04/24/23Alignment of the VDS

M. Bräuer

Alignment of the VDS

M. 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

04/24/23Alignment of the VDS

M. Bräuer

The problem VDS

Major Problems:1. The bare size2. No tracks into VDS

existing• Today: Matching

without VDS track?• Momentum?• Magnet tracking?

3. Survey/setup:

Needed: <200µm !

mm2..1r

2 m

04/24/23Alignment of the VDS

M. Bräuer

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:

04/24/23Alignment of the VDS

M. Bräuer

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)

04/24/23Alignment of the VDS

M. Bräuer

Coarse alignment: The VDS IPovh (MPI-K): Like Lord Münchhausen got out of the swamp..

Never align a plane included in tracking !

04/24/23Alignment of the VDS

M. Bräuer

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)

04/24/23Alignment of the VDS

M. Bräuer

Why coarse alignment ?How can this be?

=> robust tracking && no coarse Alignment !

reality

reco

Only tracks from the upper quadrant !

04/24/23Alignment of the VDS

M. Bräuer

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 !)

detector

xzy

move in x,y,zrotate around z

shear in x,y

scale in z

04/24/23Alignment of the VDS

M. Bräuer

Alignment = Minimisation

Change the geometry to minimise the residuals between hits and tracks.

Coarse alignment: move along the axis in parameter space: always a good idea?

Linear least squares: Parameters needed

Measurements measured

Design Matrix your problem (linear)

Covariance matrix of measurementsWeight MatrixResiduals

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 yV

aAyr

a

1 yVW

04/24/23Alignment of the VDS

M. Bräuer

Least Squares Minimisation

The principle: Minimise

AWA

AWAWA

WW

T

TT

TT

aS

ayaS

aAyaAyrraS

2

2

2

2

Gives directly the parameterand their covariance matrix 1

1

)(

)(

AWAVWAAWA

T

T

a

ya

i

i

iiTi

ii b

t

a

β00

0000

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) ! 50 GByte Matrix..

Ax=b :

..but quite sparse!

04/24/23Alignment of the VDS

M. Bräuer

Least Squares Minimisation II

Make use of „inversion by partitioning“:

• For each track

• For the alignment parameter with

i

i

iiTi

ii b

t

a

β00

0000

ΓG

GC

=> Thats it! (Numerical Inversion remains..)

iiiiii tt ββ 1 ΓΓ

'' ba

C

iiii

ii

i

Tiii

ii bb β',' 11

ΓGGΓGCC

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

04/24/23Alignment of the VDS

M. Bräuer

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

i

i=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

u i

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 !“

04/24/23Alignment of the VDS

M. Bräuer

Undefined parameters I

Replace Mr. X by some defined quantity. (I strongly prefere to work with mathematical / operational definitions!)

Common sense on toy problem: „You can cary your detector around.„- One global parameter- Moving chamber 1 by x => move chamber 1..n by xBetter: (math)The Correlation–matrix has not full rank.(..really numerics comes later ..)

=> Degenerated ellipsoid described by covariance

matrix-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

04/24/23Alignment of the VDS

M. Bräuer

Undefined parameters II

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

Blobel: Special, fast system (pivoting) Constraints are applied=> Matrices might look nice:

04/24/23Alignment of the VDS

M. Bräuer

Towards the full systemGoing to reality: Face the non-linear problem

Using:

Gives to first order:

tpwdldzdutpu

ttwdldzpttppduuTT

TT

α

α

sincos

00

,

cossin

cossin

l

zz

w

Something is missing:

track residuals

unbiased residuals (explicit exclusion!)

Non-gaussian residuals !

04/24/23Alignment of the VDS

M. Bräuer

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

track residuals

Paw fit and robust technique (MAD)

Policy:

No Minuit calls to non-lin fits in

system !

04/24/23Alignment of the VDS

M. Bräuer

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!

04/24/23Alignment of the VDS

M. Bräuer

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

Residuals:

Reproduce:(simulated tracks)

text files..(precision)

04/24/23Alignment of the VDS

M. Bräuer

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

04/24/23Alignment of the VDS

M. Bräuer

Results of the system IIIDoes it find artificial shifts?• Align• Move two opposite pots to keep cog. (global) fix!

• Align• Plot differences: HOLMES cut: 200 µm !

04/24/23Alignment of the VDS

M. Bräuer

Results of the system IVArtificial shifts: z,

The VDS alignment system is working !

04/24/23Alignment of the VDS

M. Bräuer

Physics ISome nice pictures ..obtained by using an aligned VDS

~30 Alignments inbut only one globaldata set

04/24/23Alignment of the VDS

M. Bräuer

Physics IIThe thermal limit ? (one month)

Distance between inner/outer and upper/lower planes seems to change

=> Frequent alignment!

04/24/23Alignment of the VDS

M. Bräuer

Physics IIIBeam Gas:Yes, it works - with the intaeraction-trigger !

The proton beam shape !

Can I get some run time ?

=>