1. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Acoustic Integrals in Water and Ice Prof. Sean Danaher University of Northumbria, Newcastle UK

2. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Introduction Velocity Potential Attenuation Acoustic Integrals in Water and Ice Large angles and Integrating Spheres Fluctuations and Neutrino Event Generator Conclusions

3. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Acoustic Integrals “classical” approach fast thermal energy deposition Band limited by Water Properties Attenuation Slow decay

4. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Is Pressure a good choice of Variable (1)? Electromagnetism B Magnetic Field (Physics) H Magnetic Field (Electrical Eng) E Electric Field D Electric Flux Density V Electric Potential A Magnetic Vector Potential I Current For Circuits the Electric Potential, V is normally the variable of choice. For fields E is normally the variable of choice. Nearly all detectors see E rather than B But very difficult to directly create large quantities of Electrical energy/Field. Virtually all “electrical” machines are magnetically driven. Transmit antennas are driven by current -> Magnetic Field -> Electric Field

5. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Is Pressure a good choice of Variable (2)? Acoustics p Pressure v particle velocity  Velocity Potential For Hydraulic circuits the pressure, p is normally the variable of choice. For fields p is normally the variable of choice. Nearly all detectors see p rather than v But in “free space” the coupling between an acoustic radiator and the medium is via velocity. For sources the velocity potential  is the driving variable rather than p. p is not a good choice of variable for transmission

6. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Using  makes things easier Attenuation free wave equation: Time rate of change of volume Thermal Energy injection Cooling term Cooling term can be ignored for neutrinos. The velocity potential is simply a scaled copy of the Energy deposition

7. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - How to do the integral? Integral can be evaluated using Green’s functions and retarded volume integrals. OR If the energy deposition is modelled using Monte Carlo points with a density proportional to energy,  (t) will simply be a scaled histogram of the flight times to the observer. p can be trivially calculated

8. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Integral of a tri-axial distribution 1.The pressure pulse is a scaled derivative of the cross-section along the line of sight to the observer. 2.Only the distribution along the line of sight is important. 3.The height of the pressure pulse will depend on 1/  O 2 4.At angles greater than a few degrees from the x-y plane the height of the pressure pulse will depend on 1/sin 2 . Gruneisen Constant c term from x=(-) ct c term from derivative

9. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Attenuation In distilled water the main energy loss mechanism is viscosity. In seawater however the attenuation in the 1-100 kHz region is dominated by the disassociation of boric acid and magnesium sulphate. 10 10 0 1 -6 -5 -4 -3 -2 0 x 10 -3 Relative frequency Velocity Change (%) a) Velocity Change 10 10 0 1 10 0 Relative frequency Relative Amplitude b) Attenuation The viscosity attenuation is real. Chemical Relaxation however is complex: frequencies lower than the resonance have a reduced velocity. Resonance frequencies c: 1kHz B(OH) 3 and 100kHz MgSO 4. These act as high pass filters. In Ice the attenuation is believed constant at frequencies below c 20 kHz above this Raleigh scattering from grain boundaries becomes dominant. Attenuation is real N.B. plots illustrative only

10. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - New Tuneable Complex attenuation parameterisation Velocity change is tiny and has never been measured All measurements to date determine the magnitude; not the phase. The “Gold Standard” work is that of Francois and Garrison. More recently Ainslie and Mc Colm have produced a simplified parameterisation. Niess and Bertin have published a complex parameterisation for “Mediterranean” conditions T=15 O C; S=37ppt; pH=7.9; z=2km; We have modified the A&McC parameterisation to give complex results whilst retaining the same magnitude Ice attenuation from Price

11. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Acoustic Integrals with Attenuation Can take the FT of the pulse, multiply by a(  ) and take the IFT OR Can take the IFT of the attenuation, and convolve with  (t) Complex attenuation delays the pulse and causes asymmetry Also called the unit impulse response

12. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Pulse Characteristic in Seawater with distance 10 11 GeV shower In pancake F max is c. 50 kHz near the source, c. 25 kHz @1km and c. 1 kHz at 100 km In the absence of attenuation P max d is a constant once far field is established Asymmetry is at a maximum close to the source falls to about 0.1 at 10 km but peaks again at 100 km where F max corresponds to the B(OH) 3 resonance.

13. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Pulse as a Function of Angle for CORSIKA Showers Seawater Outside a few degrees the asymmetry is constant; dictated by the longitudinal distribution. This flips for +/- angles What is happening here? Outside a few degrees the pulse shape remains the same but gets longer in time as sin  and shorter in amplitude as 1/sin 2  Asymmetry

14. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Mixing of different ages of shower The curious behaviour at small angles is caused by the time delays to the observer being an interaction of radial and longitudinal delays Because  can be created by producing a histogram of flight times, this can also be considered a probability density function PDF. Age of shower colour coded from black to white bins

15. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Variation with Energy in Seawater The pulse height in seawater grows marginally more rapidly with energy than direct proportionality. This is because as the showers get longer with energy the pancake gets more focussed. The pulse height is approx 1 pPa/GeV in the pancake

16. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Integrating Sphere Seawater @1km Energy increases as E O 2 Very inefficient Attenuation reduces acoustic energy to c 43% >100% at 10 22.8 GeV Temp rise c10  K for 10 12 GeV, 0.6 MK at 10 22.8 GeV k= 22.8 GeV without attenuation 23.17 GeV with attenuation.

17. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Pulses in Ice @1km in pancake Pulses higher and shorter than water 5-6 pPa/GeV. (1 pPa/GeV) Frequency higher c 39 kHz (24 kHz) Asymmetry nearly zero (Far field established sooner and real attenuation) Ringing in pulse

18. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Dominant Frequency Seawater + angles above the shower, - angles below Frequency falls with angle and distance 10 11 GeV primary

19. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Maximum Pressure Seawater + angles above the shower, - angles below 10 11 GeV primary Pressure falls with angle and distance

20. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Asymmetry Seawater + angles above the shower, - angles below 10 11 GeV primary Asymmetry max (c 0.6) on two areas caused by geometry In the pancake settles to c 0.2 after c 1km

21. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Dominant Frequency Ice + angles above the shower, - angles below 10 11 GeV primary Higher frequencies than in water. Real attenuation Asymmetry at +/- angles caused by longitudinal distribution Wasp waist?

22. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Max Pressure in Ice + angles above the shower, - angles below 10 11 GeV primary Far field established sooner

23. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Asymmetry in Ice + angles above the shower, - angles below 10 11 GeV primary Again two patches of high asymmetry Asymmetry goes to zero on axis

24. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Coping with Fluctuations W L V T()()() Decompose the Data matrix into 3 matrices. When multiplied we get back the original data. W and V are unitary. L contains the contribution from each of the eigenvectors in descending order along main diagonal. We can get an approximation of the original data by setting the L values to zero below a certain threshold Similar techniques are used in statistics CVA, PCA and Factor Analysis Based on Eigenvector Techniques Good SVD algorithms exist in ROOT and MATLAB Sean Vernon Danaher Using Singular Value Decomposition (SVD)

25. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - SVD II Somewhere between vector five and seven we get the “Best” representation of the data Better than the observation as noise filtered out Highly compressed (only 1-2% of original size e.g. 6/500x1.5) Have basis vectors for data so can produce “similar” data+++ SVD done on Noisy data

26. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Coping with 3 dimensions ( ) Showers binned radially into 20 bins and longitudinally into 100 bins 100 showers at each of seven decades from 10 15 to 10 21 eV (produced in Corsika - Lancaster). SVD applied and four eigenvectors taken. O Matrix 70,000x20 W matrix 70,000x4 ( ) Reshaped into four 100x700 matrices and a SVD is done on each in turn

27. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Sample Eigenvectors Figure 4. The first three radial vectors and associated longitudinal vectors.

28. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Fits Showers well Accuracy within 5% on average for all showers. Original shows “shot” noise due to thinning Longitudinal Radial Longitudinal Radial

29. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Dealing with Fluctuations With a covariance matrix it is possible to model inter- shower fluctuations Fluctuations decrease with energy so probably not important for acoustic detection

30. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Comparison of Parameterised and directly generated showers New neutrino event generator is available at the ACORNE web site Note mesokurtic distribution assumed but stats appear to be slightly leptokurtic Our previous parameterisation is fine if fluctuations are not needed (highly sensitive plot) Sensitive plot 10 05 is 1.12

31. Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM + - + - + - + - Conclusions Questions?