KM3NeT detector optimization with HOU simulation and reconstruction softwareA. G. TsirigotisIn the framework of the KM3NeT Design StudyWP2 - Paris, 10-11 December 2008

The HOU software chainUnderwater DetectorGeneration of atmospheric muons and neutrino events (F77)Detailed detector simulation (GEANT4-GDML) (C++)Optical noise and PMT response simulation (F77)Filtering Algorithms (F77 C++)Muon reconstruction (C++)Calibration (Sea top) DetectorAtmospheric Shower Simulation (CORSIKA) Unthinning Algorithm (F77)Detailed Scintillation Counter Simulation (GEANT4) (C++) Fast Scintillation Counter Simulation (F77) Reconstruction of the shower direction (F77)Muon Transportation to the Underwater Detector (C++)Estimation of: resolution, offset (F77)

Event Generation Flux ParameterizationNeutrino Interaction EventsAtmospheric Muon Generation(CORSIKA Files, Parametrized fluxes )Atmospheric Neutrinos(Bartol Flux)Cosmic Neutrinos(AGN GRB GZK and more)EarthSurvival probabilityShadowing of neutrinos by Earth

GEANT4 Simulation Detector Description Any detector geometry can be described in a very effective wayUse of Geomery Description Markup Language (GDML-XML) software packageAll the relevant physics processes are included in the simulation (NO SCATTERING)Fast SimulationEM Shower ParameterizationNumber of Cherenkov Photons Emitted (~shower energy)Angular and Longitudal profile of emitted photonsVisualizationDetector componentsParticle Tracks and Hits

Simulation of the PMT response to optical photonsStandard electrical pulse for a response to a single p.e.

Prefit, filtering and muon reconstruction algorithmsLocal (storey) Coincidence (Applicable only when there are more than one PMT looking towards the same hemisphere)Global clustering (causality) filterLocal clustering (causality) filterPrefit and Filtering based on clustering of candidate track segments (Direct Walk)2 fit without taking into account the charge (number of photons)

Kalman Filter (novel application in this area)

Charge Direction Likelihood

State vectorInitial estimationUpdate EquationsKalman Gain MatrixUpdated residual and chi-square contributionKalman Filter application to track reconstruction(timing uncertainty)

Kalman Filter Muon Track Reconstruction - AlgorithmExtrapolate the state vectorExtrapolate the covariance matrixCalculate the residual of predictionsDecide to include or not the measurement (rough criterion)Update the state vectorUpdate the covariance matrixCalculate the contribution of the filtered pointDecide to include or not the measurement (precise criterion)PredictionFilteringInitial estimates for the state vector and covariance matrix

Charge LikelihoodHit charge in PEsMean expected number of Pes (depends on distance form track and PMT orientation)Not a poisson distribution)

10,8,6,3m20 floors per tower30m seperationBetween floors30mTower GeometryFloor Geometry45o45oGeometry: 10920 OMs in a hexagonal grid.91 Towers seperated by 130m, 20 floors each. 30m between floors

Optical Module10 inch PMT housed in a 17inch benthos sphere35%Maximum Quantum EfficiencyGenova ANTARES Parametrization for angular acceptance50KHz of K40 optical noise

Neutrino Angular resolution (no cuts applied)Results

Neutrino effective area (no cuts applied)Results

ResultsEfficiency of cutsMin compatible tracks = 10Min compatible tracks = 25Min compatible tracks = 401 hour of generated atmospheric muonsNumber of misreconstructed muons per dayNumber of reconstructed atmospheric neutrinos per dayRatio = (muons)/(neutrinos) (%)

ResultsEffective Area and angular resolution :Without applying any cutsApplying the cuts that give zero misreconstructed atmospheric muons per day:Likelihod < 1.0 and Minimum number of combatible tracks 40

ResultsWithout applying any cutsApplying the cuts that give zero misreconstructed atmospheric muons per day:Likelihod < 1.0 and Minimum number of combatible tracks 40Number of reconstructed atmospheric neutrinos per day vs Neutrino Energy:

ConclusionsKalman Filter is a promising new way for filtering and reconstruction for KM3NeTHowever for the rejection of badly misreconstructed tracks additional cuts must be appliedPresented by Apostolos G. TsirigotisEmail: tsirigotis@eap.gr

Kalman Filter Basics (Linear system)Equation describing the evolution of the state vector (System Equation):Measurement equation:DefinitionsEstimated state vector after inclusion of the kth measurement (hit) (a posteriori estimation)Measurement kVector of parameters describing the state of the system (State vector)Track propagatorProcess noise (e.g. multiple scattering)Measurement noiseProjection (in measurement space) matrixa priori estimation of the state vector based on the previous (k-1) measurements

Kalman Filter Basics (Linear system)Prediction (Estimation based on previous knowledge)Extrapolation of the state vectorExtrapolation of the covariance matrixResidual of predictionsCovariance matrix of predicted residuals(criterion to decide the quality of the measurement)

Kalman Filter Basics (Linear system)Filtering (Update equations)where,is the Kalman Gain MatrixFiltered residuals:Contribution of the filtered point:(criterion to decide the quality of the measurement)

Kalman Filter (Non-Linear system)Extended Kalman Filter (EKF)Unscented Kalman Filter (UKF)A new extension of the Kalman Filter to nonlinear systems, S. J. Julier and J. K. Uhlmann (1997)

Kalman Filter Extensions Gaussian Sum Filter (GSF)t-texpectedApproximation of proccess or measurement noise by a sum of Gaussians

Run several Kalman filters in parallel one for each Gaussian component

Kalman Filter Muon Track ReconstructionPseudo-vertexZenith angleAzimuth angleState vectorMeasurement vector Hit Arrival timeHit chargeSystem Equation:Track Propagator=1 (parameter estimation)No Process noise (multiple scattering negligible for E>1TeV)Measurement equation:

Event reconstruction in underwater neutrino telescopes suffers from a high background noise. Adaptive algorithms are able to suppress automatically such a noise and therefore are considered as good candidates for track fitting at the KM3NeT environment. Adaptive algorithms, based on Kalman Filter methods, are extensively used in accelerator particle physics experiments, for event filtering, track reconstruction and vertex definition.In this note we describe an iterative event filtering and track reconstruction technique, employing Kalman Filter methods and we present results from a detailed simulation study concerning the KM3NeT detector.We evaluate the accuracy of this technique and we compare its efficiency with other standard track reconstruction methods.

The algorithm of muon track reconstruction is the following:First we must provide the algorithm with initial estimates fot the state vector and coavariance matrixAt each step we extrapolate the state vector and each covariance matrix using previous knowledge and we calculate the residuals of the predictions. Using these residuals we decide to include or not the current measurement.At the stage of the filtering we update the state vector and its covariance matrix using the information provided by the current hit, and calculating the chi-square contribution of the hit we have a more accurate criterion for the qauality of the hit.

The conclusion is that the operation of 3 such stations for 10 days will provide:The determination of a possible offset with an accuracy ~ 0.05 degThe determination of the absolute position of the -detector with an accuracy ~ 0.6 m

Calibration system and assuming that the n-Telescope resolution in determining the zenith angle of a muon track is about 0.1 degrees and the impact about 2 meters, In this techqnique the state vector of the system is estimated using progressively one measurement after the other.The apriori estimation of the state vector is based on the previous (k-1) measurements, while the a posteriori estimation of the state vector is update after inclusion of the next kth measurement.The equation ..The measurement In the first stage of prediction we extrapolate the state vector using previous knowledge, meaning all the previous (k-1) measurementsAnd we also extrapolate its covariance matrix.The estimated residual of the prediction of the kth measurement can be used as a rough criterion to decide if we will include or not the measurement.In the next stage of the filtering we update the state vector and its covariance matrix to its current value using the next kth measurementThe filtered residuals, which is the difference of the kth measurement with the one predicted using the update state vector, can be used to calculate the chisquare contribution of this measurement. This is used as a more precise criterion to decide the quality of the measurement and to include it or not

Know the equations I quoted are valid for linear systems, where the state vector propagate linearly from one measurement to the next, and the projection of the state vector to the measurement space is a linear transfromation.In the case our system is not linear several extension of the KF have been developed the previous years.One approach is to aproximate the system and measurement equation with linear ones using Taylor expansion around the estimated state vector at the previous step.One other more accurate approach is the one called UKF, a method for calculating the statistics of a random variable which undergoes a non-linear transformation

In the application of the KF to the muon track reconstruction the state vector is a 5-dimensional vector consisting from the pseudovertex, zenith and azimuth abgle,While the measurement vector includes the hit arrival time and charge.The system equation is an identity equation, in this case of parameter estimation, meaning that the state vector does not change theoretically from one measurement to the next one. We also do not include proces noise since multiple .The measurement equation is a non-linear one , and we use the method of UKF.