Download Cellar_2015nrgw_GGN_Lecture 2

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….