1. Alignment of the VDS Outline: The problem VDS Crucial: Coarse
M. Bräuer
Alignment of the VDS
Outline:
The problem VDS
Crucial:
Coarse
alignment
Titel, inh
Precise alignment
Alignment parameters
Least squares as a solution
Toy systems to align
The full system
System behaviour
Results
Alignment of the VDS

2. The problem VDS Major Problems: The bare size
No tracks into VDS existing
Today: Matching without VDS track?
Momentum?
Magnet tracking?
Survey/setup:
Needed: <200µm !
Alignment of the VDS

3. Coarse alignment: Before tracking
Assume some module-positions to be known:
Position of pots wrt. to each other unknown
Double modules within a pot not optimal adjusted
beam
1 SL u
=> Start without tracking !
Alignment of the VDS

4. Coarse alignment: 1st step
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)
=> 4 ds modules adjusted => 8 hits/track
Alignment of the VDS

5. Coarse alignment: The VDS I
Povh (MPI-K): Like Lord Münchhausen got out of the swamp..
Never align a plane included in tracking !
Alignment of the VDS

6. Coarse alignment: The VDS II
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)
=> 4 Quadrants aligned wrt. to each other
Alignment of the VDS

7. Why coarse alignment ? How can this be?
Only tracks from the upper quadrant !
reality
reco
=> robust tracking && no coarse Alignment !
Alignment of the VDS

8. Tracking and alignment
Master formula:
relating hits, tracks and geometry
Known parameters: (assumption!)
pitch
p/n strip
angle
wafer
thickness
planes perpendicular
wrt. z-axis
Undefined parameters: (now: a guess !)
shear in x,y
move in x,y,z
rotate
around z
detector y z
scale in z x
=> 7 undefined parameters  cp. later !
Alignment of the VDS

9. 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? x X
Linear least squares:
Parameters needed
Measurements measured
Design Matrix your problem (linear)
Covariance matrix of measurements
Weight Matrix
Residuals
Alignment of the VDS

10. Least Squares Minimisation
The principle: Minimise
Gives directly the parameter
and their covariance matrix
=> Solve only Ax=b to align the VDS ??
Yes, but track-parameter and alignment parameter correlate!
250 Alignment parameter, tracks (nice fit)
dim(A) = O(80000) ! GByte Matrix..
Ax=b :
..but quite sparse!
Alignment of the VDS

11. Least Squares Minimisation II
Make use of „inversion by partitioning“:
For each track
For the alignment parameter with
=> Thats it! (Numerical Inversion remains..)
Why so complex? explained using a simple toy-problem:
reality (unkonwn)
Assumption
Only parallel tracks
Quite simple to align but the real missalignment not found !
Alignment of the VDS

12. Less Complex Idea.. - Minimise with LLSQ
Something you know (Mr. X: „First plane is okay“) :
Do not touch it:
„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 !“
Alignment of the VDS

13. 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 x
Better: (math)
The Correlation–matrix has not full rank.
(..really numerics comes later ..)
=>
Degenerated ellipsoid
described by covariance matrix
Alignment of the VDS

14. 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:
Alignment of the VDS

15. Towards the full system
Going to reality: Face the non-linear problem
Using:
Gives to first order:
Something is missing:
track residuals
unbiased residuals (explicit exclusion!)
Non-gaussian residuals !
Alignment of the VDS

16. Non Gaussian residuals
Robust 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 !
Alignment of the VDS

17. The system Moreover: Hit/track association is alignment dependent!
Linear Alignment as one
block of a complex system!
Typicals:
Needs (good) tracks in VDS (1/MB event)
3 outer iterations: 1.5 h (full reco !)
2..3 innermost iterations
4 Quality iterations
Alignment of the VDS

18. Results of the system I Worst parameter: z of SL 8 Residuals:
=> 100 µm with tracks!
Residuals:
Reproduce:
(simulated tracks)
text files..
(precision)
Alignment of the VDS

19. Results of the system II
Are 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
Alignment of the VDS

20. Results of the system III
Does it find artificial shifts?
Align
Move two opposite pots to keep cog. (global) fix!
Plot differences: HOLMES cut: 200 µm !
Alignment of the VDS

21. Results of the system IV
Artificial shifts: z,
The VDS alignment system is working !
Alignment of the VDS

22. Physics I Some nice pictures ..obtained by using an aligned VDS
~30 Alignments in
but only one global
data set
Alignment of the VDS

23. Physics II The thermal limit ? (one month)
Distance between inner/outer and upper/lower planes seems to change
=> Frequent alignment!
Alignment of the VDS

24. Physics III Beam Gas: The proton beam shape ! =>
Yes, it works - with the intaeraction-trigger !
The proton
beam shape !
Can I get some run
time ?
=>
Alignment of the VDS