Seismic gravity-gradient (Newtonian) noise at the LIGO sites: site-specific determination and quantitative feasibility of monitoring-and-correction and moat shielding

Hughes and Thorne (Phys. Rev. D 58, 122002, 1998) computed the gravity-gradient noise that ambient seismic waves impose on interferometric gravitational-wave detectors — now called Newtonian noise (NN) — for the layered geology of the LIGO Hanford and Livingston sites, and left open the site-specific questions their analysis could not settle: the seismic mode mixture actually present at each site at quiet and noisy times (times of low and high ground-motion amplitude, respectively), the uncertain multilayer velocity profiles, the feasibility of monitoring-and-correction with seismometer and borehole-dilatometer arrays, the effectiveness of evacuated or water-filled moats, the influence of topography and concrete foundations, and the noisy-time contribution of higher Rayleigh overtones and basalt-guided modes. The field has built a rich mitigation methodology (Wiener and adaptive filters, optimized and borehole arrays, tiltmeters, distributed acoustic sensing, neural networks) and implemented the first cancellation system at Virgo, but has not unambiguously measured the gravitational coupling in any detector's gravitational-wave channel, nor quantified the LIGO foundations, topography, or moats. The problem therefore remains open in its original, site-specific form: determine by measurement and validated modeling the NN at the LIGO sites at quiet and noisy times, and establish quantitatively whether monitoring-and-correction and/or passive shielding can reduce it below the advanced-detector noise floor below roughly 30 Hz.

Background

Ground-based gravitational-wave detectors such as LIGO (the Laser Interferometer Gravitational-wave Observatory, with 4-km interferometers at Hanford, Washington and Livingston, Louisiana) and Virgo detect gravitational waves through the differential displacement of freely suspended, mirrored test masses at the ends of perpendicular vacuum arms. A passing gravitational wave changes the relative arm length by of order $10^{-19}$ m, and every other disturbance of the test masses is noise. Mechanical suspensions and seismic-isolation platforms suppress the direct transmission of ground vibration, but they cannot shield gravity: a seismic wave deforms the soil and perturbs its density, and the fluctuating Newtonian gravitational field of that deformed soil accelerates the test masses directly. This coupling, which bypasses all mechanical isolation, was called seismic gravity-gradient noise by Hughes and Thorne and is now generally called Newtonian noise (NN).

The effect was recognized in the earliest laser-interferometer design studies and first estimated semi-quantitatively by Saulson (1984). Hughes and Thorne (1998) — together with a nearly simultaneous Virgo analysis by Beccaria et al. (1998) — gave the first wave-based calculation. They modeled each LIGO site as a stack of elastic layers (at Hanford: roughly 220 m of alluvium, dry in the top 40 m and water-saturated below, overlying a several-kilometer-thick sequence of Columbia River basalt; at Livingston: several hundred meters of alluvium) and expanded the ambient seismic field in horizontally propagating surface modes. The elastic body waves are P-waves (compressional) and S-waves (shear); Rayleigh waves are surface waves with coupled vertical and horizontal motion, and Love waves are horizontally polarized surface waves that produce no density perturbation and hence no NN to leading order. The Rayleigh modes are labeled RF for the fundamental mode, which is confined to within roughly 10 m of the surface at 10 Hz, and RS$_1$, RS$_2$, ... and RP$_1$, RP$_2$, ... for the overtones, which propagate by bouncing between the earth's surface and the layer interfaces; RS overtones are dominated by vertically polarized shear waves, RP overtones by compressional waves, and successively higher overtones penetrate more deeply. In this formalism a mode of horizontal wave vector $\mathbf{k}$ produces the fractional density perturbation $\delta\rho/\rho = [\xi_V\delta(z) + R(z)]e^{i(\mathbf{k}\cdot\mathbf{x}-\omega t)}$, where $\xi_V$ is the vertical displacement at the surface, $\delta(z)$ is the Dirac delta placing the surface mass term, and $R(z)$ describes the subsurface density perturbation. The resulting differential test-mass displacement $\tilde{x}(f)$ per unit ground displacement $\tilde{W}(f)$ is written $\tilde{x}(f)/\tilde{W}(f) \simeq \beta(f)\cdot 4\pi G\rho(2\pi f)^{-2}$ at $f\gtrsim 3$ Hz, where $\rho$ is the soil density, $G$ is Newton's constant, and $\beta(f) = \gamma(f)\Gamma(f)\beta'(f)$ is a dimensionless reduced transfer function, with order-unity corrections $\gamma$ and $\Gamma$ for the weak correlation between the interferometer's two corner test masses and for the test masses' height above the ground.

Hughes and Thorne concluded that at quiet times $\beta' \simeq 0.35$–$0.6$ at the LIGO sites and at noisy times $\beta' \simeq 0.15$–$1.4$ (Saulson's simpler model had given $1/\sqrt{3} \approx 0.58$), so that quiet-time NN lies below the benchmark noise level of advanced LIGO interferometers at all frequencies — though not by much near 10 Hz — while at noisy times NN may significantly exceed that level near 10 Hz. They cautioned that their mode-by-mode values of $\beta'$ should not be trusted (except perhaps for the shallow RF mode), because the input velocity profiles were uncertain and even discrepant between the two available measurement campaigns at Hanford. Their Section V.D set out two mitigation ideas. The first is monitoring-and-correction: deploying three-dimensional arrays of vertical surface seismometers together with borehole dilatometers (instruments measuring volumetric strain) around each test mass, in order to determine the surface and subsurface components of $\delta\rho/\rho$ — that is, $\xi_V$ and $R(z)$ — with sufficient spatial and temporal resolution to compute the gravity-gradient force and subtract it from the data. The second is moats: narrow trenches around each test mass, either evacuated or water-filled, to reflect the RF-mode Rayleigh waves. For moats they estimated a required depth comparable to the S-wave e-folding length $Z_S \simeq 9$–$15$ m at 10 Hz and a radius of order a Rayleigh wavelength (roughly 20–35 m at 10 Hz), and they left open whether a water-filled moat at Livingston would work (water shields S-waves but transmits P-waves, and the degree of S-wave regeneration in the moat-surrounded cavity was unknown), whether the moat-surrounded ground could resonantly ring, and whether Rayleigh overtones, which propagate by bouncing between layer interfaces, would simply travel beneath the moat.

The noise is now a central design consideration for advanced and next-generation detectors: it is predicted to appear below about 30 Hz in advanced-generation detectors and to matter even more for future ones, and it helped motivate the underground designs of KAGRA (the Japanese underground detector) and the Einstein Telescope (a planned European underground detector), while entering the low-frequency noise budget of the planned US surface detector Cosmic Explorer. A substantial methodology literature now exists — coherent subtraction with Wiener filters (linear filters minimizing the mean-square error between an estimate formed from reference sensor channels and the target channel), optimized sensor arrays, tiltmeters (sensors of ground tilt), and more recently machine-learning predictors and distributed acoustic sensing (fiber-optic cables that measure strain along their length) — and the first Newtonian-noise cancellation system has been implemented at Virgo. The gravitational coupling itself has never been unambiguously measured in any interferometer's gravitational-wave channel, and the LIGO-site-specific determination and mitigation-feasibility questions posed by Hughes and Thorne remain unanswered.

Problem Statement

Determine, from site seismic measurements and validated site-specific modeling, the seismic gravity-gradient (Newtonian) noise at the LIGO Hanford and Livingston sites at both quiet and noisy times, and establish quantitatively whether monitoring-and-correction arrays and/or passive shielding (moats) can reduce this noise below the noise floor of the advanced interferometers in the band below roughly 30 Hz where NN is predicted to matter (most critically near 10 Hz at noisy times). Concretely, this requires resolving the five site-specific issues that Hughes and Thorne (1998) left open: (i) the composition of the ambient seismic mode mixture at each site at quiet and noisy times, which the surface seismometer data available to them did not constrain; (ii) the multilayer velocity profiles used in the site models — especially the near-surface compressional-wave speed $c_P(z)$ and the Poisson ratios, including measurements below the top few tens of meters and adequate P-wave profiles at Livingston; (iii) whether realistic monitoring arrays (surface seismometers plus borehole dilatometers) can measure the surface displacement $\xi_V$ and the subsurface density profile $R(z)$ with sufficient spatial and temporal fidelity to compute and subtract the gravity-gradient noise at the required level, and whether moats (evacuated or water-filled) would substantially suppress the fundamental Rayleigh mode without resonant ringing of the moat-surrounded ground or leakage of Rayleigh overtones beneath the moat; (iv) the influence of the sites' topography and of the LIGO facilities' concrete foundations on the local modal structure and on the gravity-gradient coupling, which the 1998 analysis could assess only at the tens-of-percent level; and (v) the contribution of higher-order RP modes and deeply propagating basalt-bedrock modes — and their time variability under cultural (human-made) excitation — to the noisy-time upper range of the reduced transfer function $\beta'$. These five issues are the concrete sub-questions of a single scientific objective, not five independent projects. The task is resolved when this site-specific Newtonian-noise estimate at the two sites and the verdict on whether monitoring-and-correction and/or moats can bring that noise below the advanced-detector floor below roughly 30 Hz are established; each issue is resolved once the quantity it names is determined to the accuracy that this single test requires, with any residual uncertainty stated.

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