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Likelihood Formulation for Waveform Reconstruction in IceCube Detectors

Explore the formulation of likelihood function for waveform reconstruction in IceCube detectors to make use of full waveform data for high priority reconstruction tasks. Formulate probabilities and apply Poisson statistics for optimal reconstruction. This project aims to develop new algorithms for better event reconstruction in neutrino experiments.

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Likelihood Formulation for Waveform Reconstruction in IceCube Detectors

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  1. Berkeley

  2. Overview • Introduction & Motivation • Likelihood Formulation • wf-llh-reco project in IceTray • Preliminary Results using Simulation V01-00-03 • Connections to other projects & future directions • Discussion Gary Hill & Sean Grullon

  3. Introduction & Motivation • All reconstruction algorithms in IceRec are ported from AMANDA. • Were originally developed for the Muon-DAQ… (TOTs, LEs, Peak Amp.) • Need new algorithm(s) to take advantage of the full waveform information Icecube provides. • A high priority since deployment has already begun. Gary Hill & Sean Grullon

  4. Likelihood Formulation • How can you formulate a likelihood function with the full waveform at your disposal? • Consider first the ideal case where the OM time response is a delta function • Ignore any possible amplitude fluctuations for the moment. • Given an expected distribution of photons μp(t), what is the probability of observing a waveform f(t)? • p(t) is normalized timing probability, μ is the total number of expected photons, given either numerically (Photonics) or analytically (e.g. Pandel) • f(t) is your observed waveform Gary Hill & Sean Grullon

  5. Probability of f(t) given p(t)? • Suppose you bin the photon distributions into k bins: Gary Hill & Sean Grullon

  6. Probability of {ni } | {μi} ? • The probability is given by Poisson statistics, as a product of Poisson probabilities over all the k bins: Gary Hill & Sean Grullon

  7. This product turns into something useful…. Gary Hill & Sean Grullon

  8. … Namely a multinomial distribution, the probability of arranging exactly N events into k bins, multiplied by the Poisson probability of these N events occurring. Gary Hill & Sean Grullon

  9. We have our Likelihood Function • Take the negative log of it Gary Hill & Sean Grullon

  10. Where is this applicable? • We assumed we knew the photon arrival times precisely, or have a waveform made from the superposition of many photons. • If we have a non-delta function time response, this form is still applicable as long as our PDF is slowly varying over the region described by the OM time response. • Should be the case for our optical modules, typical pulse widths are narrow relative to the scale of expected photon arrival time distribution. Gary Hill & Sean Grullon

  11. wf-llh-reco module • The icetray implementation of this likelihood reconstruction. • Currently looking at (non-directional) cascades. • Uses UPandel for the timing PDF and PHit-PNoHit to get the expected PEs. • Uses the MIGRAD minimizer in ROOT’s TMinuit class. • Module in the sandbox area of SVN, named wf-llh-reco. Gary Hill & Sean Grullon

  12. Preliminary Results using Simulation V01-00-03 • Newest version of Simulation meta-project just released over the weekend. • 500 100-GeV cascade events were simulated with the vertex at the origin. • Three reconstruction modules compared in icetray: CFirst, cscd-llh and wf-llh-reco. Gary Hill & Sean Grullon

  13. Preliminary Results Using Simulation V01-00-03: Vertex X Gary Hill & Sean Grullon

  14. Preliminary Results using Simulation V01-00-03: Vertex Y Gary Hill & Sean Grullon

  15. Preliminary Results using Simulation V01-00-03: Vertex Z Gary Hill & Sean Grullon

  16. Preliminary Results using Simulation V01-00-03: Vertex T Gary Hill & Sean Grullon

  17. Future Directions • First and foremost, continue development and check the algorithm. • Incorporate Photonics as the “PDF of choice” for waveform reconstruction. • Look at other event types (directional cascades, muon tracks, high energy tracks, etc) • Optimize the minimization, perhaps use another minimizer? Gary Hill & Sean Grullon

  18. Discussion • Thoughts and Comments • Possible connection with other waveform based algorithms? Gary Hill & Sean Grullon

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