1. Track Fitting (Kalman Filter)

2. Least Squares Fitting Generally accepted solution: Kalman filter
at Gaussian level optimal correction of multiple scattering (“process noise”)
energy loss can be incorporated similarly
with “smoother”, full information at every point of trajectory
convenient for matching with other components

3. What the Kalman filter is
A progressive way of performing a least-squares fit
Mathematically equivalent to the latter
What it is not:
a pattern recognition method (though it can be efficiently used within one)
a “robust” fitting method

4. Information Flow in the Track Fit
Origin
Effects influencing the amount of information contained in the measurements
Information that the fit has to take into account
Dilution of information
Increase of information

5. Kalman Filter
The Kalman filter process is a successive approximation scheme to estimate parameters
Simple Example: 2 parameters - intercept and slope: x = x0 + Sx * z; P = (x0 , Sx)
Errors on parameters x0 & Sx (covariance matrix): C =
Cx-x Cx-s
Cs-x Cs-s
Cx-x = <(x-xm)(x-xm)>
In general
C = <(P - Pm)(P-Pm)T>
Propagation:
x(k+1) = x(k)+Sx(k)*(z(k+1)-z(k))
Pm(k+1) = F(dz) * P(k) where
F(dz) =
Pm(k+1)
P(k)
z(k+1)-z(k)
Noise: Q(k)
(Multiple Scattering)
k+1
Cm(k+1) = F(dz) *C(k) * F(dz)T + Q(k) k

6. Kalman Filter Form the weighted average of the k+1 measurement and
Noise
(Multiple Scattering)
Form the weighted average
of the k+1 measurement and
the propagated track model:
Weights given by inverse of
Error Matrix: C-1
Pm(k+1)
Hit: X(k+1) with errors V(k+1)
Cm-1(k+1)*Pm(k+1)+ V-1(k+1)*X(k+1)
P(k+1) =
and C(k+1) =
(Cm-1(k+1) + V-1(k+1))-1
Cm-1(k+1) + V-1(k+1)
Now its repeated for the k+2 planes and so - on. This is called
FILTERING - each successive step incorporates the knowledge
of previous steps as allowed for by the NOISE and the aggregate
sum of the previous hits.

7. Kalman Filter We start the FILTER process at the conversion point
BUT… We want the best estimate of the track parameters
at the conversion point.
Must propagate the influence of all the subsequent Hits backwards
to the beginning of the track - Essentially running the FILTER in
reverse.
This is call the SMOOTHER & the linear algebra is similar.
Residuals & c2:
Residuals: r(k) = X(k) - Pm(k)
Covariance of r(k): Cr(k) = V(k) - C(k)
Then: c2 = r(k)TCr(k)-1r(k) for the kth step

8. How the Kalman Filter Works -- details
Trajectory until point (k-1)
point k-1

9. How the Kalman Filter Works
Trajectory until point (k-1)
Prediction (without process noise)
Prediction
point k-1

10. How the Kalman Filter Works
Trajectory until point (k-1)
Prediction (with process noise = mult. scattering)
Filter
Prediction
point k-1
Filtering of k-th point

11. How the Kalman Filter Works
Trajectory until point (k-1)
Prediction (with process noise = mult. scattering)
Multiple scattering
Prediction
point k-1

12. How the Kalman Filter Works
Trajectory until point (k-1)
Prediction (with process noise = mult. scattering)
Filter
Multiple scattering
Prediction
point k-1
Filtering of k-th point

13. Some Math: Prediction Parameters & covariance matrix at (k-1)
Prediction equations
Transport matrix
Process noise
Transports the information up to the (k-1)-th hit to the location of the k-th hit
Process noise takes random perturbations into account (e.g. multiple scattering, radiation)

14. Prediction vs. Measurement
Measurement & covariance matrix at (k)
Measurement equations
Residual
Projection matrix
Projection matrix Hk connects parameter vector (e.g. 5D) and the actual measurement (e.g. 1D)

15. Some Math: Filter
“Gain matrix”
In this formulation (“gain matrix formalism”), the matrix that needs to be inverted has only the dimension of the measurement (here: 1)
Filter equations
Filtered parameters & covariance matrix at (k)
In this formulation (“gain matrix formalism”), the matrix that needs to be inverted has only the dimension of the measurement (here: 1)

16. Along the Trajectory
production vertex
direction of flight 
production vertex
 direction of filter
Traditionally, the Kalman filter proceeds in the direction opposite to the particle’s flight
parameter estimate near point of origin contains information of all hits & is most precise

17. Along the Trajectory (cont’d)
production vertex
direction of filter 1 
production vertex
 direction of filter 2
If precise parameters at both ends are needed, two filters in opposite directions can be combined

18. Along the Trajectory (cont’d)
production vertex
direction of flight 
production vertex
 direction of filter
direction of smoother 
The orthodox method of propagating the full information to all points of the trajectory is the “Kalman smoother”
Excellent, but computing intensive
one parameter vector size matrix to invert per step

19. Process Noise & How to Calculate It
Important: multiple scattering model
evaluate contribution to covariance matrix
depends on track model (example is for tx = tan x, ty = tan y)
angular elements of Q
(t = thickness in terms of radiation length)

20. Extended (“thick”) Scatterers
In this case, also the spatial components of the process noise matrix Q are non-zero
(l = thickness in terms of radiation length, D=direction)

21. R. Mankel, Kalman Filter Techniques
Nonlinear fit
With non-linear transport or measurement equation, generalizations are necessary
Optimal properties are retained if linear expansion is made in the right places
in general, this requires iteration
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R. Mankel, Kalman Filter Techniques

22. Nonlinear fit (cont’d)
Knowledge of derivatives important
for helical tracks, calculate analytically
for a parameterized inhomogeneous field, transport & calculation of derivatives are usually done numerically
e.g. embedded Runge-Kutta method (adaptive step size)
see T. Oest, HERA-B notes and , and A. Spiridonov, HERA-B note
In case derivatives depend on parameters, iteration may be needed
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23. R. Mankel, Kalman Filter Techniques
Outlier Removal
In least squares fitting, outlier hits have bad influence on the parameter estimate
outliers should be removed
The traditional method of removing outliers is based on the 2 contribution of the hit to the fit
in Kalman filter language: smoothed 2s
Problems:
good hits can have a worse 2 than bad hits nearby that are causing the problem
“digital” decisions may result in bad convergence
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24. R. Mankel, Kalman Filter Techniques
Robust Estimation
Least squares fitting (& thereby Kalman filtering) reaches its limits when underlying statistics are far from Gaussian
typical example: 2 distributions in presence of multiple scattering
This problem is more pressing in electron fitting with plenty of material  radiation
for general treatment, see Stampfer et al, Comp.Phys.Comm. 79, 157
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25. Kalman Filter & Pattern Recognition
Kalman filter can be used very efficiently at the core of track following methods
“Concurrent track evolution”
“Combinatorial Kalman filter”
within & without magnetic field
see for example Nucl. Instr. Meth. A395, 169; Nucl. Instr. Meth. A426, 268
will not be discussed in detail here
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26. R. Mankel, Kalman Filter Techniques
Further Reading
Many excellent papers exist, which unfortunately cannot be done justice by listing them all here
A review of tracking methods with many references to the original literature can be found in R. Mankel, Rep. Prog. Phys. 67 (2004) 553— (online at
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27. Cosmic Rays

28. History
1785 Charles Coulomb, 1900 Elster and Geitel Charged body in air becomes discharged – there are ions in the atmosphere
1902 Rutherford, McLennan, Burton: air is traversed by extremely penetrating radiation (g rays excluded later)
1912 Victor Hess Discovery of “Cosmic Radiation” in 5350m balloon flight, 1936 Nobel Prize
1933 Anderson Discovery of the positron in CRs – shared 1936 Nobel Prize with Hess
1933 Sir Arthur Compton Radiation intensity depends on magnetic latitude
1937 Street and Stevenson Discovery of the muon in CRs (207 times heavier than electron)
1938 Pierre Auger and Roland Maze Rays in detectors separated by 20 m (later 200m) arrive simultaneously
1985 Sekido and Elliot
First correct explanation:
very energetic ions impinging on top of atmosphere
“Somewhat” open question today: where do they come from ?

29. Victor Hess, return from his decisive flight 1912
(reached 5350 m !) radiation increase > 2500m

31. Satellite observations of primaries
Primaries: energetic ions of all stable isotopes:
~85% protons, ~12% a particles
Similar to solar elemental abundance distribution but differences due to spallation during travel through space (smoothed pattern)
Li, Be, or B
Cosmic Ray p or a
C,N, or O (He in early universe)
Major source of 6Li, 9Be, 10B in the Universe (some 7Li, 11B)

32. NSCL Experiment for Li, Be, and B production by a+a collisions
Mercer et al. PRC 63 (2000)
MeV
Identify and count Li,Be,B particles
Measure cross section: how many nuclei are made per incident a particle

33. p+ po p- po p+ p- e+ e- m+ (~4 GeV, ~150/s/cm2) nm e- g g g
Ground based observations
Space
Cosmic Ray (Ion, for example proton)
Earth’s atmosphere
Atmospheric Nucleus
(about 50 secondaries after first collision) p+ po p- po g g p+ p- e+ e-
m+ (~4 GeV, ~150/s/cm2) nm e- g
Electromagnetic Shower
Hadronic Shower
Plus some: Neutrons 14C (1965 Libby)
(on earth mainly muons
and neutrinos)
(mainly g-rays)

34. Cosmic ray muons on earth
Lifetime: 2.2 ms – then decay into electron and neutrino
Travel time from production in atmosphere (~15 km): ~50 ms why do we see them ?
Average energy: ~4 GeV (remember: 1 eV = 1.6e-19 J)
Typical intensity: 150 per square meter and second
Modulation of intensity with sun activity and atmospheric pressure ~0.1%

35. Ground based observations
Advantage: Can build larger detectors can therefore see rarer cosmic rays
Disadvantage: Difficult to learn about primary
Observation methods:
1) Particle detectors on earth surface Large area arrays to detect all particles in shower
2) Use Air as detector (Nitrogen fluorescence  UV light)
Observe fluorescence with telescopes
Particles detectable across ~6 km Intensity drops by factor of 10 ~500m away from core

36. Atmospheric Showers and their Detection
electrons
-rays
muons
Fly’s Eye technique measures
fluorescence emission
The shower maximum is given by
Xmax ~ X0 + X1 log Ep
where X0 depends on primary type
for given energy Ep
Ground array measures lateral distribution
Primary energy proportional to density 600m from shower core

37. Air Shower Physics The actors: Nuclei composed of nucleons N (p,n)
Pions: π+, π-, π0
Muons: μ+, μ-
Electrons, positrons: e+, e-
Gamma rays [photons]: γ
The actions:
N + N  lots of hadronic particles and anti-particles
(mostly pions, equal mix of π+,π-,π0)
π±+ N  lots of hadronic particles and anti-particles
π±  μ± + ν (decay lifetime is 1/100 muon lifetime)
π0  γ + γ immediate decay (10-16 sec)
γ  e+ + e (and recoiling nucleus) [“pair production”]
e±  e± + γ (and recoiling nucleus) [“bremsstrahlung” or “brake radiation”]

38. Air shower building block: The electromagnetic cascade e+ e-
Each π0 decay produces two photons (γ’s), which transfers energy from the “hadronic cascade” to the “electromagnetic cascade.” γ
Air shower building block:
The electromagnetic cascade
Pair production and bremsstrahlung
In this simplified picture, the
particle number doubles in each
generation.
Each generation takes one
radiation length (37 g/cm2 in air).
The cascade continues to grow until the average energy per particle is less than an electron loses to ionization in one radiation length (81 MeV). It is then at its maximum “size,” and the number of particles then decreases. e+ e- γ e- e+ γ γ e+ e- e+ e+ e- e- γ

39. Particle detector arrays
Largest, prior to Pierre Auger project: AGASA (Japan) 111 scintillation detectors over 100 km2
Other example: Casa Mia, Utah:

40. Air Scintillation detector
1981 – 1992: Fly’s Eye, Utah : HiRes, same site
2 detector systems for stereo view
42 and 22 mirrors a 2m diameter
each mirror reflects light into 256 photomultipliers
see’s showers up to km height

42. Fly’s eye

43. Fly’s Eye

44. Fly’s eye principle

45. Pierre Auger Project
Combination of both techniques
Site: Argentina + ?. Construction started, 18 nations involved Largest detector ever: 3000 km2, 1600 detectors
40 out of 1600 particle detectors setup (30 run) 2 out of 26 fluorescence telescopes run

46. Other planned next generation observatories
Idea: observe fluorescence from space to use larger detector volume
OWL (NASA) (Orbiting Wide Angle Light Collectors)
EUSO (ESA for ISS) (Extreme Universe Space Observatory)

47. Energies of primary cosmic rays
Observable by satellite
~E-3.0
~E-3.3
Lower energies do not reach earth (but might get collected)
~E-2.7
UHECR’s:
40 events > 4e19 eV 7 events > 1e20 eV Record: October 15, 1991 Fly’s Eye: 3e20 eV
many Joules in one particle!
Man made accelerators

48. Origin of cosmic rays with E < 1018 eV
Direction cannot be determined because of deflection in galactic magnetic field
Galactic magnetic field
M83 spiral galaxy

50. Precollapse structure of massive star
Iron core collapses and triggers supernova explosion

51. Supernova 1987A by Hubble Space Telescope Jan 1997

52. Supernova 1987A seen by Chandra X-ray observatory, 2000
Shock wave hits inner ring of material and creates intense X-ray radiation

53. Cosmic ray acceleration in supernova shockfronts
No direct evidence but model works up to 1018 eV:
acceleration up to 1015 eV in one explosion, 1018eV multiple remnants
correct spectral index, knee can be explained by leakage of light particles out of Galaxy (but: hint of index discrepancy for H,He ???)
some evidence that acceleration takes place from radio and X-ray observations
explains galactic origin that is observed (less cosmic rays in SMC)

54. Ultra high energy cosmic rays (UHECR) E > 5 x 1019 eV
Record event: 3 x 1020 eV 1991 with Fly’s eye About 14 events with E > 1020 known Spectrum seems to continue – limited by event rate, no energy cutoff
Good news: sufficiently energetic so that source direction can be reconstructed (true ?)
Isotropic, not correlated with mass of galaxy or local super cluster

55. The Mystery
Isotropy implies UHECR’s come from very far away
But – UHECR’s cannot come from far away because collisions with the cosmic microwave background radiation would slow down or destroy them (most should come from closer than 20 MPc or so – otherwise cutoff at 1020 eV (FOR
PROTONS…!)
Other problem: we don’t know of any place in the cosmos that could accelerate particles to such energies (means: no working model)
Speculations include:
Colliding Galaxies
Rapidly spinning giant black holes
Highly magnetized, spinning neutron stars
New, unknown particles that do not interact with cosmic microwave background
Related to gamma ray bursts ?
Easy explanations: the highest energy UHECRs might not be protons, but rather
heavier nuclei (heavier nuclei have a somewhat higher cutoff).
Systematics on energy scale measurements from AGASA…

56. Possible Solutions to the Puzzle
1. Maybe the non-observation of the GZK cutoff is an artefact ?
AGASA Data
HIRES Data
cutoff seen ?
problem with systematic errors in energy determination ?
2. Maybe intergalactic magnetic fields as high as ~micro Gauss
then even UHECR from nearby galaxies would appear isotropic

57. The structure of the spectrum and scenarios of its origin
supernova remnants
pulsars, galactic wind
AGN, top-down ??
knee
ankle
toe ?