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A track fitting method for multiple
scattering
Peter Kvasnička,
5th SILC meeting, Prague 2007
Introduction


This talk is about a track fitting method that explicitly
takes into account multiple scattering.
IT IS NOT NEW: It was invented long time ago and,
apparently re-invented several times.
–
–
–
–

Helmut Eichinger & Manfred Regler, 1981
Gerhard Lutz 1989
Volker Blobel 2006
A.F.Zarniecki, 2007 (EUDET report)
… as I learned after I re-invented it myself.
Peter Kvasnicka, 5th SILC
meeting, Prague 2007
Motivation is obvious

Multiple scattering is a notorious complication and is
particularly serious
–
–
–
For low-energy particles
For very precise detectors
For systems of many detectors
Peter Kvasnicka, 5th SILC
meeting, Prague 2007
Motivation (continued)

Remedies:
–
–
–
–

The Kalman filter is the best known method able to account for
multiple scattering, yet it is used relatively little:
–
–

Thinner detectors
Higher energies
Extrapolation of fits to infinite energies
Better methods of fitting (Kalman filter)
High cost / benefit ratio
The Kalman filter requires just the parameters that one would want
to compute (detector resolutions), and does not offer an affordable
way of computing them.
I will introduce a different method, which is simpler and
behaves similarly to the Kalman filter in many repsects.
Peter Kvasnicka, 5th SILC
meeting, Prague 2007
Outline





Fitting tracks with lines
Probabilistic model of a particle track
Scattering algebra and statistics
Some results
Conclusions
Peter Kvasnicka, 5th SILC
meeting, Prague 2007
Fitting tracks with lines
xi  azi  b   i
X  F
ˆ  ( F T | F ) 1 FX
 i  N [0, d i2 ] cov(  i ,  j )   ij d i2
To keep things simple, I only
consider a 2D situation
 x1 
 1 z1 

 

 b
 x2 
 1 z2 
X   F 
   

...
.. ...
a



 


x
1
z
N 
 N

Peter Kvasnicka, 5th SILC
meeting, Prague 2007
Fitting tracks with lines (cont’d)
cov ˆ   T  ( F T F ) 1 s02
Xˆ  Fˆ
cov Xˆ  ( Xˆ   X  )( Xˆ   X  )T 
(1  F ( F T F ) 1 F T )  (1  F ( F T F ) 1 F T ) cov 
1
s 
( X  Xˆ )( X  Xˆ )T
N 2
2
0
Peter Kvasnicka, 5th SILC
meeting, Prague 2007
Fitting tracks with lines (cont’d)


Information on detector resolutions can be
used to weight the fit
Multiple scattering violates the assumption of
independence of regression residuals – the
covariance matrix is no longer diagonal.
Therefore, direct calculation of detector
resolutions is not possible.
Peter Kvasnicka, 5th SILC
meeting, Prague 2007
Behind the lines

As a rule, information on multiple scattering
is at hand:
–
–
–
Formulas describing the distribution of scattering
angles are well-known
Simulations (GEANT) are commonplace in
particle experiments
Last but not least, we can try to intelligently use
the data to estimate scattering distributions
Peter Kvasnicka, 5th SILC
meeting, Prague 2007
Outline





Fitting tracks by line
Probabilistic model of a particle track
Scattering algebra and statistics
Some results
Conclusions
Peter Kvasnicka, 5th SILC
meeting, Prague 2007
Probabilistic model of a particle track
z0
z1
φ0
x  x0
We use paraxial
approximation
tg φ ~ φ
z2
z3
z4
φ3
φ1
φ2
x  x0  ( z1  z0 )0
x  x0  ( z2  z0 )0  ( z2  z1 )1
x  x0  ( z3  z0 )0  ( z3  z1 )1  ( z3  z2 )2
cov(i , j )   ij i2
…
and the distribution of φ’s is Moliere, i.e., approximately Gaussian
Peter Kvasnicka, 5th SILC
meeting, Prague 2007
Probabilistic model of a particle track

To put this to some use, we set up the task as
follows:
We have a system of n scatterers and N particle tracks.
k of the scatterers are in fact detectors that provide us with
information about the impact point xi of a particle, alas with
errors di.
The task is to estimate impact points on the scatterers (some,
or all).
Peter Kvasnicka, 5th SILC
meeting, Prague 2007
Outline





Fitting tracks by line
Probabilistic model of a particle track
Scattering algebra and statistics
Some results
Conclusions
Peter Kvasnicka, 5th SILC
meeting, Prague 2007
Scattering algebra and statistics

We have to reconstruct impact points on scatterers, given
n
by
xi  x0   ( zi  z j )(zi  z j ) j
j 1
i  1,2,...n
0, z  0
( z )  
1 z  0

Our observables are
n
 I  x0   ( z I  z j )(z I  z j ) j  d I
j 1
I  1,2,...k
cov( d I , d J )   IJ 2I
Peter Kvasnicka, 5th SILC
meeting, Prague 2007
Scattering algebra and statistics

In matrix form, this can be written as
X  AX    A 
 x1 
 .1 
 
 
 x2 
 
X     2
...
...
 
 
 K 
 xn 
0
1

 1 z1  z0
AX  
...
...

1 z z
n
0

0
1

 1 z1  z0
A  
...
...

1 z z
K
0

0
...
0
...
...
...
zn  z1 ...
0
...
0
...
...
...
zK  z1 ...
 x0 
 
 1 
 
 2
 ... 
 

 n
 d1 
 
 ... 
d 
 n


0

... 

zn  zn 1
0 

1

1

... 

zK  z K 1
1 
0
Both the hidden parameters and
observables are expressed as
products of some matrices with the
(approximately jointly Gaussian)
vector of random variables.
Note that Aξ are selected rows from
AX, with 1’s added for measurement
errors.
We want to estimate X based on ξ.
Peter Kvasnicka, 5th SILC
meeting, Prague 2007
Scattering algebra and statistics
We estimate rows of Aξ as the best linear combinations of rows
of AX, that is, we seek a matrix T such that

Tr( A

Tr ( X  T )( X  T )T 
X

 TA )T ( AX  TA )T  min
T
The solution is
T T  ( A AT )1 A ATX
with   T
Covariance of weights:
cov T  TT T  ATX AX 1  AT ( A AT )1 A 
Peter Kvasnicka, 5th SILC
meeting, Prague 2007
Outline





Fitting tracks by line
Probabilistic model of a particle track
Scattering algebra and statistics
Some results
Conclusions
Peter Kvasnicka, 5th SILC
meeting, Prague 2007
Comparison with line fit
– DEPFET simulation






Zbynek will say more
on simulations in his
talk
Data: GEANT4 simulations of DEPFET detectors (Zbynek
Drasal)
5 identical detectors with identical distances of 36, 45, or 120
mm, the middle DEPFET is the DUT
Detector resolution simulated by Gaussian randomization of
impact points (sigma 0.5, 1.2 a 3 micron)
Particles: 80, 140, and 250 GeV pions
Fit without the use of DUT data
RMS residuals plotted against
scattering parameter =
(RMS scattering angle)*(jtypical distance between detectors) /
detector resolution
Peter Kvasnicka, 5th SILC
meeting, Prague 2007
Comparison - DEPFET (cont’d)
* Line fit
* Kinked fit
Peter Kvasnicka, 5th SILC
meeting, Prague 2007
Comparison: SILC TB simulation



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Same type of simulation by Zbynek, two geometries:
The one we use in October (3 telescopes 32 and 8
cm apart, DUT (CMS) 1 meter behind
The one planned for the June testbeam, with DUT in
between the more distant telescopes.
Beam energies 1, 2, … 6 GeV
Resolutions 1.5 um (tels) and 9 um (DUT)
Scattering parameter defined from the point of view
of the DUT
Peter Kvasnicka, 5th SILC
meeting, Prague 2007
Comparison – SILC TB simulation
* Line fit
* Kinked fit
Peter Kvasnicka, 5th SILC
meeting, Prague 2007
Outline





Fitting tracks by line
Probabilistic model of a particle track
Scattering algebra and statistics
Some results
Conclusions
Peter Kvasnicka, 5th SILC
meeting, Prague 2007
Conclusions





The method switches between linear regression and
interpolation between points – similarly to the Kalman filter
The method is useful in the moderate scattering regime – at low
scattering, line fits give basically the same results, and at high
scattering, there is little help. So use line fits where applicable.
We can plug in experimental uncertainties of impact point
measurements (e.g., from eta-correction)
We can also estimate other combinations of parameters, for
example, the scattering angles or detector resolutions
themselves. Nice, but so far not too useful…
Alignment: Work in progress
Peter Kvasnicka, 5th SILC
meeting, Prague 2007