Simulation:
particle trajectory     ->   intersection with layer's surface
                                 ->   physical response as closest approach distance wire-track and associated error
                                         for each hit

For the reconstruction we need the track parameters in the point of closest approach track-beam axis.

It is necessary to estimate the parameters that characterize the track.

This could be done in two different ways:
 

• From the point in space -> track and variance matrix  ->  evaluation in the point of closest approach track-beam axis  ->  smearing
• From the point in space -> Billoir fitter ->  value of the track parameters and variance matrix in the point of closest approach track-beam axis. We have to take into account the Multiple Scattering contribution in the silicon wafers.

• • • • • • • • • BILLOIR   FITTER

This parameters are:

    - K  (signed curvature)
    - phi (azimuth of the momentum)
    - d (distance of closest approach of the track to the z-axis)
    - t  (tangent of dipole angle)
    - z (the value of z at the point of closest approach of the track to the z axis)

We want to best-estimate this set of parameters trough some measurement along the track.
This is done adding the information stored in a hit one at time, that is every time the measurement is done.
 

The procedure we are going to describe is the so called "Billoir fitter" and consists in two central points:

                - updating  :  every measurement provides a hit (a set of ordered physical quantities) who
                                       gets added to the track
 
                - transport  :  the track parameters are given at a certain point in space and then expressed
                                      & nbsp; in another point
 
 

Suppose we have a hit, that is a set of physical quantities d₁^meas. With the information stored in that "array" we can estimate the set of parameters e and the variance matrix V that characterize the track.

Then we have a second hit in a different point of space.
This hit provides a different measurement of the same quantities d₂^meas.

From the first hit we can predict the value of the measured quantity as a function of the estimated parameters d₂(e) in that point of space in which the second measurement is done. So we can form a combined chi-squared which combines the old estimate and the new measurement.
 
 

We then minimize this variable and obtain a better estimate of the parameters e^opt and V_e^opt .
 

The operation of  transport is made in two step:

                - change of variables  :  the same track is parametrized in two different points in the space
                                      & nbsp;                 that correspond to the point in which the measurement is performed
 
                - update                       : the track parameters are modified taking into account effects effects
                                      & nbsp;                from interactions with material the particle passes trough.
                                      & nbsp;                The interactions include both Multiple Scattering and Energy Loss.
 
 

The measurement we have from the simulation are:

                -  distance of closest approach
                -  error associated to that distance
                -  cell
                -  position of the cell
                -  layer parameters ( r and phi )

We want to obtain from these the track parameters in the point of closest approach to the wire (this is needed from the reconstruction).

From the measurements we estimate the parameters and the variance matrix at a certain point in space.

Then adding hit's information one  at time with the Billoir technique we obtain a better estimate of these parameters:
 
           
            

To do this is necessary to explain the relation between the measurement and the parameters d(e)
and to know how to calculate the D =dd/de^t.
This is done in a local reference system for each wire. So the parameters are transported from each system to the next one.

T is called transport matrix and is simply: