DETECTOR ALIGNMENT with tracks

Catania November 2006 Adam Jacholkowski 1

DETECTOR ALIGNMENTwith tracks


OUTLINE

Why do we need software alignment Principle of Chi2 minimization Local, iterative approach with 5-6 parameters solved

at a time Example – the experiment NA57 silicon telescope Global alignment method with ALL parameters

solved simultaneously Utility of cosmic muons for detector pre-alignment,

example of the ALICE ITS detector Summary and discussion


TRACK RECONSTRUCTION PRECISION

measNpts

pts

measNpts p

Np

XpK

termsdiagonalnon

LK

L

LKL

pL

K

LBp

322

0

3

2

2

2

202

22

202

22

4

2

4

20

22

)/1(4

7)/1(

10015.0

,

cos6

1

)cos(

26

3

1

)cos(

26

tan

cos3

4

)cos(

96

0003.0

cos)/1(

(3 points parabolic approximation)

B in kGs, p in GeV/c, L in cm

σ0 = pitch/sqrt(12)

R.L. GLUCKSTERN

Nucl. Instr. & Methods 24(1963) 381

Plus(unknown)Mis-align-mentterms !!!


Why do we need software alignment Hardware alignment techniques (optical, photogrametry,

X rays etc) have technological limits 50 -100 microns Geometrical resolution of the modern detectors (like

pixels) as good as about 10 -15 microns High tracking precision necessary in order to reach

physics goals of the experiments ( for ex. heavy flavours) The only practical method to reach the required precision

– use of (high momentum) tracks A general principle – minimize the track residues but

keeping some external (physics wise) constraints


Example - Impact Parameter resolution

better than 40 µm for pT > 2.3 GeV/c~20 µm at high pT

impact parameter d0 (r)

Impact parameter resolution is crucialfor the detection of short-lived particles:charm and beauty mesons and baryons.Determined by pixel detectors:at least one component has to be better than 100 m (c for D0 meson is 123 m)

Mass 1.864 GeV/c2 c=124 m


LOCAL ALIGNMENT method (1)

x

zy

(z)

(y)

residual

trac

k


Local sensor alignment(2)


Local sensor alignment (3)


Local sensor alignment (4a)

Will come backto it


Local sensor alignment (4b)


Warning – different rotation conventions

angleangleangle )sin(1)cos(

:assumption angle small


Slightly more MATH(1)

Note – 3rd component of qxc = 0 !



tR

qq

qq

qq

q

q

q

qR

vu

vu

vu

w

v

u

for similar

1

1

1

qw= 0 as we are in the sensor/local reference system ! But tw ≠ 0

wvu

wvu

wvu

ttt

ttt

ttt

tR


w

v

defw

u

defuv

corrw

uvw

wuvuv

corrv

uvw

wvuvu

corru

t

t

t

tqqw

q

vttt

tttqqq

uttt

tttqqq

tan,tan,

0 that showHomework


With good approximation the 2 ratios ~tanψ and ~tanθ





uv

vu

uv

uu

vv

vu

vu

vu

vu

vuuv

vuvu

qq

qq

qq

qq

qq

qq

w

q

w

qv

q

v

qu

q

u

q

vqqwqq

uqqwqq

,

tan,tan

tan,tan

tan,tan

1,0

0,1

:columns) 2 rows (6Jacobian theof elements - sderivative

tan)(

tan)(

Needed for solvingthe Chi2 minimizationproblem giving as a solution6 correction parameters






NA57 SETUP (Pb - Pb run)

1.4 T

Target: 1% Pb

Scintillator

Petals: centrality trigger

MSD: multiplicity silicon detector

Tracking device:

silicon pixel planes

(5 x 5 cm2 cross section)

Lever arm: double side strips

5 cm X

(~ 1.0 M pixels)


HYPERON DETECTION

30 cm

5 cm

5 cm

by

by

Plus many other associated tracks

X


NA57 Alignment plots

Single Y (vertical) ladderY-plane tilt test

mm

Z

Z

Y

mic

rons


p-Be 40 GeV/c

Ξ event

[cm]

Y

X

Ω3YΩ3ZΩ2YΩ2ZΩ3YΩ2YΩ2ZΩ2Y Ω2Z Ω3YΩ3Y planes sequence

aspect ratio ≈ 9 !


p-Be 40 GeV/c

Ξ event

[cm]

Z

X

Ω3YΩ3ZΩ2YΩ2ZΩ3YΩ2YΩ2ZΩ2Y Ω2Z Ω3YΩ3Y planes sequence

aspect ratio ≈ 9 !


Mass Resolution: Ξ

158 A GeV/c 40 A GeV/c


Global Alignment Approach

Limitations of the local method Correlations not (fully) taken into account Convergence not always guaranteed Constraints not easy to be included

Possible solution – simultaneous fit of ALL the parameters (tracks and sensors) problem of inverting huge matrices !

Millepede Algorithm developed in DESY by Volker Blobel (http://www.desy.de/~blobel)

Numerical limitations an attempt to overcome the problem Millepede II


Global Alignment (1)


Global alignment (2)














A word on constraints


A simple, explicit example(1)

equationitrackxby kkiik

xx

xx

xy

kk

kkk

kkmeask

kk

kkkmeask

kk

Sb

S

S

xw

yxwb

xbxywb

wbxyw

ignoringfittrack

1

0)(2

/1,)(

2

3

1

2

223

1

2

(mini telescope of 3 planes)

point sourceof particles

Local & global parameters

Millipede

Simultaneousfit of globalAND local parameters

02

k

add


Polynomial Parameterization

Fit (example)


0)(2

0)(2

)(

3

1

2

2

1

2

22

1

3

1

2

mkkmkk

km

big

illili

ll

big

iki k

kikkbig

yxbwb

yxbw

yxbw


System of5 linear equations

The resulting matrix equation looks like



kiii

ixyiii

ixx

xy

xy

i

i

i

xx

xx

yxwkSandxxwSwhere

S

S

yw

yw

yw

b

b

Sxwxwxw

Sxwxwxw

xwxww

xwxww

xwxww

3

1

3

1

33

22

11

2

1

3

2

1

332211

332211

33333

22222

11111

)(

)2(

)1(

0

0

200

020

002

No problem to invert 5x5 matrix but let’s see the reduction method 3x3



)2(

)1(1,

)()(

*

33

22

11

*12

121

1221211

xy

xy

xxi

i

i

ik

S

S

Sb

yw

yw

yw

matrixbigtheofpartsCwhere

bCCCCC

The key point is that update of the matrix to be inverted(C11-C12C22

-1C21) can be done on the track by track basisdue to the quasi diagonal, symmetric form of C22



xxxxxx

xxxxxx

xxxxxx

SxwwS

xwxwS

xwxwS

xwxwS

xwwSxwxw

Sxwxw

Sxwxw

Sxww

invertedbetoMatrix

233

333223311

33222

222

2211

331122112

111

)())(())((

))(()())((

))(())(()(

Actually the matrix inversion algorithms fail for more than 50000 d.o.f.(even when using quadruple precision !!)

The next and the last step would be inclusion of constraints in orderto avoid bad collective modes like global displacement and/orshearing


A simple, explicit example(6): constraints

Forcing the fit to conform to physics principles and/or to external knowledge not known by the internal variables of the fit - 2 methods: Elimination of unknowns by direct substitution, but

equations cannot be always solved analytically, covariance matrix is calculated only for the reduced set of variables

Method of Lagrange multipliers – a preferred one

)ˆ(2 022 DdT



)ˆ(2 022 DdT

, WHERE

Jacobianˆ

s,constraint

),,(

),(),,(

smultiplier Lagrange),,,(

1

1

1

1

1

,12

11

21

n

rr

n

nr

n

n

rT

dd

dd

D

d

dd

d

initial set of parameters

Let’s assume in our toy exampleone constraint equation like β1+ β2+ β3 = 0 with d0 = 0(i. e. no global dis-placement)



3,2,11 case lour triviain

equations newr and

in termNew

22

11

0l

2

22

11

k

2

iford

dddd

ddd

i

n

llll

k

rr

kk

Now we have a (final) matrix of (n+r) x (n+r) size, like



r

n

r

n

n

rr

n

n

r

n

r

d

d

b

b

dd

dd

dd

dd

0

01

1

1

1

1

1

1

1

1

11

1

00

00

Original matrix C11

In our case we have just one extra column and row of 1s, one λ, d0 = 0 !!


ALICE coordinates


ALICE Inner Tracking System (ITS)

Alignable elements:SPD -- 240SDD -- 260SSD - 1698Total – 2198 * 6 d.o.f + ~12 collective dof


SUMMARY

All modern particle detectors need software (track) alignment methods in order to reach the design precision

Two main approaches: Local with many iterations Global needing inversion of huge matrices

We have looked into MATH involved in these 2 methods, discussed some approximations and tricks

Computing and bookkeeping very challenging in real life, especially in the LHC experiments under preparation

Alignment is part of the art of detector calibration (MATH is not ALL)

48

COSMICS as a tool in the detector pre-alignment in ALICE

Before the (true) beam becomes available…


Hadronic interaction models in cosmic rays

It is a paragraph describing the importance of the knowledge of hadronic interactions at energies involved in cosmic rays E > 1014 eV

The LHC contributions and in particular the ALICE possibilities to study p-p p-A and A-A interactions

PPR part II $6.11


Location of ALICE set-up

Element H C O Na Mg Al Si K Ca Fe

% 0.8 4.3 48.5 0.7 4.2 3.7 21.5 2.3 10.0 4.0

Rock composition over Alice

Effects on muons of the Alice environment

• N15 GeV)

• Direction

• Energy


Energy threshold of muons

Energy loss by muons crossing the rock above Alice.The effect of the shafts is to decrease the energy loss for some particular muon directions.


Utility of cosmics for alignment When B=0, equivalent to the laser beams When B≠0, equivalent to two tracks going back

to back with equal momentum (a la Z0 decay) strong constraints

Given for free ! But some weak points:

Momentum not known Limited zenithal angle range (up to 60 deg.) Limited flux

Used for alignment by most of the experiments


Silicon Pixel Detector 60 staves, 240 modules 40960 chs. per module cell size (r,z): 50 x 425 µm2

spatial resolution (r) 12 µm spatial resolution (z) 100 µm


B = 0.4 T

ITS DISPLAY (old one)

X

Y

ρφ

center of LHC

Alignable elements:SPD -- 240SDD -- 260SSD - 1698Total – 2198 * 6 d.o.f + ~12 collective dof



Examples of alignment applications(only SPD case) Test for azimuthal distortions

Non centered layers non parallel, displaced tracklets

Z shifted layers

2C fits (4 points – 2 param) residues minimization

trackletsacquire an IP


CRT muon simulation

Muon generation according to Hebbeker et al Config.C with AliGenCRT as generator Activated only ITS, CRT and some passive

materials Several “bad features”

CRT package not updated since 2 years, no documentation Multi muon (close) hits in the sensors (4-5/sensor) Resulting (abnormally) large clusters Generation at surface - propagation through the rock

above ALICE (no use of L3 measurements) No absolute normalization – not usable for flux estimate

First test simulation (low statistics) done


SPD clusters from the CRT simulation

Relatively large clusters needs some investigation (?)


Cosmic muons rec. points: 100 evts

Additional problem – noisy strips

Single clean muon


Muon distributions (1000 muons)

Momentum distribution(selected range 10-1000 GeV/c)

zenith angle in degs

Comment: only muons reaching SPD are plotted


What should we get from thesimulation

Reference plots for all tracklets consistency checks Relative IPs in all projections Angular differences (XY and YZ/XZ projections) Same for different alignment levels

Study of the FAST-OR trigger efficiency Select events with a muon crossing SPD1,2 Apply trigger algorithm efficiency

Time estimate to accumulate useful samples of data (only 1-2 muons per minute in SPD !)

Preparing and testing simple (re)alignment tools


Examples of control/reference plots

(units – microns and mrads)

(only clean, 4-rec. points in SPD)

Relevant for IP resolutionstudy


Influence of misalignment on IP/test

Preliminary test,Needs more Statistics !mradmicrons


General ITS alignment strategy Three categories of the alignment procedures

Local , iterative sensor alignment (tracks based) based on the residues (many small, 6X6 matrices inversions (framework exists – Cvetan Cheshkov)

Global approach like Millepede of Volker Blobel inversion of huge matrices (12200X12200 in case of the ITS), used by the MUON (AliMillepede - Javier Castillo)

Poor physicist’s method – propagating alignment from small sub-detectors to bigger – outside detectors ( in our case from pixels outside)


‘How I Would Align an LHC Detector’ Assemble a complementary set of events

Muons, pairs, cosmics, survey, … Align the innermost (most sensitive) detector first

Align internal DOFs with complimentary data Rigid body parameters plus non-planar distortions Use sanitized outer-tracking constraint (on curvature, …)

Align the next detector outwards next Include (aligned) innermost detector in track fit Align using standard techniques

Track self-consistency, survey, … Continue outwards

Include calorimeter, muon chambers Repeat (if necessary)

Applicable to ITS !

(Summary talk of David Brown at the LHC Detector Alignment Workshop)


OUTLOOK

Need to know what and how precisely will be measured during and after the installation (Torino) initial (mis)-alignment

Further cleaning/preparation of the alignment process infrastructure

Development and testing of the alignment procedures (highest priority for the barrel) using cosmics then pp

TPC calibration then inter-alignment with ITS Creation of the Alignment Task Force Preparation for different scenario (problems, data type) visualization (EVE) and monitoring tools (MOOD)


Backup Slides

EXEMPLE OF AN ASSEMBLY


NA57 case : λ ≈ 0, L ≈ 30cm, B ≈ 14kGs,X0 ≈ 30cm/(9x 0.012) = 277cm, pitch = 50μm

)/(0.2)(,25.0)(

/045.0)/(,0037.0)/(6

126

3

496

)0003.0(

1)/1(

22

202

2

4

20

22

pmradmrad

ppppp

LKL

L

K

LBp

mscmeas

mscmeas

Δp/p meas and msc errors equal at p ≈ 12.2 GeV/c (4.5%)

p = 12.2 Δφ = 0.25 & 0.17mrad → 0.3 mrad

meas msc.


Will we need pre-alignment ?

What to do if messy events at the start - no tracks found by the standard methods ??

Many possible problems: DAQ, trigger, channels mapping, software bugs…

Then careful debugging using display (well adapted) , histogramming, looking for hit correlations, exploiting detector overlaps etc.

REMEMBER – hard competition with ATLAS and CMS (first pp run) stay ready to face all sort of possible problems and be able to solve them quickly


RHO-PHI slice (zoom)

φ[rad]

ρ[cm]

fairly straighttracks


RHO-Z (PHI slice)

ρ[cm]

z[cm]


TPCTPC

PHOSPHOS

Muon armMuon arm

TOFTOF

TRDTRDHMPIDHMPID

PMDPMD

ITSITS

ACCORDEACCORDE

ALICE


Iterative local CHI2 alignment approachfor the ATLAS SCT detector

Diploma Thesis, Roland Härtel, November 2005

130 k selected (generated) tracks ~ 250 hits per module

None of the alignment accuracy limits meets the required alignmentaccuracy (comparable with the survey precision only). NEXT: better tracking, geometry, analytical derivatives, statistics…

Resolution16 x 580 μ


ITS HITS (6000 tracks)

PIXELS

Si DRIFTS

STRIPSρ[cm]

φ[rad]


RHO-PHI slice zoom

φ[rad]

ρ[cm]

fairly straighttracks


PHI SLICE in INVERSE GEOMETRY

Xinv

Yinv

first layer

last layer


V (not V0 !) track representation

φ[rad]

η +- kD

η – pseudo-rapidity k - a constant D - | ρ - ρ max |

track diagnostics

(simple design,intense content)

H. Drevermann fromALEPH/ATLAS


An (artificial pileup) high multiplicity ALEPH event

221 tracks


MILESTONES

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DETECTOR ALIGNMENT with tracks