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Sean Grullon with Gary Hill
Maximum likelihood reconstruction of events using waveforms
Sean Grullon w/ Gary Hill 2
VLVnT2
November 8th-11th 2005
Overview
• Introduction & Motivation• Likelihood Formulation• waveform-loglikelihood-reco project in
IceCube software framework• Preliminary Results using IceCube
simulated Data• Current Development and Future
Directions
Sean Grullon w/ Gary Hill 3
VLVnT2
November 8th-11th 2005
Introduction & Motivation
• All reconstruction algorithms in IceCube are ported from AMANDA.
• Originally developed for AMANDA’s primary DAQ. – records Time over Threshold (TOT), the leading edge
time (LE), and the Peak Amplitude.• The Full waveform is not captured• Incorporates Leading Edge time & peak amplitude information
only. • Uses the Pandel function which analytically parameterizes
timing PDF in ice.• Ice assumed to be homogeneous. • Full detail regarding the AMANDA reconstruction algorithms
can be found at Nuc. Ins. Meth. A 524 169(2004)• Focus of talk is on the development of a new reconstruction
aglorithm using the full waveform
Sean Grullon w/ Gary Hill 4
VLVnT2
November 8th-11th 2005
IceCube waveforms
• IceCube’s Analog Transient Waveform Digitizer (ATWD) captures and digitizes full waveform in situ with a ~ 420 ns time window
• Should prove powerful for high energy & non-contained events.
• FWHM of Waveform Depends linearly on the distance from the event to the optical module
• New algorithms need to be developed to take advantage of full waveform information
• A high priority since deployment has already begun.
ATWD Sample #
Voltage (m
V)
Sean Grullon w/ Gary Hill 5
VLVnT2
November 8th-11th 2005
Example Extracted waveform
• Event generated by Nitrogen laser located at a depth of 1850 m in AMANDA Array. Pulse Shapes recorded at 3 distances from laser. (45m, 115m, and 167m)
Sean Grullon w/ Gary Hill 6
VLVnT2
November 8th-11th 2005
Likelihood Formulation
• How can you formulate a likelihood function with the full waveform at your disposal?
• With the full waveform, we know:– arrival time distribution of the photons
– the probability of these arrival times.
• Given an expected distribution of photons μp(t), what is the probability of observing a waveform f(t)?– p(t) is normalized timing PDF, μ is the total number of
expected photons, given either numerically or analytically.
– f(t) is your observed waveform
Sean Grullon w/ Gary Hill 7
VLVnT2
November 8th-11th 2005
Probability of f(t) given p(t)?
}{)(
}{)(
1
i
k
ii
i
tp
Nn
ntf
•Suppose you bin the photon distributions into k time bins:
•The probability is given by Poisson statistics, as a product of Poisson probabilities over all the k bins:
i
i
en
nPk
i i
ni
ii
1 !}){|}({
Sean Grullon w/ Gary Hill 8
VLVnT2
November 8th-11th 2005
This product turns into something useful….
k
ik
i
n
k
i
n
k
i i
ni
iii
i
i
i
en
nP1
1
1
1 !}){|}({
e
NnNnP
Nn
k
i i
i
ii
i
!!!}){|}({
1
Sean Grullon w/ Gary Hill 9
VLVnT2
November 8th-11th 2005
We have our LikelihoodFunction
• Take the negative log of it
log!loglog
!loglog!loglog!log
!!!log}){|}({log
1 1
1 1
1
Nnn
NNnnN
eNn
NnP
i
k
i
k
iii
i
k
i
k
iii
Nn
k
i i
i
ii
i
Sean Grullon w/ Gary Hill 10
VLVnT2
November 8th-11th 2005
Our likelihood function – cont.
loglog}){|}({log1
NpnnPk
iiiii
•Likelihood minimized for every optical module
Sean Grullon w/ Gary Hill 11
VLVnT2
November 8th-11th 2005
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.
Sean Grullon w/ Gary Hill 12
VLVnT2
November 8th-11th 2005
IceCube software framework
• The IceCube software framework is called IceTray.• unified object oriented C++ framework for handling
online filtering and offline-software for reconstruction, analysis, and simulation.
• IceTray modules operate on the IceCube data stream.
• Modules perform specialized tasks such as reconstructions, calibrations, etc.
• Uses boost C++ libraries for offline data. Data can be saved into a binary format or XML format.
Sean Grullon w/ Gary Hill 13
VLVnT2
November 8th-11th 2005
IceCube data stream
Sean Grullon w/ Gary Hill 14
VLVnT2
November 8th-11th 2005
Waveform loglikelihood reconstruction project
• This likelihood reconstruction algorithm is currently implemented in IceTray.
• Currently reconstructing electromagnetic cascades.
• User has the option of selecting an analytical PDF (Pandel function) or a numerical PDF.
• The numerical PDF in IceCube is Photonics, a numerical framework that simulates photon propagation in the ice.
• Uses the SIMPLEX minimizer. • Uses calibrated ATWD waveform directly.
Sean Grullon w/ Gary Hill 15
VLVnT2
November 8th-11th 2005
Preliminary Results
• 2500 cascade events with an energy of 100 TeV generated
• Generated with a random vertex position and direction.
• Full IceCube simulation used. • ~¼ of the events are not contained in the
array (Up to 50 m away) • Free parameters of fit are the vertex, the
energy, and the time. • Results compared to the AMANDA style
cascade reconstruction algorithm.
Sean Grullon w/ Gary Hill 16
VLVnT2
November 8th-11th 2005
Sean Grullon w/ Gary Hill 17
VLVnT2
November 8th-11th 2005
Preliminary Results Using Full Simulation: Vertex X
RMS: 38.15 RMS: 49.62
•Accurate Vertex Reconstruction requires directional fit
•Results not a final performance indication
Sean Grullon w/ Gary Hill 18
VLVnT2
November 8th-11th 2005
Preliminary Results Using Full Simulation: Vertex Y
RMS: 36.32 RMS: 48.62
Sean Grullon w/ Gary Hill 19
VLVnT2
November 8th-11th 2005
Preliminary Results Using Full Simulation: Vertex Z
RMS: 29.65 RMS: 51.23
Sean Grullon w/ Gary Hill 20
VLVnT2
November 8th-11th 2005
Preliminary Results Using Full Simulation: Energy
Sean Grullon w/ Gary Hill 21
VLVnT2
November 8th-11th 2005
Current Development and Future Directions
• Currently testing new Photon tables with 3-D photon tracking as the PDF for reconstruction
• Investigate reconstructing the cascade direction.• Make the project part of the official IceTray
release. • Look at some sort of hit cleaning to improve
results• Improve algorithm performance for non-
contained events.• Look at other event types • Optimize the performance
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