Near optimal solution to the inverse problem for gravitational-wave bursts

We develop a method for determining the source direction (\ensuremath{\theta},\ensuremath{\varphi}) and the two waveforms ${h}_{+}$(t), ${h}_{\ifmmode\times\else\texttimes\fi{}}$(t) of a gravitational-wave burst using noisy data from three wideband gravitational-wave detectors running in coincidence. The scheme does not rely on any assumptions about the waveforms and in fact it works for gravitational-wave bursts of any kind. To improve the accuracy of the solution for (\ensuremath{\theta},\ensuremath{\varphi}), ${h}_{+}$(t), ${h}_{\ifmmode\times\else\texttimes\fi{}}$(t), we construct a near optimal filter for the noisy data which is deduced from the data themselves. We implement the method numerically using simulated data for detectors that operate, with white Gaussian noise, in the frequency band of 500--2500 Hz. We show that for broadband signals centered around 1 kHz with a conventional signal-to-noise ratio of at least 10 in each detector we are able to locate the source within a solid angle of 1\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}5}$ sr. If the signals and the detectors' band were scaled downwards in frequency by a factor \ensuremath{\iota}, at fixed signal-to-noise ratio, then the solid angle of the source's error box would increase by a factor ${\ensuremath{\iota}}^{2}$. The simulated data are assumed to be produced by three detectors: one on the east coast of the United States of America, one on the west coast of the United States of America, and the third in Germany or Western Australia. For conventional signal-to-noise ratios significantly lower than 10 the method still converges to the correct combination of the relative time delays but it is unable to distinguish between the two mirror-image directions defined by the relative time delays. The angular spread around these points increases as the signal-to-noise ratio decreases. For conventional signal-to-noise ratios near 1 the method loses its resolution completely.