Jag Tim Track Gsi 20 Nov09 Short
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Transcript of Jag Tim Track Gsi 20 Nov09 Short
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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
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
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. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
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. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
T0
Vz
y
xL
z=0Y0
X0
X’
Y’
z
V
Saeta
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
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. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
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. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
y
x
z=0
z
z=zi
TimesExample Strip-like detector
X-type plane
T T’
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
0
0
T0
Y’
y
x
z=0Y0
X0
X’
z
V
z=zi
Times
X-type plane
T’T
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
T0
Y’
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
y
x
z=0Y0
X0
X’
z
z=zi
Times
X-type plane
V
T’
T
Vz
T0
Y’
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
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
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
T0
Y’
y
x
z=0Y0
X0
X’
z
V
z=zi
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
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!
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrack2nd. Step- To build the function S to be minimized
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
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
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrack2nd. Step- The expansion of the S function is:
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrack
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
2nd. Step- That can be written in a more compact way:
where:Saeta
About timtrack
K (configuration Matrix): depend on the detector layout
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrack
a (vector of reduced data): depend on the data(They are just weighted sums and differences of the measurements)
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
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. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrack3rd. Step- Set of solutions (is just the Cramer rule):
where:
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrackError analysis- The error matrix is
- Incertitudes can be easily calculated from the K matrix elements
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
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. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrack
Other strip-like detector layouts (with the same K-matrix structure)
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrackStrip-like detectors with any shape:
x
ymin YBack y
(X,Y)XBack
XFront
y
x
vs2
vs1
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrackStrip-like detectors with any shape:
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrackStrip-like detectors with any shape:
where:
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrackPads or pixel detectors :
Y0
X0
zi
z
Y
X
z=0
Xi
Yi
∆Xi
∆Yi
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrackPads or pixel detectors :
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrackPads or pixel detectors :
where:
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrack
Other strip-like detector layouts (with different K-matrix structure)
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
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. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
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. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
Other strip-like detector layouts (with different K-matrix structure)
About timtrack
Remember:
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
ii
ii
sc
ϕϕ
sincos
==
About timtrackOther strip-like detector layouts (with different K-matrix structure)
Again:
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrackOther strip-like detector layouts (with different K-matrix structure)
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrackOther strip-like detector layouts (with different K-matrix structure)
The solution of is (Cramer rules):
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
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. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
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. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrackA typical example: 2 parallel scintillators: different properties
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrackA typical example: 2 parallel scintillators: identical properties
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrack
Drift Chambers
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrack
Drift Chambers
y
x
z=0Y0
X0
X’
Y’
z
T0
V
s
dh
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
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
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
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
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
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:
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
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
1.Time of flight of the particle from z=0 to z=zplane
(Approach without slope correction)
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrack
Drift Chambers
1.Time of flight of the particle from z=0 to z=zplane
(Approach with slope correction)
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrackDrift Chambers
2nd. Step- S is a sum over n planes:
)vf
vd
Vs(
sd++
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
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
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrackDrift Chambers
Cut
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrackDrift Chambers
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrackDrift Chambers
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrackDrift Chambers
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
About timtrack3rd. Step- Set of solutions (is just the Cramer rule):
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
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….
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto
The END
Thanks!
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009 Proyecto