Subtraction of Newtonian noise using optimized sensor arrays

Fluctuations in the local Newtonian gravitational field present a limit to high precision measurements, including searches for gravitational waves using laser interferometers. In this work, we present a model of this perturbing gravitational field and evaluate schemes to mitigate the effect by estimating and subtracting it from the interferometer data stream. Information about the Newtonian noise is obtained from simulated seismic data. The method is tested on causal as well as acausal implementations of noise subtraction. In both cases it is demonstrated that broadband mitigation factors close to 10 can be achieved removing Newtonian noise as a dominant noise contribution. The resulting improvement in the detector sensitivity will substantially enhance the detection rate of gravitational radiation from cosmological sources.

Figure 1

Strain noise spectral density of a second generation terrestrial detector—Advanced LIGO (black, bold line). The sensitivity of the first generation Initial LIGO (pink, dashed line) is shown for comparison. The Newtonian noise (green downward-pointing triangles) is dominating the Advanced LIGO sensitivity near 10 Hz. Other traces shown are other, nongravitational, limits to the sensitivity: direct seismic vibrations (brown circles), quantum radiation pressure and shot noise (purple diamonds), mirror thermal noise (red squares), and mirror suspension thermal noise (blue upward-pointing triangles).

Figure 2

Seismic 90th percentile NN estimates for the LIGO Livingston (LLO) and Hanford (LHO) sites (green and red lines), third generation strain noise model (blue line), and additional NN estimates from vibrations of walls (cyan line), building tilts (magenta line), exhaust fans (beige line), and sound waves inside buildings (black line). Seismic contributions are the only NN source significant for third generation detectors and earlier. Building tilt will be important for detectors beyond the third generation, but is not a dominating noise source at this time.

Figure 3

Histogram of one year of unaveraged 128 s seismic spectra measured during the last LIGO science run inside the corner station of the Livingston detector. The black curve is the spectral density of the simulated seismic field. The spectral histogram of the Hanford site is very similar for the frequencies plotted here.

Figure 4

Spectrum of simulated Newtonian noise (red line), reference third generation sensitivity curve described in Eq. (1) (blue line), and simulated interferometer noise based on this noise model (green line). Since other noise sources such as mirror suspension thermal noise and direct seismic vibrations will be the limiting noise sources for second and third generation detectors below $\sim 8\mathrm{Hz}$, we do not need to consider NN at these low frequencies. We therefore do not include low frequency information in our NN estimate, which creates a sharp cutoff when the simulated data is viewed in the frequency domain.

Figure 5

Comparison between the theoretical model for seismic correlation of isotropic plane-wave surface fields as described by Eq. (7) (solid line), and the correlation calculated by the simulation of a field composed of wavelets and locally generated waves (dotted line). Contributions from local sources cause variations of the correlation curve at larger distances between simulation runs (see Sec. 3 for details). Therefore, theoretical and simulated correlations match at small distances and deviate more strongly at larger distances. Since seismic fields in the context of NN subtraction only matter very near the test mass, the match between simulated and theoretical correlations at small distances means that the optimal array determined analytically by minimizing $\sqrt{R}$ in Eq. (6) should also perform well in simulation, and more importantly that the simulation should be representative of our real subtraction ability.

Figure 6

Locations of 10 sensors resulting from numerically minimizing the subtraction residual. The optimal array should be symmetric about the test mass located at (0,0), but the subtraction residual is less than ${10}^{-6}$ at 10 Hz for this array. The colors indicate the normalized seismic correlation between seismometer 1 and all other seismometers.

Figure 7

Subtraction residual as defined in Eq. (6) vs frequency for the array shown in Fig. 6, and three different spiral configurations. The ‘$N=10$, $r=8\mathrm{m}$’ array is shown in Fig. 9, and the ‘$N=10$, $r=2\mathrm{m}$’ array is the same, but with all seismometer coordinates scaled down by a factor of 4. The ‘$N=5$, $r=8\mathrm{m}$’ array has two sensors at the same positions as numbers 1 and 10 in Fig. 9, and 3 other sensors distributed along the two-turn spiral in between these two. It is assumed that the seismometers measure ground motion with $\mathrm{SNR}=100$ at all frequencies. The Rayleigh-wave speed is $200\mathrm{m}/\mathrm{s}$.

Figure 8

Offline Newtonian noise subtraction efficiency for third generation detectors. Spectrum of simulated Newtonian noise (green line), proposed third generation sensitivity curve (blue line), and NN residuals of postsubtraction for a spiral array (red line), circular array (cyan line), linear array (magenta), and the optimal array (beige line). Filters derived from all four arrays reduce the simulated NN to a level below other sources of interferometer noise as represented by the noise model.

Figure 9

The left plot shows the configuration of the spiral array. The colors correspond to the normalized seismic correlation between all seismometers and seismometer 1. The numbering of seismometers corresponds to the traces in the right plot, which shows the magnitude of the FIR filter for each sensor in units of test mass (NN) displacement over seismic displacement.

Figure 10

Spectrum of simulated Newtonian noise (green line), proposed third generation sensitivity curve (blue line), and NN residuals of FF subtraction on the training set (red line), and on a second set of time series using the same filter (cyan) using the 10-sensor optimal array.

Figure 11

Feed-forward Newtonian noise subtraction efficiency for third generation LIGO detectors. Simulated NN before subtraction (green), expected strain sensitivity (blue line), NN residuals after subtraction using postprocessing (red line), online feed forward (cyan line), and both methods combined (magenta line). Note that the combination of methods is close to the same level as either method individually. This indicates that we can safely apply feed-forward subtraction in real time, and clean up any leftover noise in postprocessing if needed.