Figure 1

Geometry of the network and travel times spent by a GWB to propagate across a three-detector network (detectors H, L, and V). The locus of constant time delay between two detectors form a ring on the sky concentric about the baseline between the two sites. For three detectors, these rings may intersect in two locations. One is the true source direction, $S$, while the other ($S^{\prime }$) is its mirror image with respect to the geometrical plane passing through the three sites. This two-fold ambiguity can be resolved by further considering the amplitudes of the responses. For four or more detectors there is a unique intersection point of all of the rings.Reuse & Permissions

Figure 2

Geometry of the null-stream construction for the 3-detector case. The null stream is obtained by projecting the data along the vector $\mathit{A}$, which is orthogonal to ${\mathit{F}}_{\mathrm{w}}^{+}$ and ${\mathit{F}}_{\mathrm{w}}^{\times }$. For $D$ nonaligned detectors $\mathit{A}$ has $D-2$ dimensions.Reuse & Permissions

Figure 3

Scatter plot of the null and incoherent energies for a simulated GWB (left) and a simulated glitch (right) seen in a network consisting of the LIGO-Hanford and LIGO-Livingston 4 km detectors and Virgo. In both cases the signal was scaled to rms SNR of 20 over the three detectors. Each point represents one trial sky position; approximately ${10}^4$ sky positions in a uniform grid were tested. The waveforms and detector network used are discussed in detail in Sec. 3. The GWB and glitch signals have the same signal-to-noise ratios and time delays in the individual detectors and so are indistinguishable to incoherent tests. Note that for both the GWB and the glitch there are sky positions for which the null energy is consistent with noise [$E_{\mathrm{null}}\simeq N(D-r)=60$ for these simulations]. However, for the GWB there are also sky positions with $E_{\mathrm{null}}\ll E_{\mathrm{inc}}$ (points above the diagonal); these are due to the fact that the GWB signal is correlated between the detectors. The glitch signal does not access this portion of the $(E_{\mathrm{null}},E_{\mathrm{inc}})$ space. This observation is the basis of our consistency test; we scan over the sky and look for directions where $E_{\mathrm{null}}<E_{\mathrm{inc}}$. The true source location is indicated by the circle on the GWB plot. Also shown on the GWB plot are the tested sky positions which have the smallest values of $E_{\mathrm{null}}-E_{\mathrm{inc}}$, $E_{\mathrm{null}}/E_{\mathrm{inc}}$, and $E_{\mathrm{null}}$. The sky maps for these same simulations are shown in Fig. 4.Reuse & Permissions

Figure 4

Sample sky hemisphere maps, normalized to $E/(N(D-r))$, for the same GWB and glitch signals used in Fig. 3. The plots on the left are for a GWB with rms SNR of 20 in the three detectors. Those on the right are for a glitch with the same relative time delays and signal energies in each detector as the GWB. (Note that these two events are indistinguishable to an incoherent analysis.) The network consists of the LIGO-Hanford (H) and LIGO-Livingston (L) 4 km detectors and Virgo (V). The null energy map (top) shows interference fringes due to the transient signal, as well as structure due to the network geometry. They are very similar for the GWB and the glitch. The incoherent energy map (middle) is constructed from the autocorrelations of the individual detector data streams and reflects variations in the network sensitivity over the sky. It is virtually identical for the GWB and the glitch because the two events have the same relative time delays and SNRs. Dividing out the incoherent energy from the null energy (bottom) removes this structure associated with the network geometry, making the signal-dependent structure clearer. The GWB hemisphere map shows sharp interference fringes (blue and red rings)where the time delays along the H–L, L–V, and H–V baselines match those of the source location; this is a realization of the sketch in Fig. 1. There is little of such structure in the glitch hemisphere map. The source location in each map is marked by a circle.Reuse & Permissions

Figure 5

Dimmelmeier—Font—Mueller waveforms used to simulate GWBs and glitches in our analysis. Left plot: time-series waveforms scaled to $\mathrm{SNR}=100$ for optimal orientation. (The linear trend present in the original waveforms has been removed.) Right plot: spectra with a 1 Hz resolution; shown for comparison is the LIGO design noise curve used for both LIGO and Virgo detectors.Reuse & Permissions

Figure 6

Scatter plot of the null and incoherent energies for the most correlated direction on the sky, defined as the direction of $\min (E_{\mathrm{null}}/E_{\mathrm{inc}})$ (i.e., the upper left limit of Fig. 3), for glitches and GWBs of various signal-to-noise ratios. Note that as the signal-to-noise ratio increases (higher $E_{\mathrm{inc}}$) the GWB and glitch distributions separate, with the GWBs remaining at relatively low null energies and the glitches having comparable null and incoherent energies. The clumping is due to the 5 distinct rms SNRs used in our simulations: 5, 10, 20, 50, 100. The null energies for some of the GWB simulations are greater than that expected (60) from the number of degrees of freedom because we are selecting sky positions from the minimum of $E_{\mathrm{null}}/E_{\mathrm{inc}}$, not from the minimum of $E_{\mathrm{null}}$.Reuse & Permissions

Figure 7

Scatter plot of the null and incoherent energies for the most correlated direction on the sky, defined as the direction of $\min (E_{\mathrm{null}}-E_{\mathrm{inc}})$ (i.e., the upper left limit of Fig. 3), for glitches and GWBs of various signal-to-noise ratios. Note that as the signal-to-noise ratio increases (higher $E_{\mathrm{inc}}$) the GWB and glitch distributions separate, with the GWBs remaining at relatively low null energies and the glitches having comparable null and incoherent energies. The clumping into horizontal bands is due to the five distinct rms SNRs used in our simulations: 5, 10, 20, 50, 100. The null energies for many of the GWB simulations are greater than that expected (60) from the number of degrees of freedom because we are selecting sky positions from the minimum of $E_{\mathrm{null}}-E_{\mathrm{inc}}$, not from the minimum of $E_{\mathrm{null}}$.Reuse & Permissions

Figure 8

Scatter plot of the null and incoherent energies for the best-fit direction on the sky, defined as the direction of $\min (E_{\mathrm{null}})$ (i.e., the left-most point in Fig. 3) for glitches and GWBs of various signal-to-noise ratios. Note that as the signal-to-noise ratio increases (higher $E_{\mathrm{inc}}$) the GWB and glitch distributions separate, with the GWBs remaining at relatively low null energies and the glitches having comparable null and incoherent energies. The clumping is due to the 5 distinct rms SNRs used in our simulations: 5, 10, 20, 50, 100. Note that a large fraction of the glitch signals produce low null energies for some sky positions. This means that GWBs and glitches are not distinguishable using only the null energy.Reuse & Permissions

Figure 9

ROC plot for $\min (E_{\mathrm{null}}/E_{\mathrm{inc}})$ as a statistic for distinguishing GWBs from noise glitches. The ROC curve value is given by the fraction of GWBs (true acceptance) and glitches (false acceptance) of given rms SNR falling to the left of a line of constant $E_{\mathrm{null}}/E_{\mathrm{inc}}$ (a diagonal line) in Fig. 6. The rapid rise of the curves at low false acceptance is indicative of the ability of the method to confidently distinguish a significant portion of GWB signals from the glitch population.Reuse & Permissions

Figure 10

ROC for $\min (E_{\mathrm{null}}-E_{\mathrm{inc}})$ as a statistic for distinguishing GWBs from noise glitches. The ROC curve value is given by the fraction of GWBs (true acceptance) and glitches (false acceptance) of given rms SNR falling to the left of a line of constant $E_{\mathrm{null}}-E_{\mathrm{inc}}$ in Fig. 7. The rapid rise of the curves at low false acceptance is indicative of the ability of the method to confidently distinguish a significant portion of GWB signals from the glitch population. This is not as powerful a statistic as $\min (E_{\mathrm{null}}/E_{\mathrm{inc}})$ (see Fig. 9).Reuse & Permissions

Figure 11

ROC plot for $\min (E_{\mathrm{null}})$ as a statistic for distinguishing GWBs from noise glitches. The ROC curve value is given by the fraction of GWBs (true acceptance) and glitches (false acceptance) of given rms SNR falling to the left of a line of constant $E_{\mathrm{null}}$ (a vertical line) in Fig. 8. The lower slope of the curves at low acceptance is due to the fact that many glitches have sky positions for which the null energy is small, even when the SNR of the glitch is high. The null energy alone is therefore not very effective for confidently distinguishing GWBs from noise glitches. While one could chose a threshold that varies with $E_{\mathrm{inc}}$ to get better performance, $\min (E_{\mathrm{null}})$ is still not as powerful a statistic as $\min (E_{\mathrm{null}}/E_{\mathrm{inc}})$.Reuse & Permissions