source: trunk/MagicSoft/TDAS-Extractor/Calibration.tex@ 6678

\section{Calibration \label{sec:calibration}}


In this section, we describe the tests performed using light pulses of different colour, 
pulse shapes and intensities with the MAGIC LED Calibration Pulser Box \cite{hardware-manual}. 
\par
The LED pulser system is able to provide fast light pulses of 2--4\,ns FWHM 
with intensities ranging from 3--4 to more than 600 photo-electrons in one inner photo-multiplier of the 
camera. These pulses can be produced in three colours {\textit {\bf green, blue}} and 
{\textit{\bf UV}}.

\begin{table}[htp]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\hline
\multicolumn{7}{|c|}{The possible pulsed light colours} \\
\hline
\hline
Colour &  Wavelength & Spectral Width & Min. Nr. &  Max. Nr. & Secondary & FWHM \\
      & [nm]         & [nm]           &  Phe's   &  Phe's    & Pulses  &  Pulse [ns]\\
\hline
Green &  520      & 40      & 6          &  120      & yes  & 3--4  \\
\hline
Blue &  460       & 30      & 6          &  600      & yes  & 3--4 \\
\hline
UV   &  375       & 12      & 3          &  50       & no   & 2--3 \\ 
\hline
\hline
\end{tabular}
\caption{The pulser colours available from the calibration system}
\label{tab:pulsercolours}
\end{table}

Table~\ref{tab:pulsercolours} lists the available colours and intensities and 
figures~\ref{fig:pulseexample1leduv} and~\ref{fig:pulseexample23ledblue} show exemplary pulses 
as registered by the FADCs.
Whereas the UV-pulse is rather stable, the green and blue pulses can show smaller secondary 
pulses after about 10--40\,ns from the main pulse.
One can see that the stable UV-pulses are unfortunately only available in such intensities as to 
not saturate the high-gain readout channel. However, the brightest combination of light pulses easily 
saturates all channels in the camera, but does not reach a saturation of the low-gain readout.
\par
Our tests can be classified into three subsections:

\begin{enumerate}
\item Un-calibrated pixels and events: These tests measure the percentage of failures of the extractor 
resulting either in a pixel declared as un-calibrated or in an event which produces a signal outside 
of the expected Gaussian distribution.
\item Number of photo-electrons: These tests measure the reconstructed numbers of photo-electrons, their 
spread over the camera and the ratio of the obtained mean values for outer and inner pixels, respectively.
\item Linearity tests: These tests measure the linearity of the extractor with respect to pulses of 
different intensity and colour.
\item Time resolution: These tests show the time resolution and stability obtained with different 
intensities and colours.
\end{enumerate}

\begin{figure}[htp]
\centering
\includegraphics[width=0.48\linewidth]{1LedUV_Pulse_Inner.eps}
\includegraphics[width=0.48\linewidth]{1LedUV_Pulse_Outer.eps}
\caption{Example of a calibration pulse from the lowest available intensity (1\,Led UV). 
The left plot shows the signal obtained in an inner pixel, the right one the signal in an outer pixel.
Note that the pulse height fluctuates much more than suggested from these pictures. Especially, a 
zero-pulse is also possible.}
\label{fig:pulseexample1leduv}
\end{figure}

\begin{figure}[htp]
\centering
\includegraphics[width=0.48\linewidth]{23LedsBlue_Pulse_Inner.eps}
\includegraphics[width=0.48\linewidth]{23LedsBlue_Pulse_Outer.eps}
\caption{Example of a calibration pulse from the highest available mono-chromatic intensity (23\,Leds Blue). 
The left plot shows the signal obtained in an inner pixel, the right one the signal in an outer pixel.
One the left side of both plots, the (saturated) high-gain channel is visible, 
on the right side from FADC slice 18 on, 
the delayed low-gain 
pulse appears. Note that in the left plot, there is a secondary pulses visible in the tail of the 
high-gain pulse. }
\label{fig:pulseexample23ledblue}
\end{figure}

We used data taken on the 7$^{\mathrm{th}}$ of June, 2004 with different pulser LED combinations, each taken with 
16384 events. 19 different calibration configurations have been tested. 
The corresponding MAGIC data run numbers range from nr. 31741 to 31772. These data have been taken 
before the latest camera repair access which resulted in a replacement of about 2\% of the pixels known to be 
mal-functioning at that time.
There is thus a lower limit to the number of un-calibrated pixels of about 1.5--2\% of known 
mal-functioning photo-multipliers.
\par
Although we had looked at and tested all colour and extractor combinations resulting from these data, 
we restrict ourselves to show here only exemplary behaviour and results of extractors. 
All plots, including those which are not displayed in this TDAS, can be retrieved from the following 
locations:

\begin{verbatim}
http://www.magic.ifae.es/~markus/pheplots/
http://www.magic.ifae.es/~markus/timeplots/
\end{verbatim}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Un-Calibrated Pixels and Events \label{sec:uncalibrated}}

The MAGIC calibration software incorporates a series of checks to sort out mal-functioning pixels. 
Except for the software bug searching criteria, the following exclusion criteria can apply:

\begin{enumerate}
\item The reconstructed mean signal $\widehat{Q}$ is less than 2.5 times the extractor resolution $R$: $\widehat{Q}<2.5\cdot R$. 
(2.5 Pedestal RMS in the case of the simple fixed window extractors, see section~\ref{sec:pedestals}). 
This criterium essentially cuts out 
dead pixels.
\item The error of the mean reconstructed signal $\delta \widehat{Q}$ is larger than the mean reconstructed signal $\widehat{Q}$: 
 $\delta \widehat{Q} > \widehat{Q}$. This criterion cuts out 
signal distributions which fluctuate so much that their RMS is bigger than its mean value. This 
criterium cuts out ``ringing'' pixels or mal-functioning extractors. 
\item The reconstructed mean number of photo-electrons lies 4.5 sigma outside 
the distribution of photo-electrons obtained with the inner or outer pixels in the camera, respectively. 
This criterium cuts out channels with apparently deviating (hardware) behaviour compared to 
the rest of the camera readout\footnote{This criteria is not applied any more in the standard analysis, 
although we kept using it here}.
\item All pixels with reconstructed negative mean signal or with a 
mean numbers of photo-electrons smaller than one. Pixels with a negative pedestal RMS subtracted 
sigma occur, especially when stars are focused onto that pixel during the pedestal run (resulting 
in a large pedestal RMS), but have moved to another pixel during the calibration run. In this case, the 
number of photo-electrons would result artificially negative. If these 
channels do not show any other deviating behaviour, their number of photo-electrons gets replaced by the 
mean number of photo-electrons in the camera, and the channel is further calibrated as normal.
\end{enumerate}

Moreover, the number of events are counted which have been reconstructed outside a 5$\sigma$ region 
from the mean signal $<\widehat{Q}>$. These events are called ``outliers''. Figure~\ref{fig:outlier} shows a typical 
outlier obtained with the digital filter applied on a low-gain signal, and figure~\ref{fig:unsuited:all}
shows the average number of all excluded pixels and outliers obtained from all 19 calibration configurations.
One can already see that the largest window sizes yield a high number of un-calibrated pixels, mostly 
due to the missing ability to recognize the low-intensity pulses (see later). One can also see that 
the amplitude extracting spline yields a higher number of outliers than the rest of the extractors.
\par
The global champion in lowest number of un-calibrated pixels results to be 
{\textit{\bf MExtractTimeAndChargeSpline}} extracting the integral over two FADC slices (extractor \#25). 
The one with the lowest number of outliers is 
{\textit{\bf MExtractFixedWindowPeakSearch}} with an extraction range of 2 slices (extractor \#11). 

\begin{figure}[htp]
\centering
\includegraphics[width=0.95\linewidth]{Outlier.eps}
\caption{Example of an event classified as ``outlier''. The histogram has been obtained 
using the digital filter (extractor \#32) applied to a high-intensity blue pulse (run 31772). 
The event marked as ``outlier'' clearly has been mis-reconstructed. It lies outside the 5$\sigma$--region from the fitted mean.}
\label{fig:outlier}
\end{figure}

\begin{figure}[htp]
\centering
\includegraphics[height=0.75\textheight]{UnsuitVsExtractor-all.eps}
\caption{Un-calibrated pixels and outlier events averaged over all available 
calibration runs.}
\label{fig:unsuited:all}
\end{figure}

The following figures~\ref{fig:unsuited:5ledsuv},~\ref{fig:unsuited:1leduv},~\ref{fig:unsuited:2ledsgreen}
and~\ref{fig:unsuited:23ledsblue} show the resulting numbers of un-calibrated pixels and events for 
different colours and intensities. Because there is a strong anti-correlation between the number of 
excluded pixels and the number of outliers per event, we have chosen to show these numbers together. 

\par

\begin{figure}[htp]
\centering
\includegraphics[height=0.95\textheight]{UnsuitVsExtractor-5LedsUV-Colour-12.eps}
\caption{Un-calibrated pixels and outlier events for a typical calibration  
pulse of UV-light which does not saturate the high-gain readout.}
\label{fig:unsuited:5ledsuv}
\end{figure}

\begin{figure}[htp]
\centering
\includegraphics[height=0.95\textheight]{UnsuitVsExtractor-1LedUV-Colour-04.eps}
\caption{Un-calibrated pixels and outlier events for a very low 
intensity pulse.}
\label{fig:unsuited:1leduv}
\end{figure}

\begin{figure}[htp]
\centering
\includegraphics[height=0.95\textheight]{UnsuitVsExtractor-2LedsGreen-Colour-02.eps}
\caption{Un-calibrated pixels and outlier events for a typical green pulse.}
\label{fig:unsuited:2ledsgreen}
\end{figure}

\begin{figure}[htp]
\centering
\includegraphics[height=0.95\textheight]{UnsuitVsExtractor-23LedsBlue-Colour-00.eps}
\caption{Un-calibrated pixels and outlier events for a high-intensity blue pulse.}
\label{fig:unsuited:23ledsblue}
\end{figure}

One can see that in general, big extraction windows raise the 
number of un-calibrated pixels and are thus less stable. Especially for the very low-intensity 
\textit{\bf 1\,Led\,UV}-pulse, the big extraction windows -- summing 8 or more slices -- cannot calibrate more 
than 50\% of the inner pixels (fig.~\ref{fig:unsuited:1leduv}). 
This is an expected behavior since big windows 
sum up more noise which in turn makes the search for the small signal more difficult.
\par
In general, one can also find that all ``sliding window''-algorithms (extractors \#17-32) discard 
less pixels than the corresponding ``fixed window''-ones (extractors \#1--16). 

The spline (extractors \#23--27) and the digital filter with the correct weights (extractors \#30-31) discard 
the least number of pixels and are also robust against slight modifications of the pulse form 
(of the weights for the digital filter). 
\par
Concerning the numbers of outliers, one can conclude that in general, the numbers are very low never exceeding
0.1\% except for the amplitude-extracting spline which seems to mis-reconstruct a certain type of events.
\par
In conclusion, already this first test excludes all extractors with too large window sizes because 
they are not able to extract cleanly small signals produced by about 4 photo-electrons. Moreover, 
the amplitude extracting spline produces a significantly higher number of outlier events.

\clearpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Number of Photo-Electrons \label{sec:photo-electrons}}

Assuming that the readout chain adds only negligible noise to the one 
introduced by the photo-multiplier itself, one can make the assumption that the variance of the 
true signal, $S$, is the amplified Poisson variance of the number of photo-electrons, 
multiplied with the excess noise of the photo-multiplier which itself is 
characterized by the excess-noise factor $F$:

\begin{equation}
Var[S] = F^2 \cdot Var[N_{phe}] \cdot \frac{<S>^2}{<N_{phe}>^2}
\label{eq:excessnoise}
\end{equation}

After introducing the effect of the night-sky background (eq.~\ref{eq:rmssubtraction}) 
and assuming that the variance of the number of photo-electrons is equal 
to the mean number of photo-electrons (because of the Poisson distribution), 
one obtains an expression to retrieve the mean number of photo-electrons  released at the photo-multiplier cathode from the 
mean extracted signal, $\widehat{S}$, and the RMS of the extracted signal obtained from 
pure pedestal runs $R$ (see section~\ref{sec:ffactor}):

\begin{equation}
<N_{phe}> \approx F^2 \cdot \frac{<\widehat{S}>^2}{Var[\widehat{S}] - R^2}
\label{eq:pheffactor}
\end{equation}

In theory, eq.~\ref{eq:pheffactor} must not depend on the extractor! Effectively, we will use it to test the 
quality of our extractors by requiring that a valid extractor yields the same number of photo-electrons 
for all pixels individually and does not deviate from the number obtained with other extractors. 
As the camera is flat-fielded, but the number of photo-electrons impinging on an inner and an outer pixel is 
different, we also use the ratio of the mean numbers of photo-electrons from the outer pixels to the one 
obtained from the inner pixels as a test variable. In the ideal case, it should always yield its central 
value of about 2.6$\pm$0.1~\cite{michele-diploma}.
\par
In our case, there is an additional complication due to the fact that the green and blue coloured light pulses 
show secondary pulses which destroy the Poisson behaviour of the number of photo-electrons. We will
have to split our sample of extractors into those being affected by the secondary pulses and those 
being immune to this effect. 
\par
Figures~\ref{fig:phe:5ledsuv},~\ref{fig:phe:1leduv},~\ref{fig:phe:2ledsgreen}~and~\ref{fig:phe:23ledsblue} show 
some of the obtained results. One can see a rather good stability for the standard 
{\textit{\bf 5\,Leds\,UV}}\ pulse, except for the extractors {\textit{\bf MExtractFixedWindowPeakSearch}}, initialized 
with an extraction window of 2 slices.
\par
There is a considerable difference for all shown non-standard pulses. Especially the pulses from green 
and blue LEDs 
show a clear dependence  of the number of photo-electrons on the extraction window. Only the largest 
extraction windows seem to catch the entire range of (jittering) secondary pulses and get the ratio 
of outer vs. inner pixels right. However, they (obviously) over-estimate the number of photo-electrons 
in the primary pulse.
\par
The strongest discrepancy is observed in the low-gain extraction (fig.~\ref{fig:phe:23ledsblue}) where all 
fixed window extractors with extraction windows smaller than 8 FADC slices fail to reconstruct the correct numbers. 
This has to do with the fact that 
the fixed window extractors fail to catch a significant part of the (larger) pulse because of the 
1~FADC slice event-to-event jitter and the larger pulse width covering about 6 FADC slices. 
Also the sliding windows smaller than 6 FADC slices and the spline smaller than 
2 FADC slices reproduce too small numbers of photo-electrons. Moreover, the digital filter shows a small dependency 
of the number of photo-electrons w.r.t. the extration window.
\par


\begin{figure}[htp]
\centering
\includegraphics[height=0.92\textheight]{PheVsExtractor-5LedsUV-Colour-12.eps}
\caption{Number of photo-electrons from a typical, not saturating calibration pulse of colour UV, 
reconstructed with each of the tested signal extractors. 
The first plots shows the number of photo-electrons obtained for the inner pixels, the second one 
for the outer pixels and the third shows the ratio of the mean number of photo-electrons for the 
outer pixels divided by the mean number of photo-electrons for the inner pixels. Points 
denote the mean of all not-excluded pixels, the error bars their RMS.}
\label{fig:phe:5ledsuv}
\end{figure}

\begin{figure}[htp]
\centering
\includegraphics[height=0.92\textheight]{PheVsExtractor-1LedUV-Colour-04.eps}
\caption{Number of photo-electrons from a typical, very low-intensity calibration pulse of colour UV, 
reconstructed with each of the tested signal extractors. 
The first plots shows the number of photo-electrons obtained for the inner pixels, the second one 
for the outer pixels and the third shows the ratio of the mean number of photo-electrons for the 
outer pixels divided by the mean number of photo-electrons for the inner pixels. Points 
denote the mean of all not-excluded pixels, the error bars their RMS.}
\label{fig:phe:1leduv}
\end{figure}

\begin{figure}[htp]
\centering
\includegraphics[height=0.92\textheight]{PheVsExtractor-2LedsGreen-Colour-02.eps}
\caption{Number of photo-electrons from a typical, not saturating calibration pulse of colour green, 
reconstructed with each of the tested signal extractors. 
The first plots shows the number of photo-electrons obtained for the inner pixels, the second one 
for the outer pixels and the third shows the ratio of the mean number of photo-electrons for the 
outer pixels divided by the mean number of photo-electrons for the inner pixels. Points 
denote the mean of all not-excluded pixels, the error bars their RMS.}
\label{fig:phe:2ledsgreen}
\end{figure}


\begin{figure}[htp]
\centering
\includegraphics[height=0.92\textheight]{PheVsExtractor-23LedsBlue-Colour-00.eps}
\caption{Number of photo-electrons from a typical, high-gain saturating calibration pulse of colour blue, 
reconstructed with each of the tested signal extractors. 
The first plots shows the number of photo-electrons obtained for the inner pixels, the second one 
for the outer pixels and the third shows the ratio of the mean number of photo-electrons for the 
outer pixels divided by the mean number of photo-electrons for the inner pixels. Points 
denote the mean of all not-excluded pixels, the error bars their RMS.}
\label{fig:phe:23ledsblue}
\end{figure}

One can see that all extractors using a large window belong to the class of extractors being affected 
by the secondary pulses, except for the digital filter. 
\par
The extractor {\textit{\bf MExtractTimeAndChargeDigitalFilter}} seems to be sufficiently stable against modifications of the 
exact form of the weights in the high-gain readout channel since all applied weights yield about 
the same number of photo-electrons and the same ratio of outer vs. inner pixels.
\par
All sliding window and spline algorithms yield a stable ratio of outer vs. inner pixels in the high and the low-gain. 
\par
Concluding, there is no fixed window extractor yielding always the correct number of photo-electrons, 
except for the extraction window of 8 FADC slices. 
Either the number of photo-electrons itself is wrong or the ratio of outer vs. inner pixels is 
not correct. All sliding window algorithms seem to reproduce the correct numbers if one takes into 
account the after-pulse behaviour of the light pulser itself. The digital filter seems to be 
stable against modifications of the intrinsic pulse width from 1~to~4\,ns. This is the expected range within which the pulses from 
realistic cosmics signals may vary. 

\clearpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Linearity \label{sec:calibration:linearity}}

\begin{figure}[htp]
\centering
\includegraphics[width=0.99\linewidth]{PheVsCharge-4.eps}
\caption{Conversion factor $c_{phe}$ for three exemplary inner pixels (upper plots) 
and three exemplary outer ones (lower plots) obtained with the extractor 
{\textit{MExtractFixedWindow}} on a window size of 8 high-gain and 8 low-gain slices 
(extractor \#4). }
\label{fig:linear:phevscharge4}
\end{figure}

In this section, we test the linearity of the conversion factors FADC counts to photo-electrons:

\begin{equation}
c_{phe} =\  <N_{phe}> / <\widehat{S}>
\end{equation}

As the photo-multiplier and the subsequent 
optical transmission devices~\cite{david} is a relatively linear device over a 
wide dynamic range, the number of photo-electrons per charge has to remain constant over the tested 
linearity region. 
\par
A first test concerns the stability of the conversion factor: mean number of averaged photo-electrons 
per FADC counts over the tested intensity region. This test includes all systematic uncertainties 
in the calculation of the number of photo-electrons and the computation of the mean signal. 
A more detailed investigation of the linearity will be shown in a 
separate TDAS~\cite{tdas-calibration}, although there, the number of photo-electrons will be calculated 
in a more independent way.

\par
Figure~\ref{fig:linear:phevscharge4} shows the conversion factor $c_{phe}$ obtained for different light intensities 
and colours for three exemplary inner and three exemplary outer pixels using a fixed window on 
8 FADC slices. The conversion factor seems to be linear to a good approximation, with the following restrictions:
\begin{itemize}
\item The green pulses yield systematically low conversion factors
\item Some of the pixels show a difference 
between the high-gain ($<$100\ phes for the inner, $<$300\ phes for the outer pixels) and the low-gain 
($>$100\ phes for the inner, $>$300\ phes for the outer pixels) region and 
a rather good stability of $c_{phe}$ for each region separately.
\end{itemize}

We conclude that, with the above restrictions,
the fixed window extractor \#4 is a linear extractor for both high-gain 
and low-gain regions, separately.
\par

Figures~\ref{fig:linear:phevscharge9} and~\ref{fig:linear:phevscharge15} show the conversion factors 
using an integrated spline and a fixed window with global peak search, respectively, over 
an extraction window of 8 FADC slices. The same behaviour is obtained as before. These extractors are 
linear to a good approximation, except for the two cases mentioned above.
\par

\begin{figure}[h!]
\centering
\includegraphics[width=0.99\linewidth]{PheVsCharge-9.eps}
\caption{Conversion factor $c_{phe}$ for three exemplary inner pixels (upper plots) 
and three exemplary outer ones (lower plots) obtained with the extractor 
{\textit{MExtractFixedWindowSpline}} 
on a window size of 8 high-gain and 8 low-gain slices (extractor \#9). }
\label{fig:linear:phevscharge9}
\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[width=0.99\linewidth]{PheVsCharge-15.eps}
\caption{Conversion factor $c_{phe}$ for three exemplary inner pixels (upper plots) 
and three exemplary outer ones (lower plots) obtained with the extractor 
{\textit{MExtractFixedWindowPeakSearch}} on a window size of 8 high-gain and 8 low-gain slices 
(extractor \#15). }
\label{fig:linear:phevscharge15}
\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[width=0.99\linewidth]{PheVsCharge-14.eps}
\caption{Example of a the development of the conversion factor FADC counts to photo-electrons for three 
exemplary inner pixels (upper plots) and three exemplary outer ones (lower plots) obtained with the extractor 
{\textit{MExtractFixedWindowPeakSearch}} 
on a window size of 6 high-gain and 6 low-gain slices (extractor \#11). }
\label{fig:linear:phevscharge11}
\end{figure}

Figure~\ref{fig:linear:phevscharge11} shows the conversion factors using a fixed window with global peak search 
integrating a window of 6 FADC slices. One can see that the linearity is completely lost above 300 photo-electrons in the 
outer pixels. Especially in the low-gain, 
the reconstructed mean charge is too low and the conversion factors bend down. We show this extractor especially because it has 
been used in the analysis and to derive a Crab spectrum with the consequence that the spectrum bends down at high energies. We 
suppose that the loss of linearity due to usage of this extractor is responsible for the encountered problems.
A similiar behaviour can be found for all extractors with window sizes smaller than 6 FADC slices, especially in the low-gain region. 
This is understandable since the low-gain pulse covers at least 6 FADC slices.
(This behaviour 
was already visible in the investigations on the number of photo-electrons in the previous section~\ref{sec:photo-electrons}).
\par
Figure~\ref{fig:linear:phevscharge20} shows the conversion factors using a sliding window of 6 FADC slices. 
The linearity is maintained like in the previous examples, except that for the smallest signals the effect
of the bias is already visible.
\par

\begin{figure}[h!]
\centering
\includegraphics[width=0.99\linewidth]{PheVsCharge-20.eps}
\caption{Example of a the development of the conversion factor FADC counts to photo-electrons for three 
exemplary inner pixels (upper plots) and three exemplary outer ones (lower plots) obtained with the extractor 
{\textit{MExtractTimeAndChargeSlidingWindow}} 
on a window size of 6 high-gain and 6 low-gain slices (extractor \#20). }
\label{fig:linear:phevscharge20}
\end{figure}

Figure~\ref{fig:linear:phevscharge23} shows the conversion factors using the amplitude-extracting spline 
(extractor \#23).
Here, the linearity is worse than in the previous examples. A very clear difference between high-gain and 
low-gain regions can be seen as well as a bigger general spread in conversion factors. In order to investigate
if there is a common, systematic effect of the extractor, we show the averaged conversion factors over all 
inner and outer pixels in figure~\ref{fig:linear:phevschargearea23}. Both characteristics are maintained 
there. Although the differences between high-gain and low-gain could be easily corrected for, we conclude 
that extractor \#23 is still unstable against the linearity tests.
\par

\begin{figure}[h!]
\centering
\includegraphics[width=0.99\linewidth]{PheVsCharge-23.eps}
\caption{Conversion factor $c_{phe}$ for three exemplary inner pixels (upper plots) 
and three exemplary outer ones (lower plots) obtained with the extractor 
{\textit{MExtractTimeAndChargeSpline}} with amplitude extraction (extractor \#23). }
\label{fig:linear:phevscharge23}
\vspace{\floatsep}
\includegraphics[width=0.9\linewidth]{PheVsCharge-Area-23.eps}
\caption{Conversion factor $c_{phe}$ averaged over all inner (left) and all outer (right) pixels 
obtained with the extractor 
{\textit{MExtractTimeAndChargeSpline}} with amplitude extraction (extractor \#23). }
\label{fig:linear:phevschargearea23}
\end{figure}

Figure~\ref{fig:linear:phevscharge24} shows the conversion factors using a spline integrating over 
one effective FADC slice in the high-gain and 1.5 effective FADC slices in the low-gain region (extractor \#24).
The same problems are found as with extractor \#23, however to a much lower extent. 
The difference between high-gain and low-gain regions is less pronounced and the spread 
in conversion factors is smaller. 
Figure~\ref{fig:linear:phevschargearea24} shows already rather good stability except for the two 
lowest intensity pulses in green and blue. We conclude that extractor \#24 is still un-stable, but 
preferable to the amplitude extractor.
\par

\begin{figure}[h!]
\centering
\includegraphics[width=0.99\linewidth]{PheVsCharge-24.eps}
\caption{Conversion factor $c_{phe}$ for three exemplary inner pixels (upper plots) 
and three exemplary outer ones (lower plots) obtained with the extractor 
{\textit{MExtractTimeAndChargeSpline}} with window size of 1 high-gain and 2 low-gain slices 
(extractor \#24). }
\label{fig:linear:phevscharge24}
\vspace{\floatsep}
\includegraphics[width=0.9\linewidth]{PheVsCharge-Area-24.eps}
\caption{Conversion factor $c_{phe}$ averaged over all inner (left) and all outer (right) pixels 
obtained with the extractor 
{\textit{MExtractTimeAndChargeSpline}} with window size of 1 high-gain and 2 low-gain slices 
(extractor \#24). }
\label{fig:linear:phevschargearea24}
\end{figure}

Looking at figure~\ref{fig:linear:phevscharge25}, one can see that raising the integration window 
by two  effective FADC slices in the high-gain and three effective FADC slices in the low-gain 
(extractor \#25), the stability is completely resumed, except for 
a systematic increase of the conversion factor above 200 photo-electrons. 
We conclude that extractor \#25 is almost as stable as the fixed window extractors. 
\par

\begin{figure}[htp]
\centering
\includegraphics[width=0.99\linewidth]{PheVsCharge-25.eps}
\caption{Conversion factor $c_{phe}$ for three exemplary inner pixels (upper plots) 
and three exemplary outer ones (lower plots) obtained with the extractor 
{\textit{MExtractTimeAndChargeSpline}} with window size of 2 high-gain and 3 low-gain slices 
(extractor \#25). }
\label{fig:linear:phevscharge25}
\vspace{\floatsep}
\includegraphics[width=0.9\linewidth]{PheVsCharge-Area-25.eps}
\caption{Conversion factor $c_{phe}$ averaged over all inner (left) and all outer (right) pixels 
obtained with the extractor 
{\textit{MExtractTimeAndChargeSpline}} with window size of 2 high-gain and 3 low-gain slices 
(extractor \#25). }
\label{fig:linear:phevschargearea25}
\end{figure}

Figure~\ref{fig:linear:phevscharge30} and~\ref{fig:linear:phevscharge31} show the conversion factors using a digital filter, 
applied on 6 FADC slices and respectively 4 FADC slices with weights calculated from the UV-calibration pulse in the 
high-gain region and from the blue calibration pulse in the low-gain region.
One can see that one or two blue  calibration pulses at low and intermediate intensity fall
out of the linear region, moreover there is a small systematic offset between the high-gain and low-gain region. 
It seems that the digital filter does not pass this test if the pulse form changes for more than 2\,ns from the 
expected one. The effect is not as problematic as it may appear here, because the actual calibration 
will not calculate the number of photo-electrons (with the F-Factor method) for every signal intensity. 
Thus, one possible reason for the instability is not relevant in the cosmics analysis. However, the limits 
of this extraction are visible here and should  be monitored further.

\par

\begin{figure}[htp]
\centering
\includegraphics[width=0.99\linewidth]{PheVsCharge-30.eps}
\caption{Conversion factor $c_{phe}$ for three exemplary inner pixels (upper plots) 
and three exemplary outer ones (lower plots) obtained with the extractor 
{\textit{MExtractTimeAndChargeDigitalFilter}}  
using a window size of 6 high-gain and 6 low-gain slices with UV-weights (extractor \#30). }
\label{fig:linear:phevscharge30}
\vspace{\floatsep}
\includegraphics[width=0.9\linewidth]{PheVsCharge-Area-30.eps}
\caption{Conversion factor $c_{phe}$ averaged over all inner (left) and all outer (right) pixels 
obtained with the extractor 
{\textit{MExtractTimeAndChargeDigitalFilter}} with window size of 6 high-gain and 6 low-gain slices and UV-weight
(extractor \#30). }
\label{fig:linear:phevschargearea30}
\end{figure}


\begin{figure}[htp]
\centering
\includegraphics[width=0.99\linewidth]{PheVsCharge-31.eps}
\caption{Conversion factor $c_{phe}$ for three exemplary inner pixels (upper plots) 
and three exemplary outer ones (lower plots) obtained with the extractor 
{\textit{MExtractTimeAndChargeDigitalFilter}} using a window size of 
4 high-gain and 4 low-gain slices (extractor \#31). }
\label{fig:linear:phevscharge31}
\vspace{\floatsep}
\includegraphics[width=0.9\linewidth]{PheVsCharge-Area-31.eps}
\caption{Conversion factor $c_{phe}$ averaged over all inner (left) and all outer (right) pixels 
obtained with the extractor 
{\textit{MExtractTimeAndChargeDigitalFilter}} with window size of 6 high-gain and 6 low-gain slices and blue weights
(extractor \#31). }
\label{fig:linear:phevschargearea3}
\end{figure}

\clearpage

\subsection{Relative Arrival Time Calibration}

The calibration LEDs
deliver fast-rising pulses, uniform over the camera in signal size and time. 
We estimate the time-uniformity to as good as about~30\,ps, a limit due to the different travel times of the light
from the light source to the inner and outer parts of the camera. For cosmics data, however, the staggering of the 
mirrors limits the time uniformity to about 600\,ps.
\par
The extractors \#17--33 are able to compute the arrival time of each pulse. 
Since the calibration does not permit a precise measurement of the absolute arrival time, we measure 
the relative arrival time for every channel with respect to a reference channel (usually pixel no.\,1):

\begin{equation}
\delta t_i = t_i - t_1
\end{equation}

where $t_i$ denotes the reconstructed arrival time of pixel number $i$ and $t_1$ the reconstructed 
arrival time of the reference pixel no. 1 (software numbering). In one calibration run, one can then fill 
histograms of $\delta t_i$ and fit them to the expected Gaussian distribution. The fits 
yield a mean $\mu(\delta t_i)$, comparable to 
systematic delays in the signal travel time, and a sigma $\sigma(\delta t_i)$, a measure of the 
combined time resolutions of pixel $i$ and pixel 1. Assuming that the PMTs and readout channels are 
of the same kind, we obtain an approximate time resolution of pixel $i$:

\begin{equation}
t^{res}_i \approx \sigma(\delta t_i)/\sqrt{2}
\end{equation}

Figures~\ref{fig:reltimesinnerleduv} show distributions of $\delta t_i$ 
for a typical inner pixel and a non-saturating calibration pulse of UV-light, 
obtained with six different extractors. 
One can see that all of them yield acceptable Gaussian distributions, 
except for the sliding window extracting 2 slices which shows a three-peak structure and cannot be fitted. 
We discarded that particular extractor from the further studies of this section.

\begin{figure}[htp]
\centering
\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor17.eps}
\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor18.eps}
\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor23.eps}
\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor24.eps}
\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor30.eps}
\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor31.eps}
\caption{Examples of a distributions of relative arrival times $\delta t_i$ of an inner pixel (no. 100) \protect\\
Top: {\textit{\bf MExtractTimeAndChargeSlidingWindow}} over 2  slices (\#17) and 4  slices (\#18) \protect\\
Center: {\textit{\bf MExtractTimeAndChargeSpline}} with maximum (\#23) and half-maximum pos. (\#24) \protect\\
Bottom: {\textit{\bf MExtractTimeAndChargeDigitalFilter}} fitted to a UV-calibration pulse over 6 slices (\#30) and 4 slices (\#31) \protect\\
A medium sized UV-pulse (5\,Leds UV) has been used which does not saturate the high-gain readout channel.}
\label{fig:reltimesinnerleduv}
\end{figure}

Figures~\ref{fig:reltimesinnerledblue1} and~\ref{fig:reltimesinnerledblue2} show 
the distributions of $\delta t_i$ for a typical inner pixel and an intense, high-gain-saturating calibration 
pulse of blue light, obtained from the low-gain readout channel.
One can see that the sliding window extractors yield double Gaussian structures, except for the 
largest window sizes of 8 and 10 FADC slices. Even then, the distributions are not exactly Gaussian.
The maximum position extracting spline also yields distributions which are not exactly Gaussian and seems 
to miss the exact arrival time in some events. Only the position of the half-maximum gives the 
expected result of a single Gaussian distribution.
A similiar problem occurs in the case of the digital filter: If one takes the correct weights 
(fig.~\ref{fig:reltimesinnerledblue2} bottom), the distribution is perfectly Gaussian and the resolution good, 
however a rather slight change from the blue calibration pulse weights to cosmics pulses weights (top) 
adds a secondary peak of events with mis-reconstructed arrival times. In principle, the $\chi^2$ of the digital filter 
fit gives an information about whether the correct shape has been used. 

\begin{figure}[htp]
\centering
\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor18_logain.eps}
\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor19_logain.eps}
\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor21_logain.eps}
\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor22_logain.eps}
\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor23_logain.eps}
\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor24_logain.eps}
\caption{Examples of a distributions of relative arrival times $\delta t_i$ of an inner pixel (no. 100) \protect\\
Top: {\textit{\bf MExtractTimeAndChargeSlidingWindow}} over 4  slices (\#18) and 6  slices (\#19) \protect\\
Center: {\textit{\bf MExtractTimeAndChargeSlidingWindow}} over 8  slices (\#20) and 10  slices (\#21)\protect\\
Bottom: {\textit{\bf MExtractTimeAndChargeSpline}} with maximum (\#23) and half-maximum pos. (\#24) \protect\\ 
A strong Blue pulse (23\,Leds Blue) has been used which does not saturate the high-gain readout channel.}
\label{fig:reltimesinnerledblue1}
\end{figure}

\begin{figure}[htp]
\centering
\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor30_logain.eps}
\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor31_logain.eps}
\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor32_logain.eps}
\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor33_logain.eps}
\caption{Examples of a distributions of relative arrival times $\delta t_i$ of an inner pixel (no. 100) \protect\\
Top: {\textit{\bf MExtractTimeAndChargeDigitalFilter}} 
fitted to cosmics pulses over 6 slices (\#30) and 4  slices (\#31) \protect\\
Bottom: {\textit{\bf MExtractTimeAndChargeDigitalFilter}} fitted to the correct blue calibration pulse over 6  slices (\#30) and 4  slices (\#31)
A strong Blue pulse (23\,Leds Blue) has been used which does not saturate the high-gain readout channel.}
\label{fig:reltimesinnerledblue2}
\end{figure}

%\begin{figure}[htp]
%\centering
%\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel400_10LedUV_Extractor32.eps}
%\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel400_10LedUV_Extractor23.eps}
%\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel400_10LedUV_Extractor17.eps}
%\caption{Example of a two distributions of relative arrival times of an outer pixel with respect to 
%the arrival time of the reference pixel no. 1. The left plot shows the result using the digital filter
% (extractor \#32), the central plot shows the result obtained with the half-maximum of the spline and the 
%right plot the result of the sliding window with a window size of 2  slices (extractor \#17). A 
%medium sized UV-pulse (10Leds UV) has been used which does not saturate the high-gain readout channel.}
%\label{fig:reltimesouter10leduv}
%\end{figure}

%\begin{figure}[htp]
%\centering
%\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel97_10LedBlue_Extractor23.eps}
%\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel97_10LedBlue_Extractor32.eps}
%\caption{Example of a two distributions of relative arrival times of an inner pixel with respect to 
%the arrival time of the reference pixel no. 1. The left plot shows the result using the half-maximum of the spline (extractor \#23), the right plot shows the result obtained with the digital filter
%(extractor \#32). A 
%medium sized Blue-pulse (10Leds Blue) has been used which saturates the high-gain readout channel.}
%\label{fig:reltimesinner10ledsblue}
%\end{figure}



%\begin{figure}[htp]
%\centering
%\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel400_10LedBlue_Extractor23.eps}
%\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel400_10LedBlue_Extractor32.eps}
%\caption{Example of a two distributions of relative arrival times of an outer pixel with respect to 
%the arrival time of the reference pixel no. 1. The left plot shows the result using the half-maximum of the spline (extractor \#23), the right plot shows the result obtained with the digital filter
%(extractor \#32). A 
%medium sized Blue-pulse (10Leds Blue) has been used which saturates the high-gain readout channel.}
%\label{fig:reltimesouter10ledsblue}
%\end{figure}

\clearpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Number of Outliers}

As in section~\ref{sec:uncalibrated}, we tested the number of outliers from the Gaussian distribution
in order to count how many times the extractor has failed to reconstruct the correct arrival time.
\par
Figure~\ref{fig:timeunsuit:5ledsuv} shows the number of outliers for the different time extractors, obtained with 
a UV pulse of about 20 photo-electrons. One can see that all time extractors yield an acceptable mis-reconstruction 
rate of about 0.5\%, except for the maximum searching spline yields three times more mis-reconstructions. 
\par
If one goes to very low-intensity pulses, as shown in figure~\ref{fig:timeunsuit:1leduv}, obtained with on average 4 photo-electrons, 
the number of mis-reconstructions increases considerably up to 20\% for some extractors. We interpret this high mis-reconstruction 
rate to the increase possibility to mis-reconstruct a pulse from the night sky background noise instead of the signal pulse from the 
calibration LEDs. One can see that the digital filter using weights on 4 FADC slices is clear inferior to the one using 6 FADC slices 
in that respect. 
\par
The same conclusion seems to hold for the green pulse of about 20 photo-electrons (figure~\ref{fig:timeunsuit:2ledsgreen}) 
where the digital filter over 6 FADC slices seems to 
yield more stable results than the one over 4 FADC slices. The half-maximum searching spline seems to be superior to the maximum-searching 
one. 
\par
In figure~\ref{fig:timeunsuit:23ledsblue}, one can see the number of outliers from an intense calibration pulse of blue light yielding about 
600 photo-electrons per inner pixel. All extractors seem to be stable, except for the digital filter with weigths over 4 FADC slices. This
is expected, since the low-gain pulse is wider than 4 FADC slices.
\par 
In all previous plots, the sliding window yielded the most stable results, however later we will see that this stability is only due to 
an increased time spread. 

\begin{figure}[htp]
\centering
\includegraphics[height=0.35\textheight]{UnsuitTimeVsExtractor-5LedsUV-Colour-12.eps}
\caption{Reconstructed arrival time resolutions from a typical, not saturating calibration pulse 
of colour UV, reconstructed with each of the tested arrival time extractors. 
The first plots shows the time resolutions obtained for the inner pixels, the second one 
for the outer pixels. Points 
denote the mean of all not-excluded pixels, the error bars their RMS.}
\label{fig:timeunsuit:5ledsuv}
\end{figure}

\begin{figure}[htp]
\centering
\includegraphics[height=0.35\textheight]{UnsuitTimeVsExtractor-1LedUV-Colour-04.eps}
\caption{Reconstructed arrival time resolutions from the lowest intensity calibration pulse 
of colour UV (carrying a mean number of 4 photo-electrons), 
reconstructed with each of the tested arrival time extractors. 
The first plots shows the time resolutions obtained for the inner pixels, the second one 
for the outer pixels. Points 
denote the mean of all not-excluded pixels, the error bars their RMS.}
\label{fig:timeunsuit:1leduv}
\end{figure}

\begin{figure}[htp]
\centering
\includegraphics[height=0.35\textheight]{UnsuitTimeVsExtractor-2LedsGreen-Colour-02.eps}
\caption{Reconstructed arrival time resolutions from a typical, not saturating calibration pulse 
of colour Green, reconstructed with each of the tested arrival time extractors. 
The first plots shows the time resolutions obtained for the inner pixels, the second one 
for the outer pixels. Points 
denote the mean of all not-excluded pixels, the error bars their RMS.}
\label{fig:timeunsuit:2ledsgreen}
\end{figure}

\begin{figure}[htp]
\centering
\includegraphics[height=0.35\textheight]{UnsuitTimeVsExtractor-23LedsBlue-Colour-00.eps}
\caption{Reconstructed arrival time resolutions from the highest intensity calibration pulse 
of colour blue, reconstructed with each of the tested arrival time extractors. 
The first plots shows the time resolutions obtained for the inner pixels, the second one 
for the outer pixels. Points 
denote the mean of all not-excluded pixels, the error bars their RMS.}
\label{fig:timeunsuit:23ledsblue}
\end{figure}

\clearpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Time Resolution \label{sec:cal:timeres}}

There are three intrinsic contributions to the timing accuracy of the signal:

\begin{enumerate}
\item The intrinsic arrival time spread of the photons on the PMT: This time spread 
can be estimated roughly by the intrinsic width $\delta t_{\mathrm{IN}}$ of the 
input light pulse.
The resulting time
resolution is given by:
\begin{equation}
\Delta t \approx \frac{\delta t_{\mathrm{IN}}}{\sqrt{Q/{\mathrm{phe}}}}
\end{equation}
The width $\delta t_{\mathrm{LED}}$ of the calibration pulses of about 2\,ns 
for the faster UV pulses and 3--4\,ns for the green and blue pulses, 
for muons it is a few hundred ps, for gammas about 1\,ns and for hadrons a few ns. 
\item The intrinsic transit time spread $\mathrm{\it TTS}$ of the photo-multiplier:
It can be of the order of a few hundreds of ps per single photo electron, depending on the 
wavelength of the incident light. As in the case of the photon arrival time spread, the total
time spread scales with the inverse of the square root of the number of photo-electrons:
\begin{equation}
\Delta t \approx \frac{\delta t_{\mathrm{TTS}}}{\sqrt{Q/{\mathrm{phe}}}}
\end{equation}
\item The reconstruction error due to the background noise and limited extractor resolution: 
This contribution is inversely proportional to the signal to square root of background light intensities.
\begin{equation}
\Delta t \approx \frac{\delta t_{\mathrm{rec}} \cdot R/\mathrm{phe}}{Q/{\mathrm{phe}}}
\end{equation}
where $R$ is the resolution defined in equation~\ref{eq:def:r}.
\item A constant offset due to the residual FADC clock jitter~\cite{florian}
\begin{equation}
\Delta t \approx \delta t_0
\end{equation}
\end{enumerate}

In the following, we show measurements of the time resolutions at different 
signal intensities in real conditions for the calibration pulses. These set upper limits to the time resolution for cosmics since their 
intrinsic arrival time spread is smaller. 

Figures~\ref{fig:time:5ledsuv} through~\ref{fig:time:23ledsblue} show the measured time resolutions for very different calibration 
pulse intensities and colours. One can see that the sliding window resolutions are always worse than the spline and digital filter 
algorithms. Moreover, the half-maximum position search by the spline is always slightly better than the maximum position search. The 
digital filter does not show notable differences with respect to the pulse form or the extraction window size, except for the low-gain 
extraction where the 4 slices seem to yield a better resolution. This is only after excluding about 30\% of the events, as shown in 
figure~\ref{fig:timeunsuit:23ledsblue}.

\begin{figure}[htp]
\centering
\includegraphics[height=0.38\textheight]{TimeResExtractor-5LedsUV-Colour-12.eps}
\caption{Reconstructed arrival time resolutions from a typical, not saturating calibration pulse 
of colour UV, reconstructed with each of the tested arrival time extractors. 
The first plots shows the time resolutions obtained for the inner pixels, the second one 
for the outer pixels. Points 
denote the mean of all not-excluded pixels, the error bars their RMS.}
\label{fig:time:5ledsuv}
\end{figure}

\begin{figure}[htp]
\centering
\includegraphics[height=0.38\textheight]{TimeResExtractor-1LedUV-Colour-04.eps}
\caption{Reconstructed arrival time resolutions from the lowest intensity calibration pulse 
of colour UV (carrying a mean number of 4 photo-electrons), 
reconstructed with each of the tested arrival time extractors. 
The first plots shows the time resolutions obtained for the inner pixels, the second one 
for the outer pixels. Points 
denote the mean of all not-excluded pixels, the error bars their RMS.}
\label{fig:time:1leduv}
\end{figure}

\begin{figure}[htp]
\centering
\includegraphics[height=0.38\textheight]{TimeResExtractor-2LedsGreen-Colour-02.eps}
\caption{Reconstructed arrival time resolutions from a typical, not saturating calibration pulse 
of colour Green, reconstructed with each of the tested arrival time extractors. 
The first plots shows the time resolutions obtained for the inner pixels, the second one 
for the outer pixels. Points 
denote the mean of all not-excluded pixels, the error bars their RMS.}
\label{fig:time:2ledsgreen}
\end{figure}

\begin{figure}[htp]
\centering
\includegraphics[height=0.38\textheight]{TimeResExtractor-23LedsBlue-Colour-00.eps}
\caption{Reconstructed arrival time resolutions from the highest intensity calibration pulse 
of colour blue, reconstructed with each of the tested arrival time extractors. 
The first plots shows the time resolutions obtained for the inner pixels, the second one 
for the outer pixels. Points 
denote the mean of all not-excluded pixels, the error bars their RMS.}
\label{fig:time:23ledsblue}
\end{figure}

\clearpage

The following figure~\ref{fig:time:dep} shows the time resolution for various calibration runs taken with different colours 
and light intensities as a funcion of the mean number of photo-electrons --
reconstructed with the F-Factor method -- for four different time extractors. The dependencies have been fit to the following 
empirical relation:

\begin{equation}
\Delta T = \sqrt{\frac{A^2}{<Q>/{\mathrm{phe}}} + \frac{B^2}{<Q>^2/{\mathrm{phe^2}}} + C^2} .
\label{eq:time:fit}
\end{equation}

The fit results are summarized in table~\ref{tab:time:fitresults}.

\begin{table}[htp]
\scriptsize{%
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\hline
\multicolumn{10}{|c|}{\large Time Fit Results} \rule{0mm}{6mm} \rule[-2mm]{0mm}{6mm} \hspace{-3mm}\\
\hline
\hline
\multicolumn{2}{|c|}{} & \multicolumn{4}{|c|}{\normalsize Inner Pixels} & \multicolumn{4}{|c|}{\normalsize Outer Pixels} \rule{0mm}{6mm} \rule[-2mm]{0mm}{4mm} \hspace{-3mm}\\
\hline
{\normalsize Nr.} & {\normalsize  Name } & {\normalsize  A}  & {\normalsize B } & {\normalsize C }& {\normalsize  $\chi^2$/NDF } 
& {\normalsize  A } &{\normalsize  B} & {\normalsize  C} &{\normalsize  $\chi^2$/NDF} \rule{0mm}{6mm} \rule[-2mm]{0mm}{4mm} \hspace{-3mm} \\
\hline
21  & Sliding Window (8,8)   & 3.5$\pm$0.4 & 29$\pm$1 & 0.24$\pm$0.05 & 10.2 &6.0$\pm$0.7 & 52$\pm$4 & 0.23$\pm$0.04 & 4.3  \\
25  & Spline Half Max.       & 1.9$\pm$0.2 & 3.8$\pm$1.0 & 0.15$\pm$0.02 & 1.6 &2.6$\pm$0.2 &8.3$\pm$1.9 & 0.15$\pm$0.01 & 2.3  \\
32  & Digital Filter (6 sl.) & 1.7$\pm$0.2 & 5.7$\pm$0.8 & 0.21$\pm$0.02 & 5.0 &2.3$\pm$0.3 &13 $\pm$2   & 0.20$\pm$0.01 & 4.0  \\
33  & Digital Filter (4 sl.) & 1.7$\pm$0.1 & 4.6$\pm$0.7 & 0.21$\pm$0.02 & 6.2 &2.3$\pm$0.2 &11 $\pm$2   & 0.20$\pm$0.01 & 5.3  \\
\hline
\hline
\end{tabular}
\caption{The fit results obtained from the fit of equation~\ref{eq:time:fit} to the time resolutions obtained for various 
intensities and colours. The fit probabilities are very small mainly because of the different intrinsic arrival time spreads of 
the photon pulses from different colours. }
\label{tab:time:fitresults}.
}
\end{table}

The low fit probabilities are partly due to the systematic differences in the pulse forms in intrinsic arrival time spreads between 
pulses of different LED colours. Nevertheless, we had to include all colours in the fit to cover the full dynamic range. In general, 
one can see that the time resolutions for the UV pulses are systematically better than for the other colours which we attribute to the fact 
the these pulses have a smaller intrinsic pulse width -- which is very close to pulses from cosmics. Moreover, there are clear differences 
visible between different time extractors, especially the sliding window extractor yields poor resolutions. The other three extractors are 
compatible within the errors, with the half-maximum of the spline being slightly better.

\par

To summarize, we find that we can obtain a time resolution of better than 1\,ns for all pulses above a threshold of 5\ photo-electrons. 
This corresponds roughly to the image cleaning threshold in case of using the best signal extractor. At the largest signals, we can 
reach a time resolution of as good as 200\,ps.
\par
The expected time resolution for inner pixels and cosmics pulses can thus be conservatively estimated to be:

\begin{equation}
\Delta T_{\mathrm{cosmics}} \approx \sqrt{\frac{4\,\mathrm{ns}^2}{<Q>/{\mathrm{phe}}} 
+ \frac{20\,\mathrm{ns}^2}{<Q>^2/{\mathrm{phe^2}}} + 0.04\,\mathrm{ns}^2} .
\label{eq:time:fitprediction}
\end{equation}

\begin{landscape}
\begin{figure}[htp]
\centering
\includegraphics[width=0.24\linewidth]{TimeResFitted-21.eps}
\includegraphics[width=0.24\linewidth]{TimeResFitted-25.eps}
\includegraphics[width=0.24\linewidth]{TimeResFitted-32.eps}
\includegraphics[width=0.24\linewidth]{TimeResFitted-33.eps}
\caption{Reconstructed mean arrival time resolutions as a function of the extracted mean number of 
photo-electrons for the weighted sliding window with a window size of 8 slices (extractor \#21, top left), 
the half-maximum searching spline (extractor~\#25, top right), 
the digital filter with correct pulse weights over 6 slices (extractor~\#30 and~\#32, bottom left) 
and the digital filter with UV calibration-pulse weights over 4 slices (extractor~\#31 and~\#33, bottom rigth).
Error bars denote the spread (RMS) of time resolutions of the investigated channels.
The marker colours show the applied 
pulser colour, except for the last (green) point where all three colours were used.}
\label{fig:time:dep}
\end{figure}
\end{landscape}

The above resolution seems to be already limited by the intrinsic resolution of the photo-multipliers and the staggering of the 
mirrors in case of the MAGIC-I telescope.

%\begin{figure}[htp]
%\centering
%\includegraphics[width=0.32\linewidth]{TimeResVsSqrtPhe-Area-24.eps}
%\includegraphics[width=0.32\linewidth]{TimeResVsSqrtPhe-Area-30.eps}
%\includegraphics[width=0.32\linewidth]{TimeResVsSqrtPhe-Area-31.eps}
%\caption{Reconstructed arrival time resolutions as a function of the square root of the 
%extimated number of photo-electrons for the half-maximum searching spline (extractor \#24, left) a
%and the digital filter with the calibration pulse weigths fitted to UV pulses over 6 FADC slices (extractor \#30, center) 
%and the  digital filter with the calibration pulse weigths fitted to UV pulses over 4 FADC slices (extractor \#31, right).
%The time resolutions have been fitted from 
%The marker colours show the applied 
%pulser colour, except for the last (green) point where all three colours were used.}
%\label{fig:time:fit2430}
%\end{figure}


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