A fast introduction to the tracking and to the Kalman filter. Alberto Rotondi Pavia

Transcription

1 A fast introduction to the tracking and to the Kalman filter Alberto Rotondi Pavia

2 The tracking To reconstruct the particle path to find the origin (vertex) and the momentum The trajectory is usually curved by the Lorentz force y 0 X B F q c v B + - Even when B is uniform, the trajectory is NOT an helix, due to energy loss multiple scattering The track is defined as a set of points usually on detector planes (real and/or virtual) x

3 The track Since on a detector plane we have two coordinates (v,w) and three momentum components (p u, p v, p w ) the track is a 5-dimensional mathematical entity 3

4 Tracking neglecting inhomogeneous magnetic field and the medium effects Tracking in inhomogeneous magnetic field neglecting the medium effects Tracking in inhomogeneous magnetic field with energy loss and multiple scattering Global fit HELIX Global fit SPLINES H. Wind, NIM 5(974)43 Progressive fit KALMAN... R. Frühwirth, NIM A6(987)444 4

5 The Helix No matter and uniform magnetic field x s x 0 R H cos Φ 0 h s cosλ R H cosφ 0 y s y 0 R H sin Φ 0 h s cosλ R H sinφ 0 z s z 0 s sinλ B // z two planes: xy : circle z - s : straight line s h R cosλ s = track length P(x 0, y 0, z 0 ) = starting point = dip angle 0 = azimuthal angle R H = radius h = -sign(qb z ) arctan RsinΦ RcosΦ y y 0 0 Φ0 0 x x0 5

6 Spline fit H. Wind, NIM 5 (974), 43 No medium effects, dishomogeneous magnetic field is taken intoaccount The spline is a smooth segmented polynomial Cubic spline through n+ points y 0,, y n : Y i t a i b i t c i t d i t 3 i = 0,,n t є [0,] parameters The parameters are found by constraining the pieces of splines to be connected in the measured points assuring the continuity up to the derivative 6

7 What is the track follower? Track extrapolation Also MC with fluctuations Switched off Track follower From one point To Another one Error Propagation (5x5 covariance Matrix) In the same point Between two Track systems 7

8 Tracking track follower vs MC MC= at each step the trajectory is sampled as a random value Tracking= at each step the trajectory is calculated as a mean value with an associate error 8

9 GEANE V. Innocente et al. Average Tracking and Error Propagation Package, CERN Program Library W503-E (99). Two main tasks: Track propagation: the same MC geometry banks are used. Error propagation: from one point to another one In the same point between different systems x y z MARS SC cosλ cosφ sinφ sinλ cosφ cosλ sinφ cosφ sinλ sinφ sinλ 0 cosλ x y z GEANE = tracking with the geometry of GEANT3 + a lot of mathematics for the transport matrix calculation 9

10 Track propagation 0

11 Track propagation II a piece of helix field along z-axis M(s) is the position vector l

12 Track propagation III

13 Track propagation IV Strandlie & Wittek, CMS note, 006/00 3

14 The first task: track propagation 4

15 The jacobian transports the erros from one step to another Here, at each step, multiple scattering and energy loss effects have to be added 5

16 Multiple scattering 0 ( k min) d u 6

17 Multiple scattering 7

18 Multiple scattering 8

19 Multiple scattering PDG: wrong GEANE 9

20 /p y z /p y z 0

21 Energy loss

22 Average energy loss de Z mec γ β max 4π N r m c z ln T β A e e dx A β I δ BETHE-BLOCH Fluctuations in energy loss k ξ E max max average energy loss energy loss in a single collision k k 0 Gaussian Vavilov σ E ξ E max β σ p σ k 0.0; N c 50 Landau are infinite!! k 0.0; N c 50 Sub-Landau is too large!!

23 Gauss and Vavilov: no problems for the track follower GEANE σ E ξ E max β Landau: problematic distribution σ p σ The track follower must be compared With the full Landau sampling ray tail Sub-Landau: what distribution? 3

24 Gauss and Vavilov: no problems for the track follower GEANE and GEANT4E contain only this 4

25 Improvements New error calculation in energy loss for heavy particles New error calculation for bremsstrahlung 5

26 Truncated Landau: Solution (GEANT3 & GEANT4): truncation of the distribution tail to have as a mean the average de/dx 6

27 Original GEANE GEANE for PANDA modified with the the -tail 7

28 8

29 Thin gaseous absorbers: The Urban distribution 9

30 -ray tail 30

31 Truncation of the Urban tail of distribution 3

32 Is the Urban distribution a good model? Comparison with an exact model in the case of a thin gas layer 3

33 p( n) ( nx) k! k e nx MIP particle: Argon CO Cluster/cm 6 35 Prymary e Effective cluster/cm: n MIP de/dx/(de/dx) min N/n ~.8 l ( x) ne n x 33

34 Urban model works well Urban Exact model Landau.5 cm of Ar/CO 90/0. GeV pions 34

35 The matching between Urban and Landau is obtained for =

36 36

37 37

38 Bremsstrahlung 38

39 Bremsstrahlung 39

40 Bremsstrahlung 40

41 =.4 =.3 =.9 =0.96 =.05 x MC σ Pull x GEANE GEANE 4

42 4 A Bayesian technique:the KALMAN filter 0 ) (, ) ( ) ( ) ( x x x x Consider the well-known weighted mean: x x x x x x x x x A simple algebraic manipulation gives the recursive form: measured point weight matrix prediction Kalman= the measurement is weighted with a model prediction (track following)

43 Example: Radar Applications /3 43/0 In a radar application, where one is interested in following a target, information about the location, speed, and acceleration of the target is measured at different moments in time with corruption by noise. State vector error of x Covariance matrix r = { x, y, z, v x, v y, v z } position velocity C = x y z vx vy vz December, 968. The Apollo 8 spacecraft has just been sent on its way to the Moon. 003:46:3 Collins: Roger. At your convenience, would you please go P00 and Accept? We're going to update to your W-matrix.

44 The original idea is very simple When m is measured at t and x(t,t ) is the prediction from t to t, the best evaluation of x at t is This is called the Kalman filter recursive form x( t ) m m x( t, t ) m m Kalman= the measurement is weighted with a model prediction (track following) 44

45 Extrapolation, filtering and smoothing X i+ Kalman filter K i+ e i+ e i k i f i k i- prefit vertex Kalman filter X i x Detector plane Measured point 45

46 Track propagation: physical effects Multiple scattering Energy loss straggling α RMS.03 Pull x MC σ x GEANE GEANE 46

47 Tracking with Kalman 47

48 Tracking with Kalman solve 48

49 49

50 x x 50

51 5

52 Track fitting tools. the GEANT3-GEANE old chain: The mathematics is that of Wittek (EMC Collaboration) The tracking banks and routines are the same as in MC. The user gives the starting and ending planes or volumes and the tracking is done automatically. It works very well (see the CERN Report W503 GEANE, 99).. Modern experiments: in the software are implemented some tracking classes: input: xi, Ti, i, step, medium, magnetic field output: new xi, Ti, i the user has to manage geometry,medium and detector interface 3. A GEANT4-GEANT4E chain there exists in the new GEANT Root framework. It is used by CMS but is not included into the official releases 5 (see Pedro Arce s talks in the Web)