M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Alignment of the VDSAlignment of the VDSM. Bräuer20.06.2001
Outline:• The problem VDS• Crucial:
• Precise alignment• Alignment parameters• Least squares as a solution• Toy systems to align• The full system • System behaviour
• Results
Coarse alignment
( )
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Teil I
HERA-B Einführung
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
HERA-B• Vorwärtsspektrometer mit festem Target• Hochraten – Experiment• Anspruchsvolles Triggersystem• ca. 650.000 Kanäle
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
HERA-B: Ansicht
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
HERA-B: Ansicht II
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
HERA-B: Ansicht III
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
The problem: VDS
2 m
Major Problems:
1. The bare size
2. No tracks into VDS existing
• Today: Matching without VDS track?
• Momentum?
• Magnet tracking?
3. Survey/setup: Needed: <200µm !
mm2..1r
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Teil II
Pre-Tracking Alignment
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Coarse alignment: Before tracking
• Position of pots wrt. to each other unknown
• Double modules within a pot not optimal adjusted
beam
1 SLu
s
=> Start without tracking !
Assume some module-positions to be known:
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Coarse alignment: 1st step
=> 4 ds modules adjusted => 8 hits/track
Assume: 2ds module-positions in different SL to be known !
=> tracks defined .. but not only tracks..
=> Tracking needed
=> Use full combinatorics for other modules in the two pots (+mild target cut)
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Alignment der Systems durch Überlapp
Povh (MPI-K): Like Lord Münchhausen got out of the swamp..
Never align a plane included in tracking !
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Coarse alignment: The VDS II
=> 4 Quadrants aligned wrt. to each other
We have seen better target spots, but this is coarse alignment !
- Searching for signals is the remaining task.- For each plane: Coarse and fine binning
10 mm 250 µm- Semi automatic procedure (´asks´ for help)
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Why coarse alignment ?How can this be?
=> robust tracking && no coarse Alignment !
reality
reco
Only tracks from the upper quadrant !
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Teil III
Fein Alignment
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Tracking: SpurmodellWesentlich:
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Tracking and alignment• Master formula:
=> 7 undefined parameters. => cp. later !
relating hits, tracks and geometry
αsinαcosαsinαcos 000 ztztyxtpuu yxT
pitch
waferthickness
p/n strip angle planes perpendicular
wrt. z-axis
• Known parameters: (assumption!)
• Undefined parameters: (now: a guess !)
detecto
rx
z
y
move in x,y,zrotate around z
shear in x,y
scale in z
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Alignment = MinimisationChange the geometry to minimise the residuals between hits and tracks.
Coarse alignment: move along the axis in parameter space:
Linear least squares: Parameters needed
Measurements measured
Design Matrix your problem (linear)
Covariance matrix of measurements
Weight Matrix
Residuals
tuur
-1 0 1 2 3 4 5 6
0
1
2
3
4
5
-1 -0.5 0 0.5 1
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
x X
y
A y
V
aAyr
a
1 y
VW
always a good idea?
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Least Squares MinimisationThe principle: Minimise
AWA
AWAWA
WW
T
TT
TT
a
S
aya
S
aAyaAyrraS
2
2
2
2
Gives directly the parameterand their covariance matrix 1
1
)(
)(
AWAV
WAAWAT
T
a
ya
i
i
iiT
i
ii b
t
a
β
00
00
00
ΓG
GC
=> Solve only Ax=b to align the VDS ??
Yes, but track-parameter and alignment parameter correlate!
250 Alignment parameter, 20000 tracks (nice fit)
dim(A) = O(80000) !
..but quite sparse!
GByte Matrix..
A x = b
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Least Squares Minimisation II
Make use of „inversion by partitioning“:
• For each track
• For the alignment parameter with
i
i
iiT
i
ii b
t
a
β
00
00
00
ΓG
GC
=> Thats it! (Numerical Inversion remains..)
iiiiii tt ββ 1
ΓΓ
'' ba
C
iiii
ii
i
Tiii
ii
bb β'
'
1
1
ΓG
GΓGCC
Ursprung: C. F. Gauß !Entdeckung in Literatur zur Landes und Erdvermessung, Grundstudium
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Least Squares Minimisation II: Toy
Why so complex? explained using a simple toy-problem:
i=1
a.) b.)
t Spur
Treffer
1
i=1
uv i
i
i=2 i=2i=3k=3
D 3
i=3i=4 i=4
• Only parallel tracks• Quite simple to align but the real missalignment not found !
reality (unkonwn) Assumption
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Less Complex Idea..Something you know (Mr. X: „First plane is okay“) :
Do not touch it:i=1
i=1 i=1
i=1
u
u u
u
i
i i
ii=2
i=2 i=2
i=2
i=3
i=3 i=3
i=3
i=4
i=4 i=4
i=4
i=1
ui
i=2 i=3 i=4
„Therefore you iterate“..
Math: .. till eternity!
What about reality ?What is the least influence wrt. to reality?
- Minimise with LLSQ - Correct treatment of global parameters: „If you can not determine, do not touch !“
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Undefined parameters I
Replace Mr. X by some defined quantity. (I strongly prefere to work with mathematical / operational definitions!)
-2
0
2
a1
-2
0
2
a2
-2
0
2
a3
-2
0
2
a3
-2
0
2
a1
-2
0
2
a2
-2
0
2
a3
-2
0
2
a3
Degenerated ellipsoid described by covariance matrix
Degenerated ellipsoid described by covariance matrix
Common sense on toy problem: „You can cary your detector around.„- One global parameter- Moving chamber 1 by x => move chamber 1..n by x
Better: (math)The Correlation–matrix has not full rank.(..really numerics comes later ..)
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Undefined parameters: RealitätThe undefined parameters need not to be guessed! => Singular Value Decomposition ..at least once
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Undefined parameters IIBlobel: Special, fast system (pivoting) Constraints are applied=> Matrices might look nice:
Linear: Log.
Praxis:Matrizen zeigen Überlapp des Detektors in DEN Daten, ohne Diskussion
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Towards the full systemGoing to reality: Face the non-linear problem
Using:
tpwdldzdutpu
ttwdldzpttppduuTT
TT
α
α
sin
cos
0
0
,
cos
sin
cos
sin
l
z
zw
track residuals
Non-gaussian residuals !
Gives to first order:
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Towards the full system
unbiased residuals (explicit exclusion!)
Achtung Falle:‚Tracking-Codes‘ können Einfluß eines Treffers ‚herausrechnen‘
Nur wenn alle Verteilungen der Annahme entsprechen! Im Alignment hochgradig falsch / gefährlich !
Nein!
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Non Gaussian residualsRobust statistics was not at hand !The way out : (Later found to be robust)
• Extend Iterations• Determine the individual resolution from
unbiased residuals• Cut on unbiased residuals
Paw fit and robust technique (MAD)
Policy:
No Minuit calls to non-lin fits in
system !
No Minuit calls to non-lin fits in
system !
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
The systemMoreover:Hit/track associationis alignment dependent!
Typicals:• Needs 20000 (good) tracks in VDS
(1/MB event)• 3 outer iterations: 1.5 h (full reco !)• 2..3 innermost iterations• 4 Quality iterations
Linear Alignment as one block of a complex system!
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Teil IV
Tests Ergebnisse
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Results of the system IWorst parameter: z of SL 8=> 100 µm with 70000 tracks!
text files..(precision)
Residuals:
Reproduce: (simulated tracks)
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Results of the system IIAre the errors from C-
Matrix OK?Bootstrap: • Have a set of data• Produce a set of fit-parameters• Draw tracks from input
sample in a random manner• Produce new fit parameters• Repeat O(500 times)• Look for RMS of fit
parameters of all sets
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Results of the system IIIDoes it find artificialshifts?• Align• Move two opposite
pots to keep cog. (global) fix!
• Align• Plot differences:
Tracker cut: 200 µm !
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Results of the system IVArtificial shifts: z,α, Wert bzgl. Original nach 1,2,3 Iterationen
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Physics ISome nice pictures ..obtained by using an
aligned VDS
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Physics ISome nice pictures ..obtained by using an
aligned VDS
~30 Alignments inbut only one globaldata set
M. Bräuer, 3.2.06Alignierung des HERA-B Vertexdetektors
Until now: align u, α, z of each (double-) sideHowever:
Alignment
z
u‘
u
track, slope: tu
β≠0
sincos ut
uu
0 utresidual u
=> Non-linear-tracking !
First glance:
Still : ToDo !