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
