Jag Tim Track Gsi 20 Nov09 Short

timtrack

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Proyecto

A Tracking Algorithm forTRASGOS

timtrack

Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto

About the TRASGO conceptA TRASGO

(TRAck reconStructinG mOdule)

is a detector able to work stand-alone offering full capabilities of timing and tracking of charged particles

DAQ ElectronicsNetwork

Power supplies


About SAETASA 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 trackIn 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)


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)


T0

Vz

y

xL

z=0Y0

X0

X’

Y’

z

V

Saeta


About timtrack

TimTrack is the algorithm developed to estimate SAETAS1. 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)


About timtrack

1st. Step

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


y

x

z=0

z

z=zi

TimesExample Strip-like detector

X-type plane

T T’


0

0

T0

Y’

y

x

z=0Y0

X0

X’

z

V

z=zi

Times

X-type plane

T’T


T0

Y’


y

x

z=0Y0

X0

X’

z

z=zi

Times

X-type plane

V

T’

T

Vz

T0

Y’


y

x

z=0Y0

X0

X’

z

V

z=zi

TiT’i

Times

X-type plane

T0

Y’

y

x

z=0Y0

X0

X’

z

V

z=zi

Coordinates

Xi

X-type plane


T0

Y’

y

x

z=0Y0

X0

X’

z

V

z=zi


Yi

Ti

T’i

Y-type plane

About timtrack1st. 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!


About timtrack2nd. Step- To build the function S to be minimized


T0

Y’y

x

Y0

X0

X’V

n planes

About timtrack2nd. Step- S is a sum over n planes:

K = X or Y

K = Y or X


About timtrack2nd. Step- The expansion of the S function is:


About timtrack


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

where:Saeta

About timtrack

K (configuration Matrix): depend on the detector layout


About timtrack

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


About timtrack3rd. 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


About timtrack3rd. Step- Set of solutions (is just the Cramer rule):

where:


About timtrackError analysis- The error matrix is

- Incertitudes can be easily calculated from the K matrix elements


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)


About timtrack

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


About timtrackStrip-like detectors with any shape:

x

ymin YBack y

(X,Y)XBack

XFront

y

x

vs2

vs1





where:


About timtrackPads or pixel detectors :

Y0

X0

zi

z

Y

X

z=0

Xi

Yi

∆Xi

∆Yi





where:


About timtrack

Other strip-like detector layouts (with different K-matrix structure)


About timtrackOther strip-like detector layouts (with different K-matrix structure)

y

x L

z=0

’

z

V

Ki

New transverse coordinates defined by an angle φ:



About timtrack

y

XBack

φYBack YFront

XFront

x

Kim

K=0

Kip

(Xp,Yp)K

+

-vs sinφ

y

XB

φ

vs

YF YB

XF

xTi’

Ti

X

Y

vs cosφ

Ki

-

K=0

K



About timtrack

Remember:


ii

ii

sc

ϕϕ

sincos

==


Again:





The solution of is (Cramer rules):


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)


A typical example2 parallel scintillators

About timtrack

vs2

vs1

z2z1

L1

➱

➱

T’1

T1 T2

T’2

z

(Yo,Y’,V,T0)➱

➱

y

L2

svT

=τ


About timtrackA typical example: 2 parallel scintillators: different properties


About timtrackA typical example: 2 parallel scintillators: identical properties


About timtrack

Drift Chambers


About timtrack

Drift Chambers

y

x

z=0Y0

X0

X’

Y’

z

T0

V

s

dh


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:

- A time measured by a TDC:

12cellwidth

K =σ

resolutionTDCT =σ


About timtrack

Drift Chambers

The time measured by a DC has 3 components:

svf

vd

VsT

d++=

1.Time of flight of the particle from z=0 to z=zplane

2.Time of drift of the electrons3.Time of the signal to the wire end


vs

θi

Xi→

h→ u

d

Zi

(Xo,Yo)

(Xi,0, Zi)

y

x

(Xp,Yp)

(Xq,Yq)

V

vd

s f

Ti

To

f0

d0s0

About timtrackSome definitions:


Y

Z

X

β

αθ

Rotation θ, around z

z=0 z=Zi

Yiy

Y0

Y’ Y’i

∆Y

Particle

d

wire

V

s

About timtrack

Drift Chambers


(Approach without slope correction)


About timtrack

Drift Chambers


(Approach with slope correction)


About timtrackDrift Chambers

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

)vf

vd

Vs(

sd++


Drift Chambers

About timtrack

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



Cut








About timtrack3rd. Step- Set of solutions (is just the Cramer rule):



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

0.101

1.005

0.0995

1287

1.18

X’ 0.1 0.098 0.1000 0.101 0.101

Y0 1.(mm) 0.998 1.000 1.005 1.005

Y’ 0.1 0.100 0.0998 0.0995 0.0995

T0 0.(ps) 3592 1555 1307 1280

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

timtrack: Simulation of a MDC track calculated with Mathlab

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 timtrackComments 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. MDCsand RPCsWall, MDCs and RICH….


The END

Thanks!


Technology

Jag Tim Track Gsi 20 Nov09 Short