Maximum likelihood reconstruction of events using waveforms Gary

Maximum likelihood reconstruction of events using waveforms Gary Hill & Sean Grullon Ice. Cube Flasher & Cascade Analysis Workshop

Overview Ice. Cube Flasher & Cascade Analysis Workshop July 23 -27, 2005 Berkeley July 26 th 2005 • • Introduction & Motivation Likelihood Formulation wf-llh-reco project in Ice. Tray Preliminary Results using Simulation V 01 -00 -03 • Connections to other projects & future directions • Discussion Gary Hill & Sean Grullon 2

Introduction & Motivation Ice. Cube Flasher & Cascade Analysis Workshop • July 23 -27, 2005 Berkeley • • July 26 th 2005 All reconstruction algorithms in Ice. Rec 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 3

Likelihood Formulation Ice. Cube Flasher & Cascade Analysis Workshop July 23 -27, 2005 Berkeley July 26 th 2005 • 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 4

Probability of f(t) given p(t)? Ice. Cube Flasher & Cascade Analysis Workshop • Suppose you bin the photon distributions into k bins: July 23 -27, 2005 Berkeley July 26 th 2005 Gary Hill & Sean Grullon 5

Probability of {ni } | {μi} ? Ice. Cube Flasher & Cascade Analysis Workshop • The probability is given by Poisson statistics, as a product of Poisson probabilities over all the k bins: July 23 -27, 2005 Berkeley July 26 th 2005 Gary Hill & Sean Grullon 6

This product turns into something useful…. Ice. Cube Flasher & Cascade Analysis Workshop July 23 -27, 2005 Berkeley July 26 th 2005 Gary Hill & Sean Grullon 7

Ice. Cube Flasher & Cascade Analysis Workshop July 23 -27, 2005 Berkeley July 26 th 2005 • … 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 8

We have our Likelihood Function Ice. Cube Flasher & Cascade Analysis Workshop • Take the negative log of it July 23 -27, 2005 Berkeley July 26 th 2005 Gary Hill & Sean Grullon 9

Where is this applicable? Ice. Cube Flasher & Cascade Analysis Workshop July 23 -27, 2005 Berkeley July 26 th 2005 • 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 10

wf-llh-reco module Ice. Cube Flasher & Cascade Analysis Workshop July 23 -27, 2005 Berkeley July 26 th 2005 • The icetray implementation of this likelihood reconstruction. • Currently looking at (non-directional) cascades. • Uses UPandel for the timing PDF and PHit. PNo. Hit 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 11

Preliminary Results using Simulation V 01 -00 -03 Ice. Cube Flasher & Cascade Analysis Workshop July 23 -27, 2005 Berkeley July 26 th 2005 • Newest version of Simulation metaproject just released over the weekend. • 500 100 -Ge. V 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 12

Preliminary Results Using Simulation V 01 -00 -03: Vertex X Ice. Cube Flasher & Cascade Analysis Workshop July 23 -27, 2005 Berkeley July 26 th 2005 Gary Hill & Sean Grullon 13

Preliminary Results using Simulation V 01 -00 -03: Vertex Y Ice. Cube Flasher & Cascade Analysis Workshop July 23 -27, 2005 Berkeley July 26 th 2005 Gary Hill & Sean Grullon 14

Preliminary Results using Simulation V 01 -00 -03: Vertex Z Ice. Cube Flasher & Cascade Analysis Workshop July 23 -27, 2005 Berkeley July 26 th 2005 Gary Hill & Sean Grullon 15

Preliminary Results using Simulation V 01 -00 -03: Vertex T Ice. Cube Flasher & Cascade Analysis Workshop July 23 -27, 2005 Berkeley July 26 th 2005 Gary Hill & Sean Grullon 16

Future Directions Ice. Cube Flasher & Cascade Analysis Workshop July 23 -27, 2005 Berkeley July 26 th 2005 • 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 17

Discussion Ice. Cube Flasher & Cascade Analysis Workshop July 23 -27, 2005 • Thoughts and Comments • Possible connection with other waveform based algorithms? Berkeley July 26 th 2005 Gary Hill & Sean Grullon 18