Track Fitting
(
Kalman
Filter)
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
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
Information Flow in the Track Fit
Effects
influencing the
amount of
information
contained in the
measurements
Information that
the fit has to
take into account
Increase of information
Dilution of information
Origin
Kalman
Filter
The Kalman filter process is a successive approximation scheme to estimate parameters
Simple Example: 2 parameters

intercept and slope: x = x
0
+ S
x
* z; P = (x
0
, S
x
)
Errors on parameters x
0
& S
x
(covariance matrix): C =
Cx

x Cx

s
Cs

x Cs

s
Cx

x = <(x

x
m
)(x

x
m
)>
In general
C = <(P

P
m
)(P

P
m
)
T
>
Propagation:
x(k+1) = x(k)+Sx(k)*(z(k+1)

z(k))
Pm(k+1) = F(
d
z) * P(k) where
F(
d
z) =
1 z(k+1)

z(k)
0 1
Cm(k+1) = F(
d
z) *C(k) * F(
d
z)
T
+ Q(k)
k
k+1
Noise: Q(k)
(Multiple Scattering)
P(k)
Pm(k+1)
Kalman
Filter
Form the weighted average
of the k+1 measurement and
the propagated track model:
Weights given by inverse of
Error Matrix: C

1
Hit: X(k+1) with errors V(k+1)
P(k+1) =
Cm

1
(k+1)*Pm(k+1)+ V

1
(k+1)*X(k+1)
Cm

1
(k+1) + V

1
(k+1)
k
k+1
Noise
(Multiple Scattering)
and C(k+1) =
(Cm

1
(k+1) + V

1
(k+1))

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.
Pm(k+1)
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 &
c
2
:
Residuals: r(k) = X(k)

Pm(k)
Covariance of r(k): Cr(k) = V(k)

C(k)
Then:
c
2
= r(k)
T
Cr(k)

1
r(k) for the k
th
step
point k

1
How the
Kalman
Filter
Works


details
1.
Trajectory until point
(k

1)
point k

1
Prediction
How the
Kalman
Filter Works
1.
Trajectory until point
(k

1)
2.
Prediction (without
process noise)
point k

1
Prediction
Filtering of
k

th
point
1.
Trajectory until point
(k

1)
2.
Prediction (with
process noise =
mult
.
scattering)
3.
Filter
How the
Kalman
Filter Works
point k

1
Prediction
Multiple scattering
1.
Trajectory until point
(k

1)
2.
Prediction (with
process noise =
mult
.
scattering)
How the
Kalman
Filter Works
point k

1
Prediction
Filtering of
k

th
point
Multiple scattering
1.
Trajectory until point
(k

1)
2.
Prediction (with
process noise =
mult
.
scattering)
3.
Filter
How the
Kalman
Filter Works
Some Math: Prediction
Parameters & covariance
matrix at (k

1)
Prediction
Transport matrix
Process noise
Prediction equations
•
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)
Prediction vs. Measurement
Measurement & covariance
matrix at (
k
)
Residual
Projection matrix
•
Projection matrix
H
k
connects parameter vector (e.g. 5D) and the
actual measurement (e.g. 1D)
Measurement
equations
Some Math: Filter
Filtered parameters &
covariance matrix at (
k
)
“Gain matrix”
Filter equations
In this formulation (“gain matrix formalism”), the
matrix that needs to be inverted has only the
dimension of the measurement (here: 1)
In this formulation (“gain matrix formalism”), the
matrix that needs to be
inverted
has only the
dimension of the measurement (here: 1)
Along the Trajectory
•
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
production
vertex
direction of flight
direction of filter
production
vertex
Along the Trajectory (cont’d)
•
If precise parameters at both ends are needed,
two
filters
in opposite directions can be combined
production
vertex
direction of filter 1
direction of filter 2
production
vertex
Along the Trajectory (cont’d)
•
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
production
vertex
direction of flight
direction of filter
direction of smoother
production
vertex
Process Noise & How to Calculate It
•
Important: multiple
scattering model
•
evaluate contribution to
covariance matrix
–
depends on
track model
(example is for
t
x
= tan
x
,
t
y
= tan
y
)
angular elements of
Q
(
t
= thickness in terms
of radiation length)
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)
28

Jul

2004
R. Mankel, Kalman Filter Techniques
21
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
28

Jul

2004
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 97

165 and 98

001, and A.
Spiridonov
, HERA

B note 98

133
•
In case derivatives depend on parameters, iteration
may be needed
28

Jul

2004
R. Mankel, Kalman Filter Techniques
23
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
c
2
contribution of the hit to the fit
–
in
Kalman
filter language:
smoothed
c
2
s
•
Problems:
–
good hits can have a worse
c
2
than bad hits nearby
that are causing the problem
–
“digital” decisions may result in
bad convergence
28

Jul

2004
R. Mankel, Kalman Filter Techniques
24
Robust Estimation
•
Least squares fitting (& thereby
Kalman
filtering) reaches its limits
when underlying statistics are
far
from Gaussian
–
typical example:
c
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
28

Jul

2004
R. Mankel, Kalman Filter Techniques
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
28

Jul

2004
R. Mankel, Kalman Filter Techniques
26
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
—
622
(online at
http://stacks.iop.org/RoPP/67/553
)
Cosmic Rays
History
1785
Charles Coulomb, 1900 Elster and Geitel
Charged body in air becomes discharged
–
there are ions in the atmosphere
1912
Victor Hess
Discovery of “Cosmic Radiation” in 5350m balloon flight, 1936 Nobel Prize
1902
Rutherford, McLennan, Burton:
air is traversed by extremely penetrating radiation (
g
rays excluded later)
1933
Sir Arthur Compton
Radiation intensity depends on magnetic latitude
1933
Anderson
Discovery of the positron in CRs
–
shared 1936 Nobel Prize with Hess
1938
Pierre Auger and Roland Maze
Rays in detectors separated by 20 m (later 200m) arrive simultaneously
1985
Sekido and Elliot
“Somewhat” open
question today: where do they come from ?
1937
Street and Stevenson
Discovery of the muon in CRs (207 times heavier than electron)
First correct explanation:
very energetic ions impinging on top of atmosphere
Victor Hess, return from his
decisive flight 1912
(reached 5350 m !)
radiation increase > 2500m
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)
Major source of
6
Li,
9
Be,
10
B in the Universe (some
7
Li,
11
B)
Cosmic Ray p or
a
C,N, or O
(He in early universe)
Li, Be, or B
NSCL Experiment for Li, Be, and B production by
a
+
a
collisions
Mercer et al. PRC 63 (2000) 065805
170

600 MeV
Measure cross section: how many nuclei are made per incident
a
particle
Identify and count Li,Be,B particles
Cosmic Ray (Ion, for example proton)
Atmospheric Nucleus
p
o
p

p
+
g
g
e
+
e

g
e

Electromagnetic
Shower
p
o
p

p
+
(mainly
g

rays)
m
+
(~4 GeV, ~150/s/cm
2
)
n
m
Hadronic Shower
(on earth mainly muons
and neutrinos)
(about 50 secondaries after first collision)
Ground based observations
Space
Earth’s atmosphere
Plus some:
Neutrons
14
C (1965 Libby)
Cosmic ray muons on earth
Lifetime: 2.2
m
s
–
then decay into electron and neutrino
Travel time from production in atmosphere (~15 km): ~50
m
s
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%
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
electrons
g

rays
muons
Ground array measures lateral distribution
Primary energy proportional to density 600m from shower core
Fly’s Eye technique measures
fluorescence emission
The shower maximum is given by
X
max
~ X
0
+ X
1
log E
p
where X
0
depends on primary type
for given energy E
p
Atmospheric Showers and their Detection
Air Shower Physics
The actors
:
Nuclei composed of nucleons N (p,n)
Pions:
π
+
,
π

,
π
0
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

,
π
0
)
π
±
+ N
lots of hadronic particles and anti

particles
(mostly pions, equal mix of
π
+
,
π

,
π
0
)
π
±
μ
±
+
ν
(decay lifetime is 1/100 muon lifetime)
π
0
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”]
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/cm
2
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.
γ
e
+
e

e
+
γ
γ
e

e
+
e
+
γ
γ
e

e

e

e
+
Each
π
0
decay produces two photons (
γ
’s),
which transfers energy from the “hadronic
cascade” to the “electromagnetic cascade.”
Particle detector arrays
Largest, prior to Pierre Auger project:
AGASA (Japan)
111
scintillation detectors over 100 km
2
Other example: Casa Mia, Utah:
Air Scintillation detector
1981
–
1992: Fly’s Eye, Utah
1999

: 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 20

30 km height
Fly’s eye
Fly’s Eye
Fly’s eye principle
Pierre Auger Project
Combination of both techniques
Site: Argentina + ?. Construction started, 18 nations involved
Largest detector ever: 3000 km
2
,
1600 detectors
40 out of 1600 particle detectors setup (30 run)
2 out of 26 fluorescence telescopes run
Other planned next generation observatories
OWL
(NASA)
(Orbiting Wide Angle Light Collectors)
Idea: observe fluorescence from space to use larger detector volume
EUSO
(ESA for ISS)
(Extreme Universe Space Observatory)
Energies of primary cosmic rays
~E

2.7
~E

3.0
~E

2.7
Observable by
satellite
Man made accelerators
~E

3.3
Lower energies
do not reach earth
(but might get
collected)
40 events > 4e19 eV
7 events > 1e20 eV
Record: October 15, 1991
Fly’s Eye: 3e20 eV
UHECR’s:
many Joules in one particle!
Origin of cosmic rays with E < 10
18
eV
Direction cannot be determined because of deflection in galactic magnetic field
Galactic magnetic field
M83 spiral galaxy
Precollapse structure of massive star
Iron core collapses and triggers supernova explosion
Supernova 1987A by Hubble Space Telescope Jan 1997
Supernova 1987A seen by Chandra X

ray observatory, 2000
Shock wave hits inner ring of material and creates intense X

ray radiation
Cosmic ray acceleration in supernova shockfronts
No direct evidence but model works up to 10
18
eV:
•
acceleration up to 10
15
eV in one explosion, 10
18
eV 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)
Ultra high energy cosmic rays (UHECR) E > 5 x 10
19
eV
Record event: 3 x 10
20
eV 1991 with Fly’s eye
About 14 events with E > 10
20
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
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 10
20
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…
Possible Solutions to the Puzzle
AGASA Data
HIRES Data
1. Maybe the non

observation of the GZK cutoff is an artefact ?
2. Maybe intergalactic magnetic fields as high as ~micro Gauss
cutoff seen ?
then even UHECR from nearby galaxies would appear isotropic
problem with systematic errors in energy determination ?
The structure of the spectrum and scenarios of its origin
supernova remnants
pulsars, galactic wind
AGN, top

down ??
knee
ankle
toe ?
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