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NEWTONIAN NOISES IN LOW FREQUENCIES 2. MITIGATION G. Cella – INFN Pisa 2015 International School on Numerical Relativity and Gravitational Waves July 26-31 2015 KISTI & KAIST, Daejeon in Korea Lectures Plan Previous lecture (Estimation) What is Gravity Gradient Noise? Relevance Seismic Gravity Gradient Noise General formulation of the seismic GGN estimation problem This lecture (Mitigation): A worked out estimation & mitigation case: seismic GGN of an homogeneous half space. Going underground Wiener subtraction Sensor placement Future developements Simplified GGN model We model the ground as an homogeneous half space. Normal modes can be obtained from the elastic wave equation (in frequency domain) z Plane longitudinal wave solutions (speed 𝑐𝐿 ): y x Transverse longitudinal wave solutions (speed 𝑐𝑇 ): 𝑧1 However these does not satisfy the boundary condition on the upper free surface: 𝜎𝑖𝑧 𝑥, 𝑦, 0 = 0 𝑧2 Simplified GGN model Why pure longitudinal or transverse plane waves does not work? A longitudinal wave, when reflected by the free surface, generate a superposition of a longitudinal and a transverse wave Similarly a transverse wave is reflected in a superposition of a transverse and a longitudinal wave An exception: a purely transverse wave with horizontal polarization and propagation Surface waves (Raileigh waves) are allowed They propagate on the horizontal direction Exponentially damped with the depth T No GGN L T+L Compression & Surface GGN T T+L Compression & Surface GGN S T T+S Compression & Surface GGN Compression & Surface GGN Simplified GGN model First approximation: We allow only for surface modes Most efficient mechanism for the propagation from far away sources (Amplitude ∝ Sources on the surface excite these modes efficiently (antropic noise) L T 1 ) 𝑟 Simplified GGN model Second approximation: We suppose isotropy (in the horizontal plane) Seismic correlations are constrained. Setting 𝐼, 𝐽 ∈ 𝑥, 𝑦 and 𝑟2 − 𝑟1 = ℎ + 𝑧2 − 𝑧1 𝑧 These properties can be checked experimentally, and the approximation validated Simplified GGN model Another consequence of horizontal isotropy: The amplitudes of surface waves are uncorrelated, and the power spectrum of each of them is direction independent: The function 𝑆 𝑘 (𝜔) tell us how much a surface wave of wave number 𝑘 is excited at the angular frequency 𝜔 The seismic correlation is now written as a relation that can be inverted to obtain 𝑆 𝑘 (𝜔) from seismic measurement. Simplified GGN model A simple procedure for inversion. A particular case of the previous relation is which is an Hankel transform that can be easily inverted We can fully characterize this model using surface seismic correlations We can study theoretically the effect of a finite correlation length, which depends on the Q factor of the mode. Finally, we can insert 𝑆 𝑘 (𝜔) in the mode expansion for GGN Simplified GGN model The final estimate can be written as The factor 𝐹(𝑘𝐿) describes coherence effects. It depends on the topology of the detector. For Virgo & LIGO we have As expected, it works as a suppression factor when 𝜆 ≫ 𝐿 For a detector on the surface 𝑧 = 0. Suppression underground Mitigation: going underground We can take advantage of the exponential dependence from 𝑧 by building the interferometer below the ground The suppression factor is frequency dependent, because the length scale is the wavelength of the relevant modes Surface -10 m -50 m -100 m -150 m These results are valid only under the assumption that surface waves are dominant ET-C ET-B Mitigation: going underground The efficiency of the mitigation depends on the material in the ground From cL=200 m/s to cL=2000 m/s Surface As expected, when the sound speed increases for a given frequency the wavelength increases and the suppression is reduced At very low frequencies going underground is not an option for mitigation -10 m -50 m -100 m -150 m Mitigation: Wiener subtraction Another possible approach to the problem of GGN mitigation is subtraction. Basic idea: exploit the expected correlation between GGN and a set of 𝑁𝑆 auxiliary sensors which are continuously monitored (for example accelerometers) We construct a «subtracted signal» 𝑌𝑠 as In the stationary case we can determine the filters 𝛼𝐾 which minimize the power spectrum of the subtracted signal. Mitigation: Wiener subtraction The basic quantities which enters in the procedure are: the spectral correlations between the signal measured by the auxiliary sensors, a 𝑁𝑠 × 𝑁𝑠 array the spectral correlations between the intrinsic noises of the auxiliary sensors, a 𝑁𝑠 × 𝑁𝑠 array which we expect to be diagonal the spectral correlations between the signal measured by the auxiliary sensors and the GGN, a vector of dimension 𝑁𝑠 the GGN power spectrum Optimal filters in the stationary case: Mitigation: Wiener subtraction The ratio between subtracted and unsubtracted GGN power spectrum is given by where A large 𝜀 means a large subtraction efficiency (0 ≤ 𝜀 ≤ 1). This can be obtained if There is a large correlation between the sensors and the GGN There is a small correlation between the sensors There is a low level of noise in the sensors Mitigation: Wiener subtraction The efficiency 𝜀 is a nonlinear function of the auxiliary sensor position and orientation An example: given two sensors located in the optimal way (on the surface), where is convenient to put the third, to optimize the subtraction at some frequency? Test mass in the origin Coordinates normalized to 𝜆 Plotted quantity is 1 − 𝜀 Mitigation: Wiener subtraction Another example: full optimization in three dimensions • Using a simple model for the correlations • 512 sensors • At a fixed frequency • Negligible auxiliary sensor noise When sensor noise is not negligible it is convenient to decrease the separation between sensors: a larger correlation can be tolerated to average the noise. Test mass here Mitigation: Wiener subtraction Efficiency estimate Several coherences Regular grid Optimal arrangement of sensors Mitigation: Wiener subtraction Optimal arrangement of sensors is frequency dependent More robust with an higher number of sensors Coherence improves the subtraction efficiency Mitigation: Wiener subtraction Specific sensor placement is not critical Detailed model needed: Volume waves Scattering effects Enough improvement for a third generation detector Good in the low frequency region From: Subtraction of Newtonian noise using optimized sensor arrays Jennifer C. Driggers, Jan Harms, and Rana X. Adhikari Phys. Rev. D 86, 102001 Atmospheric GGN Thermal turbulence Infrasonic From: T. Chreighton, Atmospheric Gravitational Noise, GWDAW 2015. LIGO-G1500688 • Relevant damping scales: 𝜆/𝑑, 𝜆𝑐𝑜 /𝑑 • Infrasound: coherent in the «high frequency» region, not in the «low frequency» one ( subtraction ) • Thermal: low coherence ( subtraction) and power law in frequency ( underground) Thermal gradient effects Atmospheric GGN Rayleigh Bernard (G.C., E. Cuoco, P. Tomassini) Thermal bubbles Several scenarios, depending on the intensity of the thermal gradient Seismic RB Bubbles Lighthill process: turbulent generation of acoustic waves (C. Cafaro, G.C.) Negligible in the «high frequency» region Can be larger at lower frequencies (to be investigated) WD-WD at 10 kpc LF earth bound detectors From: J. Harms et al, Phys. Rev. D 88, 122003 (2013) IMBH-IMBH at z=1 𝟏𝟎𝟒 solar masses Conclusions Good perspectives for beating GGN Quiet site Going underground Subtraction Lot of work to do More investigation of atmospheric GGN Atmospheric GGN subtraction? «Realistic» estimates needed (FEM approach?) «Realistic» study of subtraction procedure (FEM approach, non stationarity,….) Thank you for your attention….