ja garzon tim trackfor trasgos

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

timtrack

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Proyecto

A Tracking Algorithm forTRASGOS

timtrack

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

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

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

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)

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

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)

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

T0

Vz

y

xL

z=0Y0

X0

X’

Y’

z

V

Saeta

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

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)

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

About timtrack

1st. Step

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

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

y

x

z=0

z

z=zi

TimesExample Strip-like detector

X-type plane

T T’

0

0

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

T0

Y’

y

x

z=0Y0

X0

X’

z

V

z=zi

Times

X-type plane

T’T

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

T0

Y’

y

x

z=0Y0

X0

X’

z

z=zi

Times

X-type plane

V

T’

T

Vz

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

T0

Y’

y

x

z=0Y0

X0

X’

z

V

z=zi

TiT’i

Times

X-type plane

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

T0

Y’

y

x

z=0Y0

X0

X’

z

V

z=zi

Coordinates

Xi

X-type plane

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

T0

Y’

y

x

z=0Y0

X0

X’

z

V

z=zi

Yi

Ti

T’i

Y-type plane

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

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!

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

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

T0

Y’y

x

Y0

X0

X’V

n planes

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

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

K = X or Y

K = Y or X

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

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

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

About timtrack2nd. Step- That can be written in a more compact way:

where:Saeta

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

About timtrack

K (configuration Matrix): depend on the detector layout

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

About timtrack

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

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

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

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

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

where:

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

About timtrackError analysis- The error matrix is

- Incertitudes can be easily calculated from the K matrix elements

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

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)

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

About timtrack

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

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

About timtrackStrip-like detectors with any shape:

x

ymin YBack y

(X,Y)XBack

XFront

y

x

vs2

vs1

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

About timtrackStrip-like detectors with any shape:

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

About timtrackStrip-like detectors with any shape:

where:

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

About timtrackPads or pixel detectors :

Y0

X0

zi

z

Y

X

z=0

Xi

Yi

∆Xi

∆Yi

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

About timtrackPads or pixel detectors :

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

About timtrackPads or pixel detectors :

where:

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

About timtrack

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

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

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:

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

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

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

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

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

About timtrack

Remember:

ii

ii

sc

ϕϕ

sincos

==

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

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

Again:

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

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

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

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

The solution of is (Cramer rules):

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

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)

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

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

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

About timtrackA typical example: 2 parallel scintillators: different properties

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

About timtrackA typical example: 2 parallel scintillators: identical properties

Juan A. Garzón: 2nd. Workshop on the Trasgo Project. S. Compostela Feb. 5th-6th. 2010

The END

Thanks!