Transcript Talk given at November 2005 very large neutrino

Maximum likelihood reconstruction
of events using waveforms
Sean Grullon with Gary Hill
Overview
VLVnT2
November 8th-11th
2005
• 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
2
Introduction & Motivation
VLVnT2
•
November 8th-11th
2005
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
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IceCube waveforms
VLVnT2
•
November 8th-11th
2005
•
•
•
Voltage (mV)
•
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.
Sean Grullon w/ Gary Hill
ATWD Sample #
4
Example Extracted waveform
VLVnT2
November 8th-11th
2005
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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
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Likelihood Formulation
VLVnT2
November 8th-11th
2005
• 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
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Probability of f(t) given p(t)?
VLVnT2
November 8th-11th
2005
•Suppose you bin the photon distributions into k
time bins:
f (t )  {ni }
k
n
i 1
i
N
 p (t )  {i }
•The probability is given by Poisson statistics, as a product of
Poisson probabilities over all the k bins:
k
P({ni } | {i })  
i 1
Sean Grullon w/ Gary Hill
i
ni
ni !
e
 i
7
This product turns into something
useful….
VLVnT2
November 8th-11th
2005
k
k
P({ni } | {i })  
i 1
i
n


i
ni
ni !
i 1
k
n


i
k
 i
e

i 1
i 1
k
P({ni } | {i })  N !
i 1
Sean Grullon w/ Gary Hill


ni !
N!
i ni

N
e

8
We have our Likelihood
Function
VLVnT2
November 8th-11th
2005
• Take the
negative log of it

i ni

N
k



 log P ({ni } | {i })   log  N ! 
e
 i 1 ni ! N !
k
k
i 1
i 1




  log N !  ni log i     log ni !  N log   log N !  
k
k
i 1
i 1
  ni log i     log ni !  N log   
Sean Grullon w/ Gary Hill
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Our likelihood function – cont.
VLVnT2
November 8th-11th
2005
k
 log P({ni } | {i })   ni log  pi   N log   
i 1
•Likelihood minimized for every optical module
Sean Grullon w/ Gary Hill
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Where is this applicable?
VLVnT2
November 8th-11th
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.
Sean Grullon w/ Gary Hill
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IceCube software framework
VLVnT2
November 8th-11th
2005
• 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
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IceCube data stream
VLVnT2
November 8th-11th
2005
Sean Grullon w/ Gary Hill
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Waveform loglikelihood
reconstruction project
VLVnT2
November 8th-11th
2005
• 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
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Preliminary Results
VLVnT2
8th-11th
November
2005
• 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
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VLVnT2
November 8th-11th
2005
Sean Grullon w/ Gary Hill
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Preliminary Results Using Full
Simulation: Vertex X
VLVnT2
November 8th-11th
2005
RMS: 38.15
RMS: 49.62
•Accurate
Vertex
Reconstruction
requires
directional fit
•Results not a
final
performance
indication
Sean Grullon w/ Gary Hill
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Preliminary Results Using Full
Simulation: Vertex Y
VLVnT2
November 8th-11th
2005
RMS: 36.32
Sean Grullon w/ Gary Hill
RMS: 48.62
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Preliminary Results Using Full
Simulation: Vertex Z
VLVnT2
November 8th-11th
2005
RMS: 29.65
Sean Grullon w/ Gary Hill
RMS: 51.23
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Preliminary Results Using Full
Simulation: Energy
VLVnT2
November 8th-11th
2005
Sean Grullon w/ Gary Hill
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Current Development and Future
Directions
VLVnT2
November 8th-11th
2005
• 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 noncontained events.
• Look at other event types
• Optimize the performance
Sean Grullon w/ Gary Hill
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