Separating gravitational wave signals from instrument artifacts

Central to the gravitational wave detection problem is the challenge of separating features in the data produced by astrophysical sources from features produced by the detector. Matched filtering provides an optimal solution for Gaussian noise, but in practice, transient noise excursions or “glitches” complicate the analysis. Detector diagnostics and coincidence tests can be used to veto many glitches which may otherwise be misinterpreted as gravitational wave signals. The glitches that remain can lead to long tails in the matched filter search statistics and drive up the detection threshold. Here we describe a Bayesian approach that incorporates a more realistic model for the instrument noise allowing for fluctuating noise levels that vary independently across frequency bands, and deterministic glitch fitting using wavelets as glitch templates, the number of which is determined by a transdimensional Markov chain Monte Carlo algorithm. We demonstrate the method’s effectiveness on simulated data containing low amplitude gravitational wave signals from inspiraling binary black-hole systems, and simulated nonstationary and non-Gaussian noise comprised of a Gaussian component with the standard LIGO/Virgo spectrum, and injected glitches of various amplitude, prevalence, and variety. Glitch fitting allows us to detect significantly weaker signals than standard techniques.

Figure 1

A time-frequency scaleogram showing a loud glitch in the output of one of the LIGO detectors (originally Fig. 7 in Ref. 45).

Figure 2

The Meyer wavelet basis function for frequency layer $i=9$ and signal duration of 16 s. (a) Time domain ${\psi }_{ij}(t)$ for arbitrary time index $j$. (b) Fourier power spectrum of ${\tilde{\psi }}_{ij}(f)$. Notice how, for $i=9$ and $T=16\mathrm{s}$, this wavelet acts as a bandpass filter for frequencies $f\sim [32,64]\mathrm{Hz}$.

Figure 3

A cartoon depicting the tiling of the time-frequency plane by a discrete wavelet transform (DWT). Each pixel has time-frequency volume $\Delta t\Delta f=1/2$.

Figure 4

Plots (a)–(c) show the log evidence for different noise models applied to signal realizations $(s_1,s_2,s_3)$ as a function of the TFV used by $N_0$ and $N_1$. Models with discernibly higher evidence are preferred. (d) shows the distribution of the “hot-pixel” number used by $G_1$ for the signal realization $s_3$. Our interpretation of these results is discussed in the text.

Figure 5

A typical residual from the glitch-search phase. The top panel shows the full 16 seconds of Virgo data while the bottom is a zoomed-in view of the glitch. The Gaussian noise $n(t)$ (black) and the residual $r(t)$ (blue) are basically indistinguishable, while the raw data $s(t)$ (red) show a large spike (upper panel) or an obvious glitch (lower panel).

Figure 6

PDF of $n_G$ for stationary, Gaussian noise with a single loud glitch injected into the Virgo data. The RJMCMC automatically prefers small values of $n_G$ for the Gaussian noise, and $\sim 25$ glitch wavelets to fit the large-amplitude noise excursion seen in Fig. 5.

Figure 7

Time-to-coalescence search chains with (blue) and without (red) glitch fitting for example 1. Without glitch fitting the search locks onto a glitch in the Virgo detector (injection time $\sim 4\mathrm{s}$) while with glitch fitting the search locks onto the signal injection ($t_c\sim 6\mathrm{s}$). The GW signal was injected with SNR of 12. The prescription for the glitch fitting is described in Sec. 6.

Figure 8

Marginalized PDFs of $\overrightarrow{\lambda }$ for $\mathrm{SNR}=12$ from example 1. The vertical line marks the injected waveform’s values. Notice the multimodal structure of the posteriors, illustrating the challenge these signals pose for any analysis method.

Figure 9

Log evidence for each model under consideration for example 1. The evidence is highest for the glitch-fitting models. The BH plus glitch model $[G_1,B_1]$ begins to distinguish itself from the glitch model $[G_1,B_0]$ above injected SNR of 8. Had we naively assumed a Gaussian noise model, the evidence calculation would have erroneously indicated a detection for arbitrarily weak signal injections (false positives in the usual parlance).

Figure 10

Log Bayes factor for models $[G_1,B_1]$ vs $[G_1,B_0]$ for example 1. The GW-signal model is favored when the injected signal has SNR above $\sim 8$.

Figure 11

Spectrograms showing the glitch injections into the three detectors and the geocenter gravitational waveform. The spectrograms have been whitened using the theoretical LIGO/Virgo noise spectra. The color map is on a logarithmic scale, and represents the Fourier power.

Figure 12

Spectrograms showing the various components that make up the simulated data in the Virgo detector for example 2. The spectrograms have been whitened using the theoretical Virgo noise spectrum. The loud, low frequency glitch at $t\sim 1\mathrm{s}$ caused havoc for noise models that did not include glitch fitting. The color map is on a logarithmic scale, and represents the Fourier power.

Figure 13

Time-to-coalescence search chains with and without glitch fitting for example 2. Without glitch fitting the search locks onto a glitch in the Virgo detector (injection time $\sim 1\mathrm{s}$) while with glitch fitting the search locks onto the signal injection ($t_c\sim 6\mathrm{s}$). The GW signal was injected with SNR of 12. The prescription for the glitch fitting is described in Sec. 6.

Figure 14

Log evidence for each model under consideration for example 2. The evidence is highest for the glitch-fitting models. The BH model plus glitch model $[G_1,B_1]$ is favored for injected SNR’s above 8.

Figure 15

Log Bayes factor for models with glitch fitting, showing the odds that a signal is present, $[G_1,B_1]$, relative to no signal being present, $[G_1,B_0]$. The signal model is favored for injected SNR’s above 8.

Figure 16

Log Bayes factor for models without glitch fitting, showing the odds that a signal is present, $[G_0,B_1]$, relative to no signal being present, $[G_0,B_0]$. The signal model is always favored, even when no signal is injected. The parameters of the recovered signal do not correspond to those injected until the SNR reaches $\sim 16$.

Figure 17

An example $⟨\log \mathcal{L}⟩$ vs $\beta$ plot showing the location of $T_{*}$ and other tips to understand how parallel tempering and thermodynamic integration work. In this example the blue curve is for the model with a higher number of degrees of freedom, and was the favored explanation of the data.

Figure 18

The same plot as is found in Fig. 17 but on a log scale in $\beta$. The log version of these plots is a valuable diagnostic for the behavior of the chains at high temperature. This also shows the “rule of thumb” value for $T_{*}$ which passes the test of being lower than the point where $M_1=M_0$.