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Nov 16, 2013 (4 years and 1 month ago)

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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 ?