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Reconstruction Algorithms for UHE Neutrino Events

in Sea Water

Simon Bevan

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Contents• Techniques

• Analytically

• Look-up Table

• Minimum Number of Hydrophones

• Unknown detector location

• Passive Detector Fitting

• The Future

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Methods

Analytically – Fast, but assumes linear propagation of waves

Look-up Table – Slow, but can incorporate non-linear propagation models

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Analyticallyv v x , v y , vz shower location vertex :UNKNOWNr i x i , y i , zi location of detector i : KNOWNt i time measured at detector i KNOWNt s time of shower:UNKNOWNc sound of speed : KNOWN

v ri2 c2 ti t s

2

distance time

speed

The basic Equation: -

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Finding the Vertex

where

P M 1RQ M 1T

dx12 dy12 dz12dx13 dy13 dz13dx14 dy14 dz14

righ

and . . ..R r i

2 r j2 c2 t i

2 t j2 i , j 1,2 ;1,3 ;1,4

T 2 c2 ti t j i , j 1,2 ;1,3 ;1,4M 2

v P t sQ

• Therefore both P and Q are known

• If we can find ts, we can find the vertex

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Singular Value Decomposition• But notice M-1

• If the system is over constrained, more than 4 hit hydrophones, M becomes non symmetric and can't be inverted • If M is singular (the determinant is zero), then M can't be inverted• But there is another way – SVD

• For every m x n matrix, if m>=n, the matrix can be written as

M = ULVT

• Where U and V are orthogonal matrices, containing the column and row eigenvectors respectively, and L is a diagonal matrix containing the eigenvalues in decreasing order. • Giving

M-1 = VL-1UT

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Nearly there…..

P M 1RQ M 1T

v P t sQ

Now we can solve for P and Q

And in the vertex equation only ts is unknown

Can we solve ts? ….of course.

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ts and

v ri2 c2 ti t s

2 P t sQ r i2

χ 2

χ 2 t i actual t i recon2

t i recon t s hi v c

But this is a quadratic, how do we pick the right solution?

•Take the positive solution, and from the calculated time of interaction, ts, and the calculated vertex, propagate an imaginary wave backwards to hydrophone one using the speed of sound in water. Now repeat for the negative solution.

• We know what time the hydrophone was actually hit, therefore take the solution that most closely matches this time, ie the lowest chi2.

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Look-Up Table• Define a grid. For each point in this grid calculate the times that the you expect the detectors to be hit.

• Perform a chi2 test, with the minimum being the vertex location.

• Slow, and the more accurate you want the vertex, the slower the method.

• But this is a very model dependant method, and hence varying propagation methods can be used (refraction, reflection).

Actual PathReconstructed Path

Minimum chi2 point

Old method reconstructed point, now gives a completely different path, and hit times on different hydrophones

Refraction

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Minimum Number of Hits

• With a plane of detectors, there are always two opposite, but equally valid solutions.

• Adding a detector out of the plane destroys this symmetry

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Unknown Detector Location

• Strings attached to see bed, and supported by buoys.

• Going to move with moving bodies of water.

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The Effect of a Meter

• Using an event, scatter every hydrophone randomly using a Gaussian with a sigma of 1.

• This gives an error on the position of ±10m.

• Can this error be improved?

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Passive Detector Fitting• How do we find the detector positions?

• Ideally have a fixed transmitter, which you know the position of, then you can work out the position of each hydrophone from time of arrival techniques.

• But what if there is no such transmitter.

• Take many noise sources, and perform a multi-dimensional fit on unknown detector and positions and unknown noise locations.

• See ‘First Results from Rona and Signal Processing Techniques - Sean Danaher’The saviour or the enemy?

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Pointingd v .c b D XC bY

CTC 1111

righ

d1

d 2

d 3

righ

x1 y1 z1

x 2 y2 z2

x3 y3 z3

righ

c1

c2

c3

righ

Y

C X 1 D bYCTC D bY T X 1 T X 1 D bY 1

X 1 T X 1 M

DT bY T M D bY 1Y T MY b2 2 DT MY b DT MD 1 0

distance detecto

r location

planescalar

In Matrix Form

Re-arrange

Define

A quadratic in b

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Finally a Telescope

C X 1 D bY

Solving for b yields two solutions – the unit vector in opposite directions. Pick one of these solutions (the positive one) and solve for C

We now have a position and a direction, so we can point back to the source.

With many telescopes, we can pinpoint the source.

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The Future of ACoRNE

• Incorporate our refraction model, using the look-up table technique

• Test the model using calibration sources.

• Improve the hydrophone location algorithm. Shameless Promotion: -

• Simulating the Sensitivity of hypothetical km3 hydrophone arrays to fluxes of UHE neutrinos – Jonathan Perkin

• Future Plans for the ACoRNE collaboration – Lee Thompson

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EndNow lets link arms and sing the Lumley song……

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