\begin{equation}
v_d=v_d(E,N(T,P),C_{CO2},C_{N2}) \\
\end{equation}
-where $E$ is the electric field in the TPC, P is the atmospheric pressure, T is a temperature inside of the TPC and $C_{co2}$ and $C_{N2}$ is the concentration and N is the gas density. We suppose that these parameters will vary in time in reasonable range and the taylor expansion of the function around the nominal values can be used.
+where $E$ is the electric field in the TPC, P is the atmospheric pressure, T is the temperature inside of the TPC and $C_{CO2}$ and $C_{N2}$ is the concentration and N is the gas density. We suppose that these parameters will vary in time in reasonable range and the Taylor expansion of the function around the nominal values can be used.
\begin{equation}
\Delta{v_d}=v_d-v_{d0}=\frac{dv}{dE}\Delta{E}+\frac{dv}{dN}\Delta{N(P,T)}+\frac{dv}{dC_{CO2}}\Delta{C_{CO2}}+\frac{dv}{dC_{N2}}\Delta{C_{N2}}
\end{equation}
-Parameters in the expansion are changing with different time constant. Significant change of the drift velocity due to the gas composition changes has a time constant of days. On the other hand the changes due to the pressure and temperature variation had to be corrected on munutes level.
+The parameters in the expansion are changing with different time constant. Significant change of the drift velocity due to the gas composition changes has a time constant of days. On the other hand the changes due to the pressure and temperature variation had to be corrected on minutes level.
In the following we will focus on the influence of the changes of the gas density, temperature and pressure.
\begin{equation}
\frac{\Delta{v_d}}{v_{d0}}= k_t(t)+k_{N}\frac{\Delta{N(P,T)}}{N_0(P,T)}
\frac{\Delta{v_d}}{v_{d0}}= k_t(t)+k_{P/T}\frac{\Delta{(P/T)}}{(P/T)_0}
\end{equation}
-The time dependent offset factor $k_t(t)$ describe the influence of the gas composition and electric field changes.
+The time dependent offset factor $k_t(t)$ describes the influence of the gas composition and electric field changes.
The correction factor $v_c=\frac{\Delta{v_d}}{v_{d0}}$ can be measured using different methods:
\begin{itemize}
-\item Matching laser tracks with the surweyed mirror position
+\item Matching laser tracks with the surveyed mirror position
\item Matching with the ITS tracks
-\item Matching of the TPC primary verteces from two halves of the TPC
-\item Using cosmic tracks - matching of the tracks from two halves of the TPC
+\item Matching of the TPC primary vertices from the two halves of the TPC
+\item Using cosmic tracks - matching tracks from two halves of the TPC
\end{itemize}
The unknown parameters $k_t(t)$ and $k_N$ can be than fitted using the Kalman filter.
and the influence of other parameters (gas composition, and electric field) are only
slowly varying in time and can be expressed by smooth function $x_{off}(t)$:
\begin{equation}
-x(t) = x_{off}(t)+k_N\frac{\Delta{P/T}}{P/T}
+x(t) = x_{off}(t)+k_N\frac{\Delta{P/T}}{P/T}
+\label{eq:KalmanTime}
\end{equation}
where x(t) is the parameter which we observe.
\begin{equation}
\end{split}
\end{equation}
-Kalman filter parameters are following:
+The Kalman filter parameters are:
\begin{itemize}
\item State vector ($x_{off}(t)$, $k_N$) at given time
\item Covariance matrix
\end{itemize}
-Kalman filter implent following functions:
+The Kalman filter implement the following functions:
\begin{itemize}
\item Prediction - adding covariance element $\sigma_{xoff}$
\item Update state vector with new measurement vector ($x_t,\frac{\Delta{P/T}}{P/T}$)
\end{itemize}
+\begin{figure}[t]
+\centering
+\includegraphics[width=80mm]{picDCS/vdriftraw_time.eps}
+\includegraphics[width=80mm]{picDCS/tpraw_tp.eps}
+\caption{
+ Drift velocity as function of time (upper plot) and as a function of $\Delta(T/P)$ (lower plot)
+}
+\label{figVDrift}
+\end{figure}
+
+\begin{figure}[t]
+\centering
+\includegraphics[width=80mm]{picDCS/vdriftptcorr_time.eps}
+\includegraphics[width=80mm]{picDCS/vdriftfdrift_time.eps}
+\caption{
+ Drift velocity corrected for T/P variation as function of time (upper plot).
+ In the lower plot the correction for time dependent offset is also applied ($x_{off}(t)$ in formula\ref {eq:KalmanTime})
+}
+\label{figVDriftCorrected}
+\end{figure}
+
+
+
+
+
+\section{ Precision of the correction}
+
+The precision of the drift velocity correction and gain correction is proportional
+to the precision of the pressure and temperature measurement and to the length of the time
+interval
+\begin{eqnarray}
+ \sigma^2_x=\sigma^2_{xoff}\Delta{t}+k^2_N\sigma^2_{P/T}
+\label{eq:sigmaX}
+\end{eqnarray}
+
+The typical relative resolution of the pressure and temperature measurement is on the level of 6.$10^{-5}$ and
+1.$10^{-5}$ respectively (see picture \ref{figDCSResol}). For cool gas the coefficient $k_N$ is close to one. The contribution of the P/T correction to the drift velocity uncertainty is on the level of 6.1.$10^{-5}$ (150 microns for the full drift length of 250 cm)
+
+The $\sigma_{xoff}$ from equation \ref{eq:sigmaX} was estimated from plot \ref{figVDriftCorrected} and is on the level of 0.001 in a four day period. This estimate was obtained for the period of largest change in the present data sample. Further investigations should be carried out for extended time periods.
+
+For the TPC drift velocity determination, the requiered relative resolution is on the level of $6.10^{-5}$.
+Entering the observed sigmas into equation (\ref{eq:sigmaX}) the minimal frequncy of the drift velocity updates were estimated (equation \ref{eq:driftUpdateTime}) to be about 1 hour.
+\begin{eqnarray}
+ \Delta{t}\le\frac{\sigma^2_x}{\sigma^2_{xoff}}\approx(\frac{6.10^{-5}}{0.001/4days})^2=0.05 day.
+\label{eq:driftUpdateTime}
+\end{eqnarray}
+
+
+\begin{figure}[t]
+\centering
+\includegraphics[width=80mm]{picDCS/deltaPoverP.eps}
+\includegraphics[width=80mm]{picDCS/deltaToverT.eps}
+\caption{
+The relative resolution of the pressure and temperature measurement.
+}
+\label{figDCSResol}
+\end{figure}
+
+
\section{ Alice TPC drift calibration using tracks}
-In the first aproximation there is a linear dependence of the z position on the drift time.
-In Alice TPC the expression on the A side and C side of the chambers have the same drift vlocity part $v_d$
+In the first approximation there is a linear dependence of the z position on the drift time.
+In the Alice TPC the expression on the A side and C side of the chambers have the same drift velocity part $v_d$
with opposite sign. The full drift length $z_0A$ and $z_0C$ are different. We suppose that
the $t_0$ offset given by trigger arrival time is the same. In reality the $t_0$ equalization is applied before,
-using the pad-by-pad calibratiom pulser measurement.
+using the pad-by-pad calibration pulser measurement.
\begin{equation}
\begin{split}
z_A = z_{0A}-v_d(t-t_0) \\
\end{split}
\end{equation}
-Than the actual z position is expressed using the starting z position mesurement $\tilde{z}_A$ reps. $\tilde{z}_C$
+Than the actual z position is expressed using the starting z position measurement $\tilde{z}_A$ reps. $\tilde{z}_C$
\begin{equation}
\begin{split}
z_A = \tilde{z}_A-\frac{\Delta v_d}{v_d}(z_{0A}-\tilde{z}_{A})+\Delta t_0 \tilde{v_d}\\= \tilde{z_{A}} -v_c(z_{0A}-\tilde{z}_{A})+\Delta{z}\\
\end{split}
\end{equation}
-Combining the z masurement the track parameters can be fitted. Let's assume linear track model:
+Combining the z measurement the track parameters can be fitted. Let's assume linear track model:
\begin{equation}
\begin{split}
\tilde{z}_A =\tilde{a}_A+\tilde{b}_Ax \\
The inclination angle correction is the same on the A and C side.
-Tracks crossing the central electrode, respectivally primary tracks can be used to
-monitor correction coeficients $\Delta{z}$ and $v_c$. For tracks crossing the central electorde the a and b parameters at the crossing point fitted form A and C side are the same. In case of primary tracks, the z position at r-$\phi$ DCA are also the same:
+Tracks crossing the central electrode, respectively primary tracks can be used to
+monitor correction coefficients $\Delta{z}$ and $v_c$. For tracks crossing the central electrode the a and b parameters at the crossing point fitted form A and C side are the same. In case of primary tracks, the z position at r-$\phi$ DCA are also the same:
\begin{equation}
\begin{split}
a_A-a_C=0 \\
-Combining infomation from A and C side the correction parameters, drift correction $v_c$ and offset correction $\Delta{z}$ can be fitted.
+Combining information from A and C side the correction parameters, drift correction $v_c$ and offset correction $\Delta{z}$ can be fitted.
In case track crossed the central electrode the track parameters of the same track on A side and C side can be fitted.
The actual track parameters $a_A$ and $a_C$ respectivally $b_A$ and $b_C$ are the same.
git://git.uio.no / u / mrichter / AliRoot.git / commitdiff