class AliTRDclusterResolution: public AliTRDrecoTask

 Constructor

 Constructor

 Destructor

 Build and post histo container.
 Actual population of the container with histo is done in function Histos.

 Steering function to retrieve performance plots

 Retrieve histograms array if already build or build it

 Fill container histograms

 Steer processing of various cluster resolution dependences :

   - process resolution dependency cluster charge
   if(HasProcess(kYRes)) ProcessCharge();
   - process resolution dependency on y displacement
   if(HasProcess(kYSys)) ProcessCenterPad();
   - process resolution dependency on drift legth and drift cell width
   if(HasProcess(kSigm)) ProcessSigma();
   - process systematic shift on drift legth and drift cell width
   if(HasProcess(kMean)) ProcessMean();

 Retrieve calibration parameters from OCDB, drift velocity and t0 for the detector region specified by
 a previous call to AliTRDclusterResolution::SetCalibrationRegion().

 Retrieve global chamber position from alignment
 a previous call to AliTRDclusterResolution::SetCalibrationRegion() is mandatory.

 Set calibration region in terms of detector and pad.
 By default detector 0 mean values are considered.

 Resolution as a function of cluster charge.

 As described in the function ProcessCenterPad() the error parameterization for clusters for phi = a_L can be
 written as:


 with the contribution in case of B=0 given by:


 which further can be simplified to:


 The results for s_y and f(q) are displayed below:

 The function has to extended to accomodate gain calibration scalling and errors.

 Author
 Alexandru Bercuci <A.Bercuci@gsi.de>

 Resolution as a function of y displacement from pad center and drift length.

 Since the error parameterization of cluster r-phi position can be written as (see AliTRDcluster::SetSigmaY2()):


 one can see that for phi = a_L one gets the following expression:


 where we have explicitely marked the remaining term in case of absence of magnetic field. Thus one can use the
 previous equation to estimate s_y for B=0 and than by comparing in magnetic field conditions one can get the s_x.
 This is a simplified method to determine the error parameterization for s_x and s_y as compared to the one
 implemented in ProcessSigma(). For more details on cluster error parameterization please see also
 AliTRDcluster::SetSigmaY2()

 The representation of dy=f(y_cen, x_drift| layer) can be also used to estimate the systematic shift in the r-phi
 coordinate resulting from imperfection in the cluster shape parameterization. From the expresion of the shift derived
 in ProcessMean() with phi=exb one gets:


 where all dependences are made explicit. This last expression can be used in two ways:
   - by average on the dz/dx we can determine directly dy (the method implemented here)
   - by plotting as a function of dzdx one can determine both dx and dy components in an independent method.

 Author
 Alexandru Bercuci <A.Bercuci@gsi.de>

 As the r-phi coordinate is the only one which is measured by the TRD detector we have to rely on it to
 estimate both the radial (x) and r-phi (y) errors. This method is based on the following assumptions.
 The measured error in the y direction is the sum of the intrinsic contribution of the r-phi measurement
 with the contribution of the radial measurement - because x is not a parameter of Alice track model (Kalman).


 In the general case


 where we have explicitely show the lorentz angle correction on y and the projection of radial component on the y
 direction through the track angle in the bending plane (phi). Also we have shown that the radial component in the
 last equation has twp terms, the drift and the misalignment (x_0). For ideal geometry or known misalignment one
 can solve the equation


 by fitting a straight line:


 the error parameterization will be given by:


 Below there is an example of such dependency.

 The error parameterization obtained by this method are implemented in the functions AliTRDcluster::GetSX() and
 AliTRDcluster::GetSYdrift(). For an independent method to determine s_y as a function of drift length check the
 function ProcessCenterPad(). One has to keep in mind that while this method return the mean s_y over the distance
 to pad center distribution the other method returns the *STANDARD* value at center=0 (maximum). To recover the
 standard value one has to solve the obvious equation:


 with "<s_y>" being the value calculated here and "sigma" the width of the s_y distribution calculated in
 ProcessCenterPad().

 Author
 Alexandru Bercuci <A.Bercuci@gsi.de>

 By this method the cluster shift in r-phi and radial directions can be estimated by comparing with the MC.
 The resolution of the cluster corrected for pad tilt with respect to MC in the r-phi (measuring) plane can be
 expressed by:


 where x_cl is the drift length attached to a cluster, y_cl is the r-phi coordinate of the cluster measured by
 charge sharing on adjacent pads and y_0 and z_0 are MC reference points (as example the track references at
 entrance/exit of a chamber). If we suppose that both r-phi (y) and radial (x) coordinate of the clusters are
 affected by errors we can write


 where the starred components are the corrected values. Thus by definition the following quantity


 has 0 average over all dependency. Using this decomposition we can write:


 which can be transformed to the following linear dependence:


 if expressed as function of dy/dx-h*dz/dx. Furtheremore this expression can be plotted for various clusters
 i.e. we can explicitely introduce the diffusion (x_cl) and drift cell - anisochronity (z_cl) dependences. From
 plotting this dependence and linear fitting it with:


 the systematic shifts will be given by:


 Below there is an example of such dependency.

 The occurance of the radial shift is due to the following conditions
   - the approximation of a constant drift velocity over the drift length (larger drift velocities close to
     cathode wire plane)
   - the superposition of charge tails in the amplification region (first clusters appear to be located at the
     anode wire)
   - the superposition of charge tails in the drift region (shift towards anode wire)
   - diffusion effects which convolute with the TRF thus enlarging it
   - approximate knowledge of the TRF (approximate measuring in test beam conditions)

 The occurance of the r-phi shift is due to the following conditions
   - approximate model for cluster shape (LUT)
   - rounding-up problems

 The numerical results for ideal simulations for the radial and r-phi shifts are displayed below and used
 for the cluster reconstruction (see the functions AliTRDcluster::GetXcorr() and AliTRDcluster::GetYcorr()).

 More details can be found in the presentation given during the TRD
 software meeting at the end of 2008 and beginning of year 2009, published on indico.cern.ch.

 Author
 Alexandru Bercuci <A.Bercuci@gsi.de>