doi:10.1016/j.nima.2004.01.065

ARTICLE IN PRESS

Nuclear Instruments and Methods in Physics Research A 524 (2004) 169–194

Muon trackreconstruction and data selection techniques

AMANDA

J. Ahrens

, X. Bai

, R. Bay

, S.W. Barwick

, T. Becka

, J.K. Becker

, K.-

H. Becker

, E. Bernardini

, D. Bertrand

, A. Biron

, D.J. Boersma

,S.B

oser

O. Botner

, A. Bouchta

, O. Bouhali

, T. Burgess

, S. Carius

, T. Castermans

D. Chirkin

, B. Collin

, J. Conrad

, J. Cooley

, D.F. Cowen

, A. Davour

C. De Clercq

, T. DeYoung

, P. Desiati

, J.-P. Dewulf

, P. Ekstr

, T. Feser

M. Gaug

, T.K. Gaisser

, R. Ganugapati

, H. Geenen

, L. Gerhardt

, A. Gro

A. Goldschmidt

, A. Hallgren

, F. Halzen

, K. Hanson

, R. Hardtke

T.Harenberg

,T.Hauschildt

,K.Helbing

,M.Hellwig

,P.Herquet

,G.C.Hill

D. Hubert

, B. Hughey

, P.O. Hulth

, K. Hultqvist

, S. Hundertmark

J. Jacobsen

, A. Karle

, M. Kestel

,L.K

opke

, M. Kowalski

, K. Kuehn

J.I. Lamoureux

, H. Leich

, M. Leuthold

, P. Lindahl

, I. Liubarsky

, J. Madsen

P. Marciniewski

, H.S. Matis

, C.P. McParland

, T. Messarius

, Y. Minaeva

P. Mio

inovi

, P.C. Mock

, R. Morse

, K.S. M

unich

, J. Nam

, R. Nahnhauer

T. Neunh

offer

, P. Niessen

, D.R. Nygren

,H.

Ogelman

, Ph. Olbrechts

C. P

erez de los Heros

, A.C. Pohl

, R. Porrata

, P.B. Price

, G.T. Przybylski

K. Rawlins

, E. Resconi

, W. Rhode

, M. Ribordy

, S. Richter

, J. Rodr

ıguez

Martino

, D. Ross

, H.-G. Sander

, K. Schinarakis

, S. Schlenstedt

, T. Schmidt

D.Schneider

,R.Schwarz

,A.Silvestri

,M.Solarz

,G.M.Spiczak

,C.Spiering

M. Stamatikos

, D. Steele

, P. Steffen

, R.G. Stokstad

, K.-H. Sulanke

O. Streicher

, I. Taboada

, L. Thollander

, S. Tilav

, W. Wagner

, C. Walck

Y.-R. Wang

, C.H. Wiebusch

*, C. Wiedemann

, R. Wischnewski

, H. Wissing

K. Woschnagg

, G. Yodh

Institute of Physics, University of Mainz, D55099 Mainz, Germany

Bartol Research Institute, University of Delaware, Newark, DE 19716, USA

Department of Physics, University of California, Berkeley, CA 94720, USA

Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA

Fachbereich 8 Physik, BU Wuppertal, Gaussstrasse 20, D42119 Wuppertal, Germany

DESYZeuthen, D15738 Zeuthen, Germany

Universit

e Libre de Bruxelles, Science Faculty, Brussels, Belgium

Division of High Energy Physics, Uppsala University, S75121 Uppsala, Sweden

*Corresponding author. Tel.: +49-0-202-439-3531; fax: +49-0-202-439-2662.

E-mail address:

wiebusch@physik.uni-wuppertal.de (C.H. Wiebusch).

0168-9002/$ - see front matter

doi:10.1016/j.nima.2004.01.065

Department of Physics, Stockholm University, SE10691 Stockholm, Sweden

Department of Technology, Kalmar University, S39182 Kalmar, Sweden

University of MonsHainaut, 7000 Mons, Belgium

Department of Physics, Pennsylvania State University, University Park, PA 16802, USA

Department of Physics, University of Wisconsin, Madison, WI 53706, USA

Vrije Universiteit Brussel, Dienst ELEM, B1050 Brussels, Belgium

Department of Physics, University of Maryland, College Park, MD 20742, USA

Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

Blackett Laboratory, Imperial College, London SW7 2BW, UK

Physics Department, University of Wisconsin, River Falls, WI 54022, USA

Department de F

ısica, Universidad Sim

on Bol

ıvar, Caracas, 1080, Venezuela

The AMANDA Collaboration

Received 24 September 2003; received in revised form 20 January 2004; accepted 27 January 2004

Abstract

The Antarctic Muon And Neutrino Detector

rray (AMANDA) is a high-energy neutrino telescope operating at the

geographic South Pole. It is a lattice of photo-multiplier tubes buried deep in the polar ice between 1500 and 2000 m

The primary goal of this detector is to discover astrophysical sources of high-energy neutrinos. A high-energy muon

neutrino coming through the earth from the Northern Hemisphere can be identified by the secondary muon moving

upward through the detector.

The muon tracks are reconstructed with a maximum likelihood method. It models the arrival times and amplitudes of

Cherenkov photons registered by the photo-multipliers. This paper describes the different methods of reconstruction,

which have been successfully implemented within

AMANDA

. Strategies for optimizing the reconstruction performance and

rejecting background are presented. For a typical analysis procedure the direction of tracks are reconstructed with

about 2

accuracy.

PACS:

95.55.Vj; 95.75.Pq; 29.40.Ka; 29.85.+c

Keywords:

AMANDA

; Trackreconstruction; Neutrino telescope; Neutrino astrophysics

1. Introduction

The Antarctic Muon And Neutrino Detector

Array

[1],

AMANDA

, is a large volume neutrino

detector at the geographic South Pole. It is a lattice

of photo-multiplier tubes (PMTs) buried deep in

the optically transparent polar ice. The primary

goal of this detector is to detect high-energy

neutrinos from astrophysical sources, and deter-

mine their arrival time, direction and energy.

When a high-energy neutrino interacts in the

polar ice via a charged current reaction with a

nucleon

;

it creates a hadronic cascade, X, and a lepton,

;

These particles generate Cherenkov

photons, which are detected by the PMTs. Each

lepton flavor generates a different signal in the

detector. The two basic detection modes are

sketched in Fig. 1.

A high-energy

charged current interaction

creates a muon, which is nearly collinear with the

neutrino direction; having a mean deviation angle

?ð

TeV

[2], which implies an

accuracy requirement of

for reconstructing

the muon direction.

The high-energy muon emits a cone of Cher-

enkov light at a fixed angle

It is determined by

cos

¼ð

;

where

32 is the index of

ARTICLE IN PRESS

J. Ahrens et al. / Nuclear Instruments and Methods in Physics Research A 524 (2004) 169–194

170

refraction in the ice. For relativistic particles,

;

and

The direction of the muon is

reconstructed from the time and amplitude in-

formation of the PMTs illuminated by the

Cherenkov cone.

Radiative energy loss processes generate sec-

ondary charged particles along the muon trajec-

tory, which also produce Cherenkov radiation.

These additional photons allow an estimate of the

muon energy. However, the resolution is limited

by fluctuations of these processes. This estimate is

a lower bound on the neutrino energy, because it is

based on the muon energy at the detector. The

interaction vertex may be far outside the detector.

The

and

channels are different. The

electron from a

will generate an electro-

magnetic cascade, which is confined to a volume

of a few cubic meters. This cascade coincides with

the hadronic cascade X of the primary interaction

vertex. The optical signature is an expanding

spherical shell of Cherenkov photons with a lar-

ger intensity in the forward direction. The tau

from a

will decay immediately and also gene-

rate a cascade. However, at energies

1 PeV this

cascade and the vertex are separated by several

tens of meters, connected by a single track. This

signature of two extremely bright cascades is

unique for high-energy

;

and it is called a

double

bang event

[3].

The measurement of cascade-like events is

restricted to interactions close to the detector,

thus requiring larger instrumented volumes than

for

detection. Also the accuracy of the direction

measurement is worse for cascades than for long

muon tracks. However, when the flux is diffuse,

the

and

channels also have clear advantages.

The backgrounds from atmospheric neutrinos are

smaller. The energy resolution is significantly

better since the full energy is deposited in or near

the detector. The cascade channel is sensitive to all

neutrino flavors because the neutral current

interactions also generate cascades. In this paper,

we focus on the reconstruction of muon tracks;

details on the reconstruction of cascades are

described in Ref. [4].

The most abundant events in

AMANDA

are atmo-

spheric muons, created by cosmic rays interacting

with the Earth’s atmosphere. At the depth of

AMANDA

their rate exceeds the rate of muons from

atmospheric neutrinos by five orders of magni-

tude. Since these muons are absorbed by the earth,

a muon trackfrom the lower hemisphere is a

unique signature for a neutrino-induced muon.

The reconstruction procedure must have good

angular resolution, good efficiency, and allow

excellent rejection of down-going atmospheric

muons.

This paper describes the methods used to

reconstruct muon tracks recorded in the

AMANDA

experiment. The

AMANDA-II

detector is introduced

in Section 2. The reconstruction algorithms and

their implementation are described in Sections 3–5.

ARTICLE IN PRESS

cascade

muon

PMTs

spherical Cherenkov front

Cherenkov cone

Fig. 1. Detection modes of the

AMANDA

detector: Left: muon tracks induced by muon-neutrinos; Right: Cascades from electron- or tau-

neutrinos.

Muon neutrinos above 1 PeV are absorbed by the Earth. At

these ultra-high-energies (UHE), however, the muon back-

ground from cosmic rays is small and UHE muons coming

from the horizon and above are most likely created by UHE

neutrinos. The search for these UHE neutrinos is described in

Refs. [5,6].

J. Ahrens et al. / Nuclear Instruments and Methods in Physics Research A 524 (2004) 169–194

171

Section 6 summarizes event classes for which the

reconstruction may fail and strategies to identify

and eliminate such events. The performance

of the reconstruction procedure is shown in

Section 7. We discuss possible improvements

in Section 8.

2. The AMANDA detector

The

AMANDA-II

detector (see

Fig. 2) has been

operating since January 2000 with 677 optical

modules (OM) attached to 19 strings. Most of the

OMs are located between 1500 and 2000 m below

the surface. Each OM is a glass pressure vessel,

which contains an 8-in. hemispherical PMT and its

electronics.

AMANDA-B10

the inner core of 302

OMs on 10 strings, has been operating since 1997.

One unique feature of

AMANDA

is that it

continuously measures atmospheric muons in

coincidence with the

South Pole Air Shower

Experiment

surface arrays

SPASE-1

and

SPASE-2

[7]. These muons are used to survey the detector

and calibrate the angular resolution (see Section 7

and

Refs. [8,9]), while providing

SPASE

with

additional information for cosmic ray composition

studies [10].

The PMT signals are processed in a counting

room at the surface of the ice. The analog signals

are amplified and sent to a majority logic trigger

[11]. There the pulses are discriminated and a

trigger is formed if a minimum number of hit

PMTs are observed within a time window of

typically 2

Typical trigger thresholds were 16

hit PMT for

AMANDA-B10

and 24 for

AMANDA-II

For each trigger the detector records the peak

amplitude and up to 16 leading and trailing edge

times for each discriminated signal. The time

resolution achieved after calibration is

5ns

for the PMTs from the first 10 strings, which are

read out via coaxial or twisted pair cables. For the

remaining PMTs, which are read out with optical

fibers the resolution is

5ns

In the cold

environment of the deep ice the PMTs have low

noise rates of typically 1 kHz

The timing and amplitude calibration, the array

geometry, and the optical properties of the ice are

determined by illuminating the array with known

optical pulses from in situ sources [11]. Time

offsets are also determined from the response to

through-going atmospheric muons[12].

ARTICLE IN PRESS

light diffuser ball

HV divider

silicon gel

Module

Optical

pressure

housing

Depth

120 m

AMANDAII

AMANDAB10

Inner 10 strings:

zoomed in on one

optical module (OM)

main cable

PMT

200 m

1000 m

2350 m

2000 m

1500 m

1150 m

Fig. 2. The

AMANDA-II

detector. The scale is illustrated by the Eiffel tower at the left.

Occasionally in the paper we will refer to this earlier

detector instead of the full

AMANDA-II

detector.

J. Ahrens et al. / Nuclear Instruments and Methods in Physics Research A 524 (2004) 169–194

172

The optical absorption length in the ice is

typically 110 m at 400 nm with a strong wave-

length dependence. The effective scattering length

at 400 nm is on average

20 m

It is defined as

cos

;

where

is the scattering length

and

is the scattering angle. The ice parameters

vary strongly with depth due to horizontal ice

layers, i.e., variations in the concentration of

impurities which reflect past geological events

and climate changes [13–19].

3. Reconstruction algorithms

The muon trackreconstruction algorithm is a

maximum likelihood procedure. Prior to recon-

struction simple pattern recognition algorithms,

discussed in Section 4, generate the initial esti-

mates required by the maximum likelihood recon-

structions.

3.1. Likelihood description

The reconstruction of an event can be general-

ized to the problem of estimating a set of unknown

parameters

;

e.g. trackparameters, given a set

of experimentally measured values

The

parameters,

;

are determined by maximizing

the likelihood

which for independent

components

reduces to

Þ¼

Þð

where

is the probability density function

(p.d.f.) of observing the measured value

for

given values of the parameters

[20].

To simplify the discussion we assume that the

Cherenkov radiation is generated by a single

infinitely long muon track(with

1) and forms

a cone. It is described by the following parameters:

¼ð

;

Þð

and illustrated inFig. 3. Here,

is an arbitrary

point on the track. At time

;

the muon passes

with energy

along a direction

The geome-

trical coordinates contain five degrees of freedom.

Along this track, Cherenkov photons are emitted

at a fixed angle

relative to

Within the

reconstruction algorithm it is possible to use a

different coordinate system, e.g.

¼ð

;

The

reconstruction is performed by minimizing

log

with respect to

The values

presently recorded by

AMANDA

are

the time

and duration

TOT

(

Time Over

Threshold

) of each PMT signal, as well as the

peakamplitude

of the largest pulse in each

PMT. PMTs with no signal above threshold are

also accounted for in the likelihood function. The

hit times give the most relevant information.

Therefore we will first concentrate on

3.1.1. Time likelihood

According to the geometry in Fig. 3, photons

are expected to arrive at OM

(at

) at time

geo

?ð

Þþ

tan

vac

with

vac

the vacuum speed of light.

It is

convenient to define a relative arrival time, or

time residual

res

hit

geo

which is the difference between the observed hit

time and the hit time expected for a ‘‘direct

photon’’, a Cherenkov photon that travels un-

delayed directly from the muon to an OM without

scattering.

ARTICLE IN PRESS

Cherenkov light

t , , E

0 0 0

PMTaxis

Fig. 3. Cherenkov light front: definition of variables.

We note that Eq. (4) neglects the effect that Cherenkov light

propagates with group velocity as pointed out in Ref. [21].It

was shown inRef. [14]that for AMANDA this approximation

is justified.

J. Ahrens et al. / Nuclear Instruments and Methods in Physics Research A 524 (2004) 169–194

173

In the ideal case, the distribution,

res

;

would be a delta function. However, in a realistic

experimental situation this distribution is broa-

dened and distorted by several effects, which are

illustrated in

Fig. 4. The PMT jitter limits the

timing resolution

Noise, e.g. darknoise of the

PMT, leads to additional hits which are random in

time. These effects can generate negative

res

values, which would mimic unphysical causality

violations. Secondary radiative energy losses along

the muon trajectory create photons that arrive

after the ideal Cherenkov cone. These processes

are stochastic, and their relative photon yield

fluctuates.

AMANDA

, the dominant effect on photon

arrival times is scattering in the ice.

The effect

of scattering depends strongly on the distance,

;

of the OM from the trackas illustrated in

Fig. 4.

Since the PMTs have a non-uniform angular

response,

res

also depends on the orientation,

;

of the OM relative to the muon track(see

Fig.

3). OMs facing away from the trackcan only see

light that scatters backtowards the PMT face. On

average this effect shifts

res

to later times and

modifies the probability of a hit.

The simplest time likelihood function is based

on a likelihood constructed from

;

the p.d.f. for

arrival times of single photons

at the locations of

the hit OMs

time

hits

res

;

Note that one OM may contribute to this product

with several hits. The function

res

;

obtained from the simulation of photon propaga-

tion through ice (see Section 3.2). However, this

description is limited, because the electrical and

optical signal channels can only resolve multiple

photons separated by a few 100 ns and

10 ns

;

respectively. Within this time window, only the

arrival time of the first pulse is recorded.

This first photon is usually less scattered than

the average single photon, which modifies the

probability distribution of the detected hit time.

The arrival time distribution of the first of

photons is given by

res

Þ¼

res

tres

Þð

res

ÞÞ

is the cumulative distribution of the single

photon p.d.f. The function

res

is called the

multi-photo-electron (MPE) p.d.f. and corre-

spondingly defines

MPE

This concept can be extended to the more

general case of

res

;

the p.d.f. for the

photon out of a total of

to arrive at

res

;

given by

res

Þ¼

res

Þð

res

ÞÞ

?ð

res

ÞÞ

res

specifies the likelihood of arrival times of

individual photoelectrons for averaged time series

photoelectrons. With waveform recording the

arrival times and amplitudes of individual pulses

can be resolved.

When the number of photoelectrons,

;

is not

measured precisely enough, multi-photon informa-

tion can be included via another method. Instead

ARTICLE IN PRESS

res

high

low

+ showers + scattering

close track

far track

+ noise

jitter

jitter jitter

jitter

Fig. 4. Schematic distributions of arrival times

tes

for different

cases: Top left: PMT jitter. Top right: the effect of jitter and

random noise. Bottom left: The effect of jitter and secondary

cascades along the muon track. Bottom right: The effect of

jitter and scattering.

In water detectors this effect is neglected[22]or treated as a

small correction[23].

J. Ahrens et al. / Nuclear Instruments and Methods in Physics Research A 524 (2004) 169–194

174

of measuring

;

the p.d.f. of the first photoelec-

tron can be calculated by convolving the MPE

p.d.f.

;

with the Poisson probability

Poisson

;

where

is the mean expected number

of photoelectrons as a function of the distance,

res

Þ¼

res

This result is called the Poisson Saturated Ampli-

tude (PSA) p.d.f.

[24,25]

and correspondingly

defines

PSA

The constant

renorma-

lizes the p.d.f. to unity.

The probability of (uncorrelated) noise hits is

small. They are further suppressed by a

hit cleaning

procedure (Section 5.3), which is applied before

reconstruction. They are included in the likelihood

function by simply adding a constant p.d.f.

3.1.2. Hit and nohit likelihood

The likelihood in the previous section relies only

on the measured arrival times of photons. How-

ever, the topology of the hits is also important.

PMTs with no hits near a hypothetical trackor

PMTs with hits far from the trackare unlikely.

A likelihood utilizing this information can be

constructed as

hit

;

hit

;

where

is the number of hit OMs and

the

number of operational OMs. The probabilities

hit

and

hit

of observing or not observing a hit

depend on the trackparameters

Additional hits

due to random noise are easily incorporated:

hit

noise

and

hit

Assuming that the probability

hit

is known for

a single photon, the hit and no-hit probabilities of

OMs for

photons can be calculated:

hit

¼ð

hit

and

hit

?ð

hit

The number of photons,

;

depends on

;

the

energy of the muon:

For a fixed track

geometry, the likelihood (Eq. (10)) can be used to

reconstruct the muon energy.

3.1.3. Amplitude likelihood

The peakamplitudes recorded by

AMANDA

can be

fully incorporated in the likelihood [26], which is

particularly useful for energy reconstruction. The

likelihood can be written as

Þð

where

is the probability that OM

observes

an amplitude

;

with

0 for unhit OMs.

and

are weight factors, which describe devia-

tions of the individual OM and the total number of

hit OMs from the expectation.

depends on the

mean number

of expected photoelectrons:

Þ¼

hit

;

The probability

is normalized to the

probability of observing the most likely amplitude

is the probability that a signal of

does not produce a pulse amplitude above the

discriminator threshold. As before,

hit

;

where the no-hit probability is given

by Poisson statistics:

hit

exp

ð?

Þð

noise

The probability of

0 is a special case:

Þ¼

hit

Energy reconstructions based on

this formulation of the likelihood will be referred

to as

Full E

reco

An alternative energy reconstruction technique

(see Section 3.2.4) uses a neural net which is fed

with energy sensitive parameters.

3.1.4. Zenith weighted (Bayesian) likelihood

Another extension of the likelihood

[27–29]

incorporates external information about the muon

flux via Bayes’ Theorem. This theorem states that

for two hypotheses

and

;

Þ¼

Identifying

with the trackparameters

and

with the observations

;

Eq. (14) gives the prob-

ability that the inferred muon track

was in fact

ARTICLE IN PRESS

J. Ahrens et al. / Nuclear Instruments and Methods in Physics Research A 524 (2004) 169–194

175

responsible for the observed event

is the

probability that

;

assumed to be true, would

generate the event

—in other words, the like-

lihood described in the previous sections.

the prior probability of observing the track

;

i.e.,

the relative frequencies of different muon tracks as

a function of their parameters.

;

which is

independent of the trackparameters

;

is a

normalization constant which ensures that

Eq. (14) defines a proper probability. Because the

likelihood is only defined up to an arbitrary

constant factor, this normalization may be ignored

in the present context.

In order to obtain

;

one thus has to

determine the prior probability distribution,

;

of how likely the various possible track directions

are a priori. The reconstruction maximizes the

product of the p.d.f.

and

the prior.

The flux of muons deep underground is reason-

ably well known from previous experiments. Any

point source of muons would be at most a small

perturbation on the flux of penetrating atmo-

spheric muons and muons created by atmospheric

neutrinos. The most striking feature of the back-

ground flux from atmospheric muons is the strong

dependence on zenith angle. For vertically down-

going tracks it exceeds the flux from neutrino

induced muons by about 5 orders of magnitude

but becomes negligible for up-going tracks. This

dependence, which is modeled by a Monte Carlo

calculation [30], acts as a zenith dependent weight

to the different muon hypotheses,

With this

particular choice, some tracks, which would

otherwise reconstruct as up-going, reconstruct as

down-going tracks. This greatly reduces the rate at

which penetrating atmospheric muons are mis-

reconstructed as up-going neutrino events [31].In

principle, a more accurate prior could be used. It

would need to include the depth and energy

dependence of the atmospheric muons as well as

the angular dependence of atmospheric neutrino-

induced muons.

Upon completion of this work, we learned that

this technique was developed independently by the

NEVOD neutrino detector collaboration[32]who

were able to extract an atmospheric neutrino from

a background of 10

atmospheric muons in a

small

surface detector.

3.1.5. Combined likelihoods

The likelihood function

time

of the hit times is

the most important for trackreconstruction.

However, it is useful to include other information

like the hit probabilities. The combined p.d.f. from

Eqs. (7)–(10) is

MPE

hit

MPE

hit

which is particularly effective. Here

is an

optional weight factor which allows the adjust-

ment of the relative weight of the two likelihoods.

This likelihood is sensitive not only to the track

geometry but also to the energy of the muon.

As discussed in Section 3.1.4, the zenith angle-

dependent prior function,

;

can be included as

a multiplicative factor. This combination

Bayes

time

has been used in the analysis of atmospheric

neutrinos

[30]. However, all of these improved

likelihoods are limited by the underlying model

assumption of a single muon track.

3.2. Likelihood implementation

The actual implementation of the likelihoods

requires detailed knowledge of the photon propa-

gation in the ice. On the other hand, efficiency

considerations and numeric problems favor a

simple and robust method.

The photon hit probabilities and arrival time

distributions are simulated as functions of all

relevant parameters with a dedicated Monte Carlo

simulation and archived in large look-up tables.

This simulation is described in Refs. [26,30,33]

The

AMANDA

Collaboration has followed differ-

ent strategies for incorporating this data into the

reconstruction. In principle the probability density

functions are taken directly from these archives.

However, one has to face several technical

difficulties due to the memory requirements of

the archived tables, as well as numeric problems

related to the normalization of interpolated bins

and the calculation of multi-photon likelihoods.

Alternatively, one can simplify the model and

parametrize these archives with analytical func-

tions, which depend only on a reduced set of

parameters. Comparisons of two independent

ARTICLE IN PRESS

J. Ahrens et al. / Nuclear Instruments and Methods in Physics Research A 524 (2004) 169–194

176

parametrizations [24,34] show that the direct and

parametrized approaches yield similar results in

terms of efficiency. This indicates that the para-

metrization itself is not limiting the reconstruction

quality; rather, as mentioned earlier, the recon-

struction is limited by the assumptions of the

model being fit. Therefore, we will concentrate on

only one parametrization.

3.2.1. Analytical parametrization

A simple parametrization of the arrival time

distributions can be achieved with the following

function, which we call

Pandel function

.Itisa

gamma distribution and its usage is motivated by

an analysis of laser light signals in the BAIKAL

experiment[35]. There, it was found that for the

case of an isotropic, monochromatic and point-

like light source,

res

can be expressed in the

form

res

Þ?

?ð

res

medium

Þ¼

medium

without special assumptions on the actual optical

parameters. Here,

medium

vac

is the speed of

light in ice,

the absorption length,

the

Gamma function and

a normalization factor,

which is given by Eq. (18). This formulation has

free parameters

and

;

which are unspecified

functions of the distance

and the other geome-

trical parameters. They are empirically determined

by a Monte Carlo model.

The Pandel function has some convenient

mathematical properties: it is normalized, it is

easy to compute, and it can be integrated

analytically over the time,

res

;

which simplifies

the construction of the multi-photon (MPE) time

p.d.f.. For small distances the function has a pole

0 corresponding to a high probability of an

unscattered photon. Going to larger values of

;

longer delay times become more likely. For

distances larger than the critical value

;

the

power index to

res

changes sign, reflecting that the

probability of undelayed photons vanishes: essen-

tially all photons are delayed due to scattering.

The large freedom in the choice of the two

parameter functions

(units of time) and

(units

of length) and the overall reasonable behavior is

the motivation to use this function to parametrize

not only the time p.d.f. for point-like sources, but

also for muon tracks [34]. The Pandel function is

fit to the distributions of delay times for fixed

distances

and angles

(between the PMT axis

and the Cherenkov cone). These distributions are

previously obtained from a detailed photon

propagation Monte Carlo for the Cherenkov light

from muons. The free fit parameters are

;

and the effective distance

eff

;

which will be

introduced next.

When investigating the fit results as a function

and angle

(see

Fig. 3), we observe that

already for a simple ansatz of constant

;

and

the optical properties in

AMANDA

are described

sufficiently well within typical distances. The

dependence on

is described by an effective

distance

eff

which replaces

in Eq. (17). This

means that the time delay distributions for back-

ward illumination of the PMT is found to be

similar to a head-on illumination at a larger

distance. The following parameters are obtained

for a specific ice model, and are currently used in

the reconstruction:

557 ns

;

eff

;

98 m

;

9 m cos

Þþ

6 m cos

A comparison of the results from this parame-

trization with the full simulation is shown inFig. 5

for two extreme distances. The simple approxima-

tion describes the behavior of the full simulation

reasonably well. However, this simple overall

description has a limited accuracy, especially for

(not shown). Reconstructions, based on the

Pandel function with different ice models, and a

generic reconstruction, that uses the full simula-

tion results, yield similar results. These compar-

isons indicate that the results of the reconstruction

do not critically depend on the fine tuning of the

ARTICLE IN PRESS

J. Ahrens et al. / Nuclear Instruments and Methods in Physics Research A 524 (2004) 169–194

177

underlying ice models, and it justifies the use of the

above simple model.

3.2.2. Extension to realistic PMT signals

Although the Pandel function is the basis of a

simple normalized likelihood, it has several defi-

ciencies. It is not defined for negative

res

;

it ignores

PMT jitter, and it has a pole at

res

;

which

causes numerical difficulties. These problems can

be resolved by convolving the Pandel function,

Eq. (17), with a Gaussian, which accounts for the

PMT jitter. Unfortunately such convolution re-

quires significant computing time.

Instead the Pandel function is modified by

extending it to negative times,

res

;

with a (half)

Gaussian of width

The effects of PMT jitter are

only relevant for small values of

res

For times

res

the original function is used, and the two

parts are connected by a spline interpolation (3rd

order polynomial). The result,

res

;

is called

upandel function

Using

ffiffiffiffiffiffi

and requiring further a

smooth interpolation and the normalization to

be unchanged, the polynomial coefficients

and

normalization of the Gaussian

can be calcu-

lated analytically

[34]. The free parameter

includes all timing uncertainties, not just the

PMT jitter. Good reconstruction results are

achieved for a large range 10

20 ns

3.2.3. P

hit

Parametrization

The normalization

in Eq. (18) is used to

construct a hit probability function,

hit

The

function

hit

with

hit

;

is fit to the hit

probability determined by the full

AMANDA

detector

simulation, as a function of distance, orientation

and muon energy. The free parameters are the

Pandel parameters

and

;

and

The

effective distance

;

is similar to the effective

distance in the Pandel parametrization. We define

n as the power index of Eq. (11), which corre-

sponds to an effective number of photons. It is

important to understand that

is not a hit

probability and

n is not just a number of photons.

They are constructs, that are calibrated with a

Monte Carlo simulation. Technically, the power

index,

in Eq. (11), factorizes into

;

Þ¼

;

The variable

;

where

;

related to the number of photons incident on the

PMT and its absolute efficiency. The factor

;

is related to the orientation dependent

PMT sensitivity but also accounts for the energy-

dependent angular emission profile of photons

with respect to the bare muon.

3.2.4. Energy reconstruction

The reconstruction of the trackgeometry is a

search for five parameters. If the muon energy is

added as a fit parameter, the minimization is

significantly slower. Therefore, the energy recon-

struction is performed in two steps. First, the track

geometry is reconstructed without the energy

parameter. Then these geometric parameters are

used in an energy reconstruction, that only

determines the energy. However, if the time

likelihood utilizes amplitude information, e.g. in

the combined likelihood, Eq. (7), or the PSA

likelihood, Eq. (9), the track parameters also

ARTICLE IN PRESS

time delay / ns

d = 8m

Delay prob / ns

time delay / ns

d = 71m

Delay prob / ns

−

0 200 400

−

0 500 1000 1500

Fig. 5. Comparison of the parametrized Pandel function (dashed curves) with the detailed simulation (blackhistograms) at two

distances

from the muon track.

J. Ahrens et al. / Nuclear Instruments and Methods in Physics Research A 524 (2004) 169–194

178

depend on the energy. In this case the energy and

geometric parameters must be reconstructed to-

gether. Currently, three different approaches are

used to reconstruct the muon energy. They are

compared in Section 7.3.

(1) The simplest method utilizes the

hit

reconstruction (see Sections 3.1.2 and 3.2.3).

(2) The

Full E

reco

method (see Section 3.1.3)

models the measured amplitudes in a like-

lihood for reconstructing the energy. This

algorithm performs better, but it is more

dependent on the quality of the amplitude

calibration of the OMs.

(3) An alternative way to measure the energy is

based on a neural network

[36]. The neural

networkuses 6-6-3-1 and 6-3-5-1 feed-forward

architecture for

AMANDA-B10

and

AMANDA-II

respectively. The energy correlated variables

which are used as input are the mean of the

measured amplitudes (ADC), the mean and

RMS of the arrival times (LE) or pulse

durations (TOT), the total number of signals,

the number of OMs hit and the number of

OMs with exactly one hit.

Less challenging than a full reconstruction, a

lower energy threshold is determined by requiring

a minimum number of hit OMs. The number of hit

OMs is correlated with the energy of the muon.

Since celestial neutrinos are believed to have a

substantially harder spectrum than atmospheric

neutrinos, an excess of high multiplicity events

would indicate that a hard celestial source exists.

Values for this parameter determined from

AMANDA

data already set a tight upper limit on the diffuse

flux of high-energy celestial neutrinos [37].

3.2.5. Cascade reconstruction

The reconstruction of

cascade like events

described in detail elsewhere

[4]. The basic

approach is similar to the trackreconstruction. It

assumes events form a point light source with

photons propagating spherically outside with a

higher intensity in the forward direction. The

cascade reconstruction also uses the Pandel func-

tion (see Eq. (17)) with parameters that are specific

for cascades.

In several muon analyses, a cascade fit is used as

a competing model. In cases where the cascade fit

achieves a better likelihood than the track

reconstruction, the trackhypothesis is rejected.

In particular this is used as a selection criterion to

reject background events which are mis-recon-

structed due to bright secondary cascades.

4. First guess pattern recognition

The likelihood reconstructions need an initial

trackhypothesis to start the minimization. The

initial trackis derived from

first guess methods

which are fast analytic algorithms that do not

require an initial track.

4.1. Direct walk

A very efficient

first guess method

is the

direct

walk

algorithm. It is a pattern recognition algo-

rithm based on carefully selected hits, which were

most likely caused by direct photons.

The four step procedure starts by selecting

track

elements

, the straight line between any two hit

OMs at distance

;

which are hit with a time

difference

vac

30 ns with

50 m

The known positions of the OMs define the track

element direction

;

The vertex position

;

is taken at the center between the two

OMs. The time at the vertex

is defined as the

average of the two hit times.

In a next step, the number of

associated hits

(AH) are calculated for each trackelement.

Associated hits are those with

res

300 ns

and

25 m

res

(

in ns), where

is the

distance between hit OM and trackelement

and

res

is the

time residual

, which is defined in

Eq. (5). After selecting these associated hits, track

elements of poor quality are rejected by requiring:

10 and

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðð

20 m

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179

Here, the ‘‘lever arm’’

is the distance between

the vertex of the trackelement and the point on

the trackelement which is closest to OM

and

is the average of all

-values. Track

elements that fulfill these criteria qualify as

track

candidates

(TC).

Frequently, more than one trackcandidate is

found. In this case, a cluster search is performed for

all trackcandidates that fulfill the quality criterion

max

where

max

and

min

;

In the cluster search, the ‘‘neighbors’’ of each track

candidate are counted, where neighbors are track

candidates with space angle differences of less than

The cluster with the largest number of track

candidates is selected.

In the final step, the average direction of all

trackcandidates inside the cluster defines the

initial trackdirection. The trackvertex and time

are taken from the central track candidate in the

cluster. Well separated clusters can be used to

identify independent muon tracks in events which

contain multiple muons (see Section 6.1).

4.2. Linefit

The

linefit

[38] algorithm produces an initial

trackon the basis of the hit times with an optional

amplitude weight. It ignores the geometry of the

Cherenkov cone and the optical properties of the

medium and assumes light traveling with a velocity

along a 1-dimensional path through the detector.

The locations of each PMT,

;

which are hit at a

time

can be connected by a line

to be minimized is defined as

hit

where

hit

is the number of hits. The

minimized by differentiation with respect to the

free fit parameters

and

This can be solved

analytically

and

where

?ð

hit

denotes the mean

of parameter

with respect to all hits.

The line-fit thus yields a vertex point

;

and a

direction

The zenith angle is given

arccos

jÞ

The time residuals (Eq. (5)) for this initial track

generally do not follow the distribution expected

for a Cherenkov model. If the

parameter of the

initial trackis shifted to better agree with a

Cherenkov model, subsequent reconstructions

converge better (see Section 5.2.3).

The absolute speed

;

of the line-fit is the

mean speed of the light propagating through the

one-dimensional detector projection. Spherical

events (cascades) and high energy muons have

low

values, and thin, long events (minimally

ionizing muon tracks) have large values.

4.3. Dipole algorithm

The

dipole algorithm

considers the

unit

vector

from one hit OM to the subsequently hit OM as an

individual dipole moment. Averaging over all

individual dipole moments yields the global mo-

ment

It is calculated in two steps. First, all hits

are sorted according to their hit times. Then a

dipolemoment

is calculated

It can be expressed via an absolute value

and two angles

and

These angles

define the initial track.

The dipole algorithm does not generate as good

an initial trackas the direct walkor the line-fit, but

it is less vulnerable to a specific class of back-

ground events: almost coincident atmospheric

muons from independent air showers in which

the first muon hits the bottom and the second

muon hits the top of the detector.

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180

4.4. Inertia tensor algorithm

The

inertia tensor algorithm

is based on a

mechanical picture. The pulse amplitude from a

PMT at

corresponds to a virtual mass

One can then define the tensor of inertia

of that

virtual mass distribution. The origin is the center

of gravity

COG

of the mass distribution. The

COG

-coordinates and the tensor of inertia com-

ponents are given by

COG

and

;

?½

?ð

The amplitude weight

0 can be chosen

arbitrarily. The most common settings are

(ignoring the amplitudes) and

1 (setting the

virtual masses equal to the amplitudes). The tensor

of inertia has three eigenvalues

;

corresponding to its three main axes

The

smallest eigenvalue

corresponds to the longest

axis

In the case of a long track-like event

;

and

approximates the direction of

the track. The ambiguity in the direction along the

axis is resolved by choosing the direction where

the average OM hit time is latest. In the case of a

cascade-like event,

The ratios between

the

can be used to determine the sphericity of the

event.

5. Aspects of the technical implementation

5.1. Reconstruction framework

The basic reconstruction procedure, sketched in

Fig. 6, is sequential. A fast reconstruction program

calculates the initial trackhypothesis for the

likelihood reconstruction. All reconstruction pro-

grams may use a reduced set of hits in order to

suppress noise hits and other detector artifacts.

Event selection criteria can be applied after each

step to reduce the event sample, and allow more

time consuming calculations at later reconstruc-

tion stages. This procedure may iterate with more

sophisticated but slower algorithms analyzing

previous results. The final step is usually the

production of

Data Summary Tape

(DST) like

information, usually in form of PAW

-tuples

[39]. A detailed description of this procedure can

be found inRef. [40].

The reconstruction frameworkis implemented

with the

recoos

program[41], which is based on the

rdmc

library [42] and the SiEGMuND software

package [43]. The

recoos

program is highly

modular, which allows flexibility in the choice

and combination of algorithms.

5.2. Likelihood maximization

The aim of the reconstruction is to find the

trackhypothesis which corresponds to the max-

imum likelihood. This is done by minimizing

log

with respect to the trackparameters.

We have implemented several minimization

procedures.

The likelihood space for

AMANDA

events is often

characterized by several minima. Local likelihood

minima can arise due to symmetries in the

detector, especially in the azimuth angle, or due

to unexpected hit times caused by scattering. In the

example, shown in Fig. 7, the reconstruction

converged on a local minimum, because of non-

optimal starting values. Several techniques, which

are used to find the global minimum, are here

presented. In particular, the iterative reconstruc-

tion, Section 5.2.2, solves the problem and

converges to the global minimum. One generally

assumes that the global minimum corresponds to

the true solution, but this is not always correct due

to stochastic nature of light emission and detec-

tion. Such events cannot be reconstructed properly

and have to be rejected using quality parameters

(see Section 6.2).

ARTICLE IN PRESS

Hitcleaning Hitcleaning Selections

↓↓

↓

Data

⇒

first guess

⇒

Likelihood

⇒

Analysis

Selections Selections

Fig. 6. Schematic principle of the reconstruction chain.

J. Ahrens et al. / Nuclear Instruments and Methods in Physics Research A 524 (2004) 169–194

181

5.2.1. Minimization algorithms

The reconstruction frameworkallows us to use

and compare these numerical minimization algo-

rithms:

Simplex

[44],

Powell’s

[44],

Minuit

[45]

(using the

minimize

method), and

Simulated

annealing

[44]. The Simplex algorithm is the fastest

algorithm. Powell’s method and Minuit are

times slower than the Simplex algorithm. The

reconstruction results from Minuit and the Sim-

plex algorithm are nearly identical and almost as

good as the Powell results. Exceptions occur in less

than 1% of the cases, when these methods fail and

stop at the extreme zenith angles

and 80

The Simulated annealing algorithm is less sensitive

to local minima than the other algorithms, but it is

much slower and requires fine-tuning.

5.2.2. Iterative reconstruction

The

iterative reconstruction

algorithm success-

fully copes with the problem of local minima and

extreme zenith angles by performing multiple

reconstructions of the same event. Each recon-

struction starts with a different initial track.

Therefore, the fast Simplex algorithm is sufficient.

The ability to find the global minimum depends

strongly on the quality of the initial track. A

systematic scan of the full parameter space for

initial seeds is not feasible. Instead the iterative

algorithm concentrates on the direction angles,

zenith and azimuth, and uses reasonable values for

the spatial coordinates. The following procedure

yields good results.

The result of a first minimization is saved as a

reference. Then both direction angles are ran-

domly selected. The trackpoint,

;

is transformed

to the point on the new track, which is closest to

the center of gravity of hits. The time,

;

of this

point is shifted to match the Cherenkov expecta-

tion (see Section 5.2.3). Then a new minimization

is started. If the minimum is less than the reference

minimum, it is saved as the new reference. This

procedure is iterated

times, and the best minima

found for zenith angles

above

and

below

the

horizon, are saved, and used to generate an

important selection parameter (see Section 6.2).

This algorithm substantially reduces the number

of false minima found, after a few iterations. For

6 roughly 95% of the results are in the vicinity

of the asymptotic optimum for

For

more than 99% of the results are the global

minimum. Despite the fast convergence, the

iterative reconstruction

requires significant CPU

time, which limits its use to reduced data sets.

5.2.3. Lateral shift and time residual

The efficiency of finding the global minimum of

the likelihood function can be improved by

translating the arbitrary vertex and/or time origin

of the output trackfrom the first guess algorithm

before application of the full maximum likelihood

method.

Transformation of

In general this ‘‘vertex’’

point is arbitrary in the infinite trackapproxima-

tion used, and first guess methods may produce

positions distant from the detector. During the

likelihood minimization, numerical errors can be

avoided and the convergence improved by shifting

this point along the direction of the tracktowards

the point closest to the

center of gravity of hits

(see

Eq. (26)). The vertex time

is transformed

accordingly:

Þ¼

Transformation of t

The time

obtained from

first guess algorithms is not calculated from a full

Cherenkov model. The efficiency of the likelihood

reconstruction can be improved by shifting the

such that the time residuals, Eq. (5), fit better to a

ARTICLE IN PRESS

180

160

140

120

100

170

175

180

185

190

Parabola fit

experimental data

−

log (likelihood)

theta [

]

Fig. 7. An example of the likelihood space (one-dimensional

projection) for a specific

AMANDA

event. Shown is

log

function of the zenith angle. Each point represents a fit, for

which the zenith angle was fixed and the other trackparameters

were allowed to vary in order to find the best minimum. A local

minimum which was found by a gradient likelihood minimiza-

tion is indicated by a fitted parabola. Improved methods that

avoid this are described in the text.

J. Ahrens et al. / Nuclear Instruments and Methods in Physics Research A 524 (2004) 169–194

182

Cherenkov model. In particular it is useful to

avoid negative

res

;

which would correspond to

causality violations. This can be achieved by

transforming

res

;

where

res

is the most

negative time residual.

5.2.4. Coordinates and restricted parameters

The trackcoordinates

;

which are used by the

likelihood, are independent of the coordinates

actually chosen for the minimization. Therefore,

the coordinate system can be chosen arbitrarily.

Any of the parameters in this coordinate system

can be kept fixed. During the minimization

parameterization

functions translate the coordi-

nates as necessary. The most commonly used

coordinates are

and the zenith and azimuth

angles

;

The freedom in the choice of coordinates can be

used to improve the numerical minimization, for

systematic studies, or to fix certain parameters

according to external knowledge. An example is

the reconstruction of coincident events with the

SPASE

surface arrays [8]. Here, we fix the location

of the trajectory to coincide with the core location

at the surface as measured by

SPASE

. Then, the

direction is determined with the

AMANDA

recon-

struction subject to this constraint.

Under certain circumstances the allowed range

of the reconstruction parameters is restricted. The

most important example here is to restrict the

reconstructed zenith angle to above or below the

horizon, to find the most likely up- or down-going

tracks, respectively. Comparing the quality of the

two solutions can be used for background rejec-

tion. Technically the constrained fit is accom-

plished by multiplying the likelihood by a prior,

which is zero outside the allowed parameter range.

5.3. Preprocessing and hit cleaning

The data must be filtered and calibrated before

reconstruction. Defective OMs are removed, and

the amplitudes and hit times are calibrated. A

hit

cleaning

procedure identifies and flags hits which

appear to be noise or electronic effects, such as

cross talkor after-pulsing. These hits are not used

in the reconstruction, but they are retained for

post-reconstruction analysis.

The hit cleaning procedure can be based on

simple and robust algorithms, because the PMTs

have low noise rates. Noise and after-pulse hits are

strongly suppressed by rejecting hits that are

isolated in time and space from other signals in

the detector. Typically a hit is considered to be

noise if there is no hit within a distance of 60–

100 m and a time of

300 to

600 ns

Cross talk

hits are identified by examining the amplitudes and

pulse widths of the individual pulses and by

analyzing the correlations of uncalibrated hit times

with hits of large amplitude in channel combina-

tions which are known to cross talk to each other.

The required cross talkcorrelation map was

determined independently in a dedicated calibra-

tion campaign.

5.4. Processing speeds

The first guess algorithms are sufficiently fast

that the execution time is dominated by file input/

output and the software framework. Typical fit

times are

20 ms per event on a 850 MHz

Pentium-III Linux PC. The processing speed of

the likelihood reconstructions can vary signifi-

cantly depending on the number of free para-

meters, the number of iterations, the minimization

algorithm, and the experimental parameters like

the number of hit OMs. These effects dominate the

differences in processing speeds due to different

reconstruction algorithms. The typical execution

time for a 16-fold iterative likelihood reconstruc-

tions using the simplex minimizer to reconstruct

the five free trackparameters is

250 ms per

event.

6. Background rejection

The performance of the reconstruction depends

strongly on the quality and background selection

criteria. The major classes of background events in

AMANDA

(see Section 6.1) are suppressed by the

quality parameters presented in Section 6.2.

Optimization strategies for the selection criteria

are summarized in Section 6.3. Finally, we

evaluate the reconstruction performance in

Section 7.

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6.1. Background classes

Most background events from atmospheric

muons are well reconstructed and can be

rejected by selecting up-going reconstructed

events. However, there is a small fraction of mis-

reconstructed events, amounting to about 10

for

the unbiased and about 10

for the zenith-

weighted reconstruction. These events are rejected

by additional selection criteria described in Section

6.2. These background events are classified as

follows.

Nearly horizontal muons

: These events have true

incident angles close to the horizon. A small error

in the reconstruction causes them to appear as up-

going. These events are not severely mis-recon-

structed, but occur due to the finite angular

resolution.

Muon bundles

: The spatial separation between

multiple muons from a single air shower, a

muon

bundle

, is usually small enough that the event can

be described by a single bright muon track. If the

separation is too large, the reconstruction fails.

Cascades

: Bright stochastic energy losses (e.g.

bremsstrahlung) produce additional light, which

distorts the Cherenkov cone from the muon.

Cascades emit most of their light with the same

Cherenkov angle as the muon, but some light is

emitted at other angles. These secondary events

can cause the reconstruction to fail, especially

when the cascade(s) produce more light than the

muon itself. A special class of these events are

muons which pass outside the detector and release

a bright cascade, which can mimic an up-going hit

pattern.

Stopping muons

: Over the depth of the detector

the muon flux changes by a factor of

;

since

muons lose their energy and stop. These muons

can create an up-going hit pattern, especially when

the muon stops just before entering the detector

from the side.

Scattering layers

: The scattering of light in the

polar ice cap varies with depth. Light from bright

events, can mimic an up-going hit pattern, in

particular when it traverses layers of higher

scattering.

Corner clippers

: These are events where the

muon passes diagonally below the detector. The

light travel upwards through the detector mimick-

ing an up-going muon.

Uncorrelated coincident muons

: Due to the large

size of the

AMANDA

detector, the probability of

muons from two independent air showers forming

a single event is small on the trigger level but not

negligible. If an initial muon traverses the bottom

of the detector and a later muon traverses the top,

the combination can be reconstructed as an up-

going muon.

Electronic artifacts

: Noise, cross talkand other

transient electronic malfunctions are generally

small effects, but they can occasionally produce

hits, which distort the time pattern. Such effects

become important after a selection process of

several orders of magnitude.

6.2. Quality parameters

Background events, which pass a zenith angle

selection, need to be rejected by applying selection

criteria on quality parameters. These parameters

usually evaluate information, which is not opti-

mally exploited in the reconstruction. The detailed

choice of quality parameters is specific to each

analysis. Here, we summarize the most important

categories.

The

number of direct hits

dir

;

is the

number of hits with small time residuals:

res

(see Eq. (5)). Un-scattered photons

provide the best information for the reconstruc-

tion, and a large number of

dir

indicates high

quality information in the event. Empirically

reasonable values are

15 ns and

between

¼þ

25 and

150 ns

;

depending on the specific

analysis.

The

length of the event L

is obtained by

projecting each hit OM onto the reconstructed

trackand taking the distance between the two

outermost of these points.

can be considered as

the ‘‘lever arm’’ of the reconstruction. Larger

values corresponding to a more robust and precise

reconstruction of the track’s direction. This para-

meter is particularly powerful when calculated for

direct hits only, and is then referred to as

dir

Length requirements are efficient

against corner clippers, stopping muons and

cascades.

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184

The absolute value of the likelihood at the

maximum is a good parameter to evaluate the

quality of a reconstruction. Here, a useful ob-

servable is the

likelihood parameter

which is

defined as

log

free

where

free

is the degrees of freedom (e.g.

free

hits

5 for a trackreconstruction). For Gaussian

probability distributions this expression corre-

sponds to the reduced chi-square.

can be used

as a selection parameter, smaller values corre-

sponding to higher quality. A selection of events

with good

hit

values is efficient against

stopping muons.

Comparing

from different reconstructions is a

powerful technique. Cascade-like events will have

a better likelihood from a cascade reconstruction

than one from a trackreconstruction.

Another efficient rejection method is to compare

for the best up-going versus the best down-going

reconstruction of a single event. If the up-going

reconstruction is not significantly better than the

down-going reconstruction, the event is rejected.

These values can be obtained from the iterative

reconstruction method (Section 5.2.2) or by

restricting the parameter space. This method is

particularly powerful when the down-going recon-

struction uses a zenith weighted likelihood (Sec-

tion 3.1.4).

The reconstruction methods consider the

p.d.f. for each hit separately but ignore correla-

tions. Therefore, the reconstructions assign the

same likelihood to tracks where all hits cluster at

one end of the reconstructed trackand tracks

where the same number of hits are smoothly

distributed along the track. The latter hit

pattern indicates a successful trackreconstruction,

while the former hit pattern may be caused

by a background event. The

smoothness

parameter

was inspired by the Kolmogorov–Smirnov

test of the consistency of two distributions.

is a measure of the consistency of the observed

hit pattern with the hypothesis of constant light

emission by a muon. The simplest definition

of the smoothness

max

;

where

max

that

;

which has the largest absolute value, and

is defined as

is the distance along the trackbetween the points

of closest approach of the trackto the first and the

th hit module, with the hits taken in order of their

projected position on the track.

is the total

number of hits. Tracks with hits clustered at the

beginning or end of the trackhave

approaching

1or

;

respectively. High-quality tracks with

close to zero, have hits equally spaced along the

track. A graphical representation of the smooth-

ness construction can be found inRef. [30].

Extensions of this smoothness parameter in-

clude the restriction of the calculation to direct hits

only or using the distribution of hit times

instead

of the distances

A particularly important extension is

hit

order to account for the granularity and asym-

metric geometry of the detector one can replace

the above formulation with one that models the hit

smoothness expectation for the actual geometry of

the assumed muon track. This can be accom-

plished by using the hit probabilities of all

;

the number of operational OMs, (ordered along

the track) as weights:

hit

max

hit

with

hit

;

hit

;

if the OM

was hit and 0 otherwise and

hit

;

is the probability for OM

to be hit given the

reconstructed track. The hit probabilities are

calculated according to the algorithm in Section

3.2.3. Smoothness selections are very efficient

against secondary cascades, stopping muons and

coincident muons from independent air showers.

Interesting

AMANDA

events are analyzed with

multiple reconstruction algorithms. An event is

most likely to have been reconstructed correctly, if

the different algorithms produce consistent results.

For two reconstructions with directions

and

;

the space angle between them is given by

arccos

;

which should be reasonably small

for successful reconstructions. This concept can be

extended to multiple reconstructions and their

angular deviations from the average direction. For

different reconstructed directions,

;

the average

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J. Ahrens et al. / Nuclear Instruments and Methods in Physics Research A 524 (2004) 169–194

185

reconstructed direction,

;

is given by

We can define the parameter

arccos

Þ?

describes the average space angle between the

individual reconstructions and

is a different

parameter, which treats the deviations between

and the

as ‘‘errors’’ and adds them quadrati-

cally. Small values of

indicate consistent

reconstruction results.

The

parameters are a mathematically correct

consistency checkonly when comparing the results

of uncorrelated reconstructions of the same

intrinsic resolutions. This is not the case when

comparing different

AMANDA

reconstructions. Irre-

spective of the validity of such an interpretation,

are very efficient selection criteria,

especially against almost horizontal muons and

wide muon bundles.

A few additional selection parameters are

closely related to first guess methods. The ratio

of the eigenvalues of the

tensor of inertia

(see

Section 4.4) are a measure of the sphericity of the

event topology, which is an efficient selection

parameter against cascade backgrounds. Tracks

reconstructed as down-going by the

dipole fit

(see

Section 4.3) that have a non-negligible

dipole

moment

;

indicate coincident muons

from independent air showers. Larger values of the

line-fit speed

(see Section 4.2) are an indication

for longer muon-like, smaller values for more

spherical cascade-like events.

Finally, two approaches evaluate the ‘‘intrinsic

resolution’’ or ‘‘stability’’ of the reconstruction of

each event. One approach quantifies the sharpness

of the minimum found by the minimizers in

log

by fitting a paraboloid to it. The fitted

parameters can then be used to classify the

sharpness of the minimum. The other approach

splits an event into sub-events (for example,

containing odd- vs. even-numbered hits) and

reconstructs the sub-events. If the reconstructed

directions of the sub-events are different, then the

reconstruction of the full event has a larger

uncertainty.

6.3. Analysis strategies

Analyses that search for neutrino induced

muons must cope with a large background of

atmospheric muons. The optimal choice of recon-

struction and selection criteria varies strongly with

different expectations for the energy and angular

distribution of the signal events. The goal is to

optimize the signal efficiency over the background

or noise (square root of the background) based on

sets of signal and background data.

The selection criteria for background sensitive

variables may be adjusted individually such

that a specified fraction of signal events pass.

After these

first level criteria

are set, the

adjustment is repeated until the desired back-

ground rejection is reached. Each iteration

defines a ‘‘cut-level’’, which corresponds to

data sets of increasing purity. This simple

method is used to derive a defined set of

selection parameters for the performance Sec-

tion 7. However, less efficient criteria are mixed

with more efficient criteria, and correlations of

the variables are not taken into account.

Therefore, this method does not achieve the

optimum signal efficiency.

An improvement to this method has been

demonstrated in an

AMANDA

point source

analysis

[46,47]. Here, a selection criterion is

only applied to the most sensitive variable, and

the most sensitive variable is determined at each

cut level. An interesting aspect of this point

source search is that the experimental data

themselves can be used as a background

sample, which reduces systematic uncertainties

from the background simulation. The selection

criteria are not optimized with respect to signal

purity but with respect to an optimal signifi-

cance of a possible signal.

Another approach is to combine the selection

parameters into a single selection parameter,

called

event quality

. This can be done by

rescaling and normalizing each of the selection

parameters according to the cumulative dis-

tribution of the signal expectation. The

AMANDA

analyses of atmospheric neutrinos [30,48] used

this technique.

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J. Ahrens et al. / Nuclear Instruments and Methods in Physics Research A 524 (2004) 169–194

186

Additional approaches use discriminant analy-

sis [49] or neural nets [6,50,51] to optimize the

efficiency while taking into account the correla-

tions between selection criteria and their

individual selectivity. However, both methods

depend critically on a good agreement between

experiment and simulation. These methods

quantify the efficiency of each parameter by

including and excluding it from the optimiza-

tion procedure.

The ‘‘C

val

’’ method finds the optimum

combination of selection parameters and cut

values by numerically maximizing a significance

function,

An example is

ffiffiffiffi

;

where

is the number of signal events, and

is the

number of background events after selection.

The implementation proceeds in several

steps. First, the most efficient selection para-

meter,

;

is the parameter that individually

maximizes

The next parameter,

;

is the

parameter that maximizes

in conjunction

with

More parameters are successively

determined until the addition of a new para-

meter fails to improve

This procedure takes

correlations between the selection criteria into

account. The final number of selection para-

meters is reduced to a minimum, while max-

imizing the efficiency. Next, the optimal

selection for this combination parameters is

computed as a function of a boundary condi-

tion (e.g. the maximum number of accepted

background events). This boundary condition

is also used to define a single quality parameter.

Such a formalized procedure has to be

carefully monitored, e.g. to handle potentially

un-simulated experimental effects. The C

E-

val

procedure is monitored by defining differ-

ent, complementary optimization functions

;

which allow real and simulated data to be

compared

[30,52–54]

7. Performance

This section describes the performance of the

reconstruction methods. It is based on illustrative

data selections, and the actual performance of a

dedicated analysis can be different. Unless noted

otherwise, the data shown is from Monte

Carlo simulations of atmospheric neutrinos for

AMANDA-II

7.1. First guess algorithms

Since the first guess algorithms are used as a

starting point for the full reconstruction, they

should provide a reasonable estimate of the track

coordinates. Also, these algorithms are used as the

basis of early level filtering, and therefore need to

be sufficiently accurate for that purpose, i.e. they

should at least reconstruct the events in the correct

hemisphere.

As an example,

Table 1

gives the passing

efficiencies with respect to the

AMANDA-II

trigger

for atmospheric neutrinos (signal) and atmo-

spheric muons (background) for the first guess

methods (see Section 4), after the selection of

events with calculated zenith angles larger 80

The

direct walk

algorithm gives the best background

suppression and the highest atmospheric neutrino

passing rate. Correspondingly, it also gives the

best initial tracks to the likelihood reconstructions.

7.2. Pointing accuracy of the track reconstruction

The angular accuracy of the reconstruction can

be expressed in terms of a point spread function,

which is given by the space angle deviation

between the true and the reconstructed direction of

a muon corrected for solid angle. The space angle

deviation is a combined result of two effects: a

systematic shift in the direction and a random

spread around this shift. In a point source analysis,

for example, it is possible to correct for systematic

ARTICLE IN PRESS

Table 1

The atmospheric muon and atmospheric neutrino detection

efficiencies for a selection at

for the first guess

algorithms

Reconstruction atm.

(%) atm.

(%)

Direct walk1.5% 93%

Line-fit 4.8% 85%

Dipole algorithm 16.8% 78%

J. Ahrens et al. / Nuclear Instruments and Methods in Physics Research A 524 (2004) 169–194

187

shifts and be limited by the point spread function

alone [47].

The zenith and space angular deviations are

shown in Figs. 8 and 9. They are obtained by the

reconstruction algorithms as used in

AMANDA-B10

The same event selection is used for all. As a

general observation, the distributions of deviations

for different reconstruction algorithms is surpris-

ingly similar after a particular selection. Larger

differences are usually seen in the selection

efficiencies. A similar behavior is observed for

AMANDA-II

The dependence of the space angle deviation for

the full

AMANDA-II

detector on the cut level

for

the LH reconstruction is shown in

Fig. 10. The

tighter the selection criteria, the better the angular

resolution. The same general trend is true for the

other reconstructions. Tight criteria select events

with unambiguous hit topologies, which are

reconstructed better. The results for cut level 6

are shown in Figs. 11–13 as function of the energy

and the zenith angle.

The angular resolution (see Fig. 11) has a weak

energy dependence. The energy of the muon is

taken at the point of its closest approach to the

detector center. Best results are achieved for

energies of 100 GeV–10 TeV

At energies

100 GeV

;

the muons have paths shorter than

the full detector, which limits the angular

resolution. At energies

10 TeV

;

more light is

emitted due to individual stochastic energy loss

processes along the muon track. Here, the hit

pattern is not correctly described by the underlying

reconstruction assumption of a bare muon track

(see Section 6.1).

The space angular resolution depends on the

incident muon zenith angle (see

Fig. 12). Again

ARTICLE IN PRESS

Fig. 8. The zenith angle deviations for various reconstructions

AMANDA-B10

. The result of an atmospheric neutrino

simulation after the selection criteria of (Ahrens et al.)

[30]

shown. The fits are a

linefit

(LF), an iterated

upandel

fit (LH),

an iterated zenith-weighted

upandel

fit and a MPE fit.

Fig. 9. The distribution of space angle deviations for various

reconstructions of

AMANDA-B10

. The result of an atmospheric

neutrino simulation after the selection criteria of (Ahrens et al.)

[30]

is shown. The fits are a

linefit

(LF), an iterated

upandel

fit

(LH), an iterated zenith-weighted

upandel

fit and a MPE fit.

The cut levels defined here are typical and intended as

demonstrating example. We use typical selection parameters

from Section 6.2: the reconstructed zenith angle,

;

dir

ð?

;

dir

ð?

;

and

;

MPE

Our goal here is to illustrate the analysis,

and we do not optimize with respect to efficiency and angular

resolution. Instead each individual criterion is enforced in such

a way that 95% of the events from the previous level would

pass, and correlations between the parameters are ignored.

Specific physics analyses will use selection criteria of higher

efficiency and will achieve better angular resolutions than the

;

shown here.

J. Ahrens et al. / Nuclear Instruments and Methods in Physics Research A 524 (2004) 169–194

188

this is shown only for the LH reconstruction, the

other reconstructions are similar. Up-going muons

with cos

7 are best reconstructed, and

horizontal muons are the worst, because of the

geometry of the

AMANDA-II

detector. Nearly

vertical events with cos

1 have a poorer

angular resolution, because they illuminate

fewer strings, which can cause ambiguities in the

azimuth.

Systematic shifts also degrade the angular resolu-

tion.

AMANDA

observes a small zenith-dependent shift

ARTICLE IN PRESS

log

/GeV)

space angle deviation

[°]

Mean

Median

1.5

2.5

3.5

4.5

Fig. 11. The dependence of the space angle deviation of the LH

fit on the muon energy for

AMANDA-II

. Shown are mean (stars)

and median (circles) for simulated atmospheric neutrinos.

Cut level

space angle deviation

[°]

Mean

Median

0246

Fig. 10. The dependence of the space angle deviation of the LH

reconstruction in

AMANDA-II

on the event selection (cut levels).

cos (

)

space angle deviation

[°]

Mean

Median

1.5

2.5

3.5

4.5

1 0.75 0.5 0.25 0

Fig. 12. The space angle deviations of the LH fit as a function

of the cosine of the incident zenith angle (for

AMANDA-II

Shown are the mean (stars) and median (circles) for simulated

atmospheric neutrinos.

cos (

)

Mean

Median

1.5

0.5

1.5

1 0.5 0

Fig. 13. The zenith angle shift of the reconstruction versus the

cosine of the incident angle. Shown are mean (stars) and

median (circles) for simulated atmospheric neutrinos.

J. Ahrens et al. / Nuclear Instruments and Methods in Physics Research A 524 (2004) 169–194

189

of the reconstructed zenith angle and no systema-

tic shift in azimuth. This is shown in Fig. 13 for

simulated atmospheric neutrinos in

AMANDA-II

The size of this shift depends on the zenith angle

itself, and it is determined by the geometry of

AMANDA

, which has a larger size in vertical than in

horizontal directions. From a comparison with

AMANDA-B10

data

[46,55], we observe that these

shifts become smaller with a larger horizontal

detector size. These shifts are confirmed by

analyzing

AMANDA

events coincident with

SPASE

(see

below).

These angular deviations have been obtained

from Monte Carlo simulations. They can be

experimentally verified by analyzing coincident

events between

AMANDA

and

SPASE

. An analysis of

data from the 10 string

AMANDA-B10

detector,

shown in

Fig. 14, confirms the estimate of

obtained from Monte Carlo studies for

AMANDA-

B10

. Unfolding the estimated

SPASE

resolution of

confirms the estimated

AMANDA-B10

resolu-

tion of

near the

SPASE-AMANDA

coincidence

direction [8–10].

A simulation-independent estimate can be ob-

tained by splitting the hits of individual events in

two parts and reconstructing each sub-event

separately. The difference in the two results gives

an estimate of the total angular resolution. Such

analyses are being performed at present and results

will be published separately.

7.3. Energy reconstruction

The energy resolution of the three methods,

described in Section 3.2.4, is shown in Fig. 15 as

function of the muon energy at its closest point

to the

AMANDA-B10

center. The resolution for

AMANDA-B10

log

;

for the interest-

ing energy range of a few TeV to 1 PeV

Below

600 GeV the energy resolution is limited,

because the amount of light emitted by a muon

is only weakly dependent on its energy. Above

1 TeV the resolution improves because radiative

energy losses become dominant. Above 100 TeV

the resolution degrades, because energy loss

fluctuations dominate.

Although these methods are quite different,

their performances are similar. The full

reco

and

hit

methods achieve similar resolutions up to

1 PeV

The

hit

method becomes worse above this

energy, because in

AMANDA-B10

almost all of the

OMs are hit, and the method saturates. In

contrast, the Neural Net method shows a slightly

poorer resolution up to 1 PeV but is better above.

Its resolution is relatively constant over several

decades of energy. This is an advantage when

reconstructing an original energy spectrum with an

unfolding procedure as in Ref. [36].

The

AMANDA-II

detector contains more than

twice as many OMs as

AMANDA-B10

, and the

ARTICLE IN PRESS

Space angle (degrees)

arbitrary units

100

120

140

0 101520253

Fig. 14. Distribution of the space angle deviations between air

shower directions assigned by

SPASE-2

and muon directions

assigned by

AMANDA-B10

for coincident events measured in

1997. The figure is not corrected for the systematic shift.

log

gen

/GeV)

(log

rec

/ E

gen

))

hit

method

Full E

reco

method

Neural Net method

0.2

0.4

0.6

0.8

1.2

1.4

2345678

Fig. 15. Comparison of the resolution of three different energy

reconstruction approaches for

AMANDA-B10

gen

is the gener-

ated energy (MC) and

rec

the reconstructed energy.

J. Ahrens et al. / Nuclear Instruments and Methods in Physics Research A 524 (2004) 169–194

190

energy resolution is better, especially at larger

energies,

log

The neural net recon-

struction results for

AMANDA-II

are shown in Fig.

16. Finally, the recently installed transient wave-

form recorders (TWR) allow better amplitude

measurements, which should significantly improve

the results of the energy reconstructions, in

particular, the

full E

reco

method [56].

As discussed in Section 1, the cascade channel

can achieve substantially better resolutions, be-

cause the full energy is deposited inside or close of

the detector. Energy resolutions in

log

2 and

15 can be achieved by

AMANDA-B10

and

AMANDA-II

. respectively [16].

7.4. Systematic uncertainties

Several parameters of the detector are calibrated

and therefore only known with limited accuracy.

These parameters include the time offsets, the OM

positions and the absolute OM sensitivities. We

have estimated the effects of these uncertainties on

the resolution of

AMANDA

reconstructions

[55].As

an example, Fig. 17 shows the effect of an

additional contribution to the time calibration

uncertainty for the 10 string

AMANDA-B10

detector.

The zenith angular resolutions for simulated

atmospheric neutrino events only degrade when

the additional timing uncertainties exceed 10 ns

Additional tests with similar results were done

with non-random systematic shifts such as a

depth-dependent shift or a string-dependent shift.

Therefore, the angular resolution is insensitive to

the uncertainties in the time calibration. The

geometry of the detector is known to better than

30 cm horizontally and to better than 1 m

vertically, which corresponds to timing uncertain-

ties of

1or 3

5ns

;

respectively. Therefore, the

geometry calibration is also sufficiently accurate.

Similarly, the effect of uncertainties on other

parameters, like the absolute PMT efficiency, has

been investigated. No indication was found that

the remaining calibration uncertainties seriously

affect the angular resolution or the systematic

zenith angle offset. The combined calibration

uncertainties are expected to affect the accuracy

of the reconstruction by less than 5

in the zenith

angle resolution and to less than 0

in the

absolute pointing offset.

ARTICLE IN PRESS

arbitrary units

log10 (Energy/GeV)

1 TeV

10 TeV

100 TeV

1 PeV

700

600

500

400

300

200

100

12 345 67

Fig. 16. Energy reconstruction for simulated muons of

different fixed energy in

AMANDA-II

, using the neural net

method.

Random timing error

[

]

RMS (

)

2.5

3.5

4.5

Fig. 17. The zenith angle deviations (RMS) as function of an

additional uncertainty in the

time calibration. Data is shown

for simulated atmospheric neutrino events in

AMANDA-B10

with

the selection of (Ahrens et al.)

[30]. The transit time of the

PMTs has been shifted without correcting for in the reconstruc-

tion. The shift is a fixed value for each PMT, obtained from a

random Gaussian distribution.

J. Ahrens et al. / Nuclear Instruments and Methods in Physics Research A 524 (2004) 169–194

191

8. Discussion and Outlook

We have developed methods to reconstruct and

identify muons induced by neutrinos

[30], inspite

of the challenges of the natural environment and

large backgrounds. These methods allow us to

establish

AMANDA

as a working neutrino telescope.

The reconstruction techniques described in this

paper are still subject to improvement in several

aspects.

The likelihood description

: The likelihood func-

tions for trackreconstruction are based on the

assumption of exactly one infinitely long muon

trackper event. Extensions of this model to

encompass

starting muon tracks

(including the

description of the hadronic vertex),

stopping

muons

muon bundles

of non-negligible width, and

multiple independent muons

will be important,

particularly in the context of larger detectors such

as Ice Cube. Initial efforts fitting multiple muons

with the direct walkalgorithm have been useful in

rejecting coincident down-going muons, and work

toward reconstructing muon bundles has begun in

the context of events coincident with

SPASE

air

showers.

The p.d.f. calculation

: The likelihood function is

based on parametrizations of probability density

functions (p.d.f.). The p.d.f. is obtained from

Monte Carlo simulations, and its accuracy is

limited by the accuracy of the simulation. Better

simulations lead directly to a better p.d.f. and

hence better reconstructions.

The p.d.f. parametrizations

: The p.d.f. is para-

metrized by functions (e.g. the

Pandel functions

)

which only approximate the full p.d.f. More

accurate parametrization functions will result in

better reconstructions. For example, the scattering

coefficient shows a significant depth dependence

(see Section 2). The current reconstruction is based

on an average p.d.f. assuming depth-independent

ice properties. While the trackreconstruction is

relatively insensitive to the accuracy of the

parametrization, we expect a depth-dependent

p.d.f. to have better energy reconstruction.

Complementary information

: The current recon-

struction algorithms do not include all available

information in an event. In particular, correlations

between detected PMT signals are ignored. For

this reason dedicated selection parameters have

been designed to exploit this information. They are

used to discriminate between well reconstructed

and poorly reconstructed events and improve the

quality of the data sample. Future workwill try to

improve these parameters and expand the present

likelihood description.

Transient waveform recorders

: At the beginning

of the year 2003, the detector readout has been

upgraded with transient waveform recorders [56].

We expect a substantial improvement of the

multiple-photon detection and the dynamic range

in particular for high muon energies.

The construction of a much larger detector, the

IceCube

detector, will start in the year 2004. It will

consist of 4800 PMT deployed on 80 vertical

strings and will surround the

AMANDA

detector [57].

The performance of

IceCube

has been studied

with realistic Monte Carlo simulations and similar

analysis techniques as described in this paper [58].

The result is a substantially improved performance

in terms of sensitivity and reconstruction accuracy.

A direction accuracy of about 0

(median) for

energies above 1 TeV is achieved. Similar to

AMANDA

, we expect a further improvement by

exploiting the full information, avaliable from

the recorded wave-forms, in the reconstruction.

Acknowledgements

This research was supported by the following

agencies: Deutsche Forschungsgemeinschaft

(DFG); German Ministry for Education and

Research; Knut and Alice Wallenberg Founda-

tion, Sweden; Swedish Research Council; Swedish

Natural Science Research Council; Fund for

Scientific Research (FNRS-FWO), Flanders In-

stitute to encourage scientific and technological

research in industry (IWT), and Belgian Federal

Office for Scientific, Technical and Cultural affairs

(OSTC), Belgium. UC-Irvine AENEAS Super-

computer Facility; University of Wisconsin

Alumni Research Foundation; U.S. National

Science Foundation, Office of Polar Programs;

U.S. National Science Foundation, Physics

Division; U.S. Department of Energy; D.F.

Cowen acknowledges the support of the NSF

ARTICLE IN PRESS

J. Ahrens et al. / Nuclear Instruments and Methods in Physics Research A 524 (2004) 169–194

192

CAREER program. I. Taboada acknowledges the

support of FVPI.

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