Jag Tim Track Gsi 20 Nov09 Short

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timtrack
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A Tracking Algorithm for
timtrack TRASGOS
timtrack
timtrack
timtrack
timtrack
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Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009

About the TRASGO concept
A TRASGO
(TRAck reconStructinG mOdule)
is a detector able to work stand-alone offering full capabilities of timing and
tracking of charged particles

DAQ Electronics
Network

Power supplies

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About SAETAS
A SAETA (SmAllest sEt of daTA) is the basic unit of information
in the timtrack algorithm and in the TRASGOs concept

A SAETA contains 6 parameters defining a charged particle track
In a cartesian coordinate system:
- X0 and Y0: 2 coordinates at a reference plane
- X’ and Y’ : 2 projected slopes in planes x-z and y-z
- T0 : The time at the reference plane respect a reference time
- V : The velocity

Saeta: s = (X0,X’,Y0,Y’,T0,V)

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About SAETAS

From the mathematical point of view will be better to use:

Saeta: s = (X0,X’,Y0,Y’,T0,1/Vz)

where:

V = Vz · Sqrt(1+X’2+Y’2)

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L
x

z

Saeta
X’ V

X0 T0

Vz
z=0 Y’
Y0 y

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About timtrack

TimTrack is the algorithm developed to estimate SAETAS
1. It is based on a Least Squares Method (LSM)
2. It works directly with the primary data provided by detectors:
- Coordinates:
- Times: it is assumed that:
all times are refered to a common t=0
(all detector are WELL synchronized)
3. It lets free the six elements of a saeta:
(X0, X’, Y0, Y’, T0 and 1/Vz)

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About timtrack

1st. Step
- To define the model, giving the cuantities that are measured
as function of the parameters of the saeta

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x Times
Example Strip-like detector

X-type plane
z

T T’
z=zi

z=0
y

0

0
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x Times

X-type plane
z
T
V T’
X’
X0 T0
z=zi
Y’
z=0
Y0 y

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x Times

X-type plane
T z

T’
X’
V
X0 T0
z=zi
Vz Y’
z=0
Y0 y

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x Times

X-type plane
Ti z
T’i
V
X’
X0 T0
z=zi
Y’
z=0
Y0 y

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x Coordinates

X-type plane
z
Xi V
X’
X0 T0
z=zi
Y’
z=0
Y0 y

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x
Ti

Y-type plane
z

V
X’
X0 T0
z=zi

z=0
Y’ T’i
Y0 Yi y

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About timtrack
1st. Step

- To define the model giving the cuantities to be measured as
function of the parameters of the saeta
Either

or

3 equations (conditions) per plane!
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About timtrack
2nd. Step
- To build the function S to be minimized

x

V
X’ n planes
X0 T0
Y’
Y0 y Proyecto


About timtrack
2nd. Step
- S is a sum over n planes:

K = X or Y

K = Y or X

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About timtrack
2nd. Step
- The expansion of the S function is:

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About timtrack
2nd. Step
- That can be written in a more compact way:

where:
Saeta

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About timtrack
K (configuration Matrix): depend on the detector layout

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About timtrack

a (vector of reduced data): depend on the data
(They are just weighted sums and differences of the measurements)

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About timtrack
3rd. Step
- To apply to LSM method.

From:

leads to:

As K is definite positive, K has an inverse and:

This equation provides the saeta directly from the data

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About timtrack
3rd. Step
- Set of solutions (is just the Cramer rule):

where:

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About timtrack
Error analysis
- The error matrix is

- Incertitudes can be easily calculated from the K matrix elements

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About timtrack

Comments
- The method can be easily extended when there are correlations
between some of the measurements (e.G.: time readouts)
- Only two planes of strip-like detectors are enough to provide
unambiguously the 6 parameters of a saeta
- The solution has a matrix form: It’s very easy and fast of
implementing on computers

-There are many detector layouts with a K matrix having the
same structure (see next examples)

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About timtrack
Other strip-like detector layouts
(with the same K-matrix structure)

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About timtrack
Strip-like detectors with any shape:

x x

XBack
(X,Y) vs2
vs1

ymin YBack y
y
XFront

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About timtrack

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About timtrack

where:

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About timtrack
Pads or pixel detectors :
∆Yi

X

∆Xi

Xi

X0 z

zi

z=0
Y0 Yi Y

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About timtrack

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About timtrack

where:

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About timtrack

(with different K-matrix structure)

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About timtrack

L
x
Ki

z

’ V

z=0
y

New transverse coordinates defined by an angle φ:

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About timtrack
Other strip-like detector layouts (with different K-matrix structure)
K x
XBack x
XB Ti’

K
(Xp,Yp) -vs sinφ vs
Kim
X
vs cosφ
Kip
φ
φ
YBack + YFront YF - YB
y Y y
Ki
XF
XFront Ti

K=0 K=0

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About timtrack

Remember:
ci = cos ϕi
si = sin ϕi
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About timtrack
Again:

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About timtrack

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About timtrack

The solution of is (Cramer rules):

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About timtrack

Comments
- The “problem” of the method is that there is an inversion of a
matrix. Sometimes it may give problems (when the matrix is not well
conditioned) but there are a lot of numerical methods to do it

(And it has to be done only once)

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About timtrack
A typical example
2 parallel scintillators
T’2
T’1

y

➱
vs1

➱

➱
vs2
(Yo,Y’,V,T0)

➱
z1 z2 z

L1 T
T1
L2 τ=
T2 vs

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About timtrack
A typical example: 2 parallel scintillators: different properties

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About timtrack
A typical example: 2 parallel scintillators: identical properties

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About timtrack

Drift Chambers

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About timtrack
Drift Chambers

x

z
d
h
s

X’ V
X0
T0 Y’
z=0
Y0 y

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About timtrack
Drift Chambers
1 Step. To build the model:
In a typical Drift Chamber each layer provides two
data:
- A coordinate: given by the cell width and orientation:
cellwidth
σK =
12

- A time measured by a TDC:
σ = TDC resolution
T

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About timtrack
Drift Chambers
The time measured by a DC has 3 components:

s d f
T= + +
V vd vs

1.Time of flight of the particle from z=0 to z=zplane
2.Time of drift of the electrons
3.Time of the signal to the wire end

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x
(Xi,0, Zi) θi
→
h u (Xp,Yp)

(Xq,Yq)
s0 d d0
→ vd
Xi
s f f0
vs
V
(Xo,Yo) Ti

To
Zi

y

About timtrack
Some definitions:
X

θ
α

β Z

Y

Rotation θ, around z

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Particle

d
y Yi
s wire

∆Y

V
Y’ Y’i

Y0

z=0 z=Zi

About timtrack

Drift Chambers
(Approach without slope correction)

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About timtrack

Drift Chambers
(Approach with slope correction)

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About timtrack
Drift Chambers
2nd. Step
- S is a sum over n planes:

s d f
( + + )
V vd vs

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About timtrack
Drift Chambers

Now, the model is not linear, and the saeta has to be
found iteratively

• Calculate a Saeta
• Substitute X’ and Y’ in the formulae
• Calculate the Saeta with corrected coefficients

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About timtrack
Drift Chambers

Cut

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About timtrack
Drift Chambers

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About timtrack
3rd. Step
- Set of solutions (is just the Cramer rule):

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timtrack: Simulation of a MDC track calculated with Mathlab

Params Generated 1. fit 1. Sl.Cor 2.Sl.Cor 3.Sl.Cor 4.Sl.Cor

X0 0.(mm) -0.07 0.06 0.06 0.06 0.06

X’ 0.1 0.098 0.1000 0.101 0.101 0.101

Y0 1.(mm) 0.998 1.000 1.005 1.005 1.005

Y’ 0.1 0.100 0.0998 0.0995 0.0995 0.0995

T0 0.(ps) 3592 1555 1307 1280 1287

1/Vz 3.3 (=c). 3.74 -3.69 1.12 1.19 1.18

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Variante-Covariance Matrix (alter 1st. Slope correction)

[0.00046, 1.7e-22, -3.8e-21, -1.32e-06, 0.399, 6.9e-19;]

[-1.2e-22, 5.05e-08, 4.8e-07, 2.5e-24, -2.3e-18, -0.0005;]

[3.16e-21, 4.86e-07, 0.00013, -1.07e-22, 1.95e-16, -0.025;]

[-1.32e-06, 2.3e-24, 1.03e-22, 2.8e-08, -0.03, -7.5e-20;]

[0.399 ,-2.95e-18, -3.47e-17, -0.03, 76162,7. 2e-14;]

[2.3e-18, -0.0005, -0.0259, -4.31e-20, 2.97e-14, 14.99;]

About timtrack
Comments and Summary
- timtrack seems to offer a promising alternative for the tracking of charge
particles in Drift Chambers
- It needs only 3 layers to define a saeta (6 parameters) candidate
- It works in the coordinate-times space making hit finding quite easy: once
several layers define a candidate it is easy to extrapolate the candidate to another
layer and to look for a signal in a given time window
- Putting constraints in the model is very easy; for instance: vertex condition (it
reduces the minimum number of planes to 2)
- Time and velocity have big incertitudes but they are highly correlated with
other parameters
- With fixed time and velocity, a reduced saeta (4 params.) can be built every
two planes allowing to analyze magnetic fields effect
- With timtrack joined fit with several detectors families is possible. E.g. MDCs
and RPCsWall, MDCs and RICH….

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The END

Thanks!

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Jag Tim Track Gsi 20 Nov09 Short

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