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A Broad-band Radio Attenuation Model for Antarctica and its effect

on ANITA sensitivity

ByAmir Javaid

University of Delaware

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Topics

● Radio Attenuation in ice and ANITA sensitivity

● Dependence of Radio Attenuation in ice on electrical conductivity and other physical and chemical properties of Antarctic ice

● Electrical conductivity calculation

● Attenuation estimation techniques

● Attenuation model Results

● Effects of new attenuation model on effective volume estimation

Attenuation in ice and ANITA sensitivity

• UHE neutrinos produce Radio pulses when they interact in Antarctic ice

• One of ANITA’s goal is to catch these Radio pulses in order to detect UHE neutrinos

• UHE neutrino interactions can be very deep inside Antarctic ice

• Radio pulses attenuate as they propagate inside ice• Proper understanding of radio attenuation in ice is

essential for better detection

Dependence of Radio Attenuation in ice on physical and chemical properties of Antarctic ice

For an electromagnetic wave propagating in z direct in a dielectric medium, the Electric Field can be described as

The wave number for a dielectric medium can be written as .

In the above equation is the attenuation length or the absorption loss which has the following from:

The equation for Radio wave attenuation can be written in an approximate form 8.686

Where is weakly dependent on temperature

Carriers of conduction in polar ice

Impure ice (MacGregor[2])

As ice as no electronic current, the carriers of conduction in Pure ice are:• Bjerrum Defects

• L-Defects • D-Defects

• Ionic Defects

The carriers for impure ice include the additional following carriers• Bjerrum Defects from impurities e.g.

and . For Antarctica is only important. • Ions from acids represented by in the

grain boundaries (veins) and triple junction.

Pure Ice(Physics of ice)

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Conductivity calculation method (pure ice)

The electrical conductivity comprises of two parts

The relation for pure ice conductivity is the following

In the above relation is the imaginary part of the permittivity. Parameterization reported by Fujita et al[1] is used to calculate given by the following formula

The parameters A, B and C for different temperatures are given in table 1.

Table 1 [1]

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Conductivity calculation method (impure ice)

There are two major type of impurities existing is Antarctic ice which are Acids and Sea Salts. The contribution of Acids are represented by [H+ ] ion and sea salt chlorine [ssCl-] ion. The impure part of the conductivity can be written as

where

The first term in both of these formulae is low frequency band term which is independent of frequency and the second term is the high frequency term which is important for ANITA because its bandwidth maximum ~ 1.2GHz, where the effects high frequency term are important.

Conductivity vs frequency

Conductivity vs Temperature

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Impurity data location maps(ITASE)(Bertler et al.)[4]

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Impurity data location maps(ITASE)

All impurity locations

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Method for conversion from major ions to [H+](acid) and [ssCl-](Sea Salt)

● [H+] concentrationThe [H+] concentration can be estimated by the following charge balance equation

[H+] = [Cl-] + [NO-3] + 2 [ssSO4

2-] + [xsSO42-]

+[CH3SO-3] - 2 [Ca2+] - [K+]-2 [Mg2+] -[Na+],

● [ssCl-] concentration

The sea salt Chlorine concentration can be calculated by using the of ratios FX of different ionic concentrations with sea salt Sodium ion in sea water. This can be represented as following

FX=[ssX]/[ssNa+]

The [ssNa+] concentration in the sample is calculated by

[ssNa+]sample =min([X]sample/FX)

Where [X]sample is the value of ionic concentration of some ion X in the sample. The sea salt concentrations of other ions in the sample can be estimated by the formula

FX=[ssX]sample / [ssNa+]sample

The non sea salt ionic concentration [xsX]samplecan be calculation by

[ssX]sample=[ssX]sample+[xsX]sample

Impurity concentration data

The ITASE data consists of major ions including sodium (Na), magnesium (Mg), calcium (Ca), potassium (K), chloride (Cl), nitrate (NO3), sulphate (SO4), and methane sulfonate (MS). Bertler et al.[4].

The data includes total of 520 points around Antarctica. For this study 326 points are used because of missing values of some major ion concentrations. To estimate some of the missing data, data fitting has been performed. The plots on the right show the fitting.

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IDW method for impurity concentration interpolation

This interpolation scheme is based on inverse square radial distance weighting. Impurity concentration for a required point over Antarctica is calculated by using the weighted contributions from all the points with measured impurity data. The weighting scheme can be written as follows

kriging• Kriging is a group of geo-statistical techniques used for estimation of value and

variance of the value of a random field at location where there is no data available by using locations with available data.

• It is a method based on linear prediction also known as Gauss-Markov estimation or Gaussian regression process.

• Major types of Kriging include Simple Kriging, ordinary kriging, cokriging and some more specialized types. We have used ordinary kriging for the present work.

• Application areas of krigingo Environmental scienceo Black box modeling in computer experimentso Hydrogeologyo Miningo Natural resourceso Remote sensing

Kriging interpolation methodIf is our unknown point and for i=0,….N-1 is our input data and denote the variogram between point i and j which is chosen to be of the form

.

Where is fitted by the un-weighted least squares over all pairs of data and .

We can write our input data into the following form

So the value of the field and its variance at the point is calculated by

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Comparison of Acid impurity IDW and Kriging

Comparison of IDW and kriging(contd ..)

To compare the two models of impurity estimation we have performed an n-1 check. This check involves taking one value out of the data and estimation of that value by using the rest of the data. The results for both IDW and kriging are shown below. It is very clear from these plots that the estimation from kriging is more accurate and it also provides an estimation of error also which is not possible by IDW.

3D Temperature Model for Antarctica

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A 3D temperature model has been used for the present study. Data for this model is provided by Dr. James Fastook, University of Maine, extracted from UMISM, the University of Maine Ice Sheet Model, a multi-component model of ice-sheet physics. A detail account of the model, reference and publications can be found at the following website

http://tulip.umcs.maine.edu/~shamis/umism/umism.html

The interpolation scheme utilized to interpolate between the grid points uses two kinds of weights which are the following

● Inverse square radial distance weights

● Inverse square ice thickness difference weights

The maximum radial distance used is 200 km and only those profiles are used which has a depth difference of less than 200m from the required location depth.

𝑊 𝑖=(1/ (𝑟0+𝑟 𝑖 ) )2 (1/ (𝑑𝑖𝑓𝑓 0+𝑑𝑖𝑓𝑓 𝑖 ))

2/∑𝑗

(1 / (𝑟0+𝑟 𝑗 ) )2 (1/ (𝑑𝑖𝑓𝑓 0+𝑑𝑖𝑓𝑓 𝑗 ) )

2;

𝑟0=5km ;𝑑𝑖𝑓𝑓 0=100m;𝑇=∑𝑇 𝑖𝑊 𝑖

Temperature Profile Sample

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Comparison of Modeled Temperature profiles with Measured profiles

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Attenuation model Results with IDW (Full Path attenuation)

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Attenuation model Results with IDW (1km(max) one way Path attenuation)

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Attenuation Results with kriging(Full path)

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Attenuation Results with kriging(1km max depth one way)

Attenuation Results with kriging(Attenuation depth profile)

Kriging vs IDW(0.6GHz)

Full depth bed return power loss Top 1km (max) power loss

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Comparison with South Pole and Ross Ice Shelf measurements

(Barwick et al [5]) David Saltsburg

Implementation of Kriging attenuation model in icemc

A grid was prepared with (latitude, longitude, depth, frequency). The grid comprises of 5329 latitude longitude bins, 100 depth bins

and 15 frequency bins. The code calculates attenuation values for only 8 set of frequencies

inside the ANITA bandwidth integrated over a vertical direct and reflected ray paths, starting from the point of the origin to the surface.

The values of attenuation for the rest of the frequencies are calculated by linear interpolation.

The estimation of the attenuation grid points are done by the weighted estimation method similar to the one used for temperature estimation.

Comparison of Effective volume*steradians (Kriging vs icemc attenuation)

Icemc Run results(both rays) (2million events)Energy(eV) (kriging attenuation)

V*Ω attenuation (Total)

V*Ω attenuation (e neutrinos)

V*Ω attenuation (μ neutrinos)

V*Ω attenuation (τ neutrinos)

# of Unweight neutrinos passing trigger Direct

# of Unweight neutrinos passing trigger Reflect

82.9293 + 32.0416 - 23.8246

133.902 + 85.9633 - 43.9553 101.825 + 79.9644 - 34.5694

13.2153 + 57.2666 - 5.63664

45 2

610.65 + 72.32 - 72.32 1106.57 + 171.253 - 171.253 397.138 + 109.801 - 109.801

327.602 + 91.7526 - 91.7526

215 5

3219.14 + 153.059 - 153.059

3692.52 + 284.389 - 284.389 3285.83 + 271.779 - 271.779

2679.35 + 239.441 - 239.441

769 24

9353.14 + 269.243 - 269.243

9124.74 + 470.58 - 470.58 8801.86 + 448.925 - 448.925

10132.4 + 479.165 - 479.165

1841 106

35299 + 518.056 - 518.056

29199.9 + 819.222 - 819.222 36158.6 + 912.061 - 912.061

40533.5 + 955.093 - 955.093

5023 660

Energy(eV)(Icemc attenuation)

V*Ω attenuation (Total)

V*Ω attenuation (e neutrinos)

V*Ω attenuation (μ neutrinos)

V*Ω attenuation (τ neutrinos)

# of Unweight neutrinos passing trigger Direct

# of Unweight neutrinos passing trigger Reflect

96.9679 + 35.1439 - 20.0495

199.815 + 88.6351 - 65.7462

62.9523 + 69.5725 - 21.597

28.3781 + 59.2781 - 9.55588

45 0

477.832 + 68.6811 - 54.1843

751.649 + 163.128 - 116.441

241.655 + 102.863 - 58.0451

439.867 + 126.725 - 83.395

197 0

2813.15 + 139.907 - 139.907

3199.54 + 259.032 - 259.032

2783.67 + 241.968 - 241.968

2456.28 + 224.774 - 224.774

707 7

8037.88 + 243.199 - 243.199

7413.41 + 411.269 - 411.269

7598.94 + 404.791 - 404.791

9101.16 + 446.427 - 446.427

1727 27

30583.3 + 471.727 - 471.727

25120.5 + 741.446 - 741.446

31765.5 + 832.454 - 832.454

34858.7 + 871.655 - 871.655

4903 348

Icemc run( between icemc and krig attenuation model(direct rays only)

Un-weighted Events passing trigger

Comparison of Effective volume*steradians (Kriging vs icemc attenuation)(Reflected Rays only)

Icemc run( between icemc and krig attenuation model(Ref rays only)

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Plan

● Submit the attenuation model paper to Journal of Geophysical research for publication.

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