Correlated noise in networks of gravitational-wave detectors: Subtraction and mitigation

One of the key science goals of advanced gravitational-wave detectors is to observe a stochastic gravitational-wave background. However, recent work demonstrates that correlated magnetic fields from Schumann resonances can produce correlated strain noise over global distances, potentially limiting the sensitivity of stochastic background searches with advanced detectors. In this paper, we estimate the correlated noise budget for the worldwide advanced detector network and conclude that correlated noise may affect upcoming measurements. We investigate the possibility of a Wiener filtering scheme to subtract correlated noise from Advanced LIGO searches, and estimate the required specifications. We also consider the possibility that residual correlated noise remains following subtraction, and we devise an optimal strategy for measuring astronomical parameters in the presence of correlated noise. Using this new formalism, we estimate the loss of sensitivity for a broadband, isotropic stochastic background search using 1 yr of LIGO data at design sensitivity. Given our current noise budget, the uncertainty with which LIGO can estimate energy density will likely increase by a factor of $\approx 12$—if it is impossible to achieve significant subtraction. Additionally, narrow band cross-correlation searches may be severely affected at low frequencies $f\lesssim 70\mathrm{Hz}$ without effective subtraction.

Figure 1

Correlated noise budget for a stochastic background search with the LIGO Hanford and Livingston detectors. The noise budget is given in terms of normalized energy density; see Eq. (8). Red is the correlated noise from coupling to the length degree of freedom. Purple and turquoise show the correlated noise from indirect coupling to the angular degree of freedom assuming a beam offset of 1 mm and 3 mm, respectively. The green curve shows the statistical uncertainty obtained through 1 yr of integration at design sensitivity with a frequency resolution of 0.25 Hz. (The peak just above 60 Hz is due to a zero in the overlap reduction function.) The black curve is the power-law integrated curve [34], which represents the broadband sensitivity of the search to an isotropic background with a power-law spectrum. The dashed black line is the sensitivity to a signal with flat energy density spectrum $\alpha =0$. Correlated noise may be safely ignored only if it falls below the power-law integrated curve. Note that the estimates shown here are based on current measurements, and the noise budget may change with continued commissioning. Coupling measurements are uncertain to a factor of $\approx 2$, and we conservatively assume that the transfer function phase is such that contamination is maximal.

Figure 2

Reduced coherence with Wiener subtraction. Top left: $\mathrm{coh}(f)$ for different values of witness signal-to-noise ratio ${\rho }_w$; see Eq. (22). The blue curve (dirty) represents no Wiener subtraction. The progressively greener curves indicate subtraction with progressively higher values of ${\rho }_w$. The dashed red $1/N$ curve indicates the expected level of coherence between two uncorrelated channels. Top right: frequency-averaged coherence as a function of ${\rho }_w$. The dashed blue line indicates the coherence with no subtraction while the dashed red $1/N$ line indicates the expected level of coherence between two uncorrelated channels. Bottom left: narrow band and broadband coherence (Eq. (24) as a function of ${\rho }_w$. The dashed red line $1/N_{\text{eff}}$ indicates the expected level of broadband coherence between two uncorrelated channels. While ${\rho }_w\gtrsim 2$ is sufficient to produce an apparently clean (narrow band) coherence spectrum, ${\rho }_w\gtrsim 4$ is necessary to eliminate correlated noise at a level that is negligible as measured by the broadband coherence. Bottom right: coherence spectrum comparing Wiener and Wiener-like filtering [Eq. (25)] with ${\rho }_w=1.8$. Blue indicates the dirty spectrum, black shows Wiener subtraction, and green is Wiener-like. Both the Wiener curve and the Wiener-like curve fall on top of the $1/N$ line when ${\rho }_w\gtrsim 5.5$.

Figure 3

Wiener-like subtraction when the transfer function is known a priori. Left: coherence spectrum for a pair of channels contaminated by correlated noise (blue), for the same channels cleaned with Wiener filtering (green), and cleaned with Wiener-like subtraction using the true transfer function; see Eq. (26). For this example, ${\rho }_w=0.56$. The dashed red line indicates the expected level of accidental correlation from two uncorrelated channels. Right: broadband coherence as a function of ${\rho }_w$. The dashed blue line indicates the value with no cleaning. The green line shows the improvement through Wiener subtraction. The solid black line demonstrates the improvement obtained using a priori subtraction with a precisely known transfer function. The dashed black lines show how this improvement is compromised when we apply a priori subtraction using a transfer function with a $\pm 25\%$ amplitude error.

Figure 4

Apparent signal power as a function of the number of data sections $N_{\text{sections}}$ used to calculate the Wiener filters. Between $N=1–400$, the observed power gradually (and undesirably) falls with increasing $N_{\text{sections}}$ due to a systematic error from Wiener filtering. The turning point at $N_{\text{sections}}=400$ is due to the growth of broadband noise injected into the strain channels through the Wiener filters. (This is also an undesirable effect and does not imply improved measurement of signal power, only an increase in noise.) We use a witness signal-to-noise ratio of ${\rho }_w=5.6$.

Figure 5

Coherence spectra for varying degrees of local environmental noise vs instrumental noise. The two rows correspond to a sign difference between the local environmental and instrumental transfer functions; see Eq. (33). The top row is case one [$\text{phase}(r)=\text{phase}(\mathfrak{r})$] while the bottom row is case two [$\text{phase}(r)=\text{phase}(\mathfrak{r})+\pi$]. The left columns shows Wiener subtraction while the right shows Wiener-like subtraction. In each panel, the dashed black line represents the coherence before subtraction and the dashed green line represents a perfectly clean spectrum. Each trace shows the coherence following subtraction. The colors show the results for different evenly spaced values of $\epsilon$ [Eq. (30)]. The bluest (darkest) hue corresponds to $\epsilon =0$ (pure instrumental noise) while the reddest (lightest) hue corresponds to $\epsilon =1$ (pure local environmental noise).

Figure 6

Top left: amplitude density spectra for uncorrelated (red) and correlated strain noise (cyan). The dashed red line shows the reduction in uncorrelated noise obtained through 1 yr of integration using a (typical [26]) frequency bin width of $\delta f=0.25\mathrm{Hz}$. The blue shows a single realization of correlated noise. Top right: coherence spectra showing the contaminated strain channels (blue), the somewhat cleaner spectrum obtained by Wiener-like filtering (black), and the spectrum obtained with true Wiener filtering (magenta). For this example, $(\kappa ,\beta )=(2,2.67)$ [see, Eq. (13)] and ${\mathrm{\Pi }}^{1/2}=0.15\mathrm{pT}\mathrm{H}{\mathrm{z}}^{-1/2}$. Again, we assume 1 yr of integration time. Bottom left: witness signal-to-noise ratio assuming ${\mathrm{\Pi }}^{1/2}=0.15\mathrm{pT}\mathrm{H}{\mathrm{z}}^{-1/2}$. This noise floor is sufficient to remove detectable correlated noise. The 60 Hz electronic line has been notched. Bottom right: the expected signal-to-noise ratio $|{\mathrm{SNR}}_M(\alpha )|$ [Eq. (11)] from correlated noise as a function of the witness noise ${\mathrm{\Pi }}^{1/2}$. The different colors show how the results vary for different gravitational-wave models. The dashed lines show dependence on ${\mathrm{\Pi }}^{1/2}$ whereas the solid lines are the asymptotic values obtained when ${\mathrm{\Pi }}^{1/2}\to \infty$ (no subtraction). All plots use the magnetic cross-power measurements from [32].

Figure 7

Left: $|{\mathrm{SNR}}_M(\alpha )|$ vs ${\mathrm{\Pi }}^{1/2}$ [as in Fig. 6] except with the 3 mm angular coupling scenario: $(\kappa ,\beta )=(0.75,1.74)$. The different colors indicate different models (different values of $\alpha$). Right: the required coupling $\kappa$ (given $\beta =2.67$) as a function of witness noise ${\mathrm{\Pi }}^{1/2}$ required to reduce the correlated noise to a level ${\mathrm{SNR}}_M(\alpha )\lesssim 0.5$. The different colors indicate different models. Both plots use the magnetic cross-power measurements from [32].

Figure 8

Left: systematic error (blue), statistical uncertainty (red), and their quadrature sum (green) as a function of the low-frequency bound for the integrals in Eq. (10). The total error is minimized at $f_{\min }=78\mathrm{Hz}$, which is marked with a black circle. Right: narrow band systematic error (blue), statistical uncertainty (red), and their quadrature sum (green). The solid lines show the uncertainty for many 0.25-Hz-wide bins. The dashed lines show the broadband uncertainty obtained by the optimal combination of every frequency bin. The broadband uncertainty with correlated noise is $12\times$ larger than the broadband uncertainty with no correlated noise. In both plots, we assume $\alpha =0$, $\kappa =2$, and $\beta =2.67$. Also, the normalization is such that ${\sigma }_0=1$. Finally, both plots use the magnetic cross-power measurements from [32].