Alignment of the VDS

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

( )


Teil I

HERA-B Einführung


HERA-B• Vorwärtsspektrometer mit festem Target• Hochraten – Experiment• Anspruchsvolles Triggersystem• ca. 650.000 Kanäle


HERA-B: Ansicht


HERA-B: Ansicht II


HERA-B: Ansicht III


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


Teil II

Pre-Tracking Alignment


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:


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)


Alignment der Systems durch Überlapp

Povh (MPI-K): Like Lord Münchhausen got out of the swamp..

Never align a plane included in tracking !


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)


Why coarse alignment ?How can this be?

=> robust tracking && no coarse Alignment !

reality

reco

Only tracks from the upper quadrant !


Teil III

Fein Alignment


Tracking: SpurmodellWesentlich:


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


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?


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


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


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


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 !“


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 ..)


Undefined parameters: RealitätThe undefined parameters need not to be guessed! => Singular Value Decomposition ..at least once


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


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:


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!


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 !


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!


Teil IV

Tests Ergebnisse


Results of the system IWorst parameter: z of SL 8=> 100 µm with 70000 tracks!

text files..(precision)

Residuals:

Reproduce: (simulated tracks)


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


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 !


Results of the system IVArtificial shifts: z,α, Wert bzgl. Original nach 1,2,3 Iterationen


Physics ISome nice pictures ..obtained by using an

aligned VDS


Physics ISome nice pictures ..obtained by using an

aligned VDS

~30 Alignments inbut only one globaldata set


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 !

Documents

Alignment of the VDS