Hughes, Scott A. and Thorne, Kip S. (1998) Seismic gravitygradient noise in interferometric gravitationalwave detectors. Physical Review D, 58 (12). Art. No. 122002. ISSN 24700010. https://resolver.caltech.edu/CaltechAUTHORS:HUGprd98

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Abstract
When ambient seismic waves pass near and under an interferometric gravitationalwave detector, they induce density perturbations in the Earth, which in turn produce fluctuating gravitational forces on the interferometer’s test masses. These forces mimic a stochastic background of gravitational waves and thus constitute a noise source. This seismic gravitygradient noise has been estimated and discussed previously by Saulson using a simple model of the Earth’s ambient seismic motions. In this paper, we develop a more sophisticated model of these motions, based on the theory of multimode Rayleigh and Love waves propagating in a multilayer medium that approximates the geological strata at the LIGO sites, and we use this model to reexamine seismic gravity gradients. We characterize the seismic gravitygradient noise by a transfer function, T(f )≡x̃(f )/W̃(f ), from the spectrum of rms seismic displacements averaged over vertical and horizontal directions, W̃(f ), to the spectrum of interferometric testmass motions, x̃(f )≡Lh̃(f ); here L is the interferometer arm length, h̃(f ) is the gravitationalwave noise spectrum, and f is frequency. Our model predicts a transfer function with essentially the same functional form as that derived by Saulson, T≃4πGρ(2πf )2β(f ), where ρ is the density of Earth near the test masses, G is Newton’s constant, and β(f )≡γ(f )Γ(f )β′(f ) is a dimensionless reduced transfer function whose components γ≃1 and Γ≃1 account for a weak correlation between the interferometer’s two corner test masses and a slight reduction of the noise due to the height of the test masses above the Earth’s surface. This paper’s primary foci are (i) a study of how β′(f )≃β(f ) depends on the various Rayleigh and Love modes that are present in the seismic spectrum, (ii) an attempt to estimate which modes are actually present at the two LIGO sites at quiet times and at noisy times, and (iii) a corresponding estimate of the magnitude of β′(f ) at quiet and noisy times. We conclude that at quiet times β′≃0.35–0.6 at the LIGO sites, and at noisy times β′≃0.15–1.4. (For comparison, Saulson’s simple model gave β=β′=1/sqrt[3]=0.58.) By folding our resulting transfer function into the “standard LIGO seismic spectrum,” which approximates W̃(f ) at typical times, we obtain the gravitygradient noise spectra. At quiet times this noise is below the benchmark noise level of “advanced LIGO interferometers” at all frequencies (though not by much at ∼10 Hz); at noisy times it may significantly exceed the advanced noise level near 10 Hz. The lower edge of our quiettime noise constitutes a limit, beyond which there would be little gain from further improvements in vibration isolation and thermal noise, unless one can also reduce the seismic gravity gradient noise. Two methods of such reduction are briefly discussed: monitoring the Earth’s density perturbations near each test mass, computing the gravitational forces they produce, and correcting the data for those forces; and constructing narrow moats around the interferometers’ corner and end stations to shield out the fundamentalmode Rayleigh waves, which we suspect dominate at quiet times.
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Additional Information:  ©1998 The American Physical Society Received 4 June 1998; published 18 November 1998 We thank Peter Saulson for triggering this research, and Kenneth Libbrecht, Rai Weiss and Stan Whitcomb for helpful comments. We thank geophysicists Hiroo Kanamori and Susan Hough and seismic engineer Ronald Scott for very helpful conversations and advice, and Alan Rohay for advice and for providing his measurements of the seismic ground motion at Hanford and Livingston. We thank Albert Lazzarini for facilitating access to Rohay’s data sets and to blueprints of the LIGO site facilities, and Fred Asiri for helping us to track down information about the geological structures at the sites (including the Skagit report). We also thank Saul Teukolsky for pointing out an analytic substitution which simplified our formula for g (y). Finally, we thank Giancarlo Cella for providing us, shortly before this paper was submitted, a copy of the VIRGOProject manuscript on seismic gravity gradient noise @12# and for a helpful discussion. This research was supported by NSF Grant PHY–9424337. S.A.H. gratefully acknowledges the support of the National Science Foundation Graduate Program. K.S.T. thanks the MaxPlanckInstitut fu¨r Gravitationsphysik for hospitality during the final weeks of writing this manuscript.  
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