Detector Alignment Thomas Naumann DESY Zeuthen n Detector alignment and calibration often imply least squares fits with many parameters : n n global parametersof calibration, n = m local parametersof the data model of an object (e.g. tracks) wherem = k x lwith k... nr of objects, large, >> 1000 l... nr of parameters, small, < 10 n Traditional method to correct by residuals of local fit is biased since correlation between local and global parameters is ignored (wrong global constants used) n Solution: Fit local and global parameters simultaneously n large nr of para’s:N = n + k x l time to solve linear problem by matrix inversion ~ N³ N = 1.000:~ 1 1 GHz with double prec. N > :> 1 day intractable Idea : n only global parameters needed for calibration and alignment n use special matrix structure to get computing time ~ n 3

1) for code and manual see and.ps contains: the normal equations of least squares of the local fits i=1,...,l  i  i  i with the local parameter vectors  i and the covariance matrices  i. Traditionally neglect correlations G i with global parameters and solve locally for  i inverting l matrices  i of rank k. the sum over the covariance matrices C i of the correlations between the vector a of the m global calibration parameters. the matrices G i of the correlations between local and global parameters. Author: V.Blobel, Hamburg University 1) Inversion by partition: Use the special shape of this matrix and rewrite the matrix equation as C’ a = b’ with new matrices C’ and b’ which sum over the local parameters and explicitly contain only the global parameters : Solve the linear least square problem independent of nr of local parameters ! Millepede The matrix equation for the simultaneous fit of global and local parameters

The H1 Backward Silicon Detector has 8 z planes along the beam axis with 16  sectors each.  needs alignment of the r detectors p needs alignment of the  detectors. n get z correlations between BST parts 1 and 2 and  correlations between sectors BST uses r and  Si detectors with 640 strips of 96 and 75  m pitch : measure scattered electron’s polar angle  and energy + momentum E and p at HERA : e e

The Alignment of the H1 BST Alignment of the  detectors : get electron charge and momentum up to the HERA beam energy of E e = 27.5 GeV from BST alone ! The ratio of the energy E of the scattered electron measured by the H1 SPACAL calorimeter to the momentum p measured by the H1 BST versus E. Both measurements agree well. E / p This gives a polar angle precision of 0.4 mrad. For the r detectors a resolution of 18  m is reached.