1. Proyecto
timtrack
timtrack
timtrack
timtrack
timtrack
timtrack timtrack
A Tracking Algorithm for
timtrack TRASGOS
timtrack
timtrack
timtrack
timtrack
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
2. 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
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
3. 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)
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
4. 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)
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
5. L
x
z
Saeta
X’ V
X0 T0
Vz
z=0 Y’
Y0 y
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
6. 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)
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
7. About timtrack
1st. Step
- To define the model, giving the cuantities that are measured
as function of the parameters of the saeta
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
8. x Times
Example Strip-like detector
X-type plane
z
T T’
z=zi
z=0
y
0
0
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
9. x Times
X-type plane
z
T
V T’
X’
X0 T0
z=zi
Y’
z=0
Y0 y
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
10. x Times
X-type plane
T z
T’
X’
V
X0 T0
z=zi
Vz Y’
z=0
Y0 y
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
11. x Times
X-type plane
Ti z
T’i
V
X’
X0 T0
z=zi
Y’
z=0
Y0 y
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
12. x Coordinates
X-type plane
z
Xi V
X’
X0 T0
z=zi
Y’
z=0
Y0 y
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
13. x
Ti
Y-type plane
z
V
X’
X0 T0
z=zi
z=0
Y’ T’i
Y0 Yi y
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
14. 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!
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
15. About timtrack
2nd. Step
- To build the function S to be minimized
x
V
X’ n planes
X0 T0
Y’
Y0 y Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
16. About timtrack
2nd. Step
- S is a sum over n planes:
K = X or Y
K = Y or X
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
17. About timtrack
2nd. Step
- The expansion of the S function is:
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
18. About timtrack
2nd. Step
- That can be written in a more compact way:
where:
Saeta
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
19. About timtrack
K (configuration Matrix): depend on the detector layout
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
20. About timtrack
a (vector of reduced data): depend on the data
(They are just weighted sums and differences of the measurements)
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
21. 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
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
22. About timtrack
3rd. Step
- Set of solutions (is just the Cramer rule):
where:
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
23. About timtrack
Error analysis
- The error matrix is
- Incertitudes can be easily calculated from the K matrix elements
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
24. 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)
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
25. About timtrack
Other strip-like detector layouts
(with the same K-matrix structure)
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
26. About timtrack
Strip-like detectors with any shape:
x x
XBack
(X,Y) vs2
vs1
ymin YBack y
y
XFront
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
28. About timtrack
Strip-like detectors with any shape:
where:
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
29. About timtrack
Pads or pixel detectors :
∆Yi
X
∆Xi
Xi
X0 z
zi
z=0
Y0 Yi Y
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
30. About timtrack
Pads or pixel detectors :
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
31. About timtrack
Pads or pixel detectors :
where:
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
32. About timtrack
Other strip-like detector layouts
(with different K-matrix structure)
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
33. About timtrack
Other strip-like detector layouts
(with different K-matrix structure)
L
x
Ki
z
’ V
z=0
y
New transverse coordinates defined by an angle φ:
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
34. 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
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
35. About timtrack
Other strip-like detector layouts
(with different K-matrix structure)
Remember:
ci = cos ϕi
si = sin ϕi
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
36. About timtrack
Other strip-like detector layouts
(with different K-matrix structure)
Again:
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
37. About timtrack
Other strip-like detector layouts
(with different K-matrix structure)
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
38. About timtrack
Other strip-like detector layouts
(with different K-matrix structure)
The solution of is (Cramer rules):
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
39. 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)
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
40. 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
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
41. About timtrack
A typical example: 2 parallel scintillators: different properties
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
42. About timtrack
A typical example: 2 parallel scintillators: identical properties
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
43. About timtrack
Drift Chambers
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
44. About timtrack
Drift Chambers
x
z
d
h
s
X’ V
X0
T0 Y’
z=0
Y0 y
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
45. 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
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
46. 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
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
47. 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
48. About timtrack
Some definitions:
X
θ
α
β Z
Y
Rotation θ, around z
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
49. Particle
d
y Yi
s wire
∆Y
V
Y’ Y’i
Y0
z=0 z=Zi
50. About timtrack
Drift Chambers
1.Time of flight of the particle from z=0 to z=zplane
(Approach without slope correction)
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
51. About timtrack
Drift Chambers
1.Time of flight of the particle from z=0 to z=zplane
(Approach with slope correction)
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
52. About timtrack
Drift Chambers
2nd. Step
- S is a sum over n planes:
s d f
( + + )
V vd vs
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
53. 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
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
54. About timtrack
Drift Chambers
Cut
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
58. About timtrack
3rd. Step
- Set of solutions (is just the Cramer rule):
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
59. 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
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
61. 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….
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009
62. The END
Thanks!
Proyecto
Juan A. Garzón. timtrack: A tracking algorithm for trasgos. GSI 20.11 2009