sean grullon with gary hill maximum likelihood reconstruction of events using waveforms

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


VLVnT2


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


VLVnT2


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)


VLVnT2


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)


VLVnT2


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


VLVnT2


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 !}){|}({


VLVnT2


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


VLVnT2


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


VLVnT2


Our likelihood function – cont.

loglog}){|}({log1

NpnnPk

iiiii

•Likelihood minimized for every optical module


VLVnT2


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.


VLVnT2


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.


VLVnT2


IceCube data stream


VLVnT2


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.


VLVnT2


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.


VLVnT2



VLVnT2


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


VLVnT2


Preliminary Results Using Full Simulation: Vertex Y

RMS: 36.32 RMS: 48.62


VLVnT2


Preliminary Results Using Full Simulation: Vertex Z

RMS: 29.65 RMS: 51.23


VLVnT2


Preliminary Results Using Full Simulation: Energy


VLVnT2


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

sean grullon with gary hill maximum likelihood reconstruction of events using waveforms

Documents