Alignment of the VDS

M. Bräuer 20.06.2001 Alignment of the VDS • Outline: • The problem VDS • Crucial: Coarse alignment • Precise alignment • Alignment parameters • Least squares as a solution • Toy systems to align • The full system • System behaviour • Results Alignment of the VDS

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 ! 2 m Alignment of the VDS

beam 1 SL u 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 => Start without tracking ! Alignment of the VDS

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

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

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

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

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

x X 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 measurements Weight Matrix Residuals Alignment of the VDS

Least Squares Minimisation The principle: Minimise Gives directly theparameter and theircovariance matrix => 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! Alignment of the VDS

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: Assumption reality (unkonwn) • Only parallel tracks • Quite simple to align but the real missalignment not found ! Alignment of the VDS

Less Complex Idea.. 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

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

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

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

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

The system Moreover: Hit/track association is alignment dependent! Linear Alignment as one block of a complex system! • 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 Alignment of the VDS

Results of the system I Worst parameter: z of SL 8 => 100 µm with 70000 tracks! Residuals: Reproduce: (simulated tracks) text files.. (precision) Alignment of the VDS

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

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

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

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

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

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

Alignment of the VDS