muontrackreconstructionanddataselectiontechniques in amanda · andy s...

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

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0168-9002/$

doi:10.1016

Muon track reconstruction and data selection techniquesin AMANDA

J. Ahrensa, X. Baib, R. Bayc, S.W. Barwickd, T. Beckaa, J.K. Beckere, K.-H. Beckere, E. Bernardinif, D. Bertrandg, A. Bironf, D.J. Boersmaf, S. B .oserf,O. Botnerh, A. Bouchtah, O. Bouhalig, T. Burgessi, S. Cariusj, T. Castermansk,D. Chirkinc, B. Collinl, J. Conradh, J. Cooleym, D.F. Cowenl, A. Davourh,

C. De Clercqn, T. DeYoungo, P. Desiatim, J.-P. Dewulfg, P. Ekstr .ome, T. Fesera,M. Gaugf, T.K. Gaisserb, R. Ganugapatim, H. Geenene, L. Gerhardtd, A. GroXe,

A. Goldschmidtp, A. Hallgrenh, F. Halzenm, K. Hansonm, R. Hardtkem,T. Harenberge, T. Hauschildtf, K. Helbingp, M. Hellwiga, P. Herquetk, G.C. Hillm,

D. Hubertn, B. Hugheym, P.O. Hulthi, K. Hultqvisti, S. Hundertmarki,J. Jacobsenp, A. Karlem, M. Kestell, L. K .opkea, M. Kowalskif, K. Kuehnd,

J.I. Lamoureuxp, H. Leichf, M. Leutholdf, P. Lindahlj, I. Liubarskyq, J. Madsenr,P. Marciniewskih, H.S. Matisp, C.P. McParlandp, T. Messariuse, Y. Minaevai,

P. MioWinovi!cc, P.C. Mockd, R. Morsem, K.S. M .uniche, J. Namd, R. Nahnhauerf,T. Neunh .offera, P. Niessenn, D.R. Nygrenp, H. .Ogelmanm, Ph. Olbrechtsn,

C. P!erez de los Herosh, A.C. Pohli, R. Porratad, P.B. Pricec, G.T. Przybylskip,K. Rawlinsm, E. Resconif, W. Rhodee, M. Ribordyk, S. Richterm, J. Rodr!ıguezMartinoi, D. Rossd, H.-G. Sandera, K. Schinarakise, S. Schlenstedtf, T. Schmidtf,D. Schneiderm, R. Schwarzm, A. Silvestrid, M. Solarzc, G.M. Spiczakr, C. Spieringf,

M. Stamatikosm, D. Steelem, P. Steffenf, R.G. Stokstadp, K.-H. Sulankef,O. Streicherf, I. Taboadas, L. Thollanderi, S. Tilavb, W. Wagnere, C. Walcki,

Y.-R. Wangm, C.H. Wiebusche,*, C. Wiedemanni, R. Wischnewskif, H. Wissingf,K. Woschnaggc, G. Yodhd

a Institute of Physics, University of Mainz, D-55099 Mainz, GermanybBartol Research Institute, University of Delaware, Newark, DE 19716, USAcDepartment of Physics, University of California, Berkeley, CA 94720, USA

dDepartment of Physics and Astronomy, University of California, Irvine, CA 92697, USAeFachbereich 8 Physik, BU Wuppertal, Gaussstrasse 20, D-42119 Wuppertal, Germany

fDESY-Zeuthen, D-15738 Zeuthen, GermanygUniversit!e Libre de Bruxelles, Science Faculty, Brussels, Belgium

hDivision of High Energy Physics, Uppsala University, S-75121 Uppsala, Sweden

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

ddress: [email protected] (C.H. Wiebusch).

- see front matter r 2004 Elsevier B.V. All rights reserved.

/j.nima.2004.01.065

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J. Ahrens et al. / Nuclear Instruments and Methods in Physics Research A 524 (2004) 169–194170

iDepartment of Physics, Stockholm University, SE-10691 Stockholm, SwedenjDepartment of Technology, Kalmar University, S-39182 Kalmar, Sweden

kUniversity of Mons-Hainaut, 7000 Mons, BelgiumlDepartment of Physics, Pennsylvania State University, University Park, PA 16802, USA

mDepartment of Physics, University of Wisconsin, Madison, WI 53706, USAnVrije Universiteit Brussel, Dienst ELEM, B-1050 Brussels, Belgium

oDepartment of Physics, University of Maryland, College Park, MD 20742, USApLawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

qBlackett Laboratory, Imperial College, London SW7 2BW, UKrPhysics Department, University of Wisconsin, River Falls, WI 54022, USAsDepartment 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 Array (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.

r 2004 Elsevier B.V. All rights reserved.

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

Keywords: AMANDA; Track reconstruction; Neutrino telescope; Neutrino astrophysics

1. Introduction

The Antarctic Muon And Neutrino DetectorArray [1], AMANDA, is a large volume neutrinodetector at the geographic South Pole. It is a latticeof photo-multiplier tubes (PMTs) buried deep inthe optically transparent polar ice. The primarygoal of this detector is to detect high-energyneutrinos from astrophysical sources, and deter-mine their arrival time, direction and energy.When a high-energy neutrino interacts in thepolar ice via a charged current reaction with anucleon N:

nc þN-cþX; ð1Þ

it creates a hadronic cascade, X, and a lepton, c ¼e; m; t: These particles generate Cherenkovphotons, which are detected by the PMTs. Eachlepton flavor generates a different signal in thedetector. The two basic detection modes aresketched in Fig. 1.A high-energy nm charged current interaction

creates a muon, which is nearly collinear with theneutrino direction; having a mean deviation angleof c ¼ 0:7� � ðEn=TeVÞ

�0:7 [2], which implies anaccuracy requirement of t1� for reconstructingthe muon direction.The high-energy muon emits a cone of Cher-

enkov light at a fixed angle yc: It is determined bycos yc ¼ ðnbÞ�1; where nC1:32 is the index of

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cascademuon

PMTs

c

spherical Cherenkov frontCherenkov cone

Fig. 1. Detection modes of the AMANDA detector: Left: muon tracks induced by muon-neutrinos; Right: Cascades from electron- or tau-

neutrinos.

1Muon 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

refraction in the ice. For relativistic particles, bC1;and ycE41�: The direction of the muon isreconstructed from the time and amplitude in-formation of the PMTs illuminated by theCherenkov 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 themuon energy. However, the resolution is limitedby fluctuations of these processes. This estimate isa lower bound on the neutrino energy, because it isbased on the muon energy at the detector. Theinteraction vertex may be far outside the detector.The ne and nt channels are different. The

electron from a ne will generate an electro-magnetic cascade, which is confined to a volumeof a few cubic meters. This cascade coincides withthe hadronic cascade X of the primary interactionvertex. The optical signature is an expandingspherical shell of Cherenkov photons with a lar-ger intensity in the forward direction. The taufrom a nt will decay immediately and also gene-rate a cascade. However, at energies >1 PeV thiscascade and the vertex are separated by severaltens of meters, connected by a single track. Thissignature of two extremely bright cascades isunique for high-energy nt; 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 thanfor nm detection. Also the accuracy of the directionmeasurement is worse for cascades than for long

muon tracks. However, when the flux is diffuse,the ne and nt channels also have clear advantages.The backgrounds from atmospheric neutrinos aresmaller. The energy resolution is significantlybetter since the full energy is deposited in or nearthe detector. The cascade channel is sensitive to allneutrino flavors because the neutral currentinteractions also generate cascades. In this paper,we focus on the reconstruction of muon tracks;details on the reconstruction of cascades aredescribed in Ref. [4].The most abundant events in AMANDA are atmo-

spheric muons, created by cosmic rays interactingwith the Earth’s atmosphere. At the depth ofAMANDA their rate exceeds the rate of muons fromatmospheric neutrinos by five orders of magni-tude. Since these muons are absorbed by the earth,a muon track from the lower hemisphere is aunique signature for a neutrino-induced muon.1

The reconstruction procedure must have goodangular resolution, good efficiency, and allowexcellent rejection of down-going atmosphericmuons.This paper describes the methods used to

reconstruct muon tracks recorded in the AMANDA

experiment. The AMANDA-II detector is introducedin Section 2. The reconstruction algorithms andtheir implementation are described in Sections 3–5.

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light diffuser ball

HV divider

silicon gel

ModuleOptical

pressurehousing

Depth

120 m

AMANDA-II AMANDA-B10

Inner 10 strings: zoomed in on one

optical module (OM)

main cable

PMT

200 m1000 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.


Section 6 summarizes event classes for which thereconstruction may fail and strategies to identifyand eliminate such events. The performanceof the reconstruction procedure is shown inSection 7. We discuss possible improvementsin Section 8.

2. The AMANDA detector

The AMANDA-II detector (see Fig. 2) has beenoperating since January 2000 with 677 opticalmodules (OM) attached to 19 strings. Most of theOMs are located between 1500 and 2000 m belowthe surface. Each OM is a glass pressure vessel,which contains an 8-in. hemispherical PMT and itselectronics. AMANDA-B10,2 the inner core of 302OMs on 10 strings, has been operating since 1997.One unique feature of AMANDA is that it

continuously measures atmospheric muons incoincidence with the South Pole Air Shower

Experiment surface arrays SPASE-1 and SPASE-2

[7]. These muons are used to survey the detectorand calibrate the angular resolution (see Section 7

2Occasionally in the paper we will refer to this earlier

detector instead of the full AMANDA-II detector.

and Refs. [8,9]), while providing SPASE withadditional information for cosmic ray compositionstudies [10].The PMT signals are processed in a counting

room at the surface of the ice. The analog signalsare amplified and sent to a majority logic trigger[11]. There the pulses are discriminated and atrigger is formed if a minimum number of hitPMTs are observed within a time window oftypically 2 ms: Typical trigger thresholds were 16hit PMT for AMANDA-B10 and 24 for AMANDA-II.For each trigger the detector records the peakamplitude and up to 16 leading and trailing edgetimes for each discriminated signal. The timeresolution achieved after calibration is stC5 nsfor the PMTs from the first 10 strings, which areread out via coaxial or twisted pair cables. For theremaining PMTs, which are read out with opticalfibers the resolution is stC3:5 ns: In the coldenvironment of the deep ice the PMTs have lownoise rates of typically 1 kHz:The timing and amplitude calibration, the array

geometry, and the optical properties of the ice aredetermined by illuminating the array with knownoptical pulses from in situ sources [11]. Timeoffsets are also determined from the response tothrough-going atmospheric muons [12].

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cθ

cθ

OM

µd

x

Cherenkov lig

PMT

-axis

η p

r

ri


The optical absorption length in the ice istypically 110 m at 400 nm with a strong wave-length dependence. The effective scattering lengthat 400 nm is on average C20 m: It is defined asls=ð1�/cos ysSÞ; where ls is the scattering lengthand ys is the scattering angle. The ice parametersvary strongly with depth due to horizontal icelayers, i.e., variations in the concentration ofimpurities which reflect past geological eventsand climate changes [13–19].

ht

t , , E0 0 0

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

3We note that Eq. (4) neglects the effect that Cherenkov light

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

was shown in Ref. [14] that for AMANDA this approximation

is justified.

3. Reconstruction algorithms

The muon track reconstruction algorithm is amaximum 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 unknownparameters fag; e.g. track parameters, given a setof experimentally measured values fxg: Theparameters, fag; are determined by maximizingthe likelihood Lðx j aÞ which for independentcomponents xi of x reduces to

LðxjaÞ ¼Y

i

pðxijaÞ ð2Þ

where pðxijaÞ is the probability density function(p.d.f.) of observing the measured value xi forgiven values of the parameters fag [20].To simplify the discussion we assume that the

Cherenkov radiation is generated by a singleinfinitely long muon track (with b ¼ 1) and formsa cone. It is described by the following parameters:

a ¼ ðr0; t0; #p;E0Þ ð3Þ

and illustrated in Fig. 3. Here, r0 is an arbitrarypoint on the track. At time t0; the muon passes r0with energy E0 along a direction #p: The geome-trical coordinates contain five degrees of freedom.Along this track, Cherenkov photons are emittedat a fixed angle yc relative to #p: Within the

reconstruction algorithm it is possible to use adifferent coordinate system, e.g. a ¼ ðd; Z;yÞ: Thereconstruction is performed by minimizing�logðLÞ with respect to a:The values fxg presently recorded by AMANDA are

the time ti and duration TOTi (Time Over

Threshold) of each PMT signal, as well as thepeak amplitude Ai of the largest pulse in eachPMT. PMTs with no signal above threshold arealso accounted for in the likelihood function. Thehit times give the most relevant information.Therefore we will first concentrate on pðtjaÞ:

3.1.1. Time likelihood

According to the geometry in Fig. 3, photonsare expected to arrive at OM i (at ri) at time

tgeo ¼ t0 þ#p � ðri � r0Þ þ d tan yc

cvacð4Þ

with cvac the vacuum speed of light.3 It isconvenient to define a relative arrival time, ortime residual

tres � thit � tgeo ð5Þ

which is the difference between the observed hittime and the hit time expected for a ‘‘directphoton’’, a Cherenkov photon that travels un-delayed directly from the muon to an OM withoutscattering.

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tres

tres

tres

tres

0 0

high

low

0 0

+ showers + scattering

close track

far track

+ noise

t

jitter

jitter jitter

jitter

Fig. 4. Schematic distributions of arrival times ttes 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 the ideal case, the distribution, pðtresjaÞ;would be a delta function. However, in a realisticexperimental situation this distribution is broa-dened and distorted by several effects, which areillustrated in Fig. 4. The PMT jitter limits thetiming resolution st: Noise, e.g. dark noise of thePMT, leads to additional hits which are random intime. These effects can generate negative tresvalues, which would mimic unphysical causalityviolations. Secondary radiative energy losses alongthe muon trajectory create photons that arriveafter the ideal Cherenkov cone. These processesare stochastic, and their relative photon yieldfluctuates.In AMANDA, the dominant effect on photon

arrival times is scattering in the ice.4 The effectof scattering depends strongly on the distance, d;of the OM from the track as illustrated in Fig. 4.Since the PMTs have a non-uniform angularresponse, pðtresÞ also depends on the orientation,Z; of the OM relative to the muon track (see Fig.3). OMs facing away from the track can only seelight that scatters back towards the PMT face. On

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

small correction [23].

average this effect shifts tres to later times andmodifies the probability of a hit.The simplest time likelihood function is based

on a likelihood constructed from p1; the p.d.f. forarrival times of single photons i at the locations ofthe hit OMs

Ltime ¼YNhits

i¼1

p1ðtres;i ja ¼ di; Zi;yÞ: ð6Þ

Note that one OM may contribute to this productwith several hits. The function p1ðtres;i jaÞ isobtained from the simulation of photon propaga-tion through ice (see Section 3.2). However, thisdescription is limited, because the electrical andoptical signal channels can only resolve multiplephotons separated by a few 100 ns and C10 ns;respectively. Within this time window, only thearrival time of the first pulse is recorded.This first photon is usually less scattered than

the average single photon, which modifies theprobability distribution of the detected hit time.The arrival time distribution of the first of N

photons is given by

p1N ðtresÞ ¼Np1ðtresÞZ

N

tres

p1ðtÞ dt

� �ðN�1Þ

¼Np1ðttresÞð1� P1ðtresÞÞðN�1Þ ð7Þ

P1 is the cumulative distribution of the singlephoton p.d.f. The function p1N ðtresÞ is called themulti-photo-electron (MPE) p.d.f. and corre-spondingly defines LMPE:This concept can be extended to the more

general case of pkN ðtresÞ; the p.d.f. for the kth

photon out of a total of N to arrive at tres; given by

pkN ðtresÞ ¼N

N � 1

k � 1

!p1ðtresÞð1� P1ðtresÞÞ

ðN�kÞ

� ðP1ðtresÞÞðk�1Þ ð8Þ

pkN ðtresÞ specifies the likelihood of arrival times ofindividual photoelectrons for averaged time seriesof N photoelectrons. With waveform recording thearrival times and amplitudes of individual pulsescan be resolved.When the number of photoelectrons, N; is not

measured precisely enough, multi-photon informa-tion can be included via another method. Instead

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of measuring N; the p.d.f. of the first photoelec-tron can be calculated by convolving the MPEp.d.f. p1Nðt; dÞ with the Poisson probabilityPPoisson

N ðmÞ; where m is the mean expected numberof photoelectrons as a function of the distance, d:

p1mðtresÞ ¼1

N

XNi¼1

mi e�m

i!p1i ðtresÞ

¼m

1� e�m p1ðtresÞ e�mP1ðtresÞ ð9Þ

This result is called the Poisson Saturated Ampli-tude (PSA) p.d.f. [24,25] and correspondinglydefines LPSA: The constant N ¼ 1� e�m 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 beforereconstruction. They are included in the likelihoodfunction by simply adding a constant p.d.f. p0:

3.1.2. Hit and no-hit likelihood

The likelihood in the previous section relies onlyon the measured arrival times of photons. How-ever, the topology of the hits is also important.PMTs with no hits near a hypothetical track orPMTs with hits far from the track are unlikely.A likelihood utilizing this information can be

constructed as

Lhit ¼YNch

i¼1

Phit;iYNOM

i¼Nchþ1

Pno�hit;i ð10Þ

where Nch is the number of hit OMs and NOM thenumber of operational OMs. The probabilities Phit

and Pno�hit of observing or not observing a hitdepend on the track parameters a: Additional hitsdue to random noise are easily incorporated:Pno�hit- *Pno�hit � Pno�hitPno�noise andPhit- *Phit ¼ 1� *Pno�hit:Assuming that the probability Phit

1 is known fora single photon, the hit and no-hit probabilities ofOMs for n photons can be calculated:

Pno�hitn ¼ ð1� Phit

1 Þn

and

Phitn ¼ 1� Pno�hit

n

¼ 1� ð1� Phit1 Þn: ð11Þ

The number of photons, n; depends on Em; theenergy of the muon: n ¼ nðEmÞ: For a fixed trackgeometry, the likelihood (Eq. (10)) can be used toreconstruct the muon energy.

3.1.3. Amplitude likelihood

The peak amplitudes recorded by AMANDA can befully incorporated in the likelihood [26], which isparticularly useful for energy reconstruction. Thelikelihood can be written as

L ¼W

NOM

YNOM

i¼1

wiPiðAiÞ ð12Þ

where PiðAiÞ is the probability that OM i observesan amplitude Ai; with Ai ¼ 0 for unhit OMs. W

and wi are weight factors, which describe devia-tions of the individual OM and the total number ofhit OMs from the expectation. Pi depends on themean number m of expected photoelectrons:

PiðAiÞ ¼ Phitð1� Pthi Þ

PðAi; mÞPð/AiS;mÞ

: ð13Þ

The probability PiðAiÞ is normalized to theprobability of observing the most likely amplitude/AiS: Pth

i ðmÞ is the probability that a signal of mdoes not produce a pulse amplitude above thediscriminator threshold. As before, Phit ¼1� Pno�hit; where the no-hit probability is givenby Poisson statistics: Pno�hit ¼ expð�mÞð1� PnoiseÞ:The probability of Ai ¼ 0 is a special case: Pið0Þ ¼Pno�hit þ PhitPth

i : Energy reconstructions based onthis formulation of the likelihood will be referredto as Full Ereco:An alternative energy reconstruction technique

(see Section 3.2.4) uses a neural net which is fedwith energy sensitive parameters.

3.1.4. Zenith weighted (Bayesian) likelihood

Another extension of the likelihood [27–29]incorporates external information about the muonflux via Bayes’ Theorem. This theorem states thatfor two hypotheses A and B;

PðAjBÞ ¼PðBjAÞPðAÞ

PðBÞ: ð14Þ

Identifying A with the track parameters a and B

with the observations x; Eq. (14) gives the prob-ability that the inferred muon track a was in fact

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responsible for the observed event x: PðxjaÞ is theprobability that a; assumed to be true, wouldgenerate the event x—in other words, the like-lihood described in the previous sections. PðaÞ isthe prior probability of observing the track a; i.e.,the relative frequencies of different muon tracks asa function of their parameters. PðxÞ; which isindependent of the track parameters a; is anormalization constant which ensures thatEq. (14) defines a proper probability. Because thelikelihood is only defined up to an arbitraryconstant factor, this normalization may be ignoredin the present context.In order to obtain PðajxÞ; one thus has to

determine the prior probability distribution, PðaÞ;of how likely the various possible track directionsare a priori. The reconstruction maximizes theproduct of the p.d.f. and the prior.The flux of muons deep underground is reason-

ably well known from previous experiments. Anypoint source of muons would be at most a smallperturbation on the flux of penetrating atmo-spheric muons and muons created by atmosphericneutrinos. The most striking feature of the back-ground flux from atmospheric muons is the strongdependence on zenith angle. For vertically down-going tracks it exceeds the flux from neutrinoinduced muons by about 5 orders of magnitudebut becomes negligible for up-going tracks. Thisdependence, which is modeled by a Monte Carlocalculation [30], acts as a zenith dependent weightto the different muon hypotheses, a: With thisparticular choice, some tracks, which wouldotherwise reconstruct as up-going, reconstruct asdown-going tracks. This greatly reduces the rate atwhich penetrating atmospheric muons are mis-reconstructed as up-going neutrino events [31]. Inprinciple, a more accurate prior could be used. Itwould need to include the depth and energydependence of the atmospheric muons as well asthe angular dependence of atmospheric neutrino-induced muons.Upon completion of this work, we learned that

this technique was developed independently by theNEVOD neutrino detector collaboration [32] whowere able to extract an atmospheric neutrino froma background of 1010 atmospheric muons in asmall ð6� 6� 7:5 m3Þ surface detector.

3.1.5. Combined likelihoods

The likelihood function Ltime of the hit times isthe most important for track reconstruction.However, it is useful to include other informationlike the hit probabilities. The combined p.d.f. fromEqs. (7)–(10) is

LMPE"PhitPno�hit ¼ LMPEðLhitÞw ð15Þ

which is particularly effective. Here w is anoptional weight factor which allows the adjust-ment of the relative weight of the two likelihoods.This likelihood is sensitive not only to the trackgeometry but also to the energy of the muon.As discussed in Section 3.1.4, the zenith angle-

dependent prior function, PðyÞ; can be included asa multiplicative factor. This combination

LBayes ¼ PðyÞLtime ð16Þ

has been used in the analysis of atmosphericneutrinos [30]. However, all of these improvedlikelihoods are limited by the underlying modelassumption of a single muon track.

3.2. Likelihood implementation

The actual implementation of the likelihoodsrequires detailed knowledge of the photon propa-gation in the ice. On the other hand, efficiencyconsiderations and numeric problems favor asimple and robust method.The photon hit probabilities and arrival time

distributions are simulated as functions of allrelevant parameters with a dedicated Monte Carlosimulation 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 thereconstruction. In principle the probability densityfunctions are taken directly from these archives.However, one has to face several technicaldifficulties due to the memory requirements ofthe archived tables, as well as numeric problemsrelated to the normalization of interpolated binsand 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 ofparameters. Comparisons of two independent

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parametrizations [24,34] show that the direct andparametrized approaches yield similar results interms of efficiency. This indicates that the para-metrization itself is not limiting the reconstructionquality; rather, as mentioned earlier, the recon-struction is limited by the assumptions of themodel being fit. Therefore, we will concentrate ononly one parametrization.

3.2.1. Analytical parametrization

A simple parametrization of the arrival timedistributions can be achieved with the followingfunction, which we call Pandel function. It is agamma distribution and its usage is motivated byan analysis of laser light signals in the BAIKALexperiment [35]. There, it was found that for thecase of an isotropic, monochromatic and point-like light source, p1ðtresÞ can be expressed in theform

pðtresÞ �1

NðdÞt�ðd=lÞt

ðd=l�1Þres

Gðd=lÞ

� e� tres

1

tþ

cmedium

la

� �þ

d

la

� �ð17Þ

NðdÞ ¼ e�d=la 1þtcmedium

la

� ��d=l

ð18Þ

without special assumptions on the actual opticalparameters. Here, cmedium ¼ cvac=n is the speed oflight in ice, la the absorption length, Gðd=lÞ theGamma function and NðdÞ a normalization factor,which is given by Eq. (18). This formulation hasfree parameters l and t; which are unspecifiedfunctions of the distance d and the other geome-trical parameters. They are empirically determinedby a Monte Carlo model.The Pandel function has some convenient

mathematical properties: it is normalized, it iseasy to compute, and it can be integratedanalytically over the time, tres; which simplifiesthe construction of the multi-photon (MPE) timep.d.f.. For small distances the function has a poleat t ¼ 0 corresponding to a high probability of anunscattered photon. Going to larger values of d;longer delay times become more likely. Fordistances larger than the critical value d ¼ l; thepower index to tres 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 t (units of time) and l (unitsof length) and the overall reasonable behavior isthe motivation to use this function to parametrizenot only the time p.d.f. for point-like sources, butalso for muon tracks [34]. The Pandel function isfit to the distributions of delay times for fixeddistances d and angles Z (between the PMT axisand the Cherenkov cone). These distributions arepreviously obtained from a detailed photonpropagation Monte Carlo for the Cherenkov lightfrom muons. The free fit parameters are t; l; la

and the effective distance deff ; which will beintroduced next.When investigating the fit results as a function

of d and angle Z (see Fig. 3), we observe thatalready for a simple ansatz of constant t; l and la

the optical properties in AMANDA are describedsufficiently well within typical distances. Thedependence on Z is described by an effectivedistance deff which replaces d in Eq. (17). Thismeans that the time delay distributions for back-ward illumination of the PMT is found to besimilar to a head-on illumination at a largerdistance. The following parameters are obtainedfor a specific ice model, and are currently used inthe reconstruction:

t ¼ 557 ns; deff ¼ a0 þ a1d

l ¼ 33:3 m; a1 ¼ 0:84

la ¼ 98 m;

a0 ¼ 3:1 m� 3:9 m cosðZÞ þ 4:6 m cos2ðZÞ: ð19Þ

A comparison of the results from this parame-trization with the full simulation is shown in Fig. 5for two extreme distances. The simple approxima-tion describes the behavior of the full simulationreasonably well. However, this simple overalldescription has a limited accuracy, especially fordEl (not shown). Reconstructions, based on thePandel function with different ice models, and ageneric reconstruction, that uses the full simula-tion results, yield similar results. These compar-isons indicate that the results of the reconstructiondo not critically depend on the fine tuning of the

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time delay / ns

d = 8m

Del

ay p

rob

/ ns

time delay / ns

d = 71m

Del

ay p

rob

/ ns

10−4

10−3

10−2

10−1

0 200 400

10−7

10−6

10−5

10−4

10−3

0 500 1000 1500

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

distances d from the muon track.


underlying ice models, and it justifies the use of theabove simple model.

3.2.2. Extension to realistic PMT signals

Although the Pandel function is the basis of asimple normalized likelihood, it has several defi-ciencies. It is not defined for negative tres; it ignoresPMT jitter, and it has a pole at tres ¼ 0; whichcauses numerical difficulties. These problems canbe resolved by convolving the Pandel function,Eq. (17), with a Gaussian, which accounts for thePMT jitter. Unfortunately such convolution re-quires significant computing time.Instead the Pandel function is modified by

extending it to negative times, treso0; with a (half)Gaussian of width sg: The effects of PMT jitter areonly relevant for small values of tres: For timestresXt1 the original function is used, and the twoparts are connected by a spline interpolation (3rdorder polynomial). The result, #PðtresÞ; is calledupandel function.Using t1 ¼

ffiffiffiffiffiffi2p

psg and requiring further a

smooth interpolation and the normalization tobe unchanged, the polynomial coefficients aj andnormalization of the Gaussian Ng can be calcu-lated analytically [34]. The free parameter sgincludes all timing uncertainties, not just thePMT jitter. Good reconstruction results areachieved for a large range 10psgp20 ns:

3.2.3. PhitPno�hit Parametrization

The normalization NðdÞ in Eq. (18) is used toconstruct a hit probability function, Phit: Thefunction Phit

n with Phit1 � NðdÞ; is fit to the hit

probability determined by the full AMANDA detectorsimulation, as a function of distance, orientationand muon energy. The free parameters are thePandel parameters t and l; la; #d and n: Theeffective distance #d; is similar to the effectivedistance in the Pandel parametrization. We definen as the power index of Eq. (11), which corre-sponds to an effective number of photons. It isimportant to understand that NðdÞ is not a hitprobability and n is not just a number of photons.They are constructs, that are calibrated with aMonte Carlo simulation. Technically, the powerindex, n � N in Eq. (11), factorizes into nðZ;EmÞ ¼eðZ;EmÞnðEmÞ: The variable n; where n ¼ nðEÞ; isrelated to the number of photons incident on thePMT and its absolute efficiency. The factoreðZ;EmÞ is related to the orientation dependentPMT sensitivity but also accounts for the energy-dependent angular emission profile of photonswith respect to the bare muon.

3.2.4. Energy reconstruction

The reconstruction of the track geometry is asearch for five parameters. If the muon energy isadded as a fit parameter, the minimization issignificantly slower. Therefore, the energy recon-struction is performed in two steps. First, the trackgeometry is reconstructed without the energyparameter. Then these geometric parameters areused in an energy reconstruction, that onlydetermines the energy. However, if the timelikelihood utilizes amplitude information, e.g. inthe combined likelihood, Eq. (7), or the PSAlikelihood, Eq. (9), the track parameters also

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depend on the energy. In this case the energy andgeometric parameters must be reconstructed to-gether. Currently, three different approaches areused to reconstruct the muon energy. They arecompared in Section 7.3.

(1)
The simplest method utilizes the PhitPno�hit
reconstruction (see Sections 3.1.2 and 3.2.3).
(2) The Full Ereco method (see Section 3.1.3)
models the measured amplitudes in a like-lihood for reconstructing the energy. Thisalgorithm performs better, but it is moredependent on the quality of the amplitudecalibration of the OMs.

(3)
An alternative way to measure the energy isbased on a neural network [36]. The neuralnetwork uses 6-6-3-1 and 6-3-5-1 feed-forwardarchitecture for AMANDA-B10 and AMANDA-II,respectively. The energy correlated variableswhich are used as input are the mean of themeasured amplitudes (ADC), the mean andRMS of the arrival times (LE) or pulsedurations (TOT), the total number of signals,the number of OMs hit and the number ofOMs with exactly one hit.
Less challenging than a full reconstruction, alower energy threshold is determined by requiringa minimum number of hit OMs. The number of hitOMs is correlated with the energy of the muon.Since celestial neutrinos are believed to have asubstantially harder spectrum than atmosphericneutrinos, an excess of high multiplicity eventswould indicate that a hard celestial source exists.Values for this parameter determined from AMANDA

data already set a tight upper limit on the diffuseflux of high-energy celestial neutrinos [37].

3.2.5. Cascade reconstruction

The reconstruction of cascade like events isdescribed in detail elsewhere [4]. The basicapproach is similar to the track reconstruction. Itassumes events form a point light source withphotons propagating spherically outside with ahigher intensity in the forward direction. Thecascade reconstruction also uses the Pandel func-tion (see Eq. (17)) with parameters that are specificfor cascades.

In several muon analyses, a cascade fit is used asa competing model. In cases where the cascade fitachieves a better likelihood than the trackreconstruction, the track hypothesis is rejected.In particular this is used as a selection criterion toreject background events which are mis-recon-structed due to bright secondary cascades.

4. First guess pattern recognition

The likelihood reconstructions need an initialtrack hypothesis to start the minimization. Theinitial track is derived from first guess methods,which are fast analytic algorithms that do notrequire 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 weremost likely caused by direct photons.The four step procedure starts by selecting track

elements, the straight line between any two hitOMs at distance d; which are hit with a timedifference

jDtjod

cvacþ 30 ns with d > 50 m: ð20Þ

The known positions of the OMs define the trackelement direction ðy;fÞ: The vertex positionðx; y; zÞ is taken at the center between the twoOMs. The time at the vertex t0 is defined as theaverage of the two hit times.In a next step, the number of associated hits

(AH) are calculated for each track element.Associated hits are those with �30otreso300 nsand do25 mðtres þ 30Þ1=4 (t in ns), where d is thedistance between hit OM and track elementand tres is the time residual, which is defined inEq. (5). After selecting these associated hits, trackelements of poor quality are rejected by requiring:

NAHX10 and

sL �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðð1=NAHÞ

XiðLi �/LSÞ2Þ

qX20 m:

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Here, the ‘‘lever arm’’ Li is the distance betweenthe vertex of the track element and the point onthe track element which is closest to OM i and/LS is the average of all Li-values. Trackelements that fulfill these criteria qualify as track

candidates (TC).Frequently, more than one track candidate is

found. In this case, a cluster search is performed forall track candidates that fulfill the quality criterion

QTCX0:7Qmax

where

Qmax ¼ maxðQTCÞ

and

QTC ¼ minðNAH; 0:3 m�1 � sL þ 7Þ: ð21Þ

In the cluster search, the ‘‘neighbors’’ of each trackcandidate are counted, where neighbors are trackcandidates with space angle differences of less than15�: The cluster with the largest number of trackcandidates is selected.In the final step, the average direction of all

track candidates inside the cluster defines theinitial track direction. The track vertex and timeare taken from the central track candidate in thecluster. Well separated clusters can be used toidentify independent muon tracks in events whichcontain multiple muons (see Section 6.1).

4.2. Line-fit

The line-fit [38] algorithm produces an initialtrack on the basis of the hit times with an optionalamplitude weight. It ignores the geometry of theCherenkov cone and the optical properties of themedium and assumes light traveling with a velocityv along a 1-dimensional path through the detector.The locations of each PMT, ri; which are hit at atime ti can be connected by a line

riErþ v � ti: ð22Þ

A w2 to be minimized is defined as

w2 �XNhit

i¼1

ðri � r� v � tiÞ2 ð23Þ

where Nhit is the number of hits. The w2 isminimized by differentiation with respect to the

free fit parameters r and v: This can be solvedanalytically

r ¼ /riS� v �/tiS

and

v ¼/ri � tiS�/riS �/tiS

/t2i S�/tiS2ð24Þ

where /xiS � ð1=NhitÞPNhit

i xi denotes the meanof parameter x with respect to all hits.The line-fit thus yields a vertex point r; and a

direction e ¼ vLF=jvLFj: The zenith angle is givenby yLF � �arccosðvz=jvLFjÞ:The time residuals (Eq. (5)) for this initial track

generally do not follow the distribution expectedfor a Cherenkov model. If the t0 parameter of theinitial track is shifted to better agree with aCherenkov model, subsequent reconstructionsconverge better (see Section 5.2.3).The absolute speed vLF � jvj; of the line-fit is the

mean speed of the light propagating through theone-dimensional detector projection. Sphericalevents (cascades) and high energy muons havelow vLF values, and thin, long events (minimallyionizing muon tracks) have large values.

4.3. Dipole algorithm

The dipole algorithm considers the unit vectorfrom one hit OM to the subsequently hit OM as anindividual dipole moment. Averaging over allindividual dipole moments yields the global mo-ment M: It is calculated in two steps. First, all hitsare sorted according to their hit times. Then adipole-moment M is calculated

M �1

Nch � 1

XNch

i¼2

ri � ri�1

jri � ri�1j: ð25Þ

It can be expressed via an absolute value MDA �jMj and two angles yDA and fDA: These anglesdefine the initial track.The dipole algorithm does not generate as good

an initial track as the direct walk or the line-fit, butit is less vulnerable to a specific class of back-ground events: almost coincident atmosphericmuons from independent air showers in whichthe first muon hits the bottom and the secondmuon hits the top of the detector.

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4.4. Inertia tensor algorithm

The inertia tensor algorithm is based on amechanical picture. The pulse amplitude from aPMT at ri corresponds to a virtual mass ai at ri:One can then define the tensor of inertia I of thatvirtual mass distribution. The origin is the centerof gravity ðCOGÞ of the mass distribution. TheCOG-coordinates and the tensor of inertia com-ponents are given by

COG �XNch

i¼1

ðaiÞw � ri

and

Ik;l �XNch

i¼1

ðaiÞw � ½dkl � ðriÞ

2 � rki � rl

i �: ð26Þ

The amplitude weight wX0 can be chosenarbitrarily. The most common settings are w ¼ 0(ignoring the amplitudes) and w ¼ 1 (setting thevirtual masses equal to the amplitudes). The tensorof inertia has three eigenvalues Ij ; jef1; 2; 3g;corresponding to its three main axes ej : Thesmallest eigenvalue I1 corresponds to the longestaxis e1: In the case of a long track-like eventI15fI2; I3g and e1 approximates the direction ofthe track. The ambiguity in the direction along thee1 axis is resolved by choosing the direction wherethe average OM hit time is latest. In the case of acascade-like event, I1EI2EI3: The ratios betweenthe Ij can be used to determine the sphericity of theevent.

5. Aspects of the technical implementation

5.1. Reconstruction framework

The basic reconstruction procedure, sketched inFig. 6, is sequential. A fast reconstruction program

Hit-cleaning Hit-cleaning Selections↓ ↓ ↓

↓ ↓

Data ⇒ first guess ⇒ Likelihood ⇒ Analysis

Selections Selections

Fig. 6. Schematic principle of the reconstruction chain.

calculates the initial track hypothesis for thelikelihood reconstruction. All reconstruction pro-grams may use a reduced set of hits in order tosuppress noise hits and other detector artifacts.Event selection criteria can be applied after eachstep to reduce the event sample, and allow moretime consuming calculations at later reconstruc-tion stages. This procedure may iterate with moresophisticated but slower algorithms analyzingprevious results. The final step is usually theproduction of Data Summary Tape (DST) likeinformation, usually in form of PAW N-tuples[39]. A detailed description of this procedure canbe found in Ref. [40].The reconstruction framework is implemented

with the recoos program [41], which is based on therdmc library [42] and the SiEGMuND softwarepackage [43]. The recoos program is highlymodular, which allows flexibility in the choiceand combination of algorithms.

5.2. Likelihood maximization

The aim of the reconstruction is to find thetrack hypothesis which corresponds to the max-imum likelihood. This is done by minimizing�logðLÞ with respect to the track parameters.We have implemented several minimizationprocedures.The likelihood space for AMANDA events is often

characterized by several minima. Local likelihoodminima can arise due to symmetries in thedetector, especially in the azimuth angle, or dueto unexpected hit times caused by scattering. In theexample, shown in Fig. 7, the reconstructionconverged on a local minimum, because of non-optimal starting values. Several techniques, whichare used to find the global minimum, are herepresented. In particular, the iterative reconstruc-tion, Section 5.2.2, solves the problem andconverges to the global minimum. One generallyassumes that the global minimum corresponds tothe true solution, but this is not always correct dueto stochastic nature of light emission and detec-tion. Such events cannot be reconstructed properlyand have to be rejected using quality parameters(see Section 6.2).

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180160140120100806040200

170

175

180

185

190Parabola fit

experimental data

−log

(lik

elih

ood)

theta [°]

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

projection) for a specific AMANDA event. Shown is �logðLÞ asfunction of the zenith angle. Each point represents a fit, for

which the zenith angle was fixed and the other track parameters

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.


5.2.1. Minimization algorithms

The reconstruction framework allows us to useand 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 fastestalgorithm. Powell’s method and Minuit are B5times slower than the Simplex algorithm. Thereconstruction results from Minuit and the Sim-plex algorithm are nearly identical and almost asgood as the Powell results. Exceptions occur in lessthan 1% of the cases, when these methods fail andstop at the extreme zenith angles y ¼ 0� and 80�:The Simulated annealing algorithm is less sensitiveto local minima than the other algorithms, but it ismuch slower and requires fine-tuning.

5.2.2. Iterative reconstruction

The iterative reconstruction algorithm success-fully copes with the problem of local minima andextreme zenith angles by performing multiplereconstructions 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. Asystematic scan of the full parameter space forinitial seeds is not feasible. Instead the iterative

algorithm concentrates on the direction angles,zenith and azimuth, and uses reasonable values forthe spatial coordinates. The following procedureyields good results.The result of a first minimization is saved as a

reference. Then both direction angles are ran-domly selected. The track point, r0; is transformedto the point on the new track, which is closest tothe center of gravity of hits. The time, t0; of thispoint is shifted to match the Cherenkov expecta-tion (see Section 5.2.3). Then a new minimizationis started. If the minimum is less than the referenceminimum, it is saved as the new reference. Thisprocedure is iterated n times, and the best minimafound for zenith angles above and below thehorizon, are saved, and used to generate animportant selection parameter (see Section 6.2).This algorithm substantially reduces the number

of false minima found, after a few iterations. Forn ¼ 6 roughly 95% of the results are in the vicinityof the asymptotic optimum for n-N: For nC20more than 99% of the results are the globalminimum. Despite the fast convergence, theiterative reconstruction requires significant CPUtime, which limits its use to reduced data sets.

5.2.3. Lateral shift and time residual

The efficiency of finding the global minimum ofthe likelihood function can be improved bytranslating the arbitrary vertex and/or time originof the output track from the first guess algorithmbefore application of the full maximum likelihoodmethod.

Transformation of r0: In general this ‘‘vertex’’point is arbitrary in the infinite track approxima-tion used, and first guess methods may producepositions distant from the detector. During thelikelihood minimization, numerical errors can beavoided and the convergence improved by shiftingthis point along the direction of the track towardsthe point closest to the center of gravity of hits (seeEq. (26)). The vertex time t0 is transformedaccordingly: Dðt0Þ ¼ Dðr0Þ=c:

Transformation of t0: The time t0 obtained fromfirst guess algorithms is not calculated from a fullCherenkov model. The efficiency of the likelihoodreconstruction can be improved by shifting the t0such that the time residuals, Eq. (5), fit better to a

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Cherenkov model. In particular it is useful toavoid negative tres; which would correspond tocausality violations. This can be achieved bytransforming t0-t0 � t�res; where t�res is the mostnegative time residual.

5.2.4. Coordinates and restricted parameters

The track coordinates a; which are used by thelikelihood, are independent of the coordinatesactually chosen for the minimization. Therefore,the coordinate system can be chosen arbitrarily.Any of the parameters in this coordinate systemcan be kept fixed. During the minimizationparameterization functions translate the coordi-nates as necessary. The most commonly usedcoordinates are r0 and the zenith and azimuthangles y; f:The freedom in the choice of coordinates can be

used to improve the numerical minimization, forsystematic studies, or to fix certain parametersaccording to external knowledge. An example isthe reconstruction of coincident events with theSPASE surface arrays [8]. Here, we fix the locationof the trajectory to coincide with the core locationat the surface as measured by SPASE. Then, thedirection is determined with the AMANDA recon-struction subject to this constraint.Under certain circumstances the allowed range

of the reconstruction parameters is restricted. Themost important example here is to restrict thereconstructed zenith angle to above or below thehorizon, to find the most likely up- or down-goingtracks, respectively. Comparing the quality of thetwo 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 beforereconstruction. Defective OMs are removed, andthe amplitudes and hit times are calibrated. A hit

cleaning procedure identifies and flags hits whichappear to be noise or electronic effects, such ascross talk or after-pulsing. These hits are not usedin the reconstruction, but they are retained forpost-reconstruction analysis.

The hit cleaning procedure can be based onsimple and robust algorithms, because the PMTshave low noise rates. Noise and after-pulse hits arestrongly suppressed by rejecting hits that areisolated in time and space from other signals inthe detector. Typically a hit is considered to benoise if there is no hit within a distance of 60–100 m and a time of 7300 to 7600 ns: Cross talkhits are identified by examining the amplitudes andpulse widths of the individual pulses and byanalyzing the correlations of uncalibrated hit timeswith hits of large amplitude in channel combina-tions which are known to cross talk to each other.The required cross talk correlation map wasdetermined independently in a dedicated calibra-tion campaign.

5.4. Processing speeds

The first guess algorithms are sufficiently fastthat the execution time is dominated by file input/output and the software framework. Typical fittimes are E20 ms per event on a 850MHzPentium-III Linux PC. The processing speed ofthe likelihood reconstructions can vary signifi-cantly depending on the number of free para-meters, the number of iterations, the minimizationalgorithm, and the experimental parameters likethe number of hit OMs. These effects dominate thedifferences in processing speeds due to differentreconstruction algorithms. The typical executiontime for a 16-fold iterative likelihood reconstruc-tions using the simplex minimizer to reconstructthe five free track parameters is C250 ms perevent.

6. Background rejection

The performance of the reconstruction dependsstrongly on the quality and background selectioncriteria. The major classes of background events inAMANDA (see Section 6.1) are suppressed by thequality parameters presented in Section 6.2.Optimization strategies for the selection criteriaare summarized in Section 6.3. Finally, weevaluate the reconstruction performance inSection 7.

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6.1. Background classes

Most background events from atmosphericmuons are well reconstructed and can berejected by selecting up-going reconstructedevents. However, there is a small fraction of mis-reconstructed events, amounting to about 10�2 forthe unbiased and about 10�4 for the zenith-weighted reconstruction. These events are rejectedby additional selection criteria described in Section6.2. These background events are classified asfollows.

Nearly horizontal muons: These events have trueincident angles close to the horizon. A small errorin the reconstruction causes them to appear as up-going. These events are not severely mis-recon-structed, but occur due to the finite angularresolution.

Muon bundles: The spatial separation betweenmultiple muons from a single air shower, a muon

bundle, is usually small enough that the event canbe described by a single bright muon track. If theseparation is too large, the reconstruction fails.

Cascades: Bright stochastic energy losses (e.g.bremsstrahlung) produce additional light, whichdistorts the Cherenkov cone from the muon.Cascades emit most of their light with the sameCherenkov angle as the muon, but some light isemitted at other angles. These secondary eventscan cause the reconstruction to fail, especiallywhen the cascade(s) produce more light than themuon itself. A special class of these events aremuons which pass outside the detector and releasea bright cascade, which can mimic an up-going hitpattern.

Stopping muons: Over the depth of the detectorthe muon flux changes by a factor of B2; sincemuons lose their energy and stop. These muonscan create an up-going hit pattern, especially whenthe muon stops just before entering the detectorfrom the side.

Scattering layers: The scattering of light in thepolar ice cap varies with depth. Light from brightevents, can mimic an up-going hit pattern, inparticular when it traverses layers of higherscattering.

Corner clippers: These are events where themuon passes diagonally below the detector. The

light travel upwards through the detector mimick-ing an up-going muon.

Uncorrelated coincident muons: Due to the largesize of the AMANDA detector, the probability ofmuons from two independent air showers forminga single event is small on the trigger level but notnegligible. If an initial muon traverses the bottomof the detector and a later muon traverses the top,the combination can be reconstructed as an up-going muon.

Electronic artifacts: Noise, cross talk and othertransient electronic malfunctions are generallysmall effects, but they can occasionally producehits, which distort the time pattern. Such effectsbecome important after a selection process ofseveral orders of magnitude.

6.2. Quality parameters

Background events, which pass a zenith angleselection, need to be rejected by applying selectioncriteria on quality parameters. These parametersusually evaluate information, which is not opti-mally exploited in the reconstruction. The detailedchoice of quality parameters is specific to eachanalysis. Here, we summarize the most importantcategories.The number of direct hits, Ndirðt1:t2Þ; is the

number of hits with small time residuals:t1otresot2 (see Eq. (5)). Un-scattered photonsprovide the best information for the reconstruc-tion, and a large number of Ndir indicates highquality information in the event. Empiricallyreasonable values are t1C� 15 ns and t2 between¼ þ25 and þ150 ns; depending on the specificanalysis.The length of the event L is obtained by

projecting each hit OM onto the reconstructedtrack and taking the distance between the twooutermost of these points. L can be considered asthe ‘‘lever arm’’ of the reconstruction. Largervalues corresponding to a more robust and precisereconstruction of the track’s direction. This para-meter is particularly powerful when calculated fordirect hits only, and is then referred to asLdirðt1:t2Þ: Length requirements are efficientagainst corner clippers, stopping muons andcascades.

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The absolute value of the likelihood at themaximum is a good parameter to evaluate thequality of a reconstruction. Here, a useful ob-servable is the likelihood parameter L which isdefined as

L � �logðLÞ

Nfreeð27Þ

where Nfree is the degrees of freedom (e.g. Nfree ¼Nhits � 5 for a track reconstruction). For Gaussianprobability distributions this expression corre-sponds to the reduced chi-square. L can be usedas a selection parameter, smaller values corre-sponding to higher quality. A selection of eventswith good LPhitPno�hit

values is efficient againststopping muons.Comparing L from different reconstructions is a

powerful technique. Cascade-like events will havea better likelihood from a cascade reconstructionthan one from a track reconstruction.Another efficient rejection method is to compare

L for the best up-going versus the best down-goingreconstruction of a single event. If the up-goingreconstruction is not significantly better than thedown-going reconstruction, the event is rejected.These values can be obtained from the iterativereconstruction method (Section 5.2.2) or byrestricting the parameter space. This method isparticularly 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 thesame likelihood to tracks where all hits cluster atone end of the reconstructed track and trackswhere the same number of hits are smoothlydistributed along the track. The latter hitpattern indicates a successful track reconstruction,while the former hit pattern may be causedby a background event. The smoothness parameterS was inspired by the Kolmogorov–Smirnovtest of the consistency of two distributions.S is a measure of the consistency of the observedhit pattern with the hypothesis of constant lightemission by a muon. The simplest definitionof the smoothness S is S ¼ Smax

j ; where Smaxj is

that Sj ; which has the largest absolute value, and

Sj is defined as

Sj �j � 1

N � 1�

lj

lN: ð28Þ

lj is the distance along the track between the pointsof closest approach of the track to the first and thejth hit module, with the hits taken in order of theirprojected position on the track. N is the totalnumber of hits. Tracks with hits clustered at thebeginning or end of the track have S approachingþ1 or �1; respectively. High-quality tracks with S

close to zero, have hits equally spaced along thetrack. A graphical representation of the smooth-ness construction can be found in Ref. [30].Extensions of this smoothness parameter in-

clude the restriction of the calculation to direct hitsonly or using the distribution of hit times ti insteadof the distances li:A particularly important extension is SPhit

: Inorder to account for the granularity and asym-metric geometry of the detector one can replacethe above formulation with one that models the hitsmoothness expectation for the actual geometry ofthe assumed muon track. This can be accom-plished by using the hit probabilities of all NOM;the number of operational OMs, (ordered alongthe track) as weights: SPhit

¼ maxðSPhit

j Þ with

SPhit

j �

Pji¼1 LiPNOM

i¼1 Li

�

Pji¼1 Phit;iPNOM

i¼1 Phit;ið29Þ

Li ¼ 1; if the OM i was hit and 0 otherwise andPhit;i is the probability for OM i to be hit given thereconstructed track. The hit probabilities arecalculated according to the algorithm in Section3.2.3. Smoothness selections are very efficientagainst secondary cascades, stopping muons andcoincident muons from independent air showers.Interesting AMANDA events are analyzed with

multiple reconstruction algorithms. An event ismost likely to have been reconstructed correctly, ifthe different algorithms produce consistent results.For two reconstructions with directions e1 and

e2; the space angle between them is given by C ¼arccosðe1 � e2Þ; which should be reasonably smallfor successful reconstructions. This concept can beextended to multiple reconstructions and theirangular deviations from the average direction. Forn different reconstructed directions, ei; the average

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reconstructed direction, E; is given by E ¼Pni ei=j

Pni eij: We can define the parameter

Cw ¼X

i

½arccosðei � EÞ�w !1=w

: ð30Þ

C1 describes the average space angle between theindividual reconstructions and E: C2 is a differentparameter, which treats the deviations between E

and the ei as ‘‘errors’’ and adds them quadrati-cally. Small values of C1 or C2 indicate consistentreconstruction results.The C parameters are a mathematically correct

consistency check only when comparing the resultsof uncorrelated reconstructions of the sameintrinsic resolutions. This is not the case whencomparing different AMANDA reconstructions. Irre-spective of the validity of such an interpretation,C1 or C2 are very efficient selection criteria,especially against almost horizontal muons andwide muon bundles.A few additional selection parameters are

closely related to first guess methods. The ratioof the eigenvalues of the tensor of inertia (seeSection 4.4) are a measure of the sphericity of theevent topology, which is an efficient selectionparameter against cascade backgrounds. Tracksreconstructed as down-going by the dipole fit (seeSection 4.3) that have a non-negligible dipole

moment, MDA � j ~MM j; indicate coincident muonsfrom independent air showers. Larger values of theline-fit speed vLF (see Section 4.2) are an indicationfor longer muon-like, smaller values for morespherical cascade-like events.Finally, two approaches evaluate the ‘‘intrinsic

resolution’’ or ‘‘stability’’ of the reconstruction ofeach event. One approach quantifies the sharpnessof the minimum found by the minimizers in�logðLÞ by fitting a paraboloid to it. The fittedparameters can then be used to classify thesharpness of the minimum. The other approachsplits an event into sub-events (for example,containing odd- vs. even-numbered hits) andreconstructs the sub-events. If the reconstructeddirections of the sub-events are different, then thereconstruction of the full event has a largeruncertainty.

6.3. Analysis strategies

Analyses that search for neutrino inducedmuons must cope with a large background ofatmospheric muons. The optimal choice of recon-struction and selection criteria varies strongly withdifferent expectations for the energy and angulardistribution of the signal events. The goal is tooptimize the signal efficiency over the backgroundor noise (square root of the background) based onsets of signal and background data.

�
The selection criteria for background sensitivevariables may be adjusted individually suchthat a specified fraction of signal events pass.After these first level criteria are set, theadjustment is repeated until the desired back-ground rejection is reached. Each iterationdefines a ‘‘cut-level’’, which corresponds todata sets of increasing purity. This simplemethod is used to derive a defined set ofselection parameters for the performance Sec-tion 7. However, less efficient criteria are mixedwith more efficient criteria, and correlations ofthe variables are not taken into account.Therefore, this method does not achieve theoptimum signal efficiency.
�
An improvement to this method has beendemonstrated in an AMANDA point sourceanalysis [46,47]. Here, a selection criterion isonly applied to the most sensitive variable, andthe most sensitive variable is determined at eachcut level. An interesting aspect of this pointsource search is that the experimental datathemselves can be used as a backgroundsample, which reduces systematic uncertaintiesfrom the background simulation. The selectioncriteria are not optimized with respect to signalpurity but with respect to an optimal signifi-cance of a possible signal.
�
Another approach is to combine the selectionparameters into a single selection parameter,called event quality. This can be done byrescaling and normalizing each of the selectionparameters according to the cumulative dis-tribution of the signal expectation. The AMANDAanalyses of atmospheric neutrinos [30,48] usedthis technique.

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�
Additional approaches use discriminant analy-sis [49] or neural nets [6,50,51] to optimize theefficiency while taking into account the correla-tions between selection criteria and theirindividual selectivity. However, both methodsdepend critically on a good agreement betweenexperiment and simulation. These methodsquantify the efficiency of each parameter byincluding and excluding it from the optimiza-tion procedure.
�
The ‘‘CutEval’’ method finds the optimumcombination of selection parameters and cutvalues by numerically maximizing a significancefunction, Q: An example is Q ¼ S=
ffiffiffiffiB

p; where S

is the number of signal events, and B is thenumber of background events after selection.The implementation proceeds in several

steps. First, the most efficient selection para-meter, C1; is the parameter that individuallymaximizes Q: The next parameter, C2; is theparameter that maximizes Q in conjunctionwith C1: More parameters are successivelydetermined until the addition of a new para-meter fails to improve Q: This procedure takescorrelations between the selection criteria intoaccount. The final number of selection para-meters is reduced to a minimum, while max-imizing the efficiency. Next, the optimalselection for this combination parameters iscomputed as a function of a boundary condi-tion (e.g. the maximum number of acceptedbackground events). This boundary conditionis also used to define a single quality parameter.Such a formalized procedure has to be

carefully monitored, e.g. to handle potentiallyun-simulated experimental effects. The CutE-val procedure is monitored by defining differ-ent, complementary optimization functions Q;which allow real and simulated data to becompared [30,52–54]

Table 1

The atmospheric muon and atmospheric neutrino detection

efficiencies for a selection at yX80� for the first guess

algorithms

Reconstruction atm. m (%) atm. n (%)

Direct walk 1.5% 93%

Line-fit 4.8% 85%

Dipole algorithm 16.8% 78%

7. Performance

This section describes the performance of thereconstruction methods. It is based on illustrativedata selections, and the actual performance of adedicated analysis can be different. Unless noted

otherwise, the data shown is from MonteCarlo simulations of atmospheric neutrinos forAMANDA-II.

7.1. First guess algorithms

Since the first guess algorithms are used as astarting point for the full reconstruction, theyshould provide a reasonable estimate of the trackcoordinates. Also, these algorithms are used as thebasis of early level filtering, and therefore need tobe sufficiently accurate for that purpose, i.e. theyshould at least reconstruct the events in the correcthemisphere.As an example, Table 1 gives the passing

efficiencies with respect to the AMANDA-II triggerfor atmospheric neutrinos (signal) and atmo-spheric muons (background) for the first guessmethods (see Section 4), after the selection ofevents with calculated zenith angles larger 80�: Thedirect walk algorithm gives the best backgroundsuppression and the highest atmospheric neutrinopassing rate. Correspondingly, it also gives thebest initial tracks to the likelihood reconstructions.

7.2. Pointing accuracy of the track reconstruction

The angular accuracy of the reconstruction canbe expressed in terms of a point spread function,which is given by the space angle deviation Cbetween the true and the reconstructed direction ofa muon corrected for solid angle. The space angledeviation is a combined result of two effects: asystematic shift in the direction and a randomspread around this shift. In a point source analysis,for example, it is possible to correct for systematic

ARTICLE IN PRESS

Fig. 8. The zenith 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 line-fit (LF), an iterated upandel fit (LH),

an iterated zenith-weighted upandel fit and a MPE fit.


shifts and be limited by the point spread functionalone [47].The zenith and space angular deviations are

shown in Figs. 8 and 9. They are obtained by thereconstruction algorithms as used in AMANDA-B10.The same event selection is used for all. As ageneral observation, the distributions of deviationsfor different reconstruction algorithms is surpris-ingly similar after a particular selection. Largerdifferences are usually seen in the selectionefficiencies. A similar behavior is observed forAMANDA-II.The dependence of the space angle deviation for

the full AMANDA-II detector on the cut level5 forthe LH reconstruction is shown in Fig. 10. Thetighter the selection criteria, the better the angularresolution. The same general trend is true for theother reconstructions. Tight criteria select eventswith unambiguous hit topologies, which arereconstructed better. The results for cut level 6are shown in Figs. 11–13 as function of the energyand the zenith angle.The angular resolution (see Fig. 11) has a weak

energy dependence. The energy of the muon istaken at the point of its closest approach to thedetector center. Best results are achieved forenergies of 100 GeV–10 TeV: At energieso100 GeV; the muons have paths shorter thanthe full detector, which limits the angularresolution. At energies > 10 TeV; more light isemitted due to individual stochastic energy lossprocesses along the muon track. Here, the hitpattern is not correctly described by the underlyingreconstruction 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

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 line-fit(LF), an iterated upandel fit

(LH), an iterated zenith-weighted upandel fit and a MPE fit.

5The cut levels defined here are typical and intended as

demonstrating example. We use typical selection parameters

from Section 6.2: the reconstructed zenith angle, yDW > 80�;yLH > 80�; Nch; NLH

dir ð�15 : 25Þ; LLHdir ð�15 : 75Þ; LLH; SLH and

C1ðDW ;LH;MPEÞ: 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

C2�; shown here.

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log10 (Eµ/GeV)

spac

e an

gle

devi

atio

n [°

]

Mean

Median

1.5

2

2.5

3

3.5

4

4.5

5

0 2 4 6

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

spac

e an

gle

devi

atio

n [°

]

Mean

Median

2

3

4

5

6

789

10

0 2 4 6

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

reconstruction in AMANDA-II on the event selection (cut levels).

cos (Θµ)

spac

e an

gle

devi

atio

n [°

]

Mean

Median

1

1.5

2

2.5

3

3.5

4

4.5

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 (Θµ)

Mean

Median

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1 -0.5 0

Θµ-

ΘL

H

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.


this is shown only for the LH reconstruction, theother reconstructions are similar. Up-going muonswith cos ymC� 0:7 are best reconstructed, andhorizontal muons are the worst, because of thegeometry of the AMANDA-II detector. Nearlyvertical events with cos ymC� 1 have a poorer

angular resolution, because they illuminatefewer strings, which can cause ambiguities in theazimuth.Systematic shifts also degrade the angular resolu-

tion. AMANDA observes a small zenith-dependent shift

ARTICLE IN PRESS

0

Space angle (degrees)

arbi

trar

y un

its

20

40

60

80

100

120

140

50 10 15 20 25 30

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.

log10(Egen/GeV)

σ (l

og10

(Ere

c / E

gen)

)

Phit method

Full Ereco method

Neural Net method

0

0.2

0.4

0.6

0.8

1

1.2

1.4

2 3 4 5 6 7 8

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

reconstruction approaches for AMANDA-B10. Egen is the gener-

ated energy (MC) and Erec the reconstructed energy.


of the reconstructed zenith angle and no systema-tic shift in azimuth. This is shown in Fig. 13 forsimulated atmospheric neutrinos in AMANDA-II.The size of this shift depends on the zenith angleitself, and it is determined by the geometry ofAMANDA, which has a larger size in vertical than inhorizontal directions. From a comparison withAMANDA-B10 data [46,55], we observe that theseshifts become smaller with a larger horizontaldetector size. These shifts are confirmed byanalyzing AMANDA events coincident with SPASE(seebelow).These angular deviations have been obtained

from Monte Carlo simulations. They can beexperimentally verified by analyzing coincidentevents between AMANDA and SPASE. An analysis ofdata from the 10 string AMANDA-B10 detector,shown in Fig. 14, confirms the estimate of C3�

obtained from Monte Carlo studies for AMANDA-

B10. Unfolding the estimated SPASE resolution ofC1� confirms the estimated AMANDA-B10 resolu-tion of C3� near the SPASE-AMANDA coincidencedirection [8–10].A simulation-independent estimate can be ob-

tained by splitting the hits of individual events intwo parts and reconstructing each sub-eventseparately. The difference in the two results givesan estimate of the total angular resolution. Suchanalyses are being performed at present and resultswill 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 asfunction of the muon energy at its closest pointto the AMANDA-B10 center. The resolution forAMANDA-B10 in D log10 E is C0:4; for the interest-ing energy range of a few TeV to 1 PeV: BelowC600 GeV the energy resolution is limited,because the amount of light emitted by a muonis only weakly dependent on its energy. Above1 TeV the resolution improves because radiativeenergy losses become dominant. Above 100 TeVthe resolution degrades, because energy lossfluctuations dominate.Although these methods are quite different,

their performances are similar. The full Ereco andPhit methods achieve similar resolutions up to1 PeV: The Phit method becomes worse above thisenergy, because in AMANDA-B10 almost all of theOMs are hit, and the method saturates. Incontrast, the Neural Net method shows a slightlypoorer resolution up to 1 PeV but is better above.Its resolution is relatively constant over severaldecades of energy. This is an advantage whenreconstructing an original energy spectrum with anunfolding procedure as in Ref. [36].The AMANDA-II detector contains more than

twice as many OMs as AMANDA-B10, and the

ARTICLE IN PRESSar

bitr

ary

units

log10 (Energy/GeV)

1 TeV10 TeV

100 TeV1 PeV

700

600

500

400

300

200

100

01 2 3 4 5 6 7

Fig. 16. Energy reconstruction for simulated muons of

different fixed energy in AMANDA-II, using the neural net

method.

Random timing error [ns] R

MS

(Θµ-

ΘL

H)

2

2.5

3

3.5

4

4.5

5

1 10

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

additional uncertainty in the t0 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.


energy resolution is better, especially at largerenergies, sðD log10 EÞC0:3: 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 amplitudemeasurements, which should significantly improvethe results of the energy reconstructions, inparticular, the full Ereco 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 ofthe detector. Energy resolutions in D log10 E ofp0:2 and p0:15 can be achieved by AMANDA-B10

and AMANDA-II. respectively [16].

7.4. Systematic uncertainties

Several parameters of the detector are calibratedand therefore only known with limited accuracy.These parameters include the time offsets, the OMpositions and the absolute OM sensitivities. Wehave estimated the effects of these uncertainties onthe resolution of AMANDA reconstructions [55]. Asan example, Fig. 17 shows the effect of anadditional contribution to the time calibrationuncertainty for the 10 string AMANDA-B10 detector.The zenith angular resolutions for simulated

atmospheric neutrino events only degrade whenthe additional timing uncertainties exceed 10 ns:Additional tests with similar results were donewith non-random systematic shifts such as adepth-dependent shift or a string-dependent shift.Therefore, the angular resolution is insensitive tothe uncertainties in the time calibration. Thegeometry of the detector is known to better than30 cm horizontally and to better than 1 mvertically, which corresponds to timing uncertain-ties of t1 or 3:5 ns; respectively. Therefore, thegeometry calibration is also sufficiently accurate.Similarly, the effect of uncertainties on other

parameters, like the absolute PMT efficiency, hasbeen investigated. No indication was found thatthe remaining calibration uncertainties seriouslyaffect the angular resolution or the systematiczenith angle offset. The combined calibrationuncertainties are expected to affect the accuracyof the reconstruction by less than 5% in the zenithangle resolution and to less than 0:5� in theabsolute pointing offset.

ARTICLE IN PRESS


8. Discussion and Outlook

We have developed methods to reconstruct andidentify muons induced by neutrinos [30], inspiteof the challenges of the natural environment andlarge backgrounds. These methods allow us toestablish AMANDA as a working neutrino telescope.The reconstruction techniques described in thispaper are still subject to improvement in severalaspects.

The likelihood description: The likelihood func-tions for track reconstruction are based on theassumption of exactly one infinitely long muontrack per event. Extensions of this model toencompass starting muon tracks (including thedescription of the hadronic vertex), stopping

muons, muon bundles of non-negligible width, andmultiple independent muons will be important,particularly in the context of larger detectors suchas Ice Cube. Initial efforts fitting multiple muonswith the direct walk algorithm have been useful inrejecting coincident down-going muons, and worktoward reconstructing muon bundles has begun inthe context of events coincident with SPASE airshowers.

The p.d.f. calculation: The likelihood function isbased on parametrizations of probability densityfunctions (p.d.f.). The p.d.f. is obtained fromMonte Carlo simulations, and its accuracy islimited by the accuracy of the simulation. Bettersimulations lead directly to a better p.d.f. andhence 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. Moreaccurate parametrization functions will result inbetter reconstructions. For example, the scatteringcoefficient shows a significant depth dependence(see Section 2). The current reconstruction is basedon an average p.d.f. assuming depth-independentice properties. While the track reconstruction isrelatively insensitive to the accuracy of theparametrization, we expect a depth-dependentp.d.f. to have better energy reconstruction.

Complementary information: The current recon-struction algorithms do not include all availableinformation in an event. In particular, correlationsbetween detected PMT signals are ignored. For

this reason dedicated selection parameters havebeen designed to exploit this information. They areused to discriminate between well reconstructedand poorly reconstructed events and improve thequality of the data sample. Future work will try toimprove these parameters and expand the presentlikelihood description.

Transient waveform recorders: At the beginningof the year 2003, the detector readout has beenupgraded with transient waveform recorders [56].We expect a substantial improvement of themultiple-photon detection and the dynamic rangein particular for high muon energies.The construction of a much larger detector, the

IceCube detector, will start in the year 2004. It willconsist of 4800 PMT deployed on 80 verticalstrings and will surround the AMANDA detector [57].The performance of IceCube has been studiedwith realistic Monte Carlo simulations and similaranalysis techniques as described in this paper [58].The result is a substantially improved performancein terms of sensitivity and reconstruction accuracy.A direction accuracy of about 0:7� (median) forenergies above 1 TeV is achieved. Similar toAMANDA, we expect a further improvement byexploiting the full information, avaliable fromthe recorded wave-forms, in the reconstruction.

Acknowledgements

This research was supported by the followingagencies: Deutsche Forschungsgemeinschaft(DFG); German Ministry for Education andResearch; Knut and Alice Wallenberg Founda-tion, Sweden; Swedish Research Council; SwedishNatural Science Research Council; Fund forScientific Research (FNRS-FWO), Flanders In-stitute to encourage scientific and technologicalresearch in industry (IWT), and Belgian FederalOffice for Scientific, Technical and Cultural affairs(OSTC), Belgium. UC-Irvine AENEAS Super-computer Facility; University of WisconsinAlumni Research Foundation; U.S. NationalScience Foundation, Office of Polar Programs;U.S. National Science Foundation, PhysicsDivision; U.S. Department of Energy; D.F.Cowen acknowledges the support of the NSF

ARTICLE IN PRESS


CAREER program. I. Taboada acknowledges thesupport of FVPI.

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