Application of Kalman filter methods to event filtering and reconstruction for Neutrino Telescopy A. G. Tsirigotis In the framework of the KM3NeT Design Study VLVnT08 - Toulon, Var, France, April 2008

125 meters IceCube Geometry: 9600 OMs looking up & down in a hexagonal grid. 80 Strings, 60 storeys each. 17m between storeys

Outside view Inside View MultiPMT Optical Module (NIKHEF Design) 20 x 3” PMTs (Photonis XP53X2) in each 17” Optical Module Single PMT Rate (dark current + K40) ~ 4kHz 120 Hz Double coincidence rate per OM (20 ns window) 6 Noise Hits per 6μsec window (9600 MultiPMT OMs in a KM3 Grid) Optical Noise

Muon Event Generation (1 TeV Muons, isotropic flux, IceCube Geometry, 9600 OMs) KM3NeT

Prefit, filtering and muon reconstruction algorithms Local (storey) Coincidence ( Applicable only when there are more than one PMT looking towards the same hemisphere) Global clustering (causality) filter Local clustering (causality) filter Prefit and Filtering based on clustering of candidate track segments Χ 2 fit without taking into account the charge (number of photons) Kalman Filter (novel application in this area) dmdm L-d m (V x,V y,V z ) pseudo-vertex dγdγ d Track Parameters θ : zenith angle φ: azimuth angle (Vx,Vy,Vz): pseudo-vertex coordinates θcθc (x,y,z)

Kalman Filter – Basics (Linear system) Equation describing the evolution of the state vector (System Equation): Measurement equation: Definitions Estimated state vector after inclusion of the k th measurement (hit) (a posteriori estimation) Measurement k Vector of parameters describing the state of the system (State vector) Track propagator Process noise (e.g. multiple scattering) Measurement noise Projection (in measurement space) matrix a 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 vector Extrapolation of the covariance matrix Residual of predictions Covariance 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 Matrix Filtered 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-t expected Approximation 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 Reconstruction Pseudo-vertex Zenith angle Azimuth angle State vector Measurement vector Hit Arrival time Hit charge System Equation: Track Propagator=1 (parameter estimation) No Process noise (multiple scattering negligible for E μ >1TeV) Measurement equation:

Kalman Filter – Muon Track Reconstruction - Algorithm Extrapolate the state vector Extrapolate the covariance matrix Calculate the residual of predictions Decide to include or not the measurement (rough criterion) Update the state vector Update the covariance matrix Calculate the contribution of the filtered point Decide to include or not the measurement (precise criterion) Prediction Filtering Initial estimates for the state vector and covariance matrix

Chi-square vs Kalman Filter – Comparison (1TeV muons isotropic flux) With initial background filtering Zenith angle resolution Space angle resolution KF degrees σ=0.074±0.004 σ=0.081±0.004 efficiency = 56%KF efficiency = 54%

Filtering Efficiency (1 TeV Muons, isotropic flux, IceCube Geometry, 9600 OMs) Events with number of hits (noise+signal) >4 Number of Active OMs Signal Noise Signal Noise Number of Active OMs Events passing after background filtering Percentage of noise hits after filtering percentage

Chi-square vs Kalman Filter – Comparison (1TeV muons isotropic flux) Without initial background filtering Zenith angle resolution Space angle resolution degrees KF efficiency = 11%KF efficiency = 48%

Conclusions Kalman Filter is a promising new way for filtering and reconstruction for KM3NeT Presented by Apostolos G. Tsirigotis