Updated document - description of the time dependent correction (Marian)

diff --git a/TPC/doc/calib/DriftV/tpcDrift.tex b/TPC/doc/calib/DriftV/tpcDrift.tex
index 1393f1887f6c446ab9ff6569701a4177282536ed..6fa8eaba426d61b7de348ac724a572ff99f3f5dc 100644 (file)
--- a/TPC/doc/calib/DriftV/tpcDrift.tex
+++ b/TPC/doc/calib/DriftV/tpcDrift.tex
@@ -3,13 +3,50 @@
 \begin{document}
 
 
+\section{Drift velocity}
+
+The TPC drift velocity is changing in time together with the change of environment variables
+\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.
+\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.
+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)} 
+\end{equation}
+\begin{equation}
+\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 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 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
+\end{itemize}
+
+The unknown parameters $k_t(t)$ and $k_N$ can be than fitted using the Kalman filter.
+
+
+
+
+
+
 \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 $v_d$ 
 with opposite sign. The full drift length $z_0A$ and $z_0C$ are different. We suppose that
-the $t_0$ offset givven by trigger arrival time is the same. In reality the $t_0$ equalization is applied before,
+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.  
 \begin{equation}
 \begin{split}
@@ -78,7 +115,6 @@ a_A-a_C=0 \\
 \end{split}
 \end{equation}
 
-The measured difference