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


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 timtrack2nd. 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


The END

Thanks!

ja garzon tim trackfor trasgos

Documents

trasgo project

reference plane x

zitititimesxtype planejuan

ziyititiytype planejuan

zitimesxtype planevttvzjuan

zitimesxtype planettjuan

function s

planes xz