Gravitational wave burst source direction estimation using time and amplitude information

In this article we study two problems that arise when using timing and amplitude estimates from a network of interferometers (IFOs) to evaluate the direction of an incident gravitational wave burst. First, we discuss an angular bias in the least squares timing-based approach that becomes increasingly relevant for moderate to low signal-to-noise ratios. We show how estimates of the arrival time uncertainties in each detector can be used to correct this bias. We also introduce a stand-alone parameter estimation algorithm that can improve the arrival time estimation and provide root-sum-squared strain amplitude ($h_{\mathrm{rss}}$) values for each site. In the second part of the paper we discuss how to resolve the directional ambiguity that arises from observations in three non co-located IFOs between the true source location and its mirror image across the plane containing the detectors. We introduce a new, exact relationship among the $h_{\mathrm{rss}}$ values at the three sites that, for sufficiently large signal amplitudes, determines the true source direction regardless of whether or not the signal is linearly polarized. Both the algorithm estimating arrival times, arrival time uncertainties, and $h_{\mathrm{rss}}$ values and the directional follow-up can be applied to any set of gravitational wave candidates observed in a network of three non co-located IFOs. As a case study we test the methods on simulated waveforms embedded in simulations of the noise of the LIGO and Virgo detectors at design sensitivity.

Figure 1

Coordinate system used for source localization. Only the polar angle $\theta$ is constrained in the two-interferometer case, while both angles $\theta$ and $\varphi$ are constrained in the three-IFO case.

Figure 2

Least squares angular bias for the L1-H1 baseline with a Gaussian arrival time error distribution and ${\sigma }_{t_H}={\sigma }_{t_L}=0.1\mathrm{ms}$, 1 ms.

Figure 3

Least squares angular bias for the L1-H1 baseline with a multimodal arrival time error distribution and ${\sigma }_{t_H}={\sigma }_{t_L}=0.1\mathrm{ms}$, 1 ms.

Figure 4

Average angular error in location reconstruction from the least squares method applied to the H1-L1-V1 network when ${\sigma }_{t1}={\sigma }_{t2}={\sigma }_{t3}=1\mathrm{ms}$. Note that only one hemisphere is shown here as we assume the correct sky patch is chosen and the result in the other atmosphere will be the mirror image of this plot across the y-z plane.

Figure 5

Magnitude of three-IFO H1, L1, V1 angular bias when ${\sigma }_{t1}={\sigma }_{t2}={\sigma }_{t3}=1\mathrm{ms}$. This magnitude is defined by (22), and the color scale is in degrees. Note that only one hemisphere of the sky is shown here, as the bias of the other hemisphere is its mirror image across the y-z plane.

Figure 6

Direction of three-IFO (H1, L1, V1) angular bias when ${\sigma }_{t1}={\sigma }_{t2}={\sigma }_{t3}=1\mathrm{ms}$.

Figure 7

Coordinate transformation from IFO to GW frame. The IFO frame is defined naturally by the x and y arms of the detector. Figure is adopted from [11].

Figure 8

Efficiency of the two amplitude checks plotted as a function of the uncertainty in recovered $h_{\mathrm{rss}}$ values. The plus check is given by (40) while the minus check is given by (41).

Figure 9

Simulated waveforms used for amplitude check testing.

Figure 10

Performance of least squares angular estimator on LIGO-Virgo Project Ib simulated waveforms for varying intrinsic inband $h_{\mathrm{rss}}$. In all cases, the effect of the bias correction was negligible due to sufficiently small arrival time uncertainties.

Figure 11

Histogram of angular errors for all waveforms with an intrinsic inband $h_{\mathrm{rss}}$ of ${10}^{-20}{\mathrm{Hz}}^{-1/2}$. Note the logarithmic scale of the y-axis.

Figure 12

Example of difficulty in arrival time estimation. The parameter estimation picks the largest peak in the whitened, bandpassed data but this may not be the primary peak due to noise.

Figure 13

Average efficiency of amplitude check (40) as a function of intrinsic inband $h_{\mathrm{rss}}$ for each of the sample waveforms. The $h_{\mathrm{rss}}$ values shown here were scaled down further by the antenna factors before entering the data. Note that these curves were generated assuming that the time delays between one-IFOs and two were known exactly.

Figure 14

Average efficiency of amplitude check (40) as above except with the last term of (40) being determined using parameter estimation peak times.