cavalier fabien on behalf lal group orsay

Cavalier Fabienon behalf LAL groupOrsay

GWDAW 10December, 14th 2005

Reconstruction of Source Location using the Virgo-LIGO network

• Presentation of the method• Toy Simulation• Influence of timing resolution and Effects of systematic errors• Geometrical Properties of Virgo-LIGO network• Effects of the Beam-Pattern functions

• With n measured arrival times ti with associated errors i, compute the best estimation of and

• Use of 2 Minimization

• 2 defined as:

where t0 is the arrival time of the signal at the center of the Earth and ti

Earth(,) is the delay between the ith ITF and the center of the Earth

Reconstruction Method

n

i i

iiEarthttt

t1

2

20

0

)))(((,(

• Symmetrical Definition

• No reference detector to compute timing differences

How to choose it ?• detector with lower error• detector leading to highest time delays• detector which gives the best relative errors on time delays• …

• Uncorrelated Errors

• Easily expendable to any set of detectors

Reconstruction Method

• Put a source at a given location

• Computes arrival times tTrue[i]

• Computes timing errors in each detector • set [i] = 10-4 s in all detectors (no beam-pattern effect)• set <SNR> = 10 (beam-pattern effect)and use [i] = / SNR[i] ms (0 = 1 ms will be used by default)

• tMeasured[i] = tTrue[i] + GaussianRandom * [i]

• Compute the angular distance (Angular Error in the following plots) between reconstructed location and true one

Toy Simulation

Example of Reconstruction for Galactic Center

Estimated errors (through covariance matrix) in agreement with RMS values

Effect of Systematic Errors on Arrival Time

tMeasured[i] = tTrue[i] + GaussianRandom * [i] + Bias (for only one ITF)

• bias in the angular reconstruction when the timing bias and [i] have the same order of magnitude • bias in the angular reconstruction proportional to the timing bias• no significant difference between the three ITF

Influence of Timing Resolution

msmsorAngularErrorAngularErr

1)1()( 0

0

Reconstruction for GC (no beam-pattern)

Mean Angular Error: 1.9Median Angular Error: 0.95o

Minimal Angular Error: 0.8o

Maximal Angular Error: 4.3o

Reconstruction for =0o (no beam-pattern)

Mean Angular Error: 1.8o

Median Angular Error: 1.5o



Error increase when source crosses 3-detector plane

• 2 detectors located at (±d/2,0,0)• source in the (x,y) plane defined by its angle with x axis

• t21 = d/c cos() (1)• if error t on t21, then error on the angle: = c/d t / |sin()|

• when approaches 0, due to statistical errors, Eq.(1) cannot be inverted• = 1/ t2 (t21

measured – d/c cos())2

• minimal and equal to 0 for = acos(t21measured *c/d)

if |t21measured *c/d|≤1

• minimal and 1 for = 0 if |t21measured *c/d|>1

• Similar effect in the 3-detector case

Reconstruction for full sky (no beam-pattern)





Reconstruction for GC (beam-pattern effect included)

Reconstruction for GC (beam-pattern effect included)

All events:




Events with all SNR > 4.5 (56%):




Reconstruction for =0o (beam-pattern effect included)

Reconstruction for =0o (beam-pattern effect included)

All events:








Reconstruction for full sky (beam-pattern effect included)

All events:








Conclusion

• 2 minimization works properly for position reconstruction

• In the case of 3 detectors, the method is able to find the two possible solutions

• The method is easily extendable to any ITF network

• Errors on parameters provided by the minimization procedure

• Accuracy of one degree can be achieved with good timing reconstruction (error proportional to timing error)

• Bias in timing estimator can easily introduce bias in reconstructed angles

cavalier fabien on behalf lal group orsay

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