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Cavalier Fabien on behalf LAL group Orsay

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. Cavalier Fabien

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Cavalier Fabien on behalf LAL group Orsay

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  1. 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 Cavalier Fabien on behalf LAL group Orsay GWDAW 10 December, 14th 2005

  2. Reconstruction Method • With n measured arrival times ti with associated errors si, compute the best estimation of a and d • Use of c2 Minimization • c2 defined as: • where t0 is the arrival time of the signal at the center of the Earth and DtiEarth(a,d) is the delay between the ith ITF and the center of the Earth

  3. Reconstruction Method • 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

  4. Toy Simulation • Put a source at a given location • Computes arrival times tTrue[i] • Computes timing errors in each detector • set s[i] = 10-4 s in all detectors (no beam-pattern effect) • set <SNR> = 10 (beam-pattern effect) • and use s[i] = s0 / SNR[i] ms (s0 = 1 ms will be used by default) • tMeasured[i] = tTrue[i] + GaussianRandom * s[i] • Compute the angular distance (Angular Error in the following plots) between reconstructed location and true one

  5. Example of Reconstruction for Galactic Center Estimated errors (through covariance matrix) in agreement with RMS values

  6. Influence of Timing Resolution Effect of Systematic Errors on Arrival Time • tMeasured[i] = tTrue[i] + GaussianRandom * s[i] + Bias (for only one ITF) • bias in the angular reconstruction when the timing bias and s[i] have the same order of magnitude • bias in the angular reconstruction proportional to the timing bias • no significant difference between the three ITF

  7. Reconstruction for GC (no beam-pattern) Mean Angular Error: 1.9 Median Angular Error: 0.95o Minimal Angular Error: 0.8o Maximal Angular Error: 4.3o

  8. Reconstruction for d=0o(no beam-pattern) Mean Angular Error: 1.8o Median Angular Error: 1.5o Minimal Angular Error: 1.3o Maximal Angular Error: 3.1o

  9. 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 q with x axis • t21 = d/c cos(q) (1) • if error dt on t21, then error dq on the angle: dq = c/d dt / |sin(q)| • when q approaches 0, due to statistical errors, Eq.(1) cannot be inverted • c2(q) = 1/ dt2 (t21measured – d/c cos(q))2 • c2(q) minimal and equal to 0 for q = acos(t21measured *c/d) • if |t21measured *c/d|≤1 • c2(q) minimal and  1 for q = 0 if |t21measured *c/d|>1 • Similar effect in the 3-detector case

  10. Reconstruction for full sky (no beam-pattern) Mean Angular Error: 1.6o Median Angular Error: 1.1o Minimal Angular Error: 0.7o Maximal Angular Error: 4.5o

  11. Reconstruction for GC (beam-pattern effect included)

  12. Reconstruction for GC (beam-pattern effect included) All events: Mean Angular Error: 4.0o Median Angular Error: 1.8o Minimal Angular Error: 0.7o Events with all SNR > 4.5 (56%): Mean Angular Error: 1.8o Median Angular Error: 1.25o Minimal Angular Error: 0.7o

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

  14. Reconstruction for d=0o(beam-pattern effect included) All events: Mean Angular Error: 3.0o Median Angular Error: 2.2o Minimal Angular Error: 1.2o Events with all SNR > 4.5 (79%): Mean Angular Error: 2.1o Median Angular Error: 1.7o Minimal Angular Error: 1.2o

  15. Reconstruction for full sky (beam-pattern effect included) All events: Mean Angular Error: 2.7o Median Angular Error: 1.7o Minimal Angular Error: 0.7o Events with all SNR > 4.5 (60%): Mean Angular Error: 1.8o Median Angular Error: 1.3o Minimal Angular Error: 0.7o

  16. Conclusion • c2 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

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