has been found to be between 80 and 90$\%$ in the range from 280 to 650 nm,
with a dependence on wavelength which is shown in figure \ref{reflec}.
%
\begin{figure}[h!]
  \begin{center}
    \epsfig{file=eps/reflec.eps,width=0.7\textwidth,height=0.4\textwidth}
    \caption{Reflectivity of the MAGIC telescope mirrors as a
function of the wavelength of the incident light. The dip around 350
nm is due to destructive interference of the light reflected on the
aluminum plate with that reflected on the protective coating of the
mirror. \label{reflec}}
  \end{center} 
\end{figure}
%
\begin{figure}[p]
  \begin{center}
    \epsfig{file=eps/coma.eps,width=0.85\textwidth}
    \caption{Reflector 0.6. Images of a point-like source at
10 km from the telescope for different incident angles (from on-axis
to 2 degrees off-axis). The quantity d50 indicates the diameter of a
circle (plotted) containing 50$\%$ of the reflected light. Note that
the z-axis scale is logarithmic, and the same in the first five
plots. The last plot shows the x-axis projections in linear scale.
 \label{coma}}
  \end{center} 
\vspace*{-1cm}
\end{figure}
%
\begin{figure}[p]
  \begin{center}
    \epsfig{file=eps/refl06images.eps,width=0.85\textwidth}
    \caption{Reflector 0.6. Images of the three gamma showers shown in
fig. \ref{evcompare}, without noise added in the reflection (left) and
with the standard noise (right) as described in section
\ref{descrip}. Note that the orientation of the images has changed as a
result of the introduction of a new camera coordinate system (see page
\pageref{newcoordi}).
 \label{refl06images}}
  \end{center} 
\vspace*{-1cm}
\end{figure}
%
\begin{figure}[p]
  \begin{center}
    \epsfig{file=eps/coma_star.eps,width=0.85\textwidth}
    \caption{Reflector 0.6. Images of a star. The quantity d50
indicates the diameter of a circle (plotted) containing 50$\%$ of the
reflected light.
 \label{coma_star}}
  \end{center} 
\vspace*{-1cm}
\end{figure}
%
\subsection{The {\itshape cermaker} program}
A test program to produce cer files (input for the reflector)
containing photons from a point-light source of light placed in any
position has been added to the Reflector package {\it
(tester/cermaker.c)}. This is the same program used to produce the
plots shown in this report. The usage is as follows:
\begin{verbatim}
cermaker source_x(cm) source_y(cm) source_z(cm) [events]
\end{verbatim}
The source position is given with respect to the telescope. The output
file is called {\it cer000001}, and can be read by the reflector program.
%
\subsection{Changes in the atmospheric absorption routines}
%
In the present version of the program, the simulation of atmospheric
absorption has undergone major changes with respect to the original
implementation by J.C. Gonz\'alez and Aitor Ibarra. Three options are
available in the simulation: ATM\_NOATMOSPHERE, ATM\_90PERCENT and
ATM\_CORSIKA (see section \ref{commands}). In the 
first two, 100$\%$ and 90$\%$ of the emitted light respectively
reaches the telescope mirror, regardless of the emission height. The
changes in the present version affect only the third option, which is
the recommended one for running reflector in the standard MAGIC MC
production. In this case, a detailed estimate of the atmospheric 
transmission is done. 
\par
Three contributions to the atmospheric opacity
are considered: Rayleigh scattering, Mie scattering by aerosols, and
absorption by ozone. Details on how the effect of these three
components is calculated are given in appendix A. In reflector
versions older than 0.6,
due to a bug, the contribution of Mie scattering and Ozone absorption
was slightly overestimated because the vertical height of the emission
point above sea level was interpreted as height above the
telescope. The difference is small, since the density of the aerosols
responsible for Mie scattering decreases very fast with height and
therefore the absorption is hardly increased by going up 2.2 km in the
region were most of the Cherenkov light is produced. In the case of
ozone, its contribution is only important in the very low end of the
Cherenkov light spectrum, and so the effect of the bug in the total
amount of light reaching the telescope is negligible. Another change
is that now the observation level is read in from the cer file, instead
of being a fixed parameter in the routines. In this way the reflector
program has become more flexible, and can now be used to process cer
files produced for a detector at a different observation level.
\par
Another problem in the old implementation of absorption was that the
variation of the optical depth of the atmosphere with the zenith angle
$\theta$ was assumed to be the same for Mie scattering, ozone
absorption and Rayleigh scattering: the so called {\it air
mass} (see appendix A for details) was therefore calculated only once,
using the overall density profile of air molecules (which does not
match that of aerosols or ozone), and used to account for the
variation of all three effects with $\theta$, while rigorously
speaking it is only valid for Rayleigh scattering. Now, the simulation
of Mie scattering and ozone absorption has been moved from {\it
attenu.f} to {\it atm.c} and is done in a more accurate way which
takes into account the vertical profile of aerosols and ozone (see
appendix A for details).
%
\subsection{Other changes in Reflector 0.6}

Some other minor improvements have been introduced in Reflector 0.6:

\begin{enumerate}

\item reflector.c: Introduced NaN (Not a Number) check in the photon
loop. If NaNs are found in a photon data block (there are some in the
Corsika output from time to time, for unkown reason), it is not processed.

\item ph2cph.c: Introduced "check of positiveness" before taking a
square root in the calculation of the photon trajectory intersection
with the paraboloid (resulted sometimes in NaNs when the photon did
not intersect the paraboloid).

\item Added an option for the wobble mode in the input card (see
section \ref{opt}).

\item New output format (see sect. \ref{out}): we have removed a null
byte that was written immediately after the ascii label containing the
reflector program version number at the begining of the file. This
byte was there for historical reasons and had no function
whatsoever. Then we have added a Run header, which is like that of
Corsika, plus a couple of variables concerning the reflector
parameters: the wobble mode and the atmospheric model 
used for the simulation. The event header has also been changed to
include all the information present in the Corsika event
header. We also added in the event header three new variables which
tell us for each event what fraction of the Cherenkov photons on the
camera plane has been produced by electrons, muons, or other
particles.
\par
Finally, the ascii files {\it magic.def}, {\it 
axisdev.dat} and {\it reflectivity.dat} which the program has used as
input are now attached at the end of the output file, so that each
output file contains all the relevant information on how it was
produced.

\end{enumerate}

%------------------------------------------------------------
\section{How to Install Reflector Program \label{installation}}

You can get the current version of the Reflector Program from the
MAGIC web page: \\
{\bf http://hegra1.mppmu.mpg.de/MAGICWeb/ }\\ 
You can find
the latest public version of this program as tarred gzipped file in
the Monte Carlo Download area (you need the usual password). You have to
download the file reflector\_0.6.tar.gz and then follow the
instructions below: 
  
\begin{description}  
\item[Decompress the file using:]
        gunzip reflector*.tar.gz   
\item[Unpack the tar file with:] 
        tar xvf reflector*.tar   
\item[Go to the directory where the source files are:]
        cd MagicProgs/Simulation/Detector/Reflector\_0.6/
\item[Make symbolic links running the script:]  
        refl-install    
\item[Check if all dependencies are fulfilled:] 
        make depend   
\item[Compile the program:] 
        make 
\end{description}

If everything goes right you should have an executable file called
{\bf reflector}. 

%------------------------------------------------------------
\section{How to Run Reflector Program \label{running}}

You need a steering card to run the Reflector program. You can find an 
example in the {\bf MagicProgs/Simulation/Detector/Reflector\_0.6/input.card}
file. You have to modify this file according to your needs (see below
for instructions about steering card) and then run the program with the 
following statement:\\

\hspace{1cm}{\bf reflector $<$ input.card}

%------------------------------------------------------------
\section{Needed Files \label{neededfiles}}

The Reflector program needs some other files to run. These files are
the following:
\begin{itemize}
\item {\bf magic.def}: contains the description of MAGIC telescope
geometry, together with some other parameters needed by the Reflector
program.
\item {\bf axisdev.dat}: contains data to simulate the possible
deviation of the spot of each single mirror on the camera plane due
to its non perfect alignment. The values are x, y coordinates
distributed at random (according to a gaussian with $\sigma =
0.5$ cm).
\item {\bf reflectivity.dat}: contains the mirror reflectivity index as
a function of the wavelength. 
\end{itemize}

All these files (included in the reflector package) are usually kept
in the {\bf MagicProgs/Simulation/Detector/Data/} directory and {\it
in principle} you should {\bf not} make any change in them to run the
program.

%------------------------------------------------------------
\section{Steering Card}

The steering card sets all the parameters and options 
to steer the reflection simulation. Each line of the steering card is
a statement with its parameters, if it is the case. Lines beginning
with \# are considered comments. The Reflector program parses all the
lines sequentially. Then if you repeat a statement with different
options only the last one will be considered.  

\subsection{Mandatory Commands \label{commands}} 

\begin{description}

\item[reflector 0.6]

        This statement must be the first line of the steering card
        file. The Reflector program checks it to verify if it is reading 
        a steering card. 

\item[output\_file /disk99/reflex/Gamma\_0\_7\_1001to1010\_w0.rfl]

        The output\_file command specifies the name and the 
        path of the output file. The path can be absolute, like in the 
        example above, or relative. Although any name can be used,
        conventionally the Reflector program 
        output file name has the .rfl extension, and starts with
        the primary particle name. The first number indicates the
        zenith angle of the incident primaries, the second one
        indicates the production site (7 is for Padua) and is related
        to the random number generator seed used by CORSIKA. Then the run
        number range is shown (10 runs in this case, from 1001 to
        1010). Each run corresponds to 10000 showers. Finally, the
        label "w0" means no wobble mode was used (telescope pointing
        at the source). Alternatively,  the "w+" or  "w-" labels (only 
        in gamma files) refer to the two pointings in the
        Wobble-observation mode (see TDAS 01-05 by W. Wittek).

\item[ct\_file /path/magic.def]

        The ct\_file statement defines where the program can find the 
        telescope characteristics. Usually, the magic.def file is kept in
        MagicProgs/Simulation/Detector/Data

\item[atm\_model ATM\_CORSIKA]

        The atm\_model statement tells the program what kind of
        atmospheric absorption model to use. Possible choices are: 
        ATM\_NOATMOSPHERE,\\ ATM\_90PERCENT and ATM\_CORSIKA,
        corresponding respectively to no absorption, a 10$\%$
        absorption and a model using the US Standard atmosphere (see
        Corsika manual, appendix C) for the Rayleigh scattering and a
        model by L. Elterman \cite{elterman64,elterman65} for the Mie
        scattering and ozone absorption (see appendix A). The third
        model should be chosen for the standard MC MAGIC production.

\item[cer\_files] 

        All the lines following this statement are considered files to
        be processed by the Reflector program, one for each line,
        eventually with their paths (see the example below). Therefore this
        command must be the last one.\\
        \\
        cer\_files\\
        /disk99/cer001001\\
        /disk99/cer001002\\
        /disk99/cer001003\\
        ........ \\
        /disk99/cer001009\\
        /disk99/cer001010\\
        \\
        The cer file name can be followed by two numbers, for example:
        \\
        /disk99/cer001001 376 5723\\
        \\      
        In this case the program processes only the events between and
      including the numbers given.
        
\end{description}

\subsection{Optional Commands \label{opt}}

\begin{description}
        
\item[verbose\_level 1]

        Sets the quantity of information printed out by Reflector
        when running. Possible values are 0 to 4

\item[max\_events 50000] 

        Fixes the maximum number of events to process. 

\item[energy\_cuts 100 1000]

        This statement forces the Reflector to process only showers
        with primary energy between the given values (GeV).
        
\item[seeds n1 n2]

        Seeds for the random number generators to used by the program
        for the simulation of the absorption (both in the atmosphere
        and on the mirror). Default values are 3141592 and
        2718182.

\item[telescope\_position x y]

        Option included in version 0.5 of Reflector. Usually it is
        not needed, since for normal MC production for MAGIC the
        telescope is placed at the origin of coordinates (0,0). But,
        if for some reason, we produce cerxxxxxx files with the
        telescope in a different position, we must inform the
        Reflector program in the input card using this option
        (otherwise Reflector will fail to {\it find} the photons
        in the cer file).

\item[reflectivity\_file /path/reflectivity.dat]

        File containing mirror reflectivity as a function of
        wavelength (see section \ref{neededfiles}). If this option is
        not supplied, the program will look for \\
        ``../Data/reflectivity.dat'' as previous versions of
        Reflector did.

\item[axisdev\_file /path/axisdev.dat]

        File containing single mirror spot deviation in {\bf x} and
        {\bf y} on the camera in cm (see section
        \ref{neededfiles}) for each mirror. If this option is not
        supplied, the  program will look for ``../Data/axisdev.dat''
        as previous versions of Reflector did.

\item[fixed\_target $\Theta$ $\Phi$] 
        
        This statement fixes the telescope axis position. The first
        number is the zenith angle $\Theta$ (deg) while the second is
        the azimuthal angle $\Phi$ (deg). This corresponds to {\it
        CORSIKA}'s definition of primary particle incident direction
        (see fig. \ref{coorsystems}, left). For instance, $\phi = 90^\circ$
        means that the telescope is pointing towards East. If this
        option is omitted the telescope will always point in the
        direction of the Corsika primary (whatever it is), or a
        slightly modified direction if the wobble\_mode option is used
        (see next point). When running the reflector over gamma data
        generated in a range of zenith angles, one should therefore
        ommit the fixed\_target option.

\item[wobble\_mode w]

        Indicates whether the reflection should be done in the wobble
        mode, that is, with shifted pointing with respect to the
        nominal telescope orientation (given by fixed\_target or
        otherwise, see above). The wobble mode is described in TDAS
        01-05. Possible values for w are 0 (no wobble mode), 1, -1
        (image shift along $x_{camera}$ axis) and 3 (image shift along
        $y_{camera}$ axis).

\end{description}  

%------------------------------------------------------------

\section{Output file \label{out}} 
The reflector output file begins with two ascii lines, the first of
which informs us of the program version with which it has been
produced (NOTE: in the following, the dollar symbol \verb"$" stands
for a carriage return):\\
\\
\verb"reflector 0.6$START---RUN$" \\ 
\\
Then there is run header which is basically the one from Corsika with
two added variables, {\it wobble\_mode} and {\it
atmospheric\_model}. Check the Corsika manual for the meaning and
units of the rest of them. All of the variables are 4-byte real numbers
except the first, which is a 4 character string containing the run
header ascii label from Corsika:
\vspace*{0.5cm}
\\
%
\begin{tabular}{lll}
\multicolumn{2}{c}{Variable} & Description \\
\hline 
&& \\
1 & ASCII Label & 'RUNH' \\
2 & RunNumber & \\
3 & date & \\
4 & Corsika\_version & \\
5 & NumObsLev & \\
6 to 15 & HeightLev[10] & \\
16 & SlopeSpec & \\
17 & ELowLim & \\
18 & EUppLim & \\
19 & EGS4\_flag & \\
20 & NKG\_flag & \\
21 & Ecutoffh & \\
22 & Ecutoffm & \\
23 & Ecutoffe & \\
24 & Ecutoffg & \\
25 to 74 & C[50] & \\
75 & wobble\_mode & Wobble mode with which the reflector was run (TDAS
01-05) \\
76 & atmospheric\_model & Atmospheric model used for the absorption
simulation \\
&& 0 = no atmosphere;  1 = atm\_90percent;  \\
&& 2 = atm\_corsika. \\
77 to 94 & dummy1[18] & not used \\
95 to 134 & CKA[40] & \\
135 to 139 & CETA[5] & \\
&\\
\hline 
\end{tabular}

\newpage

\begin{tabular}{lll}
\multicolumn{2}{c}{Variable} & \parbox{11cm}{Description} \\
\hline 
&& \\
140 to 140 & CSTRBA[11] & \\
151 to 254 & dummy2[104] & not used \\
255 to 259 & AATM[5] & \\
260 to 264 & BATM[5] & \\
265 to 269 & CATM[5] & \\
270 to 273 & NFL[4] & \\
&\\
\hline 
\end{tabular}
\vspace*{0.5cm}
\\
%
Then there comes a carriage return followed by the ascii flag which
indicates the start of an event, and again a carriage return:\\
\\
\verb"$START-EVENT$"\\
\\
and then the binary event header. Each variable is a 4-byte
float number except for the first one which is the event header label
from Corsika (a string of 4 characters). Some of of the variables from
Corsika are not explained here (see Corsika manual instead).
\vspace*{0.5cm}
\\
\begin{tabular}{lll}
\multicolumn{2}{c}{Variable} & Description \\
\hline
&& \\ 
1 & ASCII label     & 'EVTH' \\
2 & EvtNumber       & Event Number \\
3 & PrimaryID   & Primary particle identification code \\
4 & Etotal              & Primary particle total energy (GeV) \\
5 & Thick0              & CORSIKA's starting altitude in g/cm2 \\
6 & FirstTarget         & CORSIKA's number of first target if fixed \\
7 & zFirstInt   & Height of first interaction in cm \\
8 to 10 & p[3]          & Primary particle momentum in x,y,-z directions (GeV) \\
11 & Theta              & Primary particle zenith angle (rad) \\
12 & Phi                & Primary particle azimuth angle (rad) \\       

13 & NumRndSeq  & Number of different CORSIKA random sequences (max. 10) \\
14 to 43 & RndData[10][3]       & RndData[i][0]: integer seed of sequence i \\
&               & RndData[i][1]: number of offset random calls (mod $10^6$) of sequence i. \\ 
&               & RndData[i][2]: number of offset random calls ($/10^6$) of sequence i. \\ 

44 & RunNumber  & Run number \\
45 & DateRun    & Date of run yymmdd \\
46 & Corsika\_version & Version of {\it CORSIKA} \\

47 & NumObsLev  & Number of observation levels (should be always 1 for
us) \\
48 to 57 & HeightLev[10]        & Height of observation levels in cm \\

58 & SlopeSpec  & Energy spectrum slope \\
59 & ELowLim    & Energy lower limit (GeV) \\
60 & EUppLim    & Energy upper limit (GeV) \\
61 & Ecutoffh        & \\
62 & Ecutoffm        & \\
63 & Ecutoffe        & \\
& \\
\hline 
\end{tabular}
%
\newpage
%

\begin{tabular}{lll}

\multicolumn{2}{c}{Variable} & \parbox{11cm}{Description} \\
\hline 
&& \\
64 & Ecutoffg        & \\
65 & NFLAIN             & \\
66 & NFLDIF             & \\
67 & NFLPI0             & \\
68 & NFLPIF             & \\
69 & NFLCHE             & \\
70 & NFRAGM             & \\
71 & Bx         & \\
72 & By         & \\
73 & EGS4yn     & \\
74 & NKGyn              & \\
75 & GHEISHAyn  & \\
76 & VENUSyn            & \\
77 & CERENKOVyn & \\
78 & NEUTRINOyn & \\
79 & HORIZONTyn & \\
80 & COMPUTER   & \\
81 & ThetaMin   & Minimum Theta of primaries (deg) \\
82 & ThetaMax   & Maximum Theta of primaries (deg) \\
83 & PhiMin     & Minimum Phi of primaries (deg) \\
84 & PhiMax     & Maximum Phi of primaries (deg) \\
85 & CBunchSize & \\
86 & CDetInX    & \\
87 & CDetInY    & \\
88 & CSpacInX   & \\
89 & CSpacInY   & \\
90 & CLenInX    & \\
91 & CLenInY    & \\
92 & COutput    & \\
93 & AngleNorthX& \\
94 & MuonInfo   & \\
95 & StepLength & \\
96 & CWaveLower & Wavelength lower limit (nm) \\
97 & CWaveUpper & Wavelength upper limit (nm) \\
98 & Multipl    & \\
99 to 138 & CorePos[2][20] & Core positions of randomized shower \\
139 to 140 & SIBYLL[2]  & \\
141 to 142 & QGSJET[2]  & \\
143 to 144 & DPMJET[2]  & \\
145 & VENUS\_cross      & \\
146 & mu\_mult\_scat    & \\
147 & NKG\_range        & \\
148 to 149 & EFRCTHN[2] & \\
150 to 151 & WMAX[2]    & \\
& \\
\hline 
\end{tabular}

\newpage

\begin{tabular}{lll}
\multicolumn{2}{c}{Variable} & \parbox{11cm}{Description} \\
\hline 
&& \\
152 & rthin\_rmax       & \\
153 to 154 & viewcone\_angles[2] & Inner and outer angles of Corsika's VIEWCONE
option. \\
155 & telescopePhi      & Telescope azimuth (rad). Measured from South, counter-clockwise \\
156 & telescopeTheta    & Telescope zenith angle (rad) \\
157 & TimeFirst         & Arrival time on camera of first photon (ns) \\
158 & TimeLast          & Arrival time on camera of last photon (ns) \\

&& 6 next variables: CORSIKA longitudinal particle fit parameters \\
&& \hspace{0.5cm} (see CORSIKA manual for precise meaning and units)\\
159 & longi\_Nmax        & Numer of charged particles at maximum \\
160 & longi\_t0          & Atmospheric depth of shower starting point (N=0) \\
161 & longi\_tmax        & Atmospheric depth of shower maximum (g/cm$^2$) \\
162 & longi\_a           & \\
163 & longi\_b           & For {\bf longi\_a}, {\bf longi\_b}, {\bf longi\_c}, see CORSIKA manual  \\
164 & longi\_c           &  \\
165 & longi\_chi2        & $\chi^2/dof$ of the fit\\
166 & CORSIKAPhs      & Original photons written by {\it CORSIKA} \\ 
167 & AtmAbsPhs       & Photons absorbed by the atmosphere  \\
168 & MirrAbsPhs      & Photons absorbed by the mirror      \\ 
169 & OutOfMirrPhs    & Photons outside the mirror          \\ 
170 & BlackSpotPhs    & Photons lost in the "black spot"    \\ 
171 & OutOfChamPhs    & Photons outside the camera         \\ 
172 & CPhotons        & Photons reaching the camera        \\ 

173 & elec\_cph\_fraction & Fraction of C-photons produced by electrons \\
174 & muon\_cph\_fraction & Fraction of C-photons produced by muons \\
175 & other\_cph\_fraction & Fraction of C-photons produced by electrons \\
176 to 182 & dummy[7] & not used \\
& \\
\hline 
\end{tabular}
%
\vspace*{0.5cm}
\\
The event header is followed by 8-word blocks, one for each Cherenkov
photon that reaches the camera. A photon block contains the following
variables (as 4-byte float numbers):
\vspace*{0.5cm}
\\
\begin{tabular}{ll}
Variable & Description \\
\hline
& \\
w       & 100000 $\times$ Particle\_type + 1000 $\times$ n + Wavelength (nm) \\
        & Particle\_type indicates what particle produced the C-photon. Its value is 2 for electrons \\
        & and positrons, and the standard codes of Corsika for the rest. \\
        & n is the index from 1 to ICERML (see Corsika manual) for the case in which each Corsika \\
        & shower is used more than once (normally, in MMCS will be just 1). \\
x, y    & Impact point in camera coordinates (cm) \\
u, v    & Director cosines of down-going versor indicating the photon direction \\
t       & Arrival time on camera (ns), measured from the time of first
interaction of the primary \\
h       & Production height (cm), measured above sea level on the
vertical of the telescope location \\
        & (it is not the {\it true} height which would be measured on
the vertical of the emitting particle!)  \\ 
phi     & Incidence angle with respect to camera plane (rad) \\
& \\
\hline
\end{tabular}
\vspace*{0.5cm}
\\
After the last photon block of an event there is a carriage return
followed by the ascii flag indicating the event end, and then two more
carriage returns before the ascii flag of the beginning of the next
event, and so on:\\
\\
\verb"$END---EVENT$$START-EVENT$Event_header....$END---EVENT$$END-----RUN$$START---RUN$..."\\
\\
The flag ``\verb$END-----RUN$'' appears after the last event in a run
(that is, the last event processed of each of the input cer
files). After the last processed run, an end of file flag is
written:\\
\\
\verb"...$END---EVENT$$END-----RUN$$END----FILE$"\\
\\
Finally, after this flag an ``ascii tail'' has been attached to the file:
it consists of the ascii files {\it magic.def}, {\it axisdev.dat} and
{\it reflectivity.dat} one after the other, separated by carriage returns:
\\
\\ 
\verb"$magic.def$axisdev.dat$reflectivity.dat"\\
\\
In this way all the relevant parameters used to produce the output are
kept together with the reflected events.
%------------------------------------------------------------
\newpage
\renewcommand{\thesubsection}{A.\arabic{subsection}}
\section*{Appendix A : atmospheric absorption}
\addcontentsline{toc}{section}{Appendix A : atmospheric absorption}
%
The simulation of the absorption of Cherenkov light in the atmosphere
has been included in the {\it Reflector} program because this feature
was not yet available in the first versions of CORSIKA used within the
MAGIC collaboration. In the latest CORSIKA versions, the atmospheric
absorption has been included as an option, but it is not
compatible with the simulation of a curved atmosphere \cite{cor02},
and hence we have kept this step as a part of our reflector
simulation. This appendix describes how the atmospheric absorption is
implemented in the program when the option ATM\_CORSIKA (see section
\ref{commands}) is chosen.
\par
The geometry of the problem is sketched in figure
\ref{fig:atmoscheme}. A Cherenkov photon is emitted in point A and
travels towards the telescope placed at B. At any moment, the height
$h$ of the photon above sea level is related to the distance $L$
between the photon and the telescope through
%
\begin{equation}
(R+h)^2 = (R+h_1)^2 + L^2 + 2 L \; (R+h_1) \; \cos \theta
\label{eq:height}
\end{equation}
%
, where $R$ is the Earth radius, $h_1$ the height (a.s.l.) of the
observation level and $\theta$ is the zenith angle of the photon
trajectory measured at the telescope site. The Cherenkov output of
CORSIKA contains for each photon the height $h_C$ of the emission
point A, measured along the vertical of the observer. The {\it true
vertical height} $h_2$ of the emission point can be obtained by
replacing $L$ by $(h_C-h_1)/\cos \theta$ in equation
(\ref{eq:height}).
%
\begin{figure}[ht]
\begin{center}
\mbox{ \epsfig{file=eps/atmoscheme.eps,width=0.8\textwidth} }
\end{center}
\caption[]
{Calculation of the true vertical height $h_2$ of the emission point
of a Cherenkov photon (point A), and the optical path traversed down to the
telescope (point B).}
\label{fig:atmoscheme}
\end{figure}
%
\par
The optical path $I(\theta, h_1, h_2)$ traversed by the photon can be
calculated integrating the air density along the trajectory
$\overline{\text{AB}}$. For the case $h/R \ll 1$, we can drop in
(\ref{eq:height}) the terms in $(h/R)^2$ and smaller, and then solve
for L. Deriving now with respect to $h$, we have:
%
\begin{equation}
\frac{dL}{dh} \simeq \sqrt{\frac{R}{2\;(h-h_1)+R\;\cos^2 \theta}}
\qquad \text{for} \qquad \frac{h}{R} \ll  1
\label{eq:dldh}
\end{equation}
%
\subsection{Rayleigh scattering}
Rayleigh scattering is the scattering of light by particles smaller
than its wavelength. These are in our case the air molecules. The
transmission coefficient due to Rayleigh scattering is a strong
function of the wavelength $\lambda$:
%
\begin{equation}
T_{\text{Rayl}} (\lambda) = \exp \Biggl[ - \frac{I(\theta, h_1, h_2)}{x_R} \; \Biggl(\frac{400 \;
\text{nm}}{\lambda}\Biggl)^4 \; \Biggl] 
\label{eq:rayleigh}
\end{equation}
%
Here $I(\theta, h_1, h_2)$ is the optical path (in g/cm$^2$) traversed
between points A and B, and $x_R = 2970$ g/cm$^2$ is the mean free path of
the Rayleigh scattering at $\lambda = 400$ nm. 
\par
A convenient way of expressing the optical path is the following:
%
\begin{equation}
I\;(\theta, h_1, h_2) = (x_1 - x_2) \cdot \mathcal{AM}\;(\theta, h_1, h_2) 
\end{equation}
Here $x_{i=1,2}$ is the mass overburden of the atmosphere above a
height $h_i$ (in the 
vertical direction) and $\mathcal{AM}$ is the so called {\it air
mass}\footnote{If we set $h_2 = \infty$, we have the usual definition
of {\it air mass} in optical astronomy.}, defined as
%
\begin{equation}
\mathcal{AM} \equiv \frac{I\;(\theta,h_1,h_2)}{I\;(0^\circ,h_1,h_2)}
\label{eq:airmass}
\end{equation}
%
, which is a function mainly of the zenith angle $\theta$ (see
fig. \ref{fig:airmass}). In our simulation, for the calculation of the
mass overburden $x_i$ we have used the U.S. standard atmosphere
parametrized by J. Linsley \cite{cor02}, the same we used in Corsika
for the shower development simulation. It consists of five layers: in
the lower four the density decreases exponentially with height, and in
the upper one the mass overburden cecreases linearly until it vanishes
at $h = 112.8$ km.
%
\begin{figure}[ht]
\begin{center}
\mbox{ \epsfig{file=eps/airmass.eps,width=0.8\textwidth} }
\end{center}
\caption[]
{Dependence with zenith angle of the air mass as defined in the
text. The air mass has been calculated for an exponential atmosphere
with scale height $H = 7.4$ km, for the observation level of MAGIC
(2.2 km a.s.l.), and for light coming from $h_2 = 10$ and at 100 km a.s.l. As
we can see the dependence with the emission height $h_2$ is very small.}
\label{fig:airmass}
\end{figure}
%
\par
For the estimate of $\mathcal{AM}$, a simpler atmospheric model has
been used, in which the vertical density profile is described by a
single exponential: $\rho = \rho_0 \; e^{-h/H}$  with scale height $H
= 7.4$ km. This simplifies the calculations, and is accurate enough
for our purposes. Using (\ref{eq:dldh}) the optical path $I(\theta,
h_1, h_2)$ can then be obtained approximately as:
%
\begin{equation}
I(\theta, h_1, h_2) = \int_A^B \rho\;(h)\; \frac{dL}{dh} \; dh \simeq
\sqrt{\frac{R}{2}} \;
\int_{h_1}^{h_2} \frac{\rho_0
\;e^{-h/H}}{\sqrt{h-h_1+\frac{1}{2}\;R\;\cos^2 \theta}} \; dh 
\label{eq:optpath}
\end{equation}
%
and finally:
% 
\begin{equation}
\mathcal{AM} \simeq e^{-\frac{R \sin^2 \theta}{2H}}
\cdot 
\frac{
\text{erfc}\;(\sqrt{\frac{R \cos^2 \theta}{2H}})
\;-\;
\text{erfc}\;(\sqrt{\frac{2 (h_2 - h_1) + R \cos^2 \theta}{2H}})
}
{
\text{erfc}\;(\sqrt{\frac{R}{2H}})
\;-\;
\text{erfc}\;(\sqrt{\frac{2(h_2 - h_1) + R}{2 H}})
}
\label{eq:airmass2}
\end{equation}
%
where we have used the complementary error function $\text{erfc}\;(x)
= \frac{2}{\sqrt{\pi}}\int_x^\infty e^{-t^2} dt$. From 
(\ref{eq:airmass2}), $\mathcal{AM}$ can be readily evaluated for any
value of $\theta$, $h_1$ and $h_2$. In fig. \ref{fig:rayleigh} the
resulting Rayleigh transmission coefficient $T_{\text{Rayl}}$ finally
obtained from (\ref{eq:rayleigh}) is plotted versus the zenith angle
for three wavelengths.
%
\begin{figure}[ht]
\begin{center}
\mbox{ \epsfig{file=eps/rayleigh.eps,width=\textwidth} }
\end{center}
\caption[]
{Rayleigh transmission coefficient as a function of zenith angle for three
different wavelengths. The solid, dashed and dotted lines correspond
respectively to light coming from 5, 10 and 20 km distance from the
telescope.}
\label{fig:rayleigh}
\end{figure}
%
\subsection{Mie scattering}
Cherenkov light also suffers Mie scattering through interaction with
small dust particles suspended in the air (aerosols), whose size is
comparable to the wavelength of the light. In our simulation of the
attenuation due to aerosols we have used the model proposed by
Elterman \cite{elterman64,elterman65}, which considers an aerosol
number density $N_p$ which (roughly) decreases exponentially up to 10
km a.s.l. with scale height $H \simeq 1.2$ km, followed by a more
tenuous layer between 10 and 30 km (see fig. \ref{fig:aerosol}a). In
this model, the aerosol size distribution is considered to be
unchanged with altitude.
%
\begin{figure}[ht]
\begin{center}
\mbox{ \epsfig{file=eps/aerosol.eps,width=0.9\textwidth} }
\end{center}
\caption[]
{Aerosol model by Elterman: in (a), number density of
aerosols as a function of height above sea level; in (b), aerosol
attenuation coefficient at sea level as a function of wavelength.}
\label{fig:aerosol}
\end{figure}
%
\par
Measured values of the aerosol attenuation coefficients at sea level
$\beta_p(0)$ for different wavelengths \cite{elterman65} are shown in
figure \ref{fig:aerosol}b. To obtain the attenuation coefficient at a
given height $h$, we simply do $\beta_p(h, \lambda) = \beta_p(0,
\lambda) \cdot N_p(h)/N_p(0)$. In the Elterman model the aerosol
transmission coefficient for the trajectory from A to B depicted in
figure \ref{fig:atmoscheme} would then be: 
%
\begin{equation}
T_{\text{Mie}}(\lambda) = e^{-\tau_{mie}} \quad \text{, with}\quad
\tau_{mie}(h_1, h_2, \theta, \lambda) = \frac{\beta_p(0, \lambda)}{N_p(0)} \;
\int_{h_1}^{h_2} 
\; N_p(h) \;\; \frac{dL}{dh}\; dh
\label{eq:aerosoltau}
\end{equation}
%
Here $\tau_{mie}$ is the aerosol optical depth of the path from A to
B. Given that the aerosol density distribution is not a simple
exponential, we have to do the integral in (\ref{eq:aerosoltau})
numerically. The integral depends on $h_2$ and on $\theta$, through
$dL/dh$ (it also 
depends on $h_1$, but this is the observation level and therefore
fixed). In this case we use the exact expression for $\;dL/dh\;$ which can be
obtained from (\ref{eq:height}). At the beginning of each simulation
run we calculate and store the results of the integral for values of
$\theta$ between 0 and 90$^\circ$ (in steps of $1^\circ$), and of
$h_2$ from $h_1$ up to 30 km (in steps of 100 m). To do the integral a
linear interpolation has been used to obtain the value of $N_p$ for
any height. During the simulation of each Cherenkov photon, we get the
corresponding precalculated value of the integral and deduce
$T_{\text{Mie}}$ from expression (\ref{eq:aerosoltau}).
%
\begin{figure}[ht]
\begin{center}
\mbox{ \epsfig{file=eps/mie.eps,width=\textwidth} }
\end{center}
\caption[]
{Aerosol transmission coefficient for three different wavelengths as a
function of the distance between the photon emission point and the
telescope. Plots for four different zenith angles between 0 and 80
degrees are shown.}
\label{fig:mie}
\end{figure}
%
\par
Since in this model the aerosols are concentrated mainly at very low
altitude, the transmission coefficient is more or less constant above
a certain height (which depends on $\theta$), as can be seen in
fig. \ref{fig:mie}. For instance, for vertically incident 300 nm light
emitted higher than 4 km above the telescope, the Mie transmission is
about 0.95.
%
\subsection{Absorption by Ozone}
%
The absorption of Cherenkov light by Ozone has been implemented
following also the Elterman standard atmosphere \cite{elterman65}. The
coefficient for ozone absorption is given by
%
\begin{equation}
\beta_3(h,\lambda) = A_v(\lambda) \cdot D_3(h)
\end{equation}
%
, where $A_v(\lambda)$ is the Vigroux \cite{vigroux53} ozone
absorption coefficient (cm$^{-1}$) and $D_3(h)$ is the ozone
concentration (cm km$^{-1}$) according to Elterman. The transmission
coefficient of Ozone in the path $\overline{\text{AB}}$ is then:
%
\begin{equation}
T_{\text{Ozone}}(\lambda) = e^{-\tau_{oz}} \quad \text{, with}\quad
\tau_{oz}(h_1, h_2, \theta, \lambda) = A_v(\lambda) \; \int_{h_1}^{h_2} 
\; D_3(h) \;\; \frac{dL}{dh}\; dh
\label{eq:ozonetau}
\end{equation}
%
\begin{figure}[ht]
\begin{center}
\mbox{ \epsfig{file=eps/ozone.eps,width=\textwidth} }
\end{center}
\caption[]
{Ozone concentration vertical profile (a) and Vigroux coefficients for
Ozone absorption (b). The Vigroux coefficients for $\lambda =$ 380 and
400 nm are zero.}
\label{fig:ozone}
\end{figure}
%
\par
Once again, like in the case of Mie scattering, the optical depth
$\tau_{oz}$ is the product of a factor which depends on $\lambda$ and
an integral which depends on $h_1$, $h_2$ and $\theta$. We proceed in
the same way as before, precalculating the values of the integral in
steps of $\Delta\theta = 1^\circ$ and $\Delta h = 100$ m, up to a
height of 50 km a.s.l. (where the ozone concentration becomes
negligible), and then reading the appropriate values for every
simulated photon.
\par
Finally, the overall atmospheric transmission coefficient is
calculated as
%
\begin{equation}
T_{total} = T_{Ray} \cdot T_{Mie} \cdot T_{Ozone}
\end{equation}
%
In figure \ref{fig:absorplot} the atmospheric transmission as a
function of the distance to the telescope for $\theta = 0^\circ$ is
shown. Ozone absorption turns out to be dominant below 320 nm, while
Rayleigh scattering is the main cause of the loss of photons at longer
wavelengths.
%
\begin{figure}[ht]
\begin{center}
\mbox{ \epsfig{file=eps/absorplot.eps,width=\textwidth} }
\end{center}
\caption[]
{Total transmission coefficient for vertically incident light as a
function of the distance between the emission point and the
telescope. The contributions of absorption by ozone and of Rayleigh
and Mie scattering are also shown for comparison.}
\label{fig:absorplot}
\end{figure}
%
\newpage
\section*{Appendix B : files in the reflector package}
\addcontentsline{toc}{section}{Appendix B : files in the reflector package}

The list of all Reflector files follows.
\begin{verbatim}

MagicProgs/Simulation/Detector/Reflector_0.6/Changelog
MagicProgs/Simulation/Detector/Reflector_0.6/Makefile
MagicProgs/Simulation/Detector/Reflector_0.6/atm.c
MagicProgs/Simulation/Detector/Reflector_0.6/atm.h
MagicProgs/Simulation/Detector/Reflector_0.6/attach.c
MagicProgs/Simulation/Detector/Reflector_0.6/attenu.f
MagicProgs/Simulation/Detector/Reflector_0.6/config.mk.linux
MagicProgs/Simulation/Detector/Reflector_0.6/config.mk.linux-gnu
MagicProgs/Simulation/Detector/Reflector_0.6/config.mk.osf1
MagicProgs/Simulation/Detector/Reflector_0.6/diag.c
MagicProgs/Simulation/Detector/Reflector_0.6/diag.h
MagicProgs/Simulation/Detector/Reflector_0.6/geometry.c
MagicProgs/Simulation/Detector/Reflector_0.6/geometry.h
MagicProgs/Simulation/Detector/Reflector_0.6/header.c
MagicProgs/Simulation/Detector/Reflector_0.6/header.h
MagicProgs/Simulation/Detector/Reflector_0.6/init.c
MagicProgs/Simulation/Detector/Reflector_0.6/init.h
MagicProgs/Simulation/Detector/Reflector_0.6/input.card
MagicProgs/Simulation/Detector/Reflector_0.6/lagrange.h
MagicProgs/Simulation/Detector/Reflector_0.6/parms.c
MagicProgs/Simulation/Detector/Reflector_0.6/parms.h
MagicProgs/Simulation/Detector/Reflector_0.6/ph2cph.c
MagicProgs/Simulation/Detector/Reflector_0.6/refl-install
MagicProgs/Simulation/Detector/Reflector_0.6/reflector.c
MagicProgs/Simulation/Detector/Reflector_0.6/version.h

MagicProgs/Simulation/Detector/Reflector_0.6/doc/Tdas0211.ps
MagicProgs/Simulation/Detector/Reflector_0.6/doc/Tdas0211.tex
MagicProgs/Simulation/Detector/Reflector_0.6/doc/magic-tdas.sty
MagicProgs/Simulation/Detector/Reflector_0.6/doc/magiclogo.eps
MagicProgs/Simulation/Detector/Reflector_0.6/doc/eps/absorplot.eps
MagicProgs/Simulation/Detector/Reflector_0.6/doc/eps/aerosol.eps
MagicProgs/Simulation/Detector/Reflector_0.6/doc/eps/airmass.eps
MagicProgs/Simulation/Detector/Reflector_0.6/doc/eps/atmoscheme.eps
MagicProgs/Simulation/Detector/Reflector_0.6/doc/eps/colimation.eps
MagicProgs/Simulation/Detector/Reflector_0.6/doc/eps/coma.eps
MagicProgs/Simulation/Detector/Reflector_0.6/doc/eps/coma_star.eps
MagicProgs/Simulation/Detector/Reflector_0.6/doc/eps/coorsystems.eps
MagicProgs/Simulation/Detector/Reflector_0.6/doc/eps/evcompare.eps
MagicProgs/Simulation/Detector/Reflector_0.6/doc/eps/mie.eps
MagicProgs/Simulation/Detector/Reflector_0.6/doc/eps/ozone.eps
MagicProgs/Simulation/Detector/Reflector_0.6/doc/eps/parabola.eps
MagicProgs/Simulation/Detector/Reflector_0.6/doc/eps/rayleigh.eps
MagicProgs/Simulation/Detector/Reflector_0.6/doc/eps/refl06images.eps
MagicProgs/Simulation/Detector/Reflector_0.6/doc/eps/reflec.eps
MagicProgs/Simulation/Detector/Reflector_0.6/doc/eps/spot10kmf1694.eps
MagicProgs/Simulation/Detector/Reflector_0.6/doc/eps/spot10kmf1700.eps
MagicProgs/Simulation/Detector/Reflector_0.6/doc/eps/spot_inf_f1697.eps
MagicProgs/Simulation/Detector/Reflector_0.6/doc/eps/telecoor.eps
MagicProgs/Simulation/Detector/Reflector_0.6/doc/eps/timing.eps

MagicProgs/Simulation/Detector/Reflector_0.6/tester/Makefile
MagicProgs/Simulation/Detector/Reflector_0.6/tester/cermaker.c

MagicProgs/Simulation/Detector/Data/axisdev.dat
MagicProgs/Simulation/Detector/Data/magic.def
MagicProgs/Simulation/Detector/Data/reflectivity.dat

MagicProgs/Simulation/Detector/lib/libranlib.a.osf1
MagicProgs/Simulation/Detector/lib/libranlib.a.linux

\end{verbatim}

%%% BIBLIOGRAPHY %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%>>>> Use the following if you are using BibTeX for bibliography
%\theBibliography

%%>>>> Or the following if you include here by hand your 
%%>>>> bibliographic entries

\begin{thebibliography}{00}

\bibitem{elterman64}
L. Elterman, Applied Optics Vol. 3, No. 6 (1964) 745.

\bibitem{elterman65}
L. Elterman, R.B. Toolin, S.L. Valley (editor), Handbook of
geophysics and space environments, McGraw-Hill, N.Y. (1965).

\bibitem{cor02}D. Heck and J. Knapp, EAS Simulation with CORSIKA: A User's
Manual, 2002.

\bibitem{vigroux53}
E. Vigroux, Contributions \`a l'étude expérimentale de l'absorption
de l'ozone, Annales de Physique, v. 8 (1953) 709.

\end{thebibliography}

\end{document}
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