1. Hellenic Open University
Physics Laboratory
School of Science and Technology
Hellenic Open University
Application of Kalman filter methods to event filtering and
reconstruction for Neutrino Telescopy
A. G. Tsirigotis
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.
VLVnT08 - Toulon, Var, France, April 2008
In the framework of the KM3NeT Design Study

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

3. MultiPMT Optical Module (NIKHEF Design)
20 x 3” PMTs (Photonis XP53X2) in each 17” Optical Module
Optical Noise
Outside view
Inside View
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)

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

5. (Vx,Vy,Vz) pseudo-vertex
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 dm
L-dm
(Vx,Vy,Vz) pseudo-vertex dγ d
Track Parameters
θ : zenith angle φ: azimuth angle (Vx,Vy,Vz): pseudo-vertex coordinates θc
(x,y,z)
Χ2 fit without taking into account the charge (number of photons)
Kalman Filter (novel application in this area)

6. Kalman Filter – Basics (Linear system)
Definitions
Vector of parameters describing the state of the system (State vector)
a priori estimation of the state vector based on the previous (k-1) measurements
Estimated state vector after inclusion of the kth measurement (hit) (a posteriori estimation)
Measurement k
Equation describing the evolution of the state vector (System Equation):
Track propagator
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 …
Process noise (e.g. multiple scattering)
Measurement equation:
Projection (in measurement space) matrix
Measurement noise

7. Kalman Filter – Basics (Linear system)
Prediction (Estimation based on previous knowledge)
Extrapolation of the state vector
Extrapolation of the covariance matrix
Residual of predictions
(criterion to decide the quality of the measurement)
In the first stage of prediction we extrapolate the state vector using previous knowledge, meaning all the previous (k-1) measurements
And 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.
Covariance matrix of predicted residuals

8. Kalman Filter – Basics (Linear system)
Filtering (Update equations)
where,
is the Kalman Gain Matrix
In the next stage of the filtering we update the state vector and its covariance matrix to its current value using the next kth measurement
The 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
Filtered residuals:
Contribution of the filtered point:
(criterion to decide the quality of the measurement)

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

10. Kalman Filter Extensions – Gaussian Sum Filter (GSF)
Approximation of proccess or measurement noise by a sum of Gaussians
Run several Kalman filters in parallel one for each Gaussian component
t-texpected

11. Kalman Filter – Muon Track Reconstruction
Pseudo-vertex
State vector
Zenith angle
Azimuth angle
Hit Arrival time
Measurement vector
Hit charge
System Equation:
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.
Track Propagator=1 (parameter estimation)
No Process noise (multiple scattering negligible for Eμ>1TeV)
Measurement equation:

12. Initial estimates for the state vector and covariance matrix
Kalman Filter – Muon Track Reconstruction - Algorithm
Filtering
Prediction
Extrapolate the state vector
Update the state vector
Update the covariance matrix
Extrapolate the covariance matrix
Calculate the residual of predictions
Decide to include or not the measurement (rough criterion)
Calculate the contribution of the filtered point
Decide to include or not the measurement (precise criterion)
The algorithm of muon track reconstruction is the following:
First we must provide the algorithm with initial estimates fot the state vector and coavariance matrix
At 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.
Initial estimates for the state vector and covariance matrix

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

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

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

16. Conclusions
Kalman Filter is a promising new way for filtering and reconstruction for KM3NeT
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 deg
The determination of the absolute position of the ν-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,
Presented by Apostolos G. Tsirigotis