Track Fitting using Kalman Filter and Smoother

Track Fitting Track Fitting using Kalman Filter and using Kalman Filter and

SmootherSmoother

S. Gorbunov S. Gorbunov DESY ZeuthenDESY Zeuthen I. KiselI. Kisel KIPKIP, Uni-Heidelberg, Uni-HeidelbergE. KryshenE. Kryshen SPbSPU, St. PetersburgSPbSPU, St. Petersburg

CBM Collaboration Meeting, GSIMarch 9-12, 2005

KIPKIP

9-12 March 2005, GSI9-12 March 2005, GSI S. Gorbunov, I. Kisel, E. KryshenS. Gorbunov, I. Kisel, E. Kryshen 22/12/12

ContentsContents

• Analytic formula of track Analytic formula of track extrapolationextrapolation

• Kalman Smoother routine Kalman Smoother routine

9-12 March 2005, GSI9-12 March 2005, GSI S. Gorbunov, I. Kisel, E. KryshenS. Gorbunov, I. Kisel, E. Kryshen 33/12/12

Extrapolation in Magnetic FieldExtrapolation in Magnetic Field

Differential equation of charged particle Differential equation of charged particle motion in a magnetic field motion in a magnetic field

Along Z direction :

9-12 March 2005, GSI9-12 March 2005, GSI S. Gorbunov, I. Kisel, E. KryshenS. Gorbunov, I. Kisel, E. Kryshen 44/12/12

Analytic ExpressionAnalytic Expression

Extrapolation error:

Coefficients: Equation of motion:

9-12 March 2005, GSI9-12 March 2005, GSI S. Gorbunov, I. Kisel, E. KryshenS. Gorbunov, I. Kisel, E. Kryshen 55/12/12

Analytic Expression FeaturesAnalytic Expression Features

Features :

• The precision of extrapolation does not depend on a shape of the magnetic field.

• One can cut off the higher-order terms in the series.

9-12 March 2005, GSI9-12 March 2005, GSI S. Gorbunov, I. Kisel, E. KryshenS. Gorbunov, I. Kisel, E. Kryshen 66/12/12

Analytic 3Analytic 3

Example: Third order extrapolation in a homogeneous By field

Small parameter:

9-12 March 2005, GSI9-12 March 2005, GSI S. Gorbunov, I. Kisel, E. KryshenS. Gorbunov, I. Kisel, E. Kryshen 77/12/12

Coefficients in the Analytic FormulaCoefficients in the Analytic Formula

ccxx == 1.13e-041.13e-04 ccyy = 5.64e-5.64e-0404

cczz = 3.09e-3.09e-0404

cxx = 3.72e-07 cxy = 2.57e-06 cxz = 1.25e-06

cyx = 3.95e-06 ccyyyy = 2.39e-2.39e-0404

ccyzyz = 3.04e-3.04e-0505

czx = 5.21e-07 czy = 4.64e-06 czz = 3.74e-06

cxxx = 5.15e-09

cxxy = 2.47e-08 cxxz = 8.73e-09

cxyx = 3.36e-08 cxyy = 3.83e-07 cxyz = 6.57e-08

cxzx = 5.28e-09 cxzy = 3.20e-08 cxzz = 1.78e-08

cyxx = 4.81e-08 cyxy = 5.46e-07 Cyx

z

= 9.34e-08

cyyx = 7.67e-07 ccyyyyyy = 5.85e-5.85e-0505

Cyy

z

= 3.75e-06

cyzx = 1.30e-07 cyzy = 1.35e-06 Cyzz = 4.03e-07

czxx = 8.33e-09 czxy = 3.35e-08 Czxz = 1.47e-08

czyx = 4.76e-08 czyy = 8.63e-07 Czyz = 1.33e-07

czzx = 1.59e-08 czzy = 9.91e-08 Czzz = 5.49e-08

Analytic Analytic 11

Analytic Analytic 22

Analytic Analytic 33

Analytic Analytic LightLight

9-12 March 2005, GSI9-12 March 2005, GSI S. Gorbunov, I. Kisel, E. KryshenS. Gorbunov, I. Kisel, E. Kryshen 88/12/12

Performance of Extrapolation Performance of Extrapolation

Method Residuals Pullsp/p x y tx ty q/p x y tx ty

Runge-Kutta 4 0.64 27 24 1.5 1.5 1.17 1.05 1.01 1.02 1.00

Analytic 3 0.64 27 24 1.5 1.5 1.18 1.05 1.00 1.02 1.00

Analytic 2 0.68 27 24 1.5 1.5 1.30 1.08 1.01 1.03 1.00

Analytic 1 0.94 30 25 1.5 1.5 1.90 1.37 1.03 1.10 1.02

Analytic Light 0.64 27 24 1.5 1.5 1.19 1.05 1.00 1.02 1.00

Analytic Central 2.49 38 25 1.7 1.5 3.77 2.23 1.03 1.33 1.00

9-12 March 2005, GSI9-12 March 2005, GSI S. Gorbunov, I. Kisel, E. KryshenS. Gorbunov, I. Kisel, E. Kryshen 99/12/12

Kalman Filter SchemeKalman Filter Scheme

Initializationr0,C0

Extrapolationrk = Ak-1 rk-1

Ck = Ak-1Ck-1ATk-1+Qk

FilteringKk= CkHT

k(Vk+HkCkHTk)-1

rk+1= rk+Kk(mk-Hkrk)Ck+1= (E-KkHk)Ck

Final resultrn,Cn

NoiseQk

Measurement

mk, Hk, Vk

rk,C

k

rk,Ck

9-12 March 2005, GSI9-12 March 2005, GSI S. Gorbunov, I. Kisel, E. KryshenS. Gorbunov, I. Kisel, E. Kryshen 1010/12/12

Kalman Smoother SchemeKalman Smoother Scheme

Final resultr0

n,C0n

Extrapolationr0

k= Ak-1r0

k+1

C0k= Ak

-1C0k+1A-1T

k

SmoothingKk= Qk(Ck+Qk)-1

r0k= r0

k+Kk(rk-r0k)

C0k= KkCk+(E-Kk)C0

k(E-Kk)T

NoiseQk

r0k+1,C0

k+1

9-12 March 2005, GSI9-12 March 2005, GSI S. Gorbunov, I. Kisel, E. KryshenS. Gorbunov, I. Kisel, E. Kryshen 1111/12/12

Kalman SmootherKalman Smoother

Optimal track parameter estimation at each measurementOptimal track parameter estimation at each measurement

Optimal track parameter estimation between measurementsOptimal track parameter estimation between measurements

Detection and suppression of outliersDetection and suppression of outliers

Method Residuals Pullsp/p x y tx ty q/p x y tx ty

Analytic Light

0.64

27 24 1.5 1.5 1.19 1.05 1.00 1.02 1.00

+ Smoother 0.64

27 24 1.5 1.5 1.19 1.05 1.00 1.02 1.00

9-12 March 2005, GSI9-12 March 2005, GSI S. Gorbunov, I. Kisel, E. KryshenS. Gorbunov, I. Kisel, E. Kryshen 1212/12/12

SummarySummary

• An analytic formula of extrapolation in magnetic field has An analytic formula of extrapolation in magnetic field has been derivedbeen derived

• A simplified formula is used in the KF fitter and CA track finder A simplified formula is used in the KF fitter and CA track finder • Kalman smoother is ready for track fitting in presence of noise Kalman smoother is ready for track fitting in presence of noise hitshits

• Tracking is well tested and availableTracking is well tested and available