# Terrestrial gravity fluctuations

## Abstract

Terrestrial gravity fluctuations are a target of scientific studies in a variety of fields within geophysics and fundamental-physics experiments involving gravity such as the observation of gravitational waves. In geophysics, these fluctuations are typically considered as signal that carries information about processes such as fault ruptures and atmospheric density perturbations. In fundamental-physics experiments, it appears as environmental noise, which needs to be avoided or mitigated. This article reviews the current state-of-the-art of modeling high-frequency terrestrial gravity fluctuations and of gravity-noise mitigation strategies. It hereby focuses on frequencies above about 50 mHz, which allows us to simplify models of atmospheric gravity perturbations (beyond Brunt–Väisälä regime) and it guarantees as well that gravitational forces on elastic media can be treated as perturbation. Extensive studies have been carried out over the past two decades to model contributions from seismic and atmospheric fields especially by the gravitational-wave community. While terrestrial gravity fluctuations above 50 mHz have not been observed conclusively yet, sensitivity of instruments for geophysical observations and of gravitational-wave detectors is improving, and we can expect first observations in the coming years. The next challenges include the design of gravity-noise mitigation systems to be implemented in current gravitational-wave detectors, and further improvement of models for future gravitational-wave detectors where terrestrial gravity noise will play a more important role. Also, many aspects of the recent proposition to use a new generation of gravity sensors to improve real-time earthquake early-warning systems still require detailed analyses.

## Keywords

Terrestrial gravity Newtonian noise Wiener filter Mitigation## Notation

- \(c=299792458\mathrm {\ m/s}\)
Speed of light

- \(G=6.674\times 10^{-11}{\mathrm {\ N\, m}}^2/{\mathrm {kg}}^2\)
Gravitational constant

- \({\mathbf {r}},\, {\mathbf {e}}_r\)
Position vector, and corresponding unit vector

- \(x,\,y,\,z\)
Cartesian coordinates

- \(r,\,\theta ,\,\phi \)
Spherical coordinates

- \(\varrho ,\,\phi ,\,z\)
Cylindrical coordinates

- \({\mathrm {d}}\varOmega \equiv {\mathrm {d}}\phi \,{\mathrm {d}}\theta \sin (\theta )\)
Solid angle

- \(\delta _{ij}\)
Kronecker delta

- \(\delta (\cdot )\)
Dirac \(\delta \) distribution

- \(\mathfrak {R}\)
Real part of a complex number

- \(\partial _x^n\)
*n*-th partial derivative with respect to*x*- \(\nabla \)
Nabla operator, e.g., \((\partial _x,\partial _y,\partial _z)\)

- \({\xi }(\mathbf {r},t)\)
Displacement field

- \(\phi _{\mathrm {s}}(\mathbf {r},t)\)
Potential of seismic compressional waves

- \(\psi _{\mathrm {s}}(\mathbf {r},t)\)
Potential of seismic shear waves

- \(\rho _0\)
Time-averaged mass density

- \(\alpha ,\,\beta \)
Compressional-wave and shear-wave speed

- \(\mu \)
Shear modulus

- \(\otimes \)
Dyadic product

- \(\mathbf {M},\,{\mathbf {v}},\,s\)
Matrix/tensor, vector, scalar

- \(P_l(x)\)
Legendre polynomial

- \(P_l^m(x)\)
Associated Legendre polynomial

- \(Y_l^m(x)\)
Scalar surface spherical harmonics

- \(J_n(x)\)
Bessel function of the first kind

- \(K_n(x)\)
Modified Bessel function of the second kind

- \(j_n(x)\)
Spherical Bessel function of the first kind

- \(Y_n(x)\)
Bessel function of the second kind

- \(y_n(x)\)
Spherical Bessel function of the second kind

- \(H_n(x)\)
Hankel function or Bessel function of the third kind

- \(h^{(2)}_n(x)\)
Spherical Hankel function of the second kind

- \(X_l^m\)
Exterior spherical multipole moment

- \(N_l^m\)
Interior spherical multipole moment

## 1 Introduction

In the past few years, researchers achieved milestones in the study of high-frequency, i.e., above tens of millihertz, gravity fluctuations [Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration) 2016; Montagner et al. 2016]. The Advanced LIGO and Virgo detectors have opened a new window to our Universe with the observation of gravitational-waves (GWs) from binary black-hole and neutron-star mergers [Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration) 2016, 2017, 2018]. Commissioning periods aim at further improving the detectors’ sensitivities, and it is predicted that terrestrial gravity noise will eventually limit their sensitivity. Terrestrial gravity noise, also known as Newtonian noise or gravity-gradient noise, becomes increasingly relevant towards lower frequencies. It is predicted to appear below 30 Hz in the advanced-generation detectors (Driggers et al. 2012b), and will play a very important role in future-generation detectors. The Einstein Telescope is a planned European detector, which targets GW observations down to a few Hertz (Punturo et al. 2010). This can only be achieved by constructing the detector underground to suppress the Newtonian-noise foreground, which is typically stronger at the surface by some orders of magnitude. This strategy was adopted for the Japanese GW detector KAGRA built underground at the Mozumi mine (Akutsu et al. (KAGRA Collaboration) 2019). There are also world-wide efforts to realize sub-Hertz GW detectors based on atom interferometry (Canuel et al. 2018), superconduction (Paik et al. 2016), and torsion bars (Shoda et al. 2014; McManus et al. 2016). Here, the issue of Newtonian noise is elevated to a potential show-stopper for ground-based versions of these detectors since Newtonian noise needs to be suppressed by several orders of magnitude, and going underground does not have a strong effect at these low frequencies. Nonetheless, low-frequency concepts are continuously improving, and it is conceivable that future detectors will be sufficiently sensitive to detect GWs well below a Hertz (Harms et al. 2013).

Strategies to mitigate Newtonian noise in GW detectors include coherent noise cancellation based on Wiener filters (Cella 2000). The idea is to monitor the sources of gravity perturbations using auxiliary sensors such as microphones and seismometers, and to use their data to generate a coherent prediction of gravity noise. The most challenging aspect of this technology is to determine the locations of a given number of sensors that optimize the cancellation performance (Coughlin et al. 2016). This is largely an unsolved problem and will remain a great practical challenge whenever sensor placement is not trivial, i.e., seismometers in boreholes or microphones many tens of meters (or higher) above ground. Detailed understanding of seismic and atmospheric fields is imperative in these cases. Experiments have recently been concluded to study the underground seismic field at the Sanford Underground Research Facility (Mandic et al. 2018), and ongoing analyses promise crucial insight into how seismic fields produce Newtonian noise. Equivalently, studies of sound fields have started at the Virgo site (Fiorucci et al. 2018), but turbulence and wind lead to additional density fluctuations, which makes the modeling of atmospheric Newtonian noise very difficult. It should be noted though that Newtonian-noise cancellation is already being applied successfully in gravimeters to reduce the foreground of atmospheric gravity noise below a few mHz using collocated pressure sensors (Neumeyer 2010).

More recently, high-precision gravity strainmeters have been considered as monitors of prompt gravity pertubations from fault ruptures (Harms et al. 2015), and consequently, it was suggested to implement gravity strainmeters in existing earthquake-early warning systems to increase warning times (Juhel et al. 2018a). Towards lower frequencies, gravity plays an increasingly important role in gravitoelastic processes (Dahlen et al. 1998; Tsuda 2014), and new effects such as self gravity need to be considered (Juhel et al. 2018b). Self gravity can lead to strong suppression of prompt gravity signals in inertial sensors like gravimeters and seismometers by causing a free-fall like response of the ground to a change in gravity, which sets the whole inertial sensor into a free fall at least for some period of time until elastic forces start to counteract this motion. This might even interfere with seismic Newtonian-noise cancellation using seismometers, but there has not been any quantitative study yet. It is therefore not only in the interest of geophysicists to further improve our understanding of prompt gravity perturbations.

This article is divided into six main sections. Section 2 serves as an introduction to gravity measurements focussing on the response mechanisms and basic properties of gravity sensors. Section 3 describes models of gravity perturbations from ambient seismic fields. The results can be used to estimate noise spectra at the surface and underground. A subsection is devoted to the problem of noise estimation in low-frequency GW detectors, which differs from high-frequency estimates mostly in that gravity perturbations are strongly correlated between different test masses. In the low-frequency regime, the gravity noise is best described as gravity-gradient noise. Section 4 is devoted to time domain models of transient gravity perturbations from seismic point sources. The formalism is applied to point forces and shear dislocations. The latter allows us to estimate gravity perturbations from earthquakes. Atmospheric models of gravity perturbations are presented in Sect. 5. This includes gravity perturbations from atmospheric temperature fields, infrasound fields, shock waves, and acoustic noise from turbulence. The solution for shock waves is calculated in time domain using the methods of Sect. 4. A theoretical framework to calculate gravity perturbations from objects is given in Sect. 6. Since many different types of objects can be potential sources of gravity perturbations, the discussion focusses on the development of a general method instead of summarizing all of the calculations that have been done in the past. Finally, Sect. 7 discusses possible passive and active noise mitigation strategies. Due to the complexity of the problem, most of the section is devoted to active noise cancellation providing the required analysis tools and showing limitations of this technique. Site selection is the main topic under passive mitigation, and is discussed in the context of reducing environmental noise and criteria relevant to active noise cancellation. Each of these sections ends with a summary and a discussion of open problems. While this article is meant to be a review of the current state of the field, it also presents original analyses especially with respect to the impact of seismic scattering on gravity perturbations (Sects. 3.3.2 and 3.3.3), active gravity noise cancellation (Sect. 7.1.3), and time-domain models of gravity perturbations from atmospheric and seismic point sources (Sects. 4.1, 4.4, and 5.3).

## 2 Gravity measurements

In this section, we describe the relevant mechanisms by which a gravity sensor can couple to gravity perturbations, and give an overview of the most widely used measurement schemes: the (relative) gravimeter (Crossley et al. 2013; Zhou et al. 2011), the gravity gradiometer (Moody et al. 2002; Ando et al. 2010; McManus et al. 2017; Canuel et al. 2018), and the gravity strainmeter, i.e., the large-scale GW detectors Virgo (Acernese et al. (Virgo Collaboration) 2015), LIGO (Aasi et al. (LIGO Scientific Collaboration) 2015), GEO600 (Lück et al. 2010), KAGRA (Akutsu et al. (KAGRA Collaboration) 2019). Strictly speaking, none of the sensors only responds to a single field quantity (such as changes in gravity acceleration or gravity strain), but there is always a dominant response mechanism in each case, which justifies to give the sensor a specific name. A clear distinction between gravity gradiometers and gravity strainmeters has never been made to our knowledge. Therefore the sections on these two measurement principles will introduce a definition, and it is by no means the only possible one. Space-borne gravity experiments such as GRACE (Wahr et al. 2004), LISA Pathfinder (Armano et al. 2016), and the future GW detector LISA (Amaro-Seoane et al. 2017) will not be included in this overview. These experiments have very similar measurement principles, all employing at least two test masses to measure changes in the tidal field (produced by Earth or associated with GWs).

The different response mechanisms to terrestrial gravity perturbations are summarized in Sect. 2.1. In Sects. 2.2 to 2.4, the different measurement schemes are explained including a brief summary of the sensitivity limitations.

### 2.1 Gravity response mechanisms

#### 2.1.1 Gravity acceleration and tidal forces

#### 2.1.2 Shapiro time delay

*c*is the speed of light, \({\mathrm {d}}s\) is the so-called line element of a path in spacetime, and \(\psi ({\mathbf {r}},t)/c^2\ll 1\). Additionally, for this metric to hold, motion of particles in the source of the gravity potential responsible for changes of the gravity potential need to be much slower than the speed of light, and also stresses inside the source must be much smaller than its mass energy density. All conditions are fulfilled in the case of Earth gravity field. Light follows

*null geodesics*with \({\mathrm {d}}s^2=0\). For the spacetime metric in Eq. (4), we can immediately write

^{1}In other words, we can integrate the time delay along a straight line as defined in Eq. (7), and so the total phase integrated over a travel distance

*L*is given by

*v*is a characteristic speed of the source, e.g., speeds of seismic waves.

#### 2.1.3 Gravity induced ground motion

It is interesting to understand the gravity-induced ground motion. This effect has been neglected in all NN calculations assuming that the signal of a seismometer is, at least ideally, a direct measurement of ground motion. This assumption is generally false, and the problem originates in the equivalence principle.

#### 2.1.4 Coupling in non-uniform, static gravity fields

*z*-axis) seismic noise \(\xi _z(t)\) coupling into the horizontal (

*x*-axis) motion of the test mass through the term \(\partial _x g_z=\partial _z g_x\) dominates over other displacement noise.

*L*/ 2. There are also two additional \(\partial _z g_x=0\) contour lines starting at the symmetry axis at heights \(\sim 0.24\) and \(\sim 0.76\). Let us assume that the test mass is at height 0.3

*L*, a distance 0.05

*L*from the cylinder axis, the total mass of the cylinder is \(M=5000\mathrm {\ kg}\), and the cylinder height is \(L=4\mathrm {\ m}\). In this case, the gravity-gradient induced vertical-to-horizontal coupling factor at 20 Hz is

^{2}). Even though the vacuum chamber was modeled with a very simple shape, and additional asymmetries in the mass distribution around the test mass may increase gravity gradients, it still seems very unlikely that the coupling would be significant. As mentioned before, one certainly needs to pay more attention when calculating the coupling at lower frequencies. The best procedure is to base the calculation on a 3D model of the near test-mass infrastructure. Accurate modeling of quasi-static gravity gradients is important in the space-borne GW detector LISA (Schumaker 2003).

### 2.2 Gravimeters

Gravimeters are instruments that measure the displacement of a test mass with respect to a non-inertial reference rigidly connected to the ground. They belong to the class of inertial sensors, i.e., sensors with an inertial reference like a suspended test mass, which also includes most seismometers. The test mass is typically supported mechanically or magnetically (atom-interferometric gravimeters are an exception), which means that the test-mass response to gravity is altered with respect to a freely falling test mass. We will use Eq. (1) as a simplified response model. There are various possibilities to measure the displacement of a test mass. The most widespread displacement sensors are based on capacitive readout, as for example used in superconducting gravimeters (see Fig. 2 and Hinderer et al. 2007). Sensitive displacement measurements are in principle also possible with optical readout systems; a method that is implemented in atom-interferometric gravimeters (Peters et al. 2001), and prototype seismometers (Berger et al. 2014) (we will explain the distinction between seismometers and gravimeters below). As will become clear in Sect. 2.4, optical readout is better suited for displacement measurements over long baselines, as required for the most sensitive gravity strain measurements, while the capacitive readout should be designed with the smallest possible distance between the test mass and the non-inertial reference (Jones and Richards 1973).

Capacitor plates are distributed around the sphere. Whenever a force acts on the sphere, the small signal produced in the capacitive readout is used to immediately cancel this force by a feedback coil. In this way, the sphere is kept at a constant location with respect to the external frame.

*additional*noise due to non-linearities and cross-coupling. As is explained further in Sect. 2.3, it is also not possible to suppress seismic noise in

*gravimeters*by subtracting the disturbance using data from a collocated seismometer. Doing so inevitably turns the gravimeter into a gravity gradiometer.

^{3}Well below resonance frequency, the response function can be approximated as \(\omega ^2/\omega _0^2\). At first, it may look as if the gravimeter should not be sensitive to very low-frequency fluctuations since the response becomes very weak. However, the strength of gravity fluctuations also strongly increases with decreasing frequency, which compensates the small response. It is clear though that if the resonance frequency were sufficiently high, then the response would become so weak that the gravity signal would not stand out above other instrumental noise anymore. The test-mass support would be too stiff. The sensitivity of the gravimeter depends on the resonance frequency of the support and the intrinsic instrumental noise. With respect to seismic noise, the stiffness of the support has no influence as explained before (the test mass can also fall freely as in atom interferometers).

For superconducting gravimeters of the Global Geodynamics Project (GGP) (Crossley and Hinderer 2010), the median spectra are shown in Fig. 3. Between 0.1 and 1 mHz, atmospheric gravity perturbations typically dominate, while instrumental noise is the largest contribution between 1 mHz and 5 mHz (Hinderer et al. 2007). The smallest signal amplitudes that have been measured by integrating long-duration signals is about \(10^{-12}\mathrm {\ m/s^2}\). A detailed study of noise in superconducting gravimeters over a larger frequency range can be found in Rosat et al. (2003). Note that in some cases, it is not fit to categorize seismic and gravity fluctuations as noise and signal. For example, Earth’s spherical normal modes coherently excite seismic and gravity fluctuations, and the individual contributions in Eq. (22) have to be understood to accurately translate data into normal-mode amplitudes (Dahlen et al. 1998).

### 2.3 Gravity gradiometers

It is not the purpose of this section to give a complete overview of the different gradiometer designs. Gradiometers find many practical applications, for example in navigation and resource exploration, often with the goal to measure static or slowly changing gravity gradients, which do not concern us here. For example, we will not discuss rotating gradiometers, and instead focus on gradiometers consisting of stationary test masses. While the former are ideally suited to measure static or slowly changing gravity gradients with high precision especially under noisy conditions, the latter design has advantages when measuring weak tidal fluctuations. In the following, we only refer to the stationary design. A gravity gradiometer measures the relative acceleration between two test masses each responding to fluctuations of the gravity field (Jekeli 2014; Moody et al. 2002). The test masses have to be located close to each other so that the approximation in Eq. (3) holds. The proximity of the test masses is used here as the defining property of gradiometers. They are therefore a special type of gravity strainmeter (see Sect. 2.4), which denotes any type of instrument that measures relative gravitational acceleration (including the even more general concept of measuring space-time strain).

It is schematically shown in Fig. 4. Let us first consider the dual-sphere design of a superconducting gradiometer. If the reference is perfectly stiff, and if we assume as before that there are no cross-couplings between degrees of freedom and the response is linear, then the subtraction of the two gravity channels cancels all of the seismic noise, leaving only the instrumental noise and the differential gravity signal given by the second line of Eq. (3). Even in real setups, the reduction of seismic noise can be many orders of magnitude since the two spheres are close to each other, and the two readouts pick up (almost) the same seismic noise (Moody et al. 2002). This does not mean though that gradiometers are necessarily more sensitive instruments to monitor gravity fields. A large part of the gravity signal (the common-mode part) is suppressed as well, and the challenge is now passed from finding a seismically quiet site to developing an instrument with lowest possible intrinsic noise.

*L*is the separation of the two atom clouds, and

*c*is the speed of light. It should be emphasized that the seismic noise remains, even if all optics and the laser source are all linked to the same infinitely stiff frame. In addition to this noise term, other coupling mechanisms may play a role, which can however be suppressed by engineering efforts. The noise-reduction factor \(\omega L/c\) needs to be compared with the common-mode suppression of seismic noise in superconducting gravity gradiometers, which depends on the stiffness of the instrument frame, and on contamination from cross coupling of degrees-of-freedom. While the seismic noise in Eq. (23) is a fundamental noise contribution in (conventional) atom-interferometric gradiometers, the noise suppression in superconducting gradiometers depends more strongly on the engineering effort (at least, we venture to claim that common-mode suppression achieved in current instrument designs is well below what is fundamentally possible).

*m*supported by a heavy mass

*M*representing the gravimeter (reference) frame, which is itself supported from a point rigidly connected to Earth. The two supports are modelled as harmonic oscillators. As before, we neglect cross coupling between degrees of freedom. Linearizing the response of the gravimeter frame and test mass for small accelerations, and further neglecting terms proportional to

*m*/

*M*, one finds the gravimeter response to gravity fluctuations:

*m*(test mass) and

*M*(reference frame) coincide, and therefore \(\delta g_1(\omega )=\delta g_2(\omega )\), then the response is again like a conventional gravimeter, but this time suppressed by the isolation function \(S(\omega ;\omega _1,\gamma _1)\).

### 2.4 Gravity strainmeters

Gravity strain is an unusual concept in gravimetry that stems from our modern understanding of gravity in the framework of general relativity. From an observational point of view, it is not much different from elastic strain. Fluctuating gravity strain causes a change in distance between two freely falling test masses, while seismic or elastic strain causes a change in distance between two test masses bolted to an elastic medium. It should be emphasized though that we cannot always use this analogy to understand observations of gravity strain (Kawamura and Chen 2004). Fundamentally, gravity strain corresponds to a perturbation of the metric that determines the geometrical properties of spacetime (Misner et al. 1973). We will briefly discuss GWs, before returning to a Newtonian description of gravity strain.

*L*is the distance between them, and \({\mathbf {e}}_{12}\) is the unit vector pointing from one to the other test mass, and \({\mathbf {e}}_{12}^\top \) its transpose. As can be seen, the gravity-strain field is represented by a \(3\times 3\) tensor. It contains the space-components of a 4-dimensional metric perturbation of spacetime, and determines all properties of GWs.

^{4}Note that the strain amplitude \(\mathbf {h}\) in Eq. (27) needs to be multiplied by 2 to obtain the corresponding amplitude of the metric perturbation (e.g., the GW amplitude).

*Throughout this article, we define gravity strain as*\(h=\varDelta L/L\),

*while the effect of a GW with amplitude*\(a_{\mathrm {GW}}\)

*on the separation of two test masses is determined by*\(a_{\mathrm {GW}}=2\varDelta L/L\).

*relative displacement*of suspended test masses typically carried out along two perpendicular baselines (arms); and (2) measurement of the

*relative rotation*between two suspended bars. Figure 6 illustrates the two cases. In either case, the response of a gravity strainmeter is obtained by projecting the gravity strain tensor onto a combination of two unit vectors, \({\mathbf {e}}_1\) and \({\mathbf {e}}_2\), that characterize the orientation of the detector, such as the directions of two bars in a rotational gravity strain meter, or of two arms of a conventional gravity strain meter. This requires us to define two different gravity strain projections. The projection for the rotational strain measurement is given by

*x*,

*y*-axes (see Fig. 5) are oriented along two perpendicular bars. The vectors \({\mathbf {e}}_1^{\,\mathrm {r}}\) and \({\mathbf {e}}_2^{\,\mathrm {r}}\) are rotated counter-clockwise by \(90^\circ \) with respect to \({\mathbf {e}}_1\) and \({\mathbf {e}}_2\). In the case of perpendicular bars \({\mathbf {e}}_1^{\,\mathrm {r}}={\mathbf {e}}_2\) and \({\mathbf {e}}_2^{\,\mathrm {r}}=-{\mathbf {e}}_1\). The corresponding projection for the conventional gravity strain meter reads

Let us now return to the discussion of the previous sections on the role of seismic isolation and its impact on gravity response. Gravity strainmeters profit from seismic isolation more than gravimeters or gravity gradiometers. We have shown in Sect. 2.2 that seismically isolated gravimeters are effectively gravity gradiometers. So in this case, seismic isolation changes the response of the instrument in a fundamental way, and it does not make sense to talk of seismically isolated gravimeters. Seismic isolation could in principle be beneficial for gravity gradiometers (i.e., the acceleration of two test masses is measured with respect to a common rigid, seismically isolated reference frame), but the common-mode rejection of seismic noise (and gravity signals) due to the differential readout is typically so high that other instrumental noise becomes dominant. So it is possible that some gradiometers would profit from seismic isolation, but it is not generally true. Let us now consider the case of a gravity strainmeter. As explained in Sect. 2.3, we distinguish gradiometers and strainmeters by the distance of their test masses. For example, the distance of the LIGO or Virgo test masses is 4 km and 3 km respectively. Seismic noise and terrestrial gravity fluctuations are insignificantly correlated between the two test masses within the detectors’ most sensitive frequency band (above 10 Hz). Therefore, the approximation in Eq. (3) does not apply. Certainly, the distinction between gravity gradiometers and strainmeters remains somewhat arbitrary since at any frequency the approximation in Eq. (3) can hold for one type of gravity fluctuation, while it does not hold for another. Let us adopt a more practical definition at this point. *Whenever the design of the instrument places the test masses as distant as possible from each other given current technology, then we call such an instrument strainmeter*. In the following, we will discuss seismic isolation and gravity response for three strainmeter designs, the laser-interferometric, atom-interferometric, and superconducting strainmeters. It should be emphasized that the atom-interferometric and superconducting concepts are still in the beginning of their development and have not been realized yet with scientifically interesting sensitivities.

*Laser-interferometric strainmeters* The most sensitive gravity strainmeters, namely the large-scale GW detectors, use laser interferometry to read out the relative displacement between mirror pairs forming the test masses. Each test mass in these detectors is suspended from a seismically isolated platform, with the suspension itself providing additional seismic isolation. Sect. 2.1.1 introduced a simplified response and isolation model based on a harmonic oscillator characterized by a resonance frequency \(\omega _0\) and viscous damping \(\gamma \).^{5} In a multi-stage isolation and suspension system as realized in GW detectors (see, e.g., Braccini et al. 2005; Matichard et al. 2015), coupling between multiple oscillators cannot be neglected, and is fundamental to the seismic isolation performance, but the basic features can still be explained with the simplified isolation and response model of Eqs. (1) and (2). The signal output of the interferometer is proportional to the relative displacement between test masses. Since seismic noise is approximately uncorrelated between two distant test masses, the differential measurement itself cannot reject seismic noise as in gravity gradiometers. Without seismic isolation, the dominant signal would be seismic strain, i.e., the distance change between test masses due to elastic deformation of the ground, with a value of about \(10^{-15}\mathrm {\ Hz^{-1/2}}\) at 50 Hz (assuming kilometer-scale arm lengths). At the same time, without seismically isolated test masses, the gravity signal can only come from the ground response to gravity fluctuations as described in Sect. 2.1.3, and from the Shapiro time delay as described in Sect. 2.1.2. These signals would lie well below the seismic noise. Consequently, to achieve the sensitivities of past science runs, the seismic isolation of the large-scale GW detectors had to suppress seismic noise by at least 7 orders of magnitude, and test masses had to be supported so that they can (quasi-)freely respond to gravity-strain fluctuations in the targeted frequency band (which, according to Eqs. (1) and (2), is achieved automatically with the seismic isolation). Stacking multiple stages of seismic isolation enhances the gravity response negligibly, while it is essential to achieve the required seismic-noise suppression. Using laser beams, long-baseline strainmeters can be realized, which increases the gravity response according to Eq. (3). The price to be paid is that seismic noise needs to be suppressed by a sophisticated isolation and suspension system since it is uncorrelated between test masses and therefore not rejected in the differential measurement. As a final note, the most sensitive torsion-bar antennas also implement a laser-interferometric readout of the relative rotation of the suspended bars (Shoda et al. 2014), and concerning the gravity response and seismic isolation, they can be modelled very similarly to conventional strainmeters. However, the suppression of seismic noise is impeded by mechanical cross-coupling, since a torsion bar has many soft degrees of freedom that can interact resonantly within the detection band. This problem spoils to some extent the big advantage of torsion bars to realize a very low-frequency torsion resonance, which determines the fundamental response and seismic isolation performance. Nonetheless, cross-coupling can in principle be reduced by precise engineering, and additional seismic pre-isolation of the suspension point of the torsion bar can lead to significant noise reduction.

*Atom-interferometric strainmeters* In this design, the test masses consist of freely-falling ultracold atom clouds. A laser beam interacting with the atoms serves as a common phase reference, which the test-mass displacement can be measured against. The laser phase is measured locally via atom interferometry by the same freely-falling atom clouds Cheinet et al. (2008). Subtraction of two of these measurements forms the strainmeter output. The gravity response is fundamentally the same as for the laser-interferometric design since it is based on the relative displacement of atom clouds. Seismic noise couples into the strain measurement through the laser. If displacement noise of the laser or laser optics has amplitude \(\xi (\omega )\), then the corresponding strain noise in atom-interferometric strainmeters is of order \(\omega \xi (\omega )/c\), where *c* is the speed of light, and \(\omega \) the signal frequency Baker and Thorpe (2012). While this noise is lower than the corresponding term \(\xi (\omega )/L\) in laser-interferometric detectors (*L* being the distance between test masses), seismic isolation is still required. As we know from previous discussions, seismic isolation causes the optics to respond to gravity fluctuations. However, the signal contribution from the optics is weaker by a factor \(\omega L/c\) compared to the contribution from distance changes between atom clouds. Here, *L* is the distance between two freely-falling atom clouds, which also corresponds approximately to the extent of the optical system. This signal suppression is very strong for any Earth-bound atom-interferometric detector (targeting sub-Hz gravity fluctuations), and we can neglect signal contributions from the optics. Here we also assumed that there are no control forces acting on the optics, which could further suppress their signal response, if for example the distance between optics is one of the controlled parameters. Nonetheless, seismic isolation is required, not only to suppress seismic noise from distance changes between laser optics, which amounts to \(\omega \xi (\omega )/c\sim 10^{-17}\mathrm {\ Hz^{-1/2}}\) at 0.1 Hz without seismic isolation (too high at least for GW detection Harms et al. (2013)), but also to suppress seismic-noise contributions through additional channels (e.g., tilting optics in combination with laser-wavefront aberrations Hogan et al. (2011)). The additional channels dominate in current experiments, which are already seismic-noise limited with strain noise many orders of magnitude higher than \(10^{-17}\mathrm {\ Hz^{-1/2}}\) Dickerson et al. (2013). It is to be expected though that improvements of the atom-interferometer technology will suppress the additional channels relaxing the requirement on seismic isolation.

*Superconducting strainmeters*The response of superconducting strainmeters to gravity-strain fluctuations is based on the differential displacement of magnetically levitated spheres. The displacement of individual spheres is monitored locally via a capacitive readout (see Sect. 2.2). Subtracting local readouts of test-mass displacement from each other constitutes the basic strainmeter scheme Paik (1976). The common reference for the local readouts is a rigid, material frame. The stiffness of the frame is a crucial parameter facilitating the common-mode rejection of seismic noise. Even in the absence of seismic noise, the quality of the reference frame is ultimately limited by thermally excited vibrations of the frame

^{6}(similar to the situation with torsion-bar antennas Harms et al. (2013)). However, since strainmeters are very large (by definition), vibrational eigenmodes of the frame can have low resonance frequencies impeding the common-mode rejection of seismic noise. In fact, it is unclear if a significant seismic-noise reduction can be achieved by means of mechanical rigidity. Therefore, seismic isolation of the strainmeter frame is necessary. In this case, each local readout is effectively a gravity-strain measurement, since the gravity response of the test mass is measured against a reference frame that also responds to gravity fluctuations (see discussion of seismically isolated gravimeters in Sect. 2.3). Another solution could be to substitute the mechanical structure by an optically rigid body as suggested in Harms et al. (2013) for a low-frequency laser-interferometric detector. The idea is to connect different parts of a structure via laser links in all degrees of freedom. The stiffness of the link is defined by the control system that forces the different parts to keep their relative positions and orientations. Optical rigidity in all degrees of freedom has not been realized experimentally yet, but first experiments known as suspension point or platform interferometers have been conducted to control some degrees of freedom in the relative orientation of two mechanical structures Aso et al. (2004); Dahl et al. (2012). This approach would certainly add complexity to the experiment, especially in full-tensor configurations of superconducting gravity strainmeters, where six different mechanical structures have to be optically linked Moody et al. (2002).

The best sensitivities achieved in the past with low-frequency GW detectors are shown in the left plot in Fig. 7. To our best knowledge, a full sensitivity spectrum has not been published yet for atom-interferometric strainmeters. Therefore the sensitivity is represented by a single dot at 1 Hz. The current record of a high-precision experiment was set more than 10 years ago with a superconducting gradiometer. Nevertheless, the sensitivity required for GW detection at low frequencies, represented by the MANGO curve in the right plot, still lies 7–9 orders of magnitude below this record sensitivity. Such sensitivity improvement not only relies on substantial technological progress concerning the strainmeter concepts, but also on a novel scheme of Newtonian-noise cancellation capable of mitigating seismic and atmospheric Newtonian noise by about 3 and 5 orders of magnitude, respectively.

### 2.5 Summary

Summary of gravity sensors

Sensor | Measurement | Sensitivity |
---|---|---|

Gravimeter | Differential displacement between ground and test mass | Superconducting gravimeters: \(\sim 1\mathrm {\ nm/s^2/\sqrt{Hz}}\) at 1 mHz |

Gradiometer | Differential displacement/rotation between two test masses. Typically compact design for measurement of static gradients or slowly varying gradients | Superconducting gradiometers: \(\sim 10^{-9}/\mathrm {\ s^2/\sqrt{Hz}}\) between 0 and 1 Hz |

Strainmeter | Differential displacement/rotation between two test masses with the aim to achieve best possible sensitivity in a certain frequency band | Laser-interferometric strainmeters (GW detectors): \(\sim 10^{-23}/\sqrt{\mathrm {Hz}}\) at 100 HzSuperconducting strainmeters: \(\sim 10^{-10}/\sqrt{\mathrm {Hz}}\) at 0.1 Hz |

## 3 Gravity perturbations from seismic fields

Already in the first design draft of a laser-interferometric GW detector laid out by Rainer Weiss, gravity perturbations from seismic fields were recognized as a potential noise contribution (Weiss 1972). He expressed the transfer function between ground motion and gravitational displacement noise of a test mass as effective isolation factor, highlighting the fact that gravitational coupling can be understood as additional link that circumvents seismic isolation. The equations that he used already had the correct dependence on ground displacement, density and seismic wavelength, but it took another decade, before Peter Saulson presented a more detailed calculation of numerical factors (Saulson 1984). He divided the half space below a test mass into volumes of correlated density fluctuations, and assigned a mean displacement to each of these volumes. Fluctuations were assumed to be uncorrelated between different volumes. The total gravity perturbation was then obtained as an incoherent sum over these volumes. The same scheme was carried out for gravity perturbations associated with vertical surface displacement. The sizes of volumes and surface areas of correlated density perturbations were determined by the length of seismic waves, but Saulson did not make explicit use of the wave nature of the seismic field that produces the density perturbations. As a result, also Saulson had to concede that certain steps in his calculation “cannot be regarded as exact”. The next step forward was marked by two papers that were published almost simultaneously by groups from the LIGO and Virgo communities (Hughes and Thorne 1998; Beccaria et al. 1998). In these papers, the wave nature of the seismic field was taken into account, producing for the first time accurate predictions of Newtonian noise. They understood that the dominant contribution to Newtonian noise would come from seismic surface waves, more specifically Rayleigh waves. The Rayleigh field produces density perturbations beneath the surface, and correlated surface displacement at the same time. The coherent summation of these effects was directly obtained, and since then, models of Newtonian noise from Rayleigh waves have not improved apart from a simplification of the formalism.

Nonetheless, Newtonian-noise models are not only important to estimate a noise spectrum with sufficient accuracy. More detailed models are required to analyze Newtonian-noise mitigation, which is discussed in Sect. 7. Especially the effect of seismic scattering on gravity perturbations needs to be quantified. A first analytical calculation of gravity perturbations from seismic waves scattered from a spherical cavity is presented in Sects. 3.3.2 and 3.3.3. In general, much of the recent research on Newtonian-noise modelling was carried out to identify possible limitations in Newtonian-noise mitigation. Among others, this has led to two major new developments in the field. First, finite-element simulations were added to the set of tools (Harms et al. 2009b; Beker et al. 2010). We will give a brief summary in Sect. 3.5. The advantage is that several steps of a complex analysis can be combined such as simulations of a seismic field, simulations of seismic measurements, and simulations of noise mitigation. Second, since seismic sources can be close to the test masses, it is clear that the seismic field cannot always be described as a superposition of propagating plane seismic waves. For this reason, analytical work has begun to base calculations of gravity perturbations on simple models of seismic sources, which can give rise to complex seismic fields (Harms et al. 2015). Since this work also inspired potential applications in geophysics and seismology, we devote Sect. 4 entirely to this new theory. Last but not least, ideas for new detector concepts have evolved over the last decade, which will make it possible to monitor gravity strain perturbations at frequencies below 1 Hz. This means that our models of seismic Newtonian noise (as for all other types of Newtonian noise) need to be extended to lower frequencies, which is not always a trivial task. We will discuss aspects of this problem in Sect. 3.6.4.

### 3.1 Seismic waves

In this section, we describe the properties of seismic waves relevant for calculations of gravity perturbations. The reader interested in further details is advised to study one of the classic books on seismology, e.g., Aki and Richards (2009). The formalism that will be introduced is most suited to describe physics in infinite or half-spaces with simple modifications such as spherical cavities, or small perturbations of a flat surface topography. At frequencies well below 10 mHz where the finite size of Earth starts to affect significantly the properties of the seismic field, seismic motion is best described by Earth’s normal modes (Dahlen et al. 1998). It should also be noted that in the approximation used in the following, the gravity field does not act back on the seismic field. This is in contrast to the theory of Earth’s normal modes, which includes the gravity potential and its derivative in the elastodynamic equations.

*primary*and means that these waves are the first to arrive after an earthquake (i.e., they are the fastest waves). These waves are characterized by a frequency \(\omega \) and a wave vector \({\mathbf {k}}^{\,\mathrm {P}}\). While one typically assumes \(\omega =k^{\mathrm {P}}\alpha \) with compressional wave speed \(\alpha \), this does not have to hold in general, and many results presented in the following sections do not require a fixed relation between frequency and wavenumber. The displacement field of a plane compressional wave can be written

*secondary*since it is the seismic phase to follow the P-wave arrival after earthquakes. The shear-wave displacement \(\xi ^{\,\mathrm {S}}({\mathbf {r}},t)\) of a single plane wave can be expressed in terms of a polarization vector \({\mathbf {e}}_p\):

*body waves*since they can propagate through media in all directions. Clearly, inside inhomogeneous media, all material constants are functions of the position vector \({\mathbf {r}}\). Another useful relation between the two seismic speeds is given by

*z*-direction, then it suffices to calculate the contribution from the scalar potential \(\psi _{\mathrm {s}}(\mathbf {r},t)\).

*z*-axis of the coordinate system, and will also be called vertical direction. The normal vector is denoted as \({\mathbf {e}}_z\). Rayleigh waves propagate along a horizontal direction \({\mathbf {e}}_k\). A wave vector \({\mathbf {k}}\) can be split into its vertical \({\mathbf {k}}_z\) and horizontal components \({\mathbf {k}}_\varrho \). The vertical wavenumbers are defined as

*higher-order Rayleigh waves*that can exist in these media (Hughes and Thorne 1998). For this reason, we will occasionally refer to Rayleigh waves as Rf-waves. According to Eqs. (34) and (40), given a shear-wave speed \(\beta \), the compressional-wave speed \(\alpha \) and Rayleigh-wave speed \(c_{\mathrm {R}}\) are functions of the Poisson’s ratio only. Figure 8 shows the values of the wave speeds in units of \(\beta \). As can be seen, for a given shear-wave speed the Rayleigh-wave speed (shown as solid line), depends only weakly on the Poisson’s ratio. The P-wave speed however varies more strongly, and in fact grows indefinitely with Poisson’s ratio approaching the value \(\nu =0.5\).

### 3.2 Basics of seismic gravity perturbations

In this section, we derive the basic equations that describe the connection between seismic fields and associated gravity perturbations. Expressions will first be derived in terms of the seismic displacement field \(\xi ({\mathbf {r}},t)\), then in terms of seismic potentials \(\phi _{\mathrm {s}}({\mathbf {r}},t),\,\psi _{\mathrm {s}}({\mathbf {r}},t)\), and this section concludes with a discussion of gravity perturbations in transform domain.

#### 3.2.1 Gravity perturbations from seismic displacement

#### 3.2.2 Gravity perturbations in terms of seismic potentials

*S*being the boundary of a body with volume

*V*). The seismic potentials vanish above surface, and therefore the gravity perturbation in empty space is the result of a surface integral. This is a very important conclusion and useful to theoretical investigations, but of limited practical relevance since the integral depends on the seismic potential \(\phi _{\mathrm {s}}({\mathbf {r}},t)\), which cannot be measured or inferred in general from measurements. The shear-wave potential enters as \(\nabla \times \psi _{\mathrm {s}}({\mathbf {r}},t)\), which is equal to the (observable) shear-wave displacement. In the absence of a surface, the solution simplifies to

#### 3.2.3 Gravity perturbations in transform domain

In certain situations, it is favorable to consider gravity perturbations in transform domain. For example, in calculations of gravity perturbations in a half space, it can be convenient to express solutions in terms of the displacement amplitudes \(\xi ({\mathbf {k}}_\varrho ,z,t)\), and in infinite space in terms of \(\xi ({\mathbf {k}},t)\). As shown in Sect. 4.4, it is also possible to obtain concise solutions for the half-space problem using cylindrical harmonics, but in the following, we only consider plane-wave harmonics.

*z*is straight-forward to calculate.

*t*is omitted. This equation is particularly useful since seismologists often define their fields in terms of seismic potentials, and it is then possible to directly write down the perturbation of the gravity potential in transform domain without solving any integrals.

### 3.3 Seismic gravity perturbations inside infinite, homogeneous media with spherical cavity

Test masses of underground detectors, as for example KAGRA (Aso et al. (KAGRA Collaboration) 2013), will be located inside large chambers hosting corner and end stations of the interferometer. Calculation of gravity perturbations based on a spherical chamber model can be carried out explicitly and provides at least some understanding of the problem. This case was first investigated by Harms et al. (2009a). In their work, contributions from normal displacement of cavity walls were taken into account, but scattering of incoming seismic waves from the cavity was neglected. In this section, we will outline the main results of their paper in Sect. 3.3.1, and present for the first time a calculation of gravity perturbations from seismic waves scattered from a spherical cavity in Sects. 3.3.2 and 3.3.3.

#### 3.3.1 Gravity perturbations without scattering

*a*can be solved. The gravity acceleration at the center \({\mathbf {r}}={\mathbf {0}}\) of the cavity is given by

If the cavity has a radius of about \(0.4\lambda \), then gravity perturbation is reduced by about a factor 2. Keeping in mind that the highest interesting frequency of Newtonian noise is about 30 Hz, and that compressional waves have a speed of about 4 km/s, the minimal cavity radius should be about 50 m to show a significant effect on gravity noise. Building such cavities would be a major and very expensive effort, and therefore, increasing cavity size does not seem to be a good option to mitigate underground Newtonian noise.

#### 3.3.2 Incident compressional wave

*a*is \((k_{\mathrm {P}}a)^3\) or higher order.

*z*-axis with the direction of propagation of the incoming wave, one obtains:

#### 3.3.3 Incident shear wave

*P*-wave to the \(l=1\) amplitude of the incident S-wave. It can be calculated using equations from Korneev and Johnson (1996) (note that the explicit solutions given in the appendix are wrong). Inserting this expression into the last equation, we finally obtain

*a*. In addition, the gravity perturbation from scattered waves is in phase with gravity perturbations of the incident wave (in the Rayleigh-scattering regime), which is beneficial for coherent noise cancellation, if necessary.

### 3.4 Gravity perturbations from seismic waves in a homogeneous half space

In this section, the gravity perturbation produced by plane seismic waves in a homogeneous half space will be calculated. The three types of waves that will be considered are compressional, shear, and Rayleigh waves. Reflection of seismic waves from the free surface will be taken into account. The purpose is to provide equations that can be used to estimate seismic Newtonian noise in GW detectors below and above surface. For underground detectors, corrections from the presence of a cavity will be neglected, but with the results of Sect. 3.3, it is straight-forward to calculate the effect of a cavity also for the half-space problem.

#### 3.4.1 Gravity perturbations from body waves

*h*being the height of the point \({\mathbf {r}}\) above surface, \(\varrho \) being the projection of \({\mathbf {r}}\) onto the surface, and \({\mathbf {k}}_\varrho \) being the horizontal wave vector (omitting superscript ’P’ to ease notation). The solution above surface can be understood as pure surface term characterized by an exponential suppression with increasing height. Also the phase term is solely a function of horizontal coordinates. These are typical characteristics for a surface gravity perturbation, and we will find similar results for gravity perturbations from Rayleigh waves. Below surface,

*h*reinterpreted as (positive valued) depth, the solution reads

*h*is the distance to the surface. These solutions can now be combined to calculate the gravity perturbation from an SV or P wave reflected from the surface. An incident compressional wave is partially converted into an SV wave and vice versa. Only waves with the same horizontal wave vector \({\mathbf {k}}_\varrho \) couple at reflection from a flat surface (Aki and Richards 2009). Therefore, the total gravity perturbation above surface in the case of an incident compressional wave can be written

#### 3.4.2 Gravity perturbations from Rayleigh waves

*h*. The density perturbations in the ground are calculated from the divergence of the Rayleigh-wave field:

*h*is the depth of the test mass. Contributions from a cavity wall need to be added, which is straight-forward at least for a very small cavity, by using results from Sect. 3.3.1 and amplitudes of shear and compressional waves dependent on depth as given in Eq. (39).

*A*by vertical surface displacement:

### 3.5 Numerical simulations

Numerical simulations have become an important tool in seismic Newtonian-noise modelling. There are two types of numerical simulations. The first will be called “kinematic” simulation. It is based on a finite-element model where each element is displaced according to an explicit, analytical expression of the seismic field. These can be easily obtained for individual seismic surface or body waves. The main work done by the kinematic simulation is to integrate gravity perturbations from a complex superposition of waves over the entire finite-element model according to Eq. (45). Today, we have explicit expressions for all types of seismic waves produced by all types of seismic sources, in infinite and half-space media. While this means in principle that many interesting kinematic simulations can be carried out, some effects are very hard to deal with. The kinematic simulation fails whenever it is impossible to provide analytical expressions for the seismic field. This is generally the case when heterogeneities of the ground play a role. Also deviations from a flat surface may make it impossible to run accurate kinematic simulations. In this case, a “dynamical” simulation needs to be employed.

A dynamical simulation only requires accurate analytical models of the seismic sources. The displacement field evolves from these sources governed by equations of motion that connect the displacement of neighboring finite elements. Even though the dynamical simulation can be considered more accurate since it does not rely on guessing solutions to the equations of motion, it is also true that not a single simulation of Newtonian noise has been carried out so far that could not have been done with a kinematic simulation. The reasons are that dynamical simulations are computationally very expensive, and constructing realistic models of the medium can be very challenging. It is clear though dynamical simulations will play an important role in future studies when effects from heterogeneities on gravity signals are investigated in detail.

*Matlab*, which facilitates analyzing sometimes complex results. Simulation results for a seismic field produced by a point force at the origin are shown in Fig. 11.

The results were presented in Beker et al. (2010). A snapshot of the displacement field is plotted on the left. The P-wavefront is relatively weak and has already passed half the distance to the boundaries of the grid. Only a spherical octant of the entire finite-element grid is shown. The true surface in this simulation is the upper face of the octant. Consequently, a strong Rayleigh-wave front is produced by the point force. Slightly faster than the Rayleigh waves, an S-wavefront spreads underground. Its maximum is close to the red marker located underground. This seismic field represents a well-known problem in seismology, the so-called Lamb’s problem, which has an explicit time-domain solution (Richards 1979). The plots on the right show the gravity perturbations evaluated at the two red markers. The P-wave, S-wave and Rf-wave arrival times are \(t_{\mathrm {P}},\,t_{\mathrm {S}}\) and \(t_{\mathrm {R}}\) respectively. The gravity perturbations are also divided into contributions from density perturbations inside the medium according to Eq. (46) and surface contributions according to Eq. (47).

A second simulation package used in the past is specfem3d. It is a free software that can be downloaded at http://www.geodynamics.org/cig/software/specfem3d. It is one of the standard simulation tools in seismology. It implements the spectral finite element method (Komatitsch and Vilotte 1998; Komatitsch and Tromp 1999). Recently, Eq. (45) has been implemented for gravity calculations (Harms et al. 2015). specfem3d simulations typically run on computer clusters, but it is also possible to execute simple examples on a modern desktop. Simulations of wave propagation in heterogeneous ground and based on realistic source models such as shear dislocations are probably easier to carry out with specfem3d than with commercial software. However, it should be noted that it is by no means trivial to run any type of simulation with specfem3d, and a large amount of work goes into defining a realistic model of the ground for specfem3d simulations.

### 3.6 Seismic Newtonian-noise estimates

The results of the analytical calculations can be used to estimate seismic Newtonian noise in GW detectors above surface and underground. The missing steps are to convert test-mass acceleration into gravity strain, and to understand the amplitudes of perturbation as random processes, which are described by spectral densities (see Sect. 5). For a precise noise estimate, one needs to measure the spectrum of the seismic field, its two-point spatial correlation or anisotropy. These properties have to be known within a volume of the medium under or around the test masses, whose size depends on the lengths of seismic waves within the relevant frequency range. Practically, since all these quantities are then used in combination with a Newtonian-noise model, one can apply simplifications to the model, which means that some of these quantities do not have to be known very accurately or do not have to be known at all. For example, it is possible to obtain good Newtonian-noise estimates based on the seismic spectrum alone. All of the published Newtonian-noise estimates have been obtained in this way, and only a few conference presentations showed results using additional information such as the anisotropy measurement or two-point spatial correlation. In the following, the calculation of Newtonian-noise spectra is outlined in detail.

#### 3.6.1 Using seismic spectra

*x*-axis. Note that all three components of acceleration are determined by vertical surface displacement. This is possible since vertical and horizontal displacements of Rayleigh waves are not independent. As we will argue in Sect. 7, expressing Newtonian noise in terms of vertical displacement is not only a convenient way to model Newtonian noise, but it is also recommended to design coherent cancellation schemes at the surface based on vertical sensor data, since horizontal sensor data can contain contributions from Love waves, which do not produce Newtonian noise. Hence, horizontal channels are expected to show lower coherence with Newtonian noise. Assuming that the Rayleigh-wave field is isotropic, one can simply average the last equation over all propagation directions. The noise spectral density of differential acceleration along a baseline of length

*L*parallel to the

*x*-axis reads

*L*. Introducing \(\lambda _{\mathrm {R}}\equiv 2\pi /k_\varrho \), the response functions, i.e., the square roots of the components of the vector in Eq. (101), divided by \(L/\lambda _{\mathrm {R}}\) are shown in Fig. 13.

*L*. In other words, increasing the length of large-scale GW detectors would decrease Newtonian noise. Rayleigh Newtonian noise is independent of

*L*for short separations. This corresponds to the regime relevant to low-frequency GW detectors (Harms et al. 2013). Equation (101) is the simplest possible seismic surface Newtonian-noise estimate. Spatial correlation of the isotropic seismic field is fully determined by the fact that all seismic waves are assumed to be Rayleigh waves. Practically one just needs to measure the spectral density of vertical surface displacement, and also an estimate of the Poisson’s ratio needs to be available [assuming a value of \(\nu =0.27\) should be a good approximation in general (Zandt and Ammon 1995)]. In GW detectors, the relevant noise component is along the

*x*-axis. Taking the square-root of the expression in Eq. (101), and using a measured spectrum of vertical seismic motion, we obtain the Newtonian-noise estimate shown in Fig. 14. Virgo’s arm length is \(L=3000\mathrm {\ m}\), and the test masses are suspended at a height of about \(h=1\mathrm {\ m}\) (although, it should be mentioned that the ground is partially hollow directly under the Virgo test masses). In order to take equal uncorrelated noise contributions from both arms into account, the single-arm strain noise needs to be multiplied by \(\sqrt{2}\). The seismic spectrum falls approximately with 1 /

*f*in units of \(\mathrm {m/s/\sqrt{Hz}}\) within the displayed frequency range, which according to Eq. (101) means that the Newtonian-noise spectrum falls with \(1/f^4\) (two additional divisions by

*f*from converting differential acceleration noise into differential displacement noise, and another division by

*f*from converting the seismic spectrum into a displacement spectrum). Note that the knee frequency of the response curve in Fig. 13 lies well below the frequency range of the spectral plots, and therefore does not influence the frequency dependence of the Newtonian-noise spectrum.

Generally, seismic spectra at the Virgo and LIGO sites show a higher grade of stationarity above 10 Hz than at lower frequencies. For example, between 1 Hz and 10 Hz, seismic spectra have pronounced diurnal variation from anthropogenic activity, and between 0.05 Hz and 1 Hz seismic spectra follow weather conditions at the oceans. These features are shown in Fig. 15. The white curves mark the 10th, 50th and 90th percentiles of the histogram. The histogram is based on a full year of 128 s spectra. Strong disturbances during the summer months from logging operations near the site increase the width of the histogram in the anthropogenic band. In general, a 90th percentile curve exceeding the global high-noise model is almost certainly a sign of anthropogenic disturbances. At lowest frequencies, strong spectra far above the 90th percentile are frequently being observed due to earthquakes. Additional examples of Newtonian-noise spectra evaluated in this way can be found in Driggers et al. (2012b) and Beker et al. (2012).

#### 3.6.2 Corrections from anisotropy measurements

Figure 16 shows the anisotropy measurement at 10 Hz and Newtonian-noise suppression of a single test mass obtained from anisotropy measurements over a range of frequencies. The seismic array was used to triangulate the source of dominant seismic waves over a period of a few hours. As shown in the left plot of Fig. 16, the waves at 10 Hz almost always come from a preferred direction approximately equal to \(100^\circ \). The same is true at almost all frequencies between 5 Hz and 30 Hz. Using the mean azimuth of waves within this range of frequencies, the Newtonian-noise suppression was calculated using Eq. (100) inserting the mean azimuth at each frequency as direction of propagation \(\phi \) of the Rayleigh waves. An azimuth of \(90^\circ \) corresponds to a direction perpendicular to the arm, which means that one expects Newtonian noise to be lower compared to the isotropic case. The suppression factor is plotted on the right of Fig. 16 with a typical value of about 2. If the situation is the same at the other end station at LIGO Hanford (which is a reasonable assumption, also for the Livingston site), and conservatively assuming that the field is isotropic in the central station, then Newtonian noise would be reduced overall by about a factor \(\sqrt{2}\).

#### 3.6.3 Corrections from two-point spatial correlation measurements

*h*, the kernel is to be substituted by the Delta-distribution \(\delta (\varrho )\). This means that for negligible test-mass height, the gravity perturbation from a homogeneous and isotropic field is determined by the seismic spectral density \(S(\xi _z;\omega )=C(\xi _z;0,\omega )\).

Eq. (109) also states that for a homogeneous, isotropic field, the values of \(\varrho \) that are most relevant to the Newtonian-noise estimate lie around \(\varrho =\sqrt{2}h\) where the kernel assumes its maximum. The kernel is plotted as solid curve in Fig. 17. For example, LIGO test masses are suspended 1.5 m above ground. Spatial correlation over distances much longer than 5 m are irrelevant to estimate Newtonian noise at the LIGO sites from homogeneous and isotropic fields. Consequently a seismic experiment designed to measure spatial correlations to improve Newtonian-noise estimates does not need to cover distances longer than this. Of course, in reality, fields are neither homogeneous nor isotropic, and seismic arrays should be designed conservatively so that all important features can be observed. The kernel of the integral transform in Eq. (104) is a function of the variables \(k_\varrho ,\,h\) with maximum at \(k_\varrho =1/(2h)\). It is displayed in Fig. 17 as dashed curve. The behavior of the two kernels with changing *h* is intuitive. The higher the test mass above ground, the larger the scales of the seismic field that dominate the gravity perturbation, which means larger values of \(\varrho \) and smaller values of \(k_\varrho \). Kernels in higher dimension can also be calculated for homogeneous seismic fields, and for the general case. The calculation is straightforward and will not be presented here.

In Fig. 18, wavenumber spectra measured at the LIGO Hanford site are shown for three different frequencies. The maxima in all three spectra correspond to Rayleigh waves (since the corresponding speeds are known to be Rayleigh-wave speeds). However, the 50 Hz spectrum contains a second mode with significant amplitude that lies much closer to the origin, which is therefore much faster than a Rayleigh wave. It can only be associated with a body wave. One can now split the integration of this map into two parts, one for the Rayleigh wave, and one for the body wave, using a different numerical factor in each case. This can work, but with the information that can be extracted from this spectrum alone, it is not possible to say what type of body wave it is. So one can either study particle motion with three-axes sensors to characterize the body wave further (which was not possible in this case since the array consisted of vertical sensors only), or instead of \(\gamma (\nu )<1\) one can use the conservative numerical factor equal to 1 to calculate at least the corresponding gravity perturbation from pure surface displacement. The latter method would neglect sub-surface density perturbations produced by a P-wave. It should be noted that one can obtain a model independent estimate of Newtonian noise with a 3D array. The numerical factor \(\gamma (\nu )\) came from a calculation of sub-surface gravity perturbations based on surface displacement. With information about the entire 3D displacement field, this step is not necessary and the noise estimate becomes model independent and does not require any other prior knowledge. An example of calculating Newtonian noise based on a 3D spatial correlation function is given in Sect. 5.2.

#### 3.6.4 Low-frequency Newtonian-noise estimates

There are qualitative differences between low- and high-frequency Newtonian noise that are worth being discussed more explicitly. First of all, we need to provide a definition of what should be considered low frequencies. There are two length scales relevant to Newtonian-noise estimates. The first is the size *L* of the GW detector. The second is the depth *h* of the detector. In this section, we will consider the scenario where both length scales are much shorter than the reduced length of seismic waves: \(h,L\ll \lambda /(2\pi )\). This should typically be the case below about 1 Hz.

*L*:

*L*along the

*x*-coordinate. The first important observation is that the strain noise is independent of the detector size. The common-mode rejection of the differential acceleration, which is proportional to \(L^2\) with respect to noise power, exactly compensates the \(1/L^2\) from the conversion into strain. This also means that Newtonian-noise is significantly weaker at low frequencies consistent with Fig. 13, which shows that gravity gradient response saturates below some test-mass distance.

Next, we discuss the role of detector depths. It should be emphasized that Eq. (113) is valid only above surface. As we have seen in Eq. (87), density changes below surface give rise to additional contributions if the test mass is located underground. We have not explicitly calculated these contributions for Rayleigh waves in this article. The point that we want to make though is that if the length of the Rayleigh wave is much longer than the depth of the detector, then the surface model in Eq. (113) is sufficiently accurate. It can be used with the parameter *h* set to 0. This is not only true for Newtonian noise from Rayleigh waves, but for all forms of seismic Newtonian noise. It should be noted though that these conclusions are not generally true in the context of coherent Newtonian-noise cancellation. If a factor 1000 noise reduction is required (as predicted for low-frequency GW detectors, see Harms et al. 2013), then much more detail has to be included into the noise models, to be able to predict cancellation performance. Here, not only the depth of the detector could matter, but also the finite thickness of the crust, the curvature of Earth, etc.

Estimates of seismic Newtonian noise at low frequencies were presented with focus on atom-interferometric GW detectors in Vetrano and Viceré (2013). The interesting aspect here is that atom interferometers in general have a more complicated response to gravity perturbations. A list of gravity couplings for atom interferometers can be found in Dimopoulos et al. (2008). So while atom-interferometric GW detectors would also be sensitive to gravity strain only, the response function may be more complicated compared to laser interferometers depending on the detector design.

### 3.7 Summary and open problems

In this section on Newtonian noise from ambient seismic fields, we reviewed basic analytical equations to calculate density perturbations in materials due to vibrations, to calculate the associated gravity perturbations, and to estimate Newtonian noise based on observations of the seismic field. Equations were given for gravity perturbations of seismic body waves in infinite and half spaces, and for Rayleigh waves propagating on a free surface. Newtonian noise above a half space can be fully characterized by surface displacement, even for body waves. It was found that analytical expressions for gravity perturbations from body and Rayleigh waves have the same form, just the numerical, material dependent conversion factor between seismic and gravity amplitudes has different values also depending on the propagation direction of a body wave with respect to the surface normal. In practice, this means that prior information such as seismic speeds of body waves is required to calculate gravity perturbations based on surface data alone. Another important difference between body and Rayleigh gravity perturbations is that the conversion factor has a material and propagation-direction dependent complex phase in the body-wave case. This has consequences on the design of a seismic surface array that one would use to coherently cancel the gravity perturbations, which will be discussed further in Sect. 7.

Scattering of body waves from spherical cavities was calculated concluding that gravity perturbations on a test mass inside a cavity are insignificantly affected by seismic scattering from the cavity. Here, “insignificantly” is meant with respect to Newtonian-noise estimates. In coherent noise cancellation schemes, scattering could be significant if the subtraction goals are sufficiently high. An open problem is to understand the impact of seismic scattering on gravity perturbations in heterogeneous materials where scattering sources are continuously distributed. This problem was studied with respect to its influence on the seismic field (Norris 1986; Liu et al. 2009), but the effect on gravity perturbations has not been investigated yet.

We also showed that the calculation of simple Newtonian-noise estimates can be based on seismic spectra alone, provided that one has confidence in prior information (e.g., that Rayleigh waves dominate seismic noise). In general, seismic arrays help to increase confidence in Newtonian-noise estimates. It was shown that either simple anisotropy measurements or measurements of 2D wavenumber spectra can be used to improve Newtonian-noise estimates. In this section, we did not discuss in detail the problem of estimating wavenumber spectra. Simply carrying out the Fourier transform in Eq. (106) is prone to aliasing. A review on this problem is given in Krim and Viberg (1996). Estimation of wavenumber spectra has also become an active field of research in GW groups, using data from the LIGO Hanford array deployed between April 2012 and February 2013, and the surface and underground arrays at the Sanford Underground Research Facility, which are currently being deployed with data to be expected in 2015. The problem of Newtonian-noise estimation using seismic arrays needs to be separated though from the problem of Newtonian-noise cancellation. The latter is based on Wiener filtering. From an information theory perspective, the Wiener filtering process is easier to understand than the noise estimation since Wiener filters are known to extract information from reference channels in an optimal way for the purpose of noise cancellation (under certain assumptions). There is no easy way to define a cost function for spectral estimation, which makes the optimal estimation of wavenumber spectra rather a philosophical problem than a mathematical or physical one. The optimal choice of analysis methods depends on which features of the seismic field are meant to be represented most accurately in a wavenumber spectrum. For example, some methods are based on the assumption that all measurement noise is stationary and effectively interpreted as isotropic seismic background. This does not have to be the case if the seismic field itself acts as a noise background for measurements of dominant features of the field. Nonetheless, designs of seismic arrays used for noise cancellation need to be based on information about wavenumber spectra. Initially, array data are certainly the only reliable sources of information, but also with Newtonian-noise observations, optimization of noise-mitigation schemes will be strongly guided by our understanding of the seismic field.

## 4 Gravity perturbations from seismic point sources

In Sect. 3, we have reviewed our understanding of how seismic fields produce gravity perturbations. We did, however, not pay attention to sources of the seismic field. In this section, gravity perturbations will be calculated based on models of seismic sources, instead of the seismic field itself. This can serve two purposes. First, a seismic source can be easier to characterize than the seismic field itself, since characterization of a seismic field requires many seismometers in general deployed in a 3D array configuration. Second, in the context of geophysical observations, source models are essential to infer source parameters. For example, it was suggested to promptly detect and characterize fault ruptures leading to earthquakes using gravimeters and gravity gradiometers (Chiba and Obata 1990; Harms et al. 2015; Juhel et al. 2018a). In this case, the analysis of gravity data from high-precision gravity gradiometers can be understood as a new development in the field of terrestrial gravimetry. First observations of prompt gravity perturbations from earthquakes have recently been achieved with gravimeters (Montagner et al. 2016; Vallée et al. 2017). Such observations require clever filtering of the data since otherwise seismic perturbations from oceanic microseisms mask the underlying gravity signal. The signals were predicted based on a theory of co-seismic gravity perturbations from fault rupture in infinite homogeneous media and homogeneous half spaces (Harms et al. 2015; Harms 2016), and longer observation times of at least a few tens of seconds were required. A new generation of high-precision gravity gradiometers will make it possible to detect these signals within a few seconds (Juhel et al. 2018a). The basics of modeling gravity fluctuations from seismic point sources are provided in this chapter. The same formalism can be applied to point sources of sound waves as shown in Sect. 5.3.

### 4.1 Gravity perturbations from a point force

*F*(

*t*). In this section, we use it to present another interesting result. It has often been conjectured that a transient source of seismic vibrations would be a problem to coherent mitigation schemes since the gravity perturbation starts to be significant before any of the seismometers can sense the first ground motion produced by this source. Therefore, it would be impossible to coherently remove a significant contribution to Newtonian noise using seismic data. Some evidence speaking against this conjecture was already found in numerical simulations of approaching wavefronts from earthquakes (Harms et al. 2009a), but there was no analytical explanation of the results. We can make up for this now. Let us make the following Gedankenexperiment. Let us assume that all seismic noise is produced by a single source. Let us assume that this source is switched on at time \(t=0\). Before this time, the entire seismic field is zero. Now the source starts to irradiate seismic waves. The waves do not reach the test mass before \(t=r_0/\alpha \), where \(r_0\) is the distance between the source and the test mass. The situation is illustrated in Fig. 19. The dashed line marks the arrival of seismic waves. From that time on, we have the usual Newtonian noise from ambient seismic fields. Interesting however is what happens before arrival. The Newtonian noise is hardly visible. Therefore, an inset plot was added to show gravity perturbations before wave arrival. Not only is the rms of the gravity perturbation much lower, but as expected, it evolves much slower than the Newtonian noise from ambient seismic fields. Equation (117) says that the source function is filtered by a double integrator to obtain the gravity acceleration. Another double integrator needs to be applied to convert gravity acceleration into test-mass displacement. Therefore, whatever the source function is, and the corresponding source spectrum \(F(\omega )\), gravity perturbations will be strongly suppressed at high frequencies. Due to the transient character of this effect, it is difficult to characterize the problem in terms of Newtonian-noise spectra, but it should be clear that a seismic source would have to be very peculiar (i.e., radiating very strongly at high frequencies and weakly at low frequencies), to cause a problem to coherent Newtonian-noise cancellation, without causing other problems to the detector such as a loss of cavity lock due to low-frequency ground disturbances.

### 4.2 Density perturbation from a point shear dislocation in infinite homogeneous media

*z*-axis being parallel to the slip direction, and the

*x*-axis perpendicular to the fault plane. Spherical coordinates \(r,\theta ,\phi \) will be used in the following that are related to the Cartesian coordinates via \(x=r\sin (\theta )\cos (\phi )\), \(y=r\sin (\theta )\sin (\phi )\), \(z=r\cos (\theta )\), with \(0<\theta <\pi \), and \(0<\phi <2\pi \). The double couple drives a displacement field that obeys conservation of linear and angular momenta. Its explicit form is given in Aki and Richards (2009). It consists of a near-field component:

### 4.3 Gravity perturbations from a point shear dislocation

Fault slip generates elastodynamic deformation (static and transient), including compression and dilation that induce local perturbations of the material density. These in turn lead to global perturbations of the gravity field. In this section, we consider an elementary problem: we develop an analytical model of time-dependent gravity perturbations generated by a point-shear dislocation in an infinite, elastic, and homogeneous medium. We are interested in frequencies higher than 0.01 Hz, for which we can ignore the effects of self-gravitation (Dahlen et al. 1998): we compute the gravity changes induced by mass redistribution caused by elastic deformation, but ignore the effect of gravity force fluctuations on the deformation. The results in this subsection were published in Harms et al. (2015). The gravity perturbation can either be obtained analogously to the case of a point force by seeking for a known solution of the P-wave potential and rewriting it as gravity potential, or by attempting a direct integration of density perturbations. First, we will show how to carry out the direct integration.

*r*into two intervals: \(0<r<r_0\) and \(r_0<r\). Over the first interval, one obtains the exterior multipole expansion:

The model of Eq. (127) can also be used to predict gravity perturbations for sources buried in half spaces at least until the seismic waves reach the surface. In reality, this typically allows us to model up to a few seconds of time series of an earthquake, but it was shown in Harms et al. (2015) using numerical simulations that the duration of the modeled gravity perturbation can be extended for some time without causing major deviations from the half-space signal. Clearly, a half-space model is favored to include surface effects.

### 4.4 Seismic sources in a homogeneous half space

It turns out that the calculation of seismic fields from point sources in a homogeneous half-space is signifciantly more complicated. This is to be expected of course since a seismic source produces compressional, shear and Rayleigh waves emerging in the far field from a rather complex near field depending on the depth of the source and how it is oriented with respect to the plane surface. It was shown that the Green’s function can be cast into a single integral using the so-called Cagniard–de Hoop method, and for arbitrary source-time functions, another convolutional integral is required (de Hoop 1962; Richards 1979; Aki and Richards 2009; Kausel 2012). The most explicit derivation of these results can be found in Watanabe (2014), where the author outlines in detail the analysis carried out in the complex plane dealing with poles and branch cuts. It is this paper that provided the starting point for a calculation of gravity perturbations from seismic point sources in a half-space (Harms 2016). In the following, we will review the most important results of this paper.

The next step is to solve the elastodynamic equation for the Green’s function of seismic potentials. This can be easily done in transform domain, which includes a Laplace transform with respect to time, and Fourier transforms with respect to the spatial coordinates. Next, the solutions for the seismic potentials obtained in this way require an inverse Fourier transform with respect to the wave-vector component \(k_z\) so that potentials depend on \({\mathbf {k}}_\varrho \) and *z* as in Eq. (128). Not only can we use Eq. (128) in this way to calculate the gravity potential, but expressing fields in terms of these variables is also an essential step for the Cagniard–de Hoop method. The remaining problem is to calculate the other three inverse transforms: two inverse Fourier transforms to obtain a dependence of fields on horizontal coordinates, and one inverse Laplace transform to obtain a dependence on time. The elegance of the Cagniard–de Hoop method is then to recast this triple integral of inverse transforms into a single integral. This remaining integral cannot be solved in general, and as mentioned before, the result is a Green’s function, which means that an additional convolution is required for arbitrary source-time functions.

It needs to be pointed out that the calculation in Harms (2016) neglects gravity induced ground motion as described in Sect. 2.1.3. This means that the calculation is not sufficient to predict a gravimeter signal. Consequently, the calculation also neglects second and higher-order effects in the gravitational constant *G*, e.g., changes in the gravity field due to the induced ground motion, which is a so-called *self-gravity effect* (Rundle 1980; Wang 2005).

^{7}For the actual calculation, one needs to translate the coordinates used for the published moment tensors into the fault-slip oriented coordinate system used for example in Harms et al. (2015) and Harms (2016).

*x*-axis, and the slip vector defines the direction of the

*z*-axis. Now, vectors are to be expressed in a new coordinate system whose axes correspond to the cardinal directions \({\mathbf {e}}_{\mathrm {E}},\,{\mathbf {e}}_{\mathrm {N}}\), and the normal vector of Earth’s surface \({\mathbf {e}}_{\mathrm {V}}\). Based on the geometry shown in Fig. 22, the following relations can be found

Figure 23 shows the result for the perturbation of gravity acceleration in vertical direction at 100 km distance to the epicenter. The curve uses the estimated source function of the Tohoku-Oki earthquake (Wei et al. 2012),^{8} which had a total rupture duration of about 300 s, with almost all of the total seismic moment, \(5\times 10^{22}\mathrm {\ Nm}\), already released after 120 s. After 16 s, the first seismic waves reach an epicentral distance of 100 km. After all waves have passed, the gravity potential settles to a new value offset by about \(6\,\mathrm \mu m/s^2\) from the previous value. It is this lasting change in gravity that was first observed by networks of gravimeters (Imanishi et al. 2004) and the satellite mission GRACE (Wahr et al. 2004; Cafaro and Ali 2009; Wang et al. 2012). In the meantime, also the prompt gravity signal during fault rupture of the Tohoku-Oki earthquake was observed with several gravimeters (Montagner et al. 2016; Vallée et al. 2017). These observations match well the predictions from numerical codes, which also include the gravity-induced ground motion (Juhel et al. 2018b).

### 4.5 Summary and open problems

We have reviewed the calculation of gravity perturbations based on models of seismic sources. The general expressions for these perturbations can be complicated, but especially when neglecting surface effects, the gravity perturbations assume a very simple form due to a fundamental equivalence between seismic and gravity potentials according to Eq. (55).

The solution of the point force was used to highlight the difference between locally generated gravity perturbations, i.e., at the test mass, and perturbations from an incident seismic wavefront. It was shown that such gravity perturbations experience additional integrations with respect to perturbations from the local seismic field at the test mass. Gravity perturbations from distant seismic wavefronts are more likely to play a role in sub-Hz GW detectors, and also there the seismic event producing the wavefront needs to be very strong. As an example, we have presented the formalism to estimate perturbations from earthquakes in Sects. 4.2 to 4.4.

These results also have important implications for coherent Newtonian-noise cancellation schemes. It was argued in the past that seismic sensors deployed around the test mass can never provide information of gravity perturbations from incident seismic disturbances that have not yet reached the seismic array. Therefore, there would be a class of gravity perturbations that cannot be subtracted with seismic sensors. While the statement is generally correct, we now understand that the gravity perturbations are significant only well below the GW detection band (of any > 1 Hz GW detector), unless the source of the seismic wavefront has untypically strong high-frequency content.

The theory of gravity perturbations from seismic point sources has just begun to be explored. In Sect. 4.4, a first calculation of gravity perturbations from point sources in half spaces was reviewed. Self-gravity effects, i.e., effects at second and higher order in *G*, are still poorly understood. The approach of Sect. 2.1.3 might provide further insight, but it has not been combined yet with the calculation of gravity perturbations from point sources.

## 5 Atmospheric gravity perturbations

The properties of the atmosphere give rise to many possible mechanisms to produce gravity perturbations. Generally, one can associate gravity fluctuations with fields of pressure, temperature, or humidity. It is often useful to categorize atmospheric processes according to a characteristic length scale describing the phenomena: global scale, synoptic scale (several 100 km to several 1000 km), mesoscale (several 1 km to several 100 km), and the microscale (up to several 1 km). Mostly the microscale phenomena are relevant to Newtonian noise modeling in GW detectors. Mesoscale phenomena might become relevant at frequencies below a few 10 mHz, i.e., below the Brunt–Väisälä frequency. One source of gravity perturbations are microscale pressure fluctuations in the planetary or atmospheric boundary layer (Elliott 1972b; McDonald and Herrin 1975), which is the lowest part of the atmosphere directly influenced by the surface. They can be divided into fluctuations of static and dynamic pressure. Fluctuations of static pressure, which include sound, are present even in the absence of wind (Albertson et al. 1998), while fluctuations of dynamic pressure are connected to anything that requires wind (Bollaert et al. 2004; Schaffarczyk and Jeromin 2018). There is also a connection between the two, for example, through the Lighthill process, which describes the generation of sound waves by turbulence (Lighthill 1952, 1954). Additional structures emerge with the presence of wind such as vortices or so-called coherent structures (Träumner et al. 2015).

The variety of phenomena also makes the monitoring of certain variables a challenging task. Sound is typically measured with microphones. However, pressure fluctuations produced by turbulent flow in the vicinity of a microphone can mask an underlying sound signal (Green 2015). This contribution is often called wind noise. Clever sensor design, averaging pressure signals over some baseline, or constructing wind shields can prove effective (Elliott 1972a; Walker and Hedlin 2009; Noble et al. 2014). However, when it comes to an order of magnitude suppression of gravity perturbations from pressure fluctuations, then even a small incoherent contribution to signals from wind noise can be detrimental. In this case, entirely new approaches need to be considered. LIDAR (derived from *light* and *radar*) technology has been applied to investigate microscale physics in the atmospheric boundary layer. It consists of a laser beam that scatters back from the atmosphere and can be operated in a scanning fashion providing volumetric information about the temporal evolution of wind velocity (Chai et al. 2004), or the temperature (Behrendt 2005; Hammann et al. 2015) and humidity fields (Späth et al. 2016). LIDAR is a promising candidate to also monitor pressure fluctuations in the frequency band relevant to GW detectors, 10 mHz–20 Hz, but it does not have the required sensitivity yet.

Atmospheric gravity perturbations have been known since long to produce noise in gravimeter data (Neumeyer 2010), where they can be observed below about 1 mHz. Creighton published the first detailed analysis of atmospheric Newtonian noise in large-scale GW detectors (Creighton 2008). It includes noise models for infrasound waves, quasi-static temperature fields advected in various modes past test masses, and shockwaves. Atmospheric gravity perturbations are expected to be the dominant contribution to ambient Newtonian noise below 1 Hz, and can still be significant at higher frequencies (Fiorucci et al. 2018). Generally, the Navier–Stokes equations need to be used to calculate density perturbations and associated gravity fluctuations (Davidson 2004). The models of gravity perturbations from sound fields are very similar to perturbations from seismic compressional waves as given in Sect. 3. Quasi-static density perturbations associated with non-uniform temperature or humidity fields can be transported past a gravity sensor and cause gravity fluctuations. One goal of Newtonian-noise modeling is to provide a strategy for noise mitigation. For this reason, it is important to understand the dependence of each noise contribution on distance between source and test mass, and also to calculate correlation functions. The former determines the efficiency of passive isolation schemes, such as constructing detectors underground, the latter determines the efficiency of coherent cancellation using sensor arrays.

Analytic models of atmospheric Newtonian noise are summarized in Sects. 5.1 and 5.2. In addition, based on the point-source formalism of Sect. 4, a new shockwave model is presented in Sect. 5.3. Preliminary work on modeling gravity perturbations from turbulence was first published in Cafaro and Ali (2009), and is reviewed and improved in Sect. 5.4.

### 5.1 Gravity perturbation from atmospheric sound waves

^{9}

### 5.2 Gravity perturbations from quasi-static atmospheric temperature perturbations

*temperature structure function*\(D(\delta T;r)\):

*v*with time \(\tau \), and Eq. (137) can be reformulated as

*l*of the smallest turbulence structures, \(\omega <v/l\), which is of the order kHz). Technically, the Fourier transform can be calculated by multiplying an exponential term \(\exp (-\epsilon \tau )\) to the integrand, and subsequently taking the limit \(\epsilon \rightarrow 0\).

*S*and \(S'\).

*s*of the two streamlines \(S,\,S'\) and

*v*are evaluated at \({\mathbf {r}}\). Together with Taylor’s hypothesis, temperature fluctuations between the two pockets are significant if \(\tau \) is sufficiently close to the time \(\tau _0\) it takes for the pocket at \({\mathbf {r}}\,'\) to reach the reference plane, and also

*s*must be sufficiently small. The temperature correlation can then be written as

*v*/

*s*, the spectrum falls exponentially since \(K_\nu (x)\rightarrow \sqrt{\pi /(2x)}\exp (-x)\) for \(x\gg |\nu ^2-1/4|\). This means that the distance between streamlines contributing to the two-point spatial correlation must be very small to push the exponential suppression above the detection band. The integral over \(V'\) in Eq. (136) can be turned into an integral over streamlines \(S'\) that lie within a bundle \(s\lesssim v/\omega \) of streamline

*S*, which allows us to approximate the volume element as cylindrical bundle \({\mathrm {d}}V'=2\pi s {\mathrm {d}}s\, {\mathrm {d}}\tau _0v({\mathbf {r}}\,)\). The form of the volume element is retained over the whole extent of the streamline since the air is nearly incompressible for all conceivable wind speeds, i.e., changes in the speed of the cylindrical pocket are compensated by changes in the radius of the pocket to leave the volume constant. Hence, the speed in the volume element can be evaluated at \({\mathbf {r}}\). With this notation, the integral can be carried out over \(0<s<\infty \) since the modified Bessel function automatically enforces the long-distance cutoff necessary for our approximations, which yields

*S*, \(S'\), whose contributions to this integral are evaluated in terms of the duration \(\tau _0\) it takes for the pocket at \({\mathbf {r}}\,'\) to reach the reference plane that goes through all streamlines, and contains the test mass at \({\mathbf {r}}_0\) and location \({\mathbf {r}}\) (as indicated in Fig. 25). Since we consider the pocket on streamline

*S*to be at the reference plane at time

*t*, we can set \(\tau _0=t-t'\), and integrating contributions from all streamlines over the reference plane with area element \({\mathrm {d}}A\), with wind speed \(v(\varrho \,)\), and \(\varrho \) pointing from the test mass to streamlines on the reference plane, we can finally write

### 5.3 Gravity perturbations from shock waves

*V*. In the theory of moment tensor sources, an explosion in air at \(t_0=0\) can be represented by a diagonal moment tensor according to

*A*being the cross-sectional area of the “source tube” around the aircraft trajectory, and

*v*the speed of the aircraft). In the case of uniform motion of the aircraft, the calculation of the integral over the trajectory is straight-forward.

### 5.4 Gravity perturbations in turbulent flow

*r*. The integrands are products of three spherical Bessel functions. An analytic solution for this type of integral was presented in Mehrem et al. (1991) where we find that the integral is non-zero only if the three wavenumbers fulfill the triangular relation \(|k'-k''|\le k\le k'+k''\) (i.e., the sum of the three corresponding wave vectors needs to vanish), and the orders of the spherical Bessel functions must fulfill \(|n'-n''|\le n\le n'+n''\). Especially the last relation is useful since many products can be recognized by eye to have zero value. In each case, the result of the integration is a rational function of the three wavenumbers if the triangular condition is fulfilled, and zero otherwise. While it may be possible to solve the integral analytically, we will stop the calculation at this point. Numerical integration as suggested in Cafaro and Ali (2009) is a valuable option. The square-roots of the noise spectra normalized to units of GW amplitude, \(2S(\delta {\mathbf {a}};{\mathbf {k}},\omega )/(L\omega ^2)^2\), are shown in Fig. 28 for \(k = 0.1\mathrm {\ m^{-1}}\), \(k = 0.67\mathrm {\ m^{-1}}\), \(k = 1.58\mathrm {\ m^{-1}}\), \(k = 3.0\mathrm {\ m^{-1}}\), where \(L=3000\mathrm {\ m}\) is the length of an interferometer arm.

Each spectrum is exponentially suppressed above the corner frequency \(1/\tau _0(k)\) with \(\tau _0= {\mathrm {3.5\,s,\,0.52\,s,\,0.22\,s,\,0.12\,s}}\). Below the corner frequency, the spectrum is proportional to \(1/\omega ^2\). In order to calculate the dissipation rate \(\epsilon \), a measured spectrum was used (Albertson et al. 1997), which has a value of about \(1{\mathrm {\ m^3\ s^{-2}}}\) at \(k=1{\mathrm { m^{-1}}}\), and wavenumber dependence approximately equal to the Kolmogorov spectrum. In this way, we avoid the implicit relation of the dissipation rate in Eq. (157), since \(\epsilon \) also determines the Kolmogorov energy spectrum. Solving the implicit relation for \(\epsilon \) gave poor numerical results, and also required us to extend the energy spectrum (valid in the inertial regime) to higher wavenumbers (the viscous regime). It is also worth noting that the energy spectrum and the scale \(\mathscr {R}\) (we used a value of 150 m) are the only required model inputs related to properties of turbulence. Any other turbulence parameter in this calculation can be calculated from these two (and a few standard parameters such as air viscosity, air pressure, \(\ldots \)). The resulting spectra show that Newtonian noise from the Lighthill process is negligible above 5 Hz, but it can be a potential source of noise in low-frequency detectors. In the future, it should be studied how strongly the Lighthill gravity perturbation is suppressed when the detector is built underground.

### 5.5 Atmospheric Newtonian-noise estimates

In the following, we present the strain-noise forms of gravity perturbations from infrasound fields and uniformly advected temperature fluctuations. While the results of the previous sections allow us in principle to estimate noise at the surface as well as underground, we will only present analytic expressions for surface-noise spectra here. An analysis of infrasound Newtonian noise in underground detectors was presented by Fiorucci et al. (2018), which uses a numerical method to average Eq. (132) over propagation directions of sound waves. Gravity perturbations associated with advection of temperature and humidity fields are still poorly understood, and predictions of how it is suppressed with increasing detector depth would be highly speculative.

*L*along \({\mathbf {e}}_x\) reads

*L*between the two test masses in units of sound wavelength \(\lambda _{\mathrm {IS}}=2\pi /k\). For short distances between the test masses, the response is independent of

*L*, and at large distances, the response falls with 1 /

*L*. The long-distance response follows from the fact that gravity noise is uncorrelated between the two test masses, which means that the spectral densities of displacement noise of the two test masses can be added and subsequently divided by their distance, \((S_1(x;\omega )+S_2(x;\omega ))/L\), to yield the strain noise. Instead, the response for small

*L*corresponds to the regime where the two test masses experience two very similar gravity accelerations, which turn into a gravity-gradient signal when subtracted. The gravity gradient across two close test masses does not depend significantly on their distance

*L*.

Figure 30 shows infrasound Newtonian noise spectra for low-frequency detector concepts (left) and for the proposed third-generation GW detector Einstein Telescope (right). For the low-frequency part, it can be seen that infrasound Newtonian noise is only weakly suppressed with increasing detector depth, since fluctuations of static pressure maintain high coherence over long distances in this band. Consequently, to reach the final sensitivity target below \(10^{-19}\,\mathrm Hz^{-1/2}\) at 0.1 Hz, several orders of magnitude reduction of sound Newtonian noise are still required. In comparison, infrasound Newtonian noise will play a minor role for the Einstein Telescope if constructed at least 100 m underground. There is a smaller, but significant contribution from sound fields inside the several tens of meters large cavities that host the vortex stations of the detector.

The slope of infrasound Newtonian noise is steeper than of seismic Newtonian noise (see Fig. 35), which can be taken as an indication that there may be a frequency below which atmospheric Newtonian noise dominates over seismic Newtonian noise. This has in fact been predicted in Harms et al. (2013). Using measured spectra of atmospheric pressure fluctuations and seismic noise, the intersection between seismic and infrasound Newtonian noise happens at about 1 Hz for a test mass at the surface. We also know (see Sect. 7.1.5) that Newtonian noise from atmospheric pressure fluctuations is the dominant ambient noise background around 1 mHz. One might be tempted to conclude that gravity perturbations from advected temperature fields may be an even stronger contribution at low frequencies. However, one has to be careful since models presented in this chapter cannot easily be extended to mHz frequencies, and temperature structure functions as in Eq. (137) can take different forms. The quasi-static approximation of the temperature field will fail at sufficiently low frequencies, and the temperature field cannot be characterized anymore as a result of turbulent mixing (Kukharets and Nalbandyan 2006).

### 5.6 Summary and open problems

In this section, we reviewed models of atmospheric gravity perturbations that are either associated with infrasound waves, or with quasi-stationary temperature fields advected by wind. For surface detectors, atmospheric Newtonian noise starts to be significant below 10 Hz according to these models, but this is only true for sound spectra representative of remote places. Sound levels in laboratory buildings can exceed sound levels at remote locations by more than an order of magnitude even without heavy machinery running, e.g., due to ventilation systems or pumps (Fiorucci et al. 2018). In this case, sound Newtonian noise can become significant above 10 Hz already, and become a potentially limiting noise contribution in advanced detectors.

According to Eqs. (161) and (162), and comparing with seismic Newtonian noise (see Fig. 35), we see that atmospheric spectra are steeper and therefore potentially the dominating gravity perturbation in low-frequency detectors. However, both models are based on approximations that may not hold at frequencies below a few Hz. A summary of approximations applied to the infrasound Newtonian noise model can be found in Harms et al. (2013). Also, the noise model of advected temperature fluctuations is likely inaccurate at sub-Hz frequencies since it is based on the assumption that the temperature field is quasi-stationary, is produced by turbulent mixing, and transported by uniform air flow.

As we have seen, few time-varying atmospheric noise models have been developed so far, which leaves plenty of space for future work in this field. For example, convection may produce atmospheric gravity perturbations, and only very simple models of gravity perturbations from turbulence have been calculated so far. While these yet poorly modeled forms of atmospheric noise are likely insignificant in GW detectors sensitive above 10 Hz, they may become important in low-frequency detectors. Improving these models will also answer to the important question how strongly atmospheric Newtonian noise is suppressed with depth. Finally, the question should be addressed whether atmospheric disturbances transmitted in the form of seismic waves into the ground can be neglected in Newtonian-noise models. As we outlined briefly in Sect. 5.1, even though transmission coefficients of sound waves into the ground are negligible with respect to their effect on seismic and infrasound fields, it seems that they may be relevant with respect to their effect on the gravity field.

## 6 Gravity perturbations from objects

In the previous sections, Newtonian-noise models were developed for density perturbations described by fields in infinite or half-infinite media. The equations of motion that govern the propagation of disturbances play an important role since they determine the spatial correlation functions of the density field. In addition, gravity perturbations can also be produced by objects of finite size, which is the focus of this section. Typically, the objects can be approximated as sufficiently small, so that excitation of internal modes do not play a role in calculations of gravity perturbations. The formalism that is presented can in principle also be used to calculate gravity perturbations from objects that experience deformations, but this scenario is not considered here. In the case of deformations, it is advisable to make use of a numerical simulation. For example, to calculate gravity perturbations from vibrations of vacuum chambers that surround the test masses in GW detectors, Pepper used a numerical simulation of chamber deformations (Pepper 2007). A first analytical study of gravity perturbations from objects was performed by Thorne and Winstein who investigated disturbances of anthropogenic origin (Thorne and Winstein 1999). The paper of Creighton has a section on gravity perturbations from moving tumbleweeds, which was considered potentially relevant to the LIGO Hanford detector (Creighton 2008). Interesting results were also presented by Lockerbie (2012), who investigated corrections to gravity perturbations related to the fact that the test masses are cylindrical and not, as typically approximated, point masses.

Section 6.1 presents rules of thumb that make it possible to estimate the relevance of perturbations from an object “by eye” before carrying out any calculation. Sections 6.2 and 6.3 review well-known results on gravity perturbations from objects in uniform motion, and oscillating objects. A generic analytical method to calculate gravity perturbations from oscillating and rotating objects based on multipole expansions is presented in Sects. 6.5 and 6.6.

### 6.1 Rules of thumb for gravity perturbations

*r*to the test mass that oscillates with amplitude \(\xi (t)\ll r\). We can use the dipole form in Eq. (45) to calculate the gravity perturbation at \({\mathbf {r}}_0={\mathbf {0}}\):

If the distance *r* is decreased to its minimum when the test mass and the perturbing mass almost touch, then the factor \(\delta V/r^3\) is of order unity. It is an interesting question if there exist geometries of disturbing mass and test mass that minimize or maximize the gravitational coupling of small oscillations. An example of a minimization problem that was first studied by Lockerbie (2012) is presented in Sect. 6.4. The maximization of gravitational coupling by varying object and test-mass geometries could be interesting in some experiments. Maybe it is possible to base a general theorem on the multipole formalism for small oscillations introduced in Sect. 6.5.

One mechanism that could potentially boost gravity perturbations from objects are internal resonances. It is conceivable that vibration amplitudes are amplified by factors up to a few hundred on resonance, and therefore it is important to investigate carefully the infrastructure close to the test mass. There is ongoing work on this for the Virgo detector where handles attached to the ground are located within half a meter to the test masses. While the rule of thumb advocated in this section rules out any significant perturbation from the handles, handle resonances may boost the gravity perturbations to a relevant level. Finally, we want to emphasize that the rule of thumb only applies to perturbative motion of objects. An object that changes location, or rotating objects do not fall under this category.

### 6.2 Objects moving with constant speed

Objects moving at constant speed produce gravity perturbations through changes in distance from a test mass. It is straight-forward to write down the gravitational attraction between test mass and object as a function of time. The interesting question is rather what the perturbation is as a function of frequency. While gravity fluctuations from random seismic or infrasound fields are characterized by their spectral densities, gravity changes from moving objects need to be expressed in terms of their Fourier amplitudes, which are calculated in this section. Since the results should also be applicable to low-frequency detectors where the test masses can be relatively close to each other, the final result will be presented as strain amplitudes.

We consider the case of an object of mass *m* that moves at constant speed *v* along a straight line that has distance \(r_1,\,r_2\) to two test masses of an arm at closest approach. The vectors \({\mathbf {r}}_1,\,{\mathbf {r}}_2\) pointing from the test mass to the points of closest approach are perpendicular to the velocity \({\mathbf {v}}\). The closest approach to the first test mass occurs at time \(t_1\), and at \(t_2\) to the second test mass.

*L*is the distance between the test masses, and \({\mathbf {e}}_{12}\) is the unit vector pointing from test mass 1 to test mass 2, and so \(t_2=t_1+L({\mathbf {e}}_{12}\cdot {\mathbf {v}})/v^2\). Another substitution that can be made is

*D*, and a car is driving directly above the test masses with \({\mathbf {v}}\) parallel \({\mathbf {e}}_{12}\) and perpendicular to \({\mathbf {r}}_1\). Therefore, \({\mathbf {r}}_1={\mathbf {r}}_2\), and \(t_2-t_1=L/v\). The corresponding strain amplitude is

*L*at frequencies \(\omega \ll v/L\). The plots in Fig. 31 show the strain amplitudes with varying speeds

*v*and arm lengths

*L*. In the former case, the arm length is kept constant at \(L=500\mathrm {\ m}\), in the latter case, the speed is kept constant at \(v=20\mathrm {\ m/s}\). The mass of the car is 1000 kg, and the depth of the test masses is 300 m.

While this form of noise is irrelevant to large-scale GW detectors sensitive above 10 Hz, low-frequency detectors could be strongly affected. According to the left plot, one should better enforce a speed limit on cars to below 10 m/s if the goal is to have good sensitivity around 0.1 Hz. Another application of these results is to calculate Newtonian noise from uniformly advected atmospheric temperature fields as discussed in Sect. 5.2. For uniform airflow, the remaining integrals in Eq. (144) are the Fourier transform of Eq. (165), whose solution was given in this section.

### 6.3 Oscillating point masses

Oscillating masses can be a source of gravity perturbations, where we understand oscillation as a periodic change in the position of the center of mass. As we have seen in Sect. 6.1, it is unlikely that these perturbations are dominant contributions to Newtonian noise, but in the case of strongly reduced seismic Newtonian noise (for example, due to coherent noise cancellation), perturbations from oscillating objects may become significant. For an accurate calculation, one also needs to model disturbances resulting from the reaction force on the body that supports the oscillation. In this section, we neglect the reaction force. Oscillation is only one of many possible modes of object motion that can potentially change the gravity field. A formalism that can treat all types of object vibrations and other forms of motion is presented in Sect. 6.4.

*m*as strain noise between two test masses at distance

*L*to each other separated along the direction of the unit vector \({\mathbf {e}}_{12}\) (Fig. 32).

Figure 33 shows the gravity strain response to an oscillating object for oscillations parallel to \({\mathbf {e}}_{12}\) (left) and perpendicular to \({\mathbf {e}}_{12}\) (right). The position of the object is parameterized by the angle \(\phi =\arccos ({\mathbf {e}}_{12}\cdot {\mathbf {e}}_{r_1})\) (which we call polar angle). Equations (172) and (173) correspond to the response on the lines \(\phi =\pi /2\). The response grows to infinity for \(\lambda =1\) and polar angle \(\phi =0\) since the object is collocated with the second test mass. Note that \(\lambda >1\) and \(\phi =0\) means that the object lies between the two test masses.

### 6.4 Interaction between mass distributions

*A*to the center of mass

*B*. Each individual mass is expanded into its multipoles according to Eq. (262) calculated in identically oriented coordinate systems, but with their origins corresponding to the two centers of mass. The situation is depicted in Fig. 34.

*M*in a coordinate system centered on its position is \(X_0^0=M\). Therefore the interaction energy can be written

*M*has a radius

*R*and a height

*H*. Aligning the

*z*-axis of the coordinate system with the symmetry axis of the cylinder, the only non-vanishing moments of the cylinder have \(m=0\) due to axial symmetry. Therefore, the relevant solid spherical harmonic expressed in cylindrical coordinates is given by

*z*-axis is defined parallel to the symmetry axis of the cylinder, the spherical angular coordinate \(\theta \) in \(I_2^0({\mathbf {R}}_{AB}\,)\) represents the angle between the symmetry axis and the separation vector \({\mathbf {R}}_{AB}\). The interaction energy of the quadrupole term can be written

*m*. The gravitational force is the negative gradient of the interaction energy, which can be calculated using Eq. (254)

### 6.5 Oscillating objects

In the previous section, we introduced the formalism of bipolar expansion to calculate gravitational interactions between two bodies. However, what we typically want is something more specific such as the change in gravity produced by translations and rotations of bodies. Translations in the form of small oscillations will be studied in this section, rotations in the following section. We emphasize that the same formalism can also be used to describe changes in the gravity field due to arbitrary vibrations of bodies by treating these as changes in the coefficients of a multipole expansion.

### 6.6 Rotating objects

*l*of spherical harmonics.

*z*-axis of the body-fixed system, then by \(\beta \) around the \(y'\)-axis of the once rotated coordinate system (following the convention in Steinborn and Ruedenberg 1973), and finally by \(\gamma \) around the \(z''\)-axis of the twice rotated coordinate system. Rotations around the

*z*-axes lead to simple complex phases being multiplied to the spherical harmonics. The rotation around the \(y'\)-axis is more complicated, and the general, explicit expressions for the components \(D_{m,m'}^{(l)^*}(\alpha ,\beta ,\gamma )\) of the rotation matrix are given by Steinborn and Ruedenberg (1973):

*k*that give non-negative factorials in the two binomial coefficients: \(\max (0,m-m')\le k\le \min (l-m',l+m)\). In the remainder of this section, we apply the rotation transformation to the simple case of a rotating ring of

*N*point masses. Its multipole moments have been calculated in Sect. 3. The goal is to calculate the gravity perturbation produced by the rotating ring, assumed to have its symmetry axis pointing towards the test mass that is now modelled as a point mass. In this case, we can take Eq. (175) as starting point. The rotation transforms the exterior multipole moments \(X_l^{m,\mathrm {B}}\). We have seen that multipole moments of the ring vanish unless \(m=0,N,2N,\ldots \) and \(l+m\) must be even. Only the first (or last) Euler rotation by an angle \(\alpha =\omega t\) is required, which yields

Since \(X_l^m\) vanishes unless \(m=0,N,2N,\ldots \), only specific multiples of the rotation frequency \(\omega \) can be found in the time-varying gravity field. The number *N* of point masses on the ring quantifies the level of symmetry of the ring, and acts as an up-conversion factor of the rotation frequency. Therefore, if gravity perturbations are to be estimated from rotating bodies such as a rotor, then the level of symmetry is important. However, the higher the up-conversion, the stronger is the decrease of the perturbation with distance from the ring. It would of course be interesting to study the effect of asymmetries of the ring on gravity perturbations. For example, the point masses can be slightly different, and their distance may not be equal among them. It is not a major effort to generalize the symmetric ring study to be able to calculate the effect of these deviations.

### 6.7 Summary and open problems

In this section, we reviewed the theoretical framework to calculate gravity perturbations produced by finite-size objects. Models have been constructed for uniformly moving objects, oscillating objects, as well as rotating objects. In all examples, the object was assumed to be rigid, but expanding a mass distribution into multipole moments can also facilitate simple estimates of gravity perturbations from excited internal vibration modes. An “external” vibration in the sense of an isolated oscillation does not exist strictly speaking since there must always be a physical link to another object to compensate the momentum change, but it is often possible to identify a part of a larger object as main source of gravity perturbations and to apply the formalism for oscillating masses.

Many forms of object Newtonian noise have been estimated (Pepper 2007; Driggers et al. 2012b). So far, none of the potential sources turned out to be relevant. In Sect. 6.1, we learned why it is unlikely that object Newtonian noise dominates over seismic Newtonian noise. Still, one should not take these rules of thumb as a guarantee. Strong vibration, i.e., with amplitudes much larger than ground motion, can in principle lead to significant noise contributions, especially if the vibration is enhanced by internal resonances of the objects. Any form of macroscopic motion including rotations (in contrast to small-amplitude vibrations) should of course be avoided in the vicinity of the test masses.

The curves are based on seismic, sound, and vibration measurements. The seismic Newtonian noise curves are modelled using Eq. (100), the sound Newtonian noise using Eq. (161), and estimates of gravity perturbations from wall panels, the buildings, and fans are modelled using equations from this section. Gravity perturbations from the buildings assume a rocking motion of walls and roof. The exhaust fan strongly vibrates due to asymmetries of the rotating parts, which was taken as source of gravity perturbations. Finally, panels attached to the structure of the buildings show relatively high amplitudes of a membrane like vibration. Nonetheless, these sources, even though very massive, do not contribute significantly to the noise budget.

Greater care is required when designing future GW detectors with target frequencies well below 10 Hz. These will rely on some form of Newtonian-noise mitigation (passive or active), which increases the relative contribution of other forms of gravity perturbations. Also, in some cases, as for the uniform motion discussed in Sect. 6.2, there is a link between the shape of the gravity perturbation spectrum and the distance between object and test mass. These classes of gravity perturbations (and we have identified only one of them so far), can be much stronger at lower frequencies.

Future work on object Newtonian noise certainly includes a careful study of this problem for low-frequency GW detectors. In general, it would be beneficial to set up a catalogue of potential sources and corresponding gravity models to facilitate the process of estimating object Newtonian noise in new detector designs. Another interesting application of the presented formalism could be in the context of experiments carried out with the intention to be sensitive to gravity perturbations produced by an object [such as the quantum-gravity experiment proposed by Feynman (Zeh 2011)]. The formalism presented in this section may help to optimize the geometrical design of such an experiment.

## 7 Newtonian-noise mitigation

In early sensitivity plots of GW detectors, Newtonian noise was sometimes included as infrastructure noise. It means that it was considered a form of noise that cannot be mitigated in a straight-forward manner, except maybe by changing the detector site or applying other major changes to the infrastructure. Today however, some form of Newtonian-noise mitigation is part of every design study and planning for future generations of GW detectors, and it is clear that mitigation techniques will have a major impact on the future direction of ground-based GW detection. The first to mention strategies of seismic Newtonian-noise mitigation “by modest amounts” were Hughes and Thorne (1998). Their first idea was to use arrays of dilatometers in boreholes, and seismometers at the surface to monitor the seismic field and use the sensor data for a coherent subtraction of Newtonian noise. The second idea was to construct narrow moats around the test masses that reflect incoming Rayleigh waves and therefore reduce seismic disturbances and associated gravity perturbations. As they already recognized in their paper, and as will be discussed in detail in Sect. 7.3, moats must be very deep (about 10 m for the LIGO and Virgo sites). They are also less effective to reduce Newtonian noise from body waves, and above all, it is assumed that the seismic sources are located outside the region protected by the moats.

Coherent cancellation of seismic Newtonian noise will be based on techniques that have already been implemented successfully in GW detectors to mitigate other forms of noise (Giaime et al. 2003; Driggers et al. 2012a; De Rosa et al. 2012; Driggers et al (The LIGO Scientific Collaboration Instrument Science Authors) 2019). These techniques are known as *active* noise mitigation. It is currently considered as a means to reduce seismic Newtonian noise, but the same scheme may also be applied to atmospheric Newtonian noise (see especially Sect. 7.1.5) and possibly also other forms of gravity perturbations. While for example active seismic isolation cancels seismic disturbances before they reach the final suspension stages of a test mass, gravity perturbations have to be canceled in the data of the GW detector. It is said that coherent cancellation comes without ultimate limitations, but this statement is likely incorrect and certainly misleading. Many issues can limit the effectiveness of noise cancellation: available number of sensors, available quality and type of sensors, sub-optimal filter design, e.g., due to statistical and numerical errors, back-action of the sensors on the monitored field, non-stationarity of noise, and even fundamental aspects like the equivalence principle can interfere (see Eq. (18)). The prediction by Hughes and Thorne of a modest noise reduction rather follows from a vision of a practicable solution at the time the paper was written. The first detailed study of coherent Newtonian-noise cancellation was carried out by Cella (2000). He studied the Wiener-filter scheme. Wiener filters are based on observed mutual correlation between environmental sensors and the target channel. The Wiener filter is the optimal linear filter to reduce variance in a target channel as explained in Sect. 7.1.1. The goal of a cancellation scheme can be different though, e.g., reduction of a stationary noise background in non-stationary data. The focus in Sect. 7.1 will also lie on Wiener filters, but limitations will be demonstrated, and the creation of optimal filters using real data is mostly an open problem.

Techniques to mitigate Newtonian noise without using environmental data are summarized under the category of *passive* Newtonian-noise mitigation. Site selection is the best understood passive mitigation strategy. The idea is to identify the quietest detector site in terms of seismic noise and possibly atmospheric noise, which obviously needs to precede the construction of the detector as part of a site-selection process. The first systematic study was carried out for the Einstein Telescope (Beker et al. 2012) with European underground sites. Other important factors play a role in site selection, and therefore one should not expect that future detector sites will be chosen to minimize Newtonian noise, but rather to reduce it to an acceptable level. Current understanding of site selection for Newtonian-noise reduction is reviewed in Sect. 7.2. Other passive noise-reduction techniques are based on building shields against disturbances that cause density fluctuations near the test masses, such as moats and recess structures against seismic Newtonian noise, which are investigated in Sect. 7.3.

### 7.1 Coherent noise cancellation

Coherent noise cancellation, also known as *active noise cancellation*, is based on the idea that the information required to model noise in data can be obtained from auxiliary sensors that monitor the sources of the noise. The noise model can then be subtracted from the data in real time or during post processing with the goal to minimize the noise. In practice, cancellation performance is limited for various reasons. Depending on the specific implementation, non-stationarity of data, sensor noise, and limited information content in sensor data can limit the performance. Sensor noise poses a limitation since the noise-cancellation filter maps sensor noise (together with the interesting signal) to the filter output, which is then added to the target channel. It follows that the auxiliary sensors must provide information about the sources with sufficiently high signal-to-noise ratio.

The best way to understand the noise-cancellation problem is to think of it as an optimization of extraction of information, subject to constraints. Constraints can exist for the maximum number of auxiliary sensors, for the possible array configurations, and for the amount of data that can be used to calculate the optimal filter. Also the type of filter and the algorithm used to calculate it can enforce constraints on information extraction. There is little understanding of how most of these constraints limit the performance. A well-explored cancellation scheme is based on *Wiener filters* (Orfanidis 2007). Wiener filters are linear filters calculated from correlation between reference and target channels. They are introduced in Sect. 7.1.1. In the context of seismic or atmospheric Newtonian-noise cancellation, the sensors monitor fields, which means that correlations between them are to be expected. In this case, if the field is wide-sense stationary (defined in Sect. 7.1.1), if the target channel is wide-sense stationary, and if all forms of noise are additive, then the Wiener filter is known to be the optimal linear filter for a given configuration of the sensor array (Rey Vega and Rey 2013). In Sects. 7.1.2 to 7.1.4, the problem is described for seismic and infrasound Newtonian noise. The focus lies on gravity perturbations from fluctuating fields. Noise cancellation from objects is expected to be far less challenging since it is easier to monitor the object’s motion or vibration. Calculating a Wiener filter does not address the problem whether the sensors have their optimal locations in the field to extract information most efficiently. The optimization of array configurations for noise cancellation is a separate problem, which is discussed in Sect. 7.1.6.

#### 7.1.1 Wiener filtering

A linear filter that produces an estimate of a random stationary (target) process minimizing the mean-square error between target and estimation is known as Wiener filter (Benesty et al. 2008). It is based on the idea that data from reference channels exhibit some form of correlation to the target channel, which can therefore be used to provide a coherent estimate of certain contributions to the target channel. Strictly speaking, the random processes only need to be wide-sense stationary, which means that noise moments are independent of time up to second order (i.e., variances and correlations). Without prior knowledge of the random processes, the Wiener filter itself needs to be estimated. In this section, we briefly review Wiener filtering, and discuss some of its limitations.

*n*represents time \(t_n=t_0+n\varDelta t\), where \(\varDelta t\) is the common sampling time of the random processes. With discretely sampled data, a straight-forward filter implementation is the convolution with a finite-impulse response filter (FIR). These filters are characterized by a filter order

*N*. Assuming that we have

*M*reference channels, the FIR filter \(\mathbf {w}\) is a \((N+1)\times M\) matrix with components \(w_{nm}\). The convolution assumes the form

*M*reference channels. This equation implies that there is only one target channel \(y_n\), in which case the FIR filter is also known as multiple-input-single-output (MISO) filter. We have marked the filter output with a hat to indicate that it should be interpreted as an estimate of the actual target channel. The coefficients of the Wiener filter can be calculated by demanding that the mean-square deviation \(\langle (y_n-\hat{y}_n)^2\rangle \) between the target channel and filter output is minimized, which directly leads to the Wiener–Hopf equations:

*NM*-dimensional vector that is obtained by concatenating the

*M*columns of the matrix \(\mathbf {w}\). The \((N+1)M\times (N+1)M\) matrix \(\mathbf {C}_{xx}\) is the cross-correlation matrix between reference channels. Correlations must be evaluated between all samples of all reference channels where sample times differ at most by \(N\varDelta t\). It contains the autocorrelations of each reference channel as \((N+1)\times (N+1)\) blocks on its diagonal:

*M*reference channels. In this form it is a symmetric Toeplitz matrix. The \((N+1)M\)-dimensional vector \({\mathbf {C}}_{xy}\) is a concatenation of correlations between each reference channel and the target channel. The components contributed by a single reference channel are

A frequency-domain version of the Wiener filter can be obtained straight-forwardly by dividing the data into segments and calculating their discrete Fourier transforms. Equation (195) translates into a segment-wise noise cancellation where *n* stands for a double index to specify the segment and the discrete frequency (also known as frequency bin). For stationary random processes, correlations between noise amplitudes at different frequencies are zero. This means that coherent noise cancellation in frequency domain can be done on each frequency bin separately, which is numerically much less demanding and more accurate since the dimensionality of the system of equations in Eq. (192) is reduced from *NM* to *M* (for *N* different frequency bins). In contrast, time-domain correlations \(c_k,\,s_k\) can be large for non-zero values of *k*. This can cause significant numerical problems to solve the Wiener–Hopf equations, and as observed in Coughlin et al. (2014), FIR filters of lower order can be more effective (even though theoretically, increasing the filter order should not make the cancellation performance worse).

#### 7.1.2 Cancellation of Newtonian noise from Rayleigh waves

Generally, there are two ways to model Newtonian-noise cancellation. First, one can use analytic models of the correlations between seismic sensors and with test-mass acceleration. Studies based on models have the advantage that they allow us to investigate in detail the impact of sensor noise and array configuration on cancellation performance. For Rayleigh-wave Newtonian noise, this was first presented in an earlier version of this review article and then in greater detail by Coughlin et al. (2016). The second method is to estimate the required correlation functions from seismic data. While it is straight-forward to calculate correlations \(\mathbf {C}_{xx}\) in Eq. (192), or equivalently, the cross-spectral densities \(\mathbf {C}_\mathrm{SS}(\omega )\) between seismometers, correlations between seismometers and Newtonian-noise test-mass acceleration are not always available. So far, we only have weak indication of the presence of Newtonian noise in data of the LIGO Hanford detector (Coughlin et al. 2018b). The best one can do in this case is to use the seismic correlations to obtain a more accurate model of correlations with the test mass. The equations providing such a model for the most general Rayleigh-wave field were first presented in Coughlin et al. (2016). In this section, we will derive simplified expressions for the homogeneous field.

Another aspect that we do not consider in this section is to make use of seismic tiltmeters for Newtonian-noise reduction. As shown in Harms and Venkateswara (2016), the tiltmeter is the ideal instrument to cancel Newtonian noise from Rayleigh waves. If the seismic field is homogeneous, then only one sufficiently sensitive tiltmeter under each test mass would be required. We choose to present methods since seismic fields can be strongly inhomogeneous as for example at the Virgo site due to subsurface laboratory space under the test masses, divided concrete platforms to support different heavy infrastructure, and anchoring of these platforms into lower layers of hard rock by several tens of meter long poles.

*x*-axis measured at height

*h*above surface, the correlation is given by (see Sect. 3.6.3 for the definition of correlation between random signals in frequency domain)

Since the purpose of this section is to evaluate and design a coherent noise cancellation of gravity perturbations in *x*-direction, one may wonder why the correlation with the vertical surface displacement is used, and not the displacement along the direction of the *x*-axis. The reason is that in general horizontal seismic motion of a flat surface correlates weakly with gravity perturbations produced at the surface. Other waves such as horizontal shear waves can produce horizontal surface displacement without perturbing gravity. Vertical surface displacement always perturbs gravity, no matter by what type of seismic wave it is produced. The situation is different underground as we will see in Sect. 7.1.3.

*L*between the test masses is much smaller than the length of the Rayleigh wave. In other words, the detector measures gravity gradients. Considering the case that direction of acceleration and direction of separation are the same, the correlation is given by

*L*. This is the typical situation for strain quantities at low frequencies since the differential signal is proportional to the distance, which then cancels in the strain variable when dividing by

*L*. We have seen this already in Sect. 3.6.4.

*x*-coordinate. Assuming a signal-to-noise ratio of \(\sigma =10\), the single seismometer residual would be 0.38. The second seismometer needs to be placed at \(x_2=-0.28\lambda ^{\mathrm {R}}\) and \(y_2=0\), with residual 0.09. The third seismometer at \(x_3=0.75\lambda ^{\mathrm {R}}\) and \(y_3=0\), with residual 0.07. The step-wise optimization described here works for a single frequency since the optimal locations depend on the length \(\lambda ^{\mathrm {R}}\) of a Rayleigh wave. In reality, the goal is to subtract over a band of frequencies, and the seismometer placement should be optimized for the entire band. The result is shown in the left of Fig. 38 for a sub-optimal spiral array, and seismometers with frequency-independent \(\sigma =100\). Rayleigh-wave speed decreases from 1.5 km/s at 1 Hz to 250 m/s at 50 Hz. There are three noteworthy features. First, the minimal relative residual lies slightly below the value of the inverse seismometer signal-to-noise ratio. It is a result of averaging of self noise from different seismometers. Second, residuals increasing with \(1/\omega \) at low frequencies is a consequence of the finite array diameter. An array cannot analyze waves much longer than its diameter. Third, the residuals grow sharply towards higher frequencies. The explanation is that the array has a finite seismometer density, and therefore, waves shorter than the typical distance between seismometers cannot be analyzed. If the seismic speed is known, then the array diameter and number of seismometers can be adjusted in this way to meet a subtraction goal in a certain frequency range.

Residuals are also shown in the right plot of Fig. 38 for the subtraction of gravity-gradient noise (i.e., the low-frequency case). The sensor signal-to-noise ratio is the same as before. Rayleigh-wave speed decreases from 3.5 km/s at 0.1 Hz to 1.5 km/s at 1 Hz. The low-frequency slope of the residual spectra has an additional \(1/\omega \) compared to the residual spectra of acceleration Newtonian noise due to an effective decrease of seismometer signal-to-noise ratio from common-mode rejection of Newtonian noise in the detector. Overall, the reduction of Newtonian noise in low-frequency detectors remains more challenging since Newtonian noise from Rayleigh waves needs to be reduced by about three orders of magnitude (Harms et al. 2013; Canuel et al. 2016). It was proposed to achieve additional mitigation by averaging locally over many independent strain measurements (Chaibi et al. 2016). A combination of measures might provide the required noise reduction.

A special case of GW detector is the full-tensor design, which is able to observe all components of the spatial part of the metric perturbation (or gravity gradient) (Paik et al. 2016). Such a detector would have uniform sensitivity over GW propagation directions provided that all channels are consistently included in a polarization-dependent analysis. The question emerges whether full-tensor detectors offer any advantage to NN cancellation, since out of 5 independent gravitational channels, only two effectively carry information about the two polarizations of a passing GW. The three remaining channels can be used for NN cancellation. Concerning Rayleigh-wave NN, it was shown that NN can be suppressed by the same amount in full-tensor detectors as in conventional detectors independent of the GW propagation direction, which results in a significant sensitivity gain due to the omnidirectional GW response (Paik and Harms 2016). Moreover, the authors also found that combining detector channels and microphones deployed in a surface array, cancellation of sound NN is more effective in full-tensor detectors compared to conventional detectors. Since channels of full-tensor detectors can only be used optimally with respect to one source of NN, it was proposed that a full-tensor detector should be operated such that it minimizes the sound NN, while NN from seismic fields is to be canceled only using seismometers.

#### 7.1.3 Cancellation of Newtonian noise from body waves

In this section, the focus lies on noise subtraction in infinite media. As we have seen in Sects. 3.2 and 3.4, any gravity perturbation can be divided into two parts, one that has the form of gravity perturbations from seismic fields in infinite space, and another that is produced by the surface. Subtraction of the surface part follows the scheme outlined in Sect. 7.1.2 using surface arrays. The additional challenge is that body waves can have a wide range of angles of incidence leading to a continuous range of apparent horizontal speeds, which could affect the array design. In the following, we introduce the formalism to describe coherent noise cancellation of the bulk contribution. We assume here that Newtonian noise is uncorrelated between test masses, which makes the results relevant only to large-scale GW detectors. In order to simplify the analysis, only homogeneous and isotropic body-wave fields are considered, without contributions from surface waves. This case was first studied in detail by Harms and Badaracco (2019). We will review their main results.

The test masses are assumed to be located underground inside cavities. We know from Sect. 3.3.1 that gravity perturbations are produced by compressional waves through density perturbations of the medium, and by shear and compressional waves due to displacement of cavity walls. Since for body waves it is interesting to consider the measurement of arbitrary displacement directions, a seismic measurement is represented by the projection \({\mathbf {e}}_n\cdot \varvec{\xi }({\mathbf {r}},\omega )\), where \({\mathbf {e}}_n\) is the direction of the axis of the seismometer. Also the gravity measurement is represented by a similar projection \({\mathbf {e}}_n\cdot \delta {\mathbf {a}}({\mathbf {r}},\omega )\). Therefore, the general two-point correlation function depends on the directions \({\mathbf {e}}_1,\,{\mathbf {e}}_2\) of two measurements, and the unit vector \({\mathbf {e}}_{12}\) that points from one measurement location at \({\mathbf {r}}_1\) to the other at \({\mathbf {r}}_2\).

*z*-axis parallel to \({\mathbf {e}}_{12}\) so that \({\mathbf {e}}_k\cdot {\mathbf {e}}_{12}=\cos (\theta )\). Instead of writing down the explicit expression of \({\mathbf {e}}_k\otimes {\mathbf {e}}_k\) and evaluating the integral over all of its independent components, one can reduce the problem to two integrals only. The point is that the matrix that results from the integration can in general be expressed in terms of two “basis” matrices \(\mathbf {1}\) and \({\mathbf {e}}_{12}\otimes {\mathbf {e}}_{12}\). For symmetry reasons, it cannot depend explicitly on any other combination of the coordinate basis vectors \({\mathbf {e}}_x\otimes {\mathbf {e}}_x\), \({\mathbf {e}}_x\otimes {\mathbf {e}}_y\), ...Expressing the integral as linear combination of basis matrices, \(P_1(\varPhi _{12})\mathbf {1}+P_2(\varPhi _{12})({\mathbf {e}}_{12}\otimes {\mathbf {e}}_{12})\) with \(\varPhi _{12}\equiv k^{\mathrm {P}}|{\mathbf {r}}_2-{\mathbf {r}}_1|\), solutions for \(P_1(\varPhi _{12}),P_2(\varPhi _{12})\) can be calculated as outlined in Flanagan (1993), Allen and Romano (1999), and the correlation function finally reads

*x*-axis. The plot only shows a plane of possible seismometer placement, and all seismometers measure along the relevant direction of gravity acceleration. Ideally, optimization should be done in three dimensions, but for the first three seismometers, the 2D representation is sufficient. In these calculations, the P-wave speed is assumed to be a factor 1.8 higher than the S-wave speed. The mixing ratio is 1/3, and the signal-to-noise ratio is 100. The optimal location of the second seismometer lies in orthogonal direction at \(x_2=z_2=0\) and \(y_2=\pm 0.33\lambda ^{\mathrm {P}}\). We choose the positive

*y*-coordinate. In this case, the third seismometer needs to be placed at \(x_3=z_3=0\) and \(y_3=- 0.33\lambda ^{\mathrm {P}}\). With three seismometers, a residual of 0.44 can be achieved.

The left plot in Fig. 41 shows the subtraction residuals of bulk Newtonian noise using a 3D spiral array with all seismometers measuring along the relevant direction of gravity acceleration. The mixing ratio is 1/3. The ultimate limit enforced by seismometer self noise, \(1/(\sigma \sqrt{N})\), is not reached. Nonetheless, residuals are strongly reduced over a wide range of frequencies. Note that residuals do not approach 1 at highest and lowest frequencies, since a single seismometer at the test mass already reduces residuals to 0.67 at all frequencies assuming constant \(\sigma =100\).

#### 7.1.4 Cancellation of Newtonian noise from infrasound

*x*-coordinate can be technically obtained by calculating the derivative \(\partial _x\), with \(x\equiv x_2-x_1\) and Eq. (216), we find

As a final remark, infrasound waves have properties that are very similar to compressional seismic waves, and the result of Sect. 7.1.2 was that broadband cancellation fo Newtonian noise from compressional waves can be achieved with primitive array designs, provided that the field is not mixed with shear waves. Air does not support the propagation of shear waves, so one might wonder why subtraction of infrasound Newtonian noise does not have these nice properties. The reason lies in the sensors. Microphones provide different information. In a way, they are more similar in their response to seismic strainmeters. According to Eq. (215), correlations between a strainmeter and gravity perturbations also vanishes if the strainmeter is located at the test mass. What this means though is that a different method to monitor infrasound waves may make a big difference. It is a “game with gradients”. One could either monitor pressure gradients, or the displacement of air particles due to pressure fluctuations. Both would restore correlations of sensors at the test mass with gravity perturbations.

#### 7.1.5 Demonstration: Newtonian noise in gravimeters

The problem of coherent cancellation of Newtonian noise as described in the previous sections is not entirely new. Gravimeters are sensitive to gravity perturbations caused by redistribution of air mass in the atmosphere (Neumeyer 2010). These changes can be monitored through their effect on atmospheric pressure. For this reason, pressure sensors are deployed together with gravimeters for a coherent cancellation of atmospheric Newtonian noise (Banka and Crossley 1999). In light of the results presented in Sect. 7.1.4, it should be emphasized that the cancellation is significantly less challenging in gravimeters since the pressure field is not a complicated average over many sound waves propagating in all directions. This does not mean though that modelling these perturbations is less challenging. Accurate calculations based on Green’s functions are based on spherical Earth models, and the model has to include the additional effect that a change in the mass of an air column changes the load on the surface, and thereby produces additional correlations with the gravimeter signal (Guo et al. 2004). Nonetheless, from a practical point of view, the full result is more similar to the coherent relations such as Eq. (132), which means that local sensing of pressure fluctuations should yield good cancellation performance.

#### 7.1.6 Optimizing sensor arrays for noise cancellation

In the previous sections, we focussed on the design and performance evaluation of an optimal noise-cancellation filter for a given set of reference sensors. In this section, we address the problem of calculating the array configuration that minimizes noise residuals given sensor noise of a fixed number of sensors. The analysis will be restricted to homogeneous fields of density perturbations. The optimization can be based on a model or measured two-point spatial correlations \(C(\delta \rho ;{\mathbf {r}},\omega )\). We start with a general discussion and later present results for the isotropic Rayleigh-wave field.

*R*defined in Eq. (204) as a function of sensor locations \({\mathbf {r}}_i\). Accordingly, the optimal sensor locations fulfill the equation

*M*sensors, i.e., \(k\in 1,\ldots ,M\). In homogeneous fields, the Newtonian-noise spectrum and seismic spectrum are independent of sensor location,

*k*of the vector \({\mathbf {C}}({\mathbf {s}};n)\) and the

*k*th row and column of \(\mathbf {C}({\mathbf {s}};{\mathbf {s}})\) depend on the coordinates of the sensor

*k*. This means that the derivative \(\nabla _k\) produces many zeros in the last equation, which allows us to simplify it into the following form:

*M*sensors. Solutions to this equation need to be calculated numerically. Optimization of arrays using Eq. (221) produces accurate solutions more quickly than traditional optimization methods, which directly attempt to find the global minimum of the residual

*R*. Traditional codes (nested sampling, particle swarm optimization) produce solutions that converge to the ones obtained by solving Eq. (221).

*R*as a function of sensor coordinates for a total of 1 to 3 sensors, from left to right. In the case of a single sensor, the axes represent its

*x*and

*y*coordinates. For more than one sensor, the axes correspond to the

*x*coordinates of two sensors. All coordinates not shown in these plots assume their optimal values.

The green and orange curves represent Eq. (221) either for the derivatives \(\partial _x,\,\partial _y\) or \(\partial _{x_1},\,\partial _{x_2}\). These curves need to intersect at the optimal coordinates. It can be seen that they intersect multiple times. The numerical search for the optimal array needs to find the intersection that belongs to the minimum value of *R*. For the isotropic case, it is not difficult though to tune the numerical search such that the global minimum is found quickly. The optimal intersection is always the one closest to the test mass at the origin. While it is unclear if this holds for all homogeneous seismic fields, it seems intuitive at least that one should search intersections close to the test mass in general.

*M*sensors. So choosing initial positions in the numerical search wisely significantly decreases computation time, and greatly reduces the risk to get trapped in local minima.

Figure 45 shows the noise residuals of Newtonian noise from an isotropic Rayleigh-wave field using optimal arrays with 1 to 6 sensors and sensor SNR = 100. The residuals are compared with the sensor-noise limit \(1/\mathrm {SNR}/\sqrt{M}\) (dashed curve). Arrays with \(M>3\) yield residuals that are close to a factor \(\sqrt{2}\) above the sensor-noise limit. The origin of the factor \(\sqrt{2}\) has not been explained yet. It does not appear in all noise residuals, for example, the noise residual of a Wiener filter using a single reference channel perfectly correlated with the target channel, see Eq. (196), is given by 1/SNR.

In many situations, it will not be possible to model the correlations \(\mathbf {C}_{\mathrm {SS}}\) and \({\mathbf {C}}_{\mathrm {SN}}\). In this case, observations of seismic correlations \(\mathbf {C}_{\mathrm {SS}}\) can be used to calculate \({\mathbf {C}}_{\mathrm {SN}}\), see Eq. (198), and also \(C_{\mathrm {NN}}\), see Eq. (107). Seismic correlations are observed with seismometer arrays. It is recommended to choose a number of seismometers for this measurement that is significantly higher than the number of seismometers foreseen for the noise cancellation. Otherwise, aliasing effects and resolution limits can severely impact the correlation estimates. Various array-processing algorithms are discussed in Krim and Viberg (1996).

Cancellation of Newtonian noise from isotropic Rayleigh-wave fields at wavelength \(\lambda \)

Sensor coordinates \([\lambda ]\) | Noise residual \(\sqrt{R}\) |
---|---|

(0.293,0) | 0.568 |

(0.087,0), (\(-\)0.087,0) | \(2.28\times 10^{-2}\) |

(0.152,\(-\)0.103), (0.152,0.103), (\(-\)0.120,0) | \(1.24\times 10^{-2}\) |

(0.194,0.112), (0.194,\(-\)0.112), (\(-\)0.194,0.112), (\(-\)0.194,\(-\)0.112) | \(7.90\times 10^{-3}\) |

(0.191,0.215), (0.299,0), (0.191,\(-\)0.215), (\(-\)0.226,0.116), (\(-\)0.226,\(-\)0.116) | \(6.69\times 10^{-3}\) |

(0.206,0.196), (0.295,0), (0.206,\(-\)0.196), (\(-\)0.206,0.196), (\(-\)0.295,0), (\(-\)0.206,\(-\)0.196) | \(6.04\times 10^{-3}\) |

The noise residuals of the stepwise optimization were \(R=0.38\), 0.09, and 0.07 for the first three seismometers, while the fully optimized residuals are \(R=0.38\), 0.014 and 0.0074, i.e., much lower for \(N\ge 2\).

#### 7.1.7 Newtonian noise cancellation using gravity sensors

In the previous sections, we have investigated Newtonian-noise cancellation using auxiliary sensors that monitor density fluctuations near the test masses. An alternative that has been discussed in the past is to use gravity sensors instead. One general concern about this scheme is that a device able to subtract gravity noise can also cancel GW signals. This fact indeed limits the possible realizations of such a scheme, but it is shown in the following that at least Newtonian noise in large-scale GW detectors from a Rayleigh-wave field can be cancelled using auxiliary gravity sensors. However, it will become clear as well that it will be extremely challenging to build a gravity sensor with the required sensitivity.

The only (conventional) type of gravity sensor that can be used to cancel Newtonian noise in GW detectors is the gravity strainmeter or gravity gradiometer.^{10} As we have discussed in Sect. 2.2, the sensitivity of gravimeters is fundamentally limited by seismic noise, and any attempt to mitigate seismic noise in gravimeters inevitably transforms its response into a gravity gradiometer type. So in the following, we will only consider gravity strainmeters/gradiometers as auxiliary sensors.

Let us first discuss a few scenarios where noise cancellation cannot be achieved. If two identical large-scale GW detectors are side-by-side, i.e., with test masses approximately at the same locations, then Newtonian-noise cancellation by subtracting their data inevitably means that GW signals are also cancelled. Let us make the arms of one of the two detectors shorter, with both detectors’ test masses at the corner station staying collocated. Already one detector being shorter than the other by a few meters reduces Newtonian-noise correlation between the two detectors substantially. The reason is that correlation of gravity fluctuations between the end test masses falls rapidly with distance according to Eq. (222). It can be verified that subtracting data of these two detectors to cancel at least gravity perturbations of the inner test masses does not lead to sensitivity improvements. Instead, it effectively changes the arm length of the combined detector to \(\varDelta L\), where \(\varDelta L\) is the difference of arm lengths of the two detectors, and correspondingly increases Newtonian noise.

If Newtonian noise is uncorrelated between two test masses of one arm, then decreasing arm length increases Newtonian strain noise. However, as shown in Fig. 13, if the detector becomes shorter than a seismic wavelength and Newtonian noise starts to be correlated between test masses, Newtonian strain noise does not increase further. Compared to the Newtonian noise in a large-scale detector with arm length *L*, Newtonian noise in the short detector is greater by (up to) a factor \(k_\varrho L\). In this regime, the small gravity strainmeter is better described as gravity gradiometer. The common-mode suppression of Newtonian noise in the gradiometer due to correlation between test masses greatly reduces Newtonian-noise correlation between gradiometer and the inner test masses of the large-scale detector. Consequently, a gravity gradiometer cannot be used for noise cancellation in this specific configuration.

*x*points along the arm of the large-scale detector, are perfectly correlated with \(\delta a_x\). This can be seen from Eq. (100), since derivatives of the acceleration \(\delta a_x\) with respect to

*z*, i.e., the vertical direction, does not change the dependence on directions \(\phi \). The coherence (normalized correlation) between \(\delta a_x\) and \(\partial _z\delta a_x\) is shown in the left of Fig. 46 making use of \(\langle \delta a_x({\mathbf {0}}\,),\partial _z\delta a_x(\varvec{\varrho }\,)\rangle _{\mathrm {norm}}=\langle \delta a_x({\mathbf {0}}\,),\delta a_x(\varvec{\varrho }\,)\rangle _{\mathrm {norm}}\).

The idea is now to place one full-tensor gravity gradiometer at each test mass of the large-scale detector, and to cancel Newtonian noise of each mass. In this way, it is also impossible to cancel GW signals since GW signals of the gradiometers cancel each other. The limitations of this scheme are determined by the distance between the test mass of the large-scale detector and test-masses of the gravity gradiometer. The smaller the distance, the better the correlation and the higher the achievable noise reduction. Using Eq. (196), the maximal noise reduction can be calculated as a function of the coherence. In Fig. 46, right plot, the achievable noise suppression is shown as a function of distance between test masses. For example, at 10 Hz, and assuming a Rayleigh-wave speed of 250 m/s, the distance needs to be smaller than 1 m for a factor 5 noise reduction. This also means that the size of the gradiometer must be of order 1 m.

*s*. A numerical factor of order unity is omitted. Given a Newtonian strain noise \(h_{\mathrm {NN}}\) of the large-scale detector with arm length

*L*, the gradiometer observes

*s*. One could raise well-justified doubts at this point if a meter-scale detector can achieve displacement sensitivity of large-scale GW detectors. Nonetheless, the analysis of this section has shown that Newtonian-noise cancellation using gravity sensors is in principle possible.

### 7.2 Site selection

An elegant way to reduce Newtonian noise is to select a detector site with weak gravity fluctuations. It should be relatively straightforward to avoid proximity to anthropogenic sources (except maybe for the sources that are necessarily part of the detector infrastructure), but it is not immediately obvious how efficient this approach is to mitigate seismic or atmospheric Newtonian noise. With the results of Sects. 3 and 5, and using numerous past observations of infrasound and seismic fields, we will be able to predict the possible gain from site selection. The aim is to provide general guidelines that can help to make a site-selection process more efficient, and help to identify suitable site candidates, which can be characterized in detail with follow-up measurements. These steps have been carried out recently in Europe as part of the design study of the Einstein Telescope (Beker et al. 2012, 2015), and promising sites were indeed identified.

Already with respect to the minimization of Newtonian noise, site selection is a complicated process. One generally needs to divide into site selection for gravity measurements at low and high frequencies. The boundary between these two regimes typically lies at a few Hz. The point here is that at sufficiently low frequencies, gravity perturbations produced at or above surface are negligibly suppressed at underground sites with respect to surface sites. At higher frequencies, a detailed site-specific study is required to quantify the gain from underground construction since it strongly depends on local geology. In general, sources of gravity perturbations have different characteristics at lower and higher frequencies. Finally, to complicate the matter even further, one may also be interested to identify a site where one can expect to achieve high noise cancellation through Wiener filtering or similar methods.

#### 7.2.1 Global surface seismicity

^{11}(archiving global seismic data), and the Japanese seismic broadband network F-Net

^{12}operated by NIED. Seismic data cannot be easily obtained from countries that have not signed the Comprehensive Nuclear-Test-Ban Treaty (which are few though). The results of their analysis were presented in the form of spectral histograms for each site, accessible through a Google Earth kmz file. An example is shown in Fig. 47 for a seismic station in the US.

Studying these maps gives an idea where to find quiet places on Earth, and helps to recognize generic patterns such as the influence of mountain ranges, and the proximity to oceans. A more detailed analysis based on these data can be found in Coughlin and Harms (2012a). It should be noted that especially in Japan, many seismic stations used in this study are built a few meters underground, which may lead to substantial reduction of observed ambient seismicity above a few Hz with respect to surface sites. Nonetheless, there are regions on all continents with very low surface seismicity above 1 Hz, approaching a global minimum often referred to as global low-noise model (Berger et al. 2004; Coughlin and Harms 2012a). This means that one should not expect that a surface or underground site can be found on Earth that is significantly quieter than the identified quietest surface sites. Of course, underground sites may still be attractive since the risk is lower that seismicity will change in the future, while surface sites can in principle change seismicity over the course of many years, because of construction or other developments. For the same reason, it may be very challenging to find quiet surface sites in densely populated countries. As a rule of thumb, a site that is at least 50 km away from heavy traffic and seismically active faults, and at least 100 km away from the ocean, has a good chance to show low ambient seismicity above a few Hz. Larger distances to seismically active zones may be necessary for other reasons such as avoiding damage to the instrument.

Below a few Hz, ambient seismicity is more uniform over the globe. Oceanic microseisms between 0.1 Hz and 1 Hz are stronger within 200 km to the coast, and then decreasing weakly in amplitude towards larger distances. This implies that it is almost impossible to find sites with a low level of oceanic microseisms in countries such as Italy and Japan. At even lower frequencies, it seems that elevated seismic noise can mostly be explained by proximity to seismically active zones, or extreme proximity to cities or traffic. Here one needs to be careful though with the interpretation of data since quality of low-frequency data strongly depends on the quality of the seismic station. A less protected seismometer exposed to wind and other weather phenomena can have significantly increased low-frequency noise. In summary, the possibility to find low-noise surface sites should not be excluded, but underground sites are likely the only seismically quiet locations in most densely populated countries (which includes most countries in Europe).

#### 7.2.2 Underground seismicity

Seismologists have been studying underground seismicity at many locations over decades, and found that high-frequency seismic spectra are all significantly quieter than at typical surface sites. This can be explained by the exponential fall off of Rayleigh-wave amplitudes according to Eq. (39), combined with the fact that high-frequency seismicity is typically generated at the surface, and most surface sites are covered by a low-velocity layer of unconsolidated soil. The last means that amplitude decreases over relatively short distances to the surface. Seismic measurements have been carried out in boreholes (Douze 1964; Sax and Hartenberger 1964), and specifically in the context of site characterization for future GW detectors at former or still active underground mines (Harms et al. 2010; Beker et al. 2012, 2015; Naticchioni et al. 2014). There are however hardly any underground array measurements to characterize the seismic field in terms of mode composition. An exception is the last underground array measurements at the former Homestake mine in South Dakota, USA (Mandic et al. 2018; Coughlin et al. 2018a). This is mostly due to the fact that these experiments are very costly, and seismic stations have to be maintained under unusual conditions (humidity, temperature, dust, ...).

The seismic underground Newtonian-noise estimate from Harms and Badaracco (2019) is shown in Fig. 49. This can be compared with the underground acoustic Newtonian noise shown in Fig. 30. According to these results, seismic surface and acoustic Newtonian noise are sufficiently suppressed if the detector is 700 m underground, and for quiet surface sites, even 300 m would likely be sufficient. The body-wave Newtonian noise however likely lies above the targeted noise level (according to the ET-D model), and cancellation of body-wave Newtonian noise would still be required.

#### 7.2.3 Site selection criteria in the context of coherent noise cancellation

An important aspect of the site selection that has not been considered much in the past is that a site should offer the possibility for efficient coherent cancellation of Newtonian noise. From Sect. 7.1 we know that the efficiency of a cancellation scheme is determined by the two-point spatial correlation of the seismic field. If it is well approximated by idealized models, then we have seen that efficient cancellation would be possible. However, if scattering is significant, or many local sources contribute to the seismic field, then correlation can be strongly reduced, and a seismic array consisting of a potentially large number of seismometers needs to be deployed. The strongest scatterer of seismic waves above a few Hz is the surface with rough topography. This problem was investigated analytically in numerous publications, see for example Gilbert and Knopoff (1960), Abubakar (1962), Hudson (1967), and Ogilvy (1987). If the study is not based on a numerical simulation, then some form of approximation needs to be applied to describe topographic scattering. The earliest studies used the Born approximation, which means that scattering of scattered waves is neglected. In practice, it leads to accurate descriptions of seismic fields when the seismic wavelength is significantly longer than the topographic perturbation, and the slope of the topography is small in all directions.

Combining the rms map with knowledge of ambient seismicity, it was in fact possible to find many sites fulfilling the two requirements. Figure 50 shows the scattering coefficients for incident Rayleigh waves at a high-rms site in Montana. Excluding the Rayleigh-to-Rayleigh scattering channel (which, as explained in the study, does not increase the complexity of a coherent cancellation), a total integrated scatter of 0.04 was calculated. Including the fact that scattering coefficients for body waves are expected to be higher even, this value is large enough to influence the design of seismic arrays used for noise cancellation. Also, it is important to realize that the seismic field in the vicinity of the surface is poorly represented by the Born approximation (which is better suited to represent the far field produced by topographic scattering), which means that spatial correlation at the site may exhibit more complicated patterns not captured by their study. As a consequence, at a high-rms site a seismic array would likely have to be 3D and relatively dense to observe sufficiently high correlation between seismometers. Heterogeneous ground may further add to the complexity, but we do not have the theoretical framework yet to address this problem quantitatively. For this, it will be important to further develop the scattering formalism introduced in Sect. 3.3.

Underground sites that were and are being studied by GW scientists are all located in high-rms regions. This is true for the sites presented in the ET design study, for the Homestake site that is currently hosting the R&D efforts in the US, and also for the Kamioka site in Japan, which hosts the KAGRA detector. Nonetheless, a careful investigation of spatial correlation and Wiener filtering in high-rms sites has never been carried out, and therefore our understanding of seismic scattering needs to be improved before we can draw final conclusions.

### 7.3 Noise reduction by constructing recess structures or moats

Even though the primary purpose of the recess is not to reflect Rayleigh waves, seismic scattering can be significant. Due to the methods chosen by the authors, scattering could not be simulated, and validity of this approximation had to be explained. Above some frequency, the wavelength is sufficiently small so that scattering from a 4 m deep recess is significant. This regime is marked red in the plot, and the prediction of noise reduction may not be accurate. Above 20 Hz it can be seen that reduction gets weaker. This is because the gravity perturbation starts to be dominated by density perturbations of the central pillar. It is possible that the recess already acts as a moat at these frequencies, and that the central pillar has less seismic noise than simulated in their study. A detailed simulation of scattering from the recess structure using dynamical finite-element methods is necessary to estimate the effect (see Sect. 3.5 for details). The Newtonian-noise spectrum calculated from the reduction curve is shown in the right of Fig. 52. The green curve models the sensitivity of a possible future version of a LIGO detector. Without noise reduction, it would be strongly limited by Newtonian noise. With recess, Newtonian noise only modestly limits the sensitivity and implementation of coherent noise cancellation should provide the missing noise reduction. It is to be expected that the idea of removing mass around test masses only works at the surface. The reason is that seismic speeds are much larger underground (by a factor 10 at least compared to 250 m/s). The idea would be to place test masses at the centers of huge caverns, but Fig. 9 tells us that the radius of such a cavern would have to be extremely large (of the order 100 m for a factor 2 Newtonian-noise reduction at 10 Hz).

### 7.4 Summary and open problems

In this section, we have described Newtonian-noise mitigation schemes including coherent noise cancellation using Wiener filters, and passive mitigation based on recess structures and site selection. While some of the mitigation strategies are well understood (for example, coherent cancellation of Rayleigh-wave Newtonian noise, or site selection with respect to ambient seismicity), others still need to be investigated in more detail. Only first steps were taken to understand the coherent cancellation of Newtonian noise from seismic body waves. For example, the study of cancellation with arrays including seismometers as well as seismic strainmeters or tiltmeters should be investigated. Also, the impact of field anisotropy and heterogeneity needs to be studied. We have also reviewed our current understanding of site-dependent effects on coherent noise cancellation in Sect. 7.2.3, which adds to the complexity of the site-selection process. In this context, sites should be avoided where significant seismic scattering can be expected. This is generally the case in complex topographies typical for mountains. It should be emphasized though that an extensive and conclusive study of the impact of scattering on coherent cancellation has not been carried out so far.

Concerning passive mitigation strategies, site selection is the preferred option and should be part of any design study of future GW detectors. The potential gain in low-frequency noise can be orders of magnitude, which cannot be guaranteed with any other mitigation strategy. This fact is of course well recognized by the community, as demonstrated by the detailed site-selection study for the Einstein Telescope and the fact that it was decided to construct the Japanese GW detector KAGRA underground. Alternative passive mitigation schemes such as the construction of recess structures around test masses are likely effective only at surface sites as explained in Sect. 7.3. The impact of these structures strongly depends on the ratio of structure size to seismic wavelength. Newtonian noise at underground sites is dominated by contributions from body waves, which can have lengths of hundreds of meters even at frequencies as high as 10 Hz. At the surface, smaller-scale structures may turn out to be sufficient since Rayleigh-wave lengths at 10 Hz can be a factor 10 smaller than the lengths of body waves underground. Results from finite-element simulations are indeed promising, and more detailed follow-up investigations should be carried out to identify possible challenges of this approach.

## Footnotes

- 1.
It should be emphasized that in general, the null constraint given by Eq. (5) cannot be obtained from the geodesic equation since the geodesic equation is valid for all freely falling objects (massive and massless). The reason that the null constraint can be derived from Eq. (8) is that we used the null constraint together with the geodesic equation to obtain Eq. (8), which is therefore valid only for massless particles.

- 2.
According to pages 2 and 25 of second attachment to https://alog.ligo-wa.caltech.edu/aLOG/index.php?callRep=6760.

- 3.
Winterflood explains in his thesis why vertical resonance frequencies are higher than horizontal, and why this does not necessarily have to be so (Winterflood 2001).

- 4.
In order to identify components of the metric perturbation with tidal forces acting on test masses, one needs to choose specific spacetime coordinates, the so-called transverse-traceless gauge (Misner et al. 1973).

- 5.
In reality, the dominant damping mechanism in suspension systems is not viscous damping, but structural damping characterized by the so-called loss angle \(\phi \), which quantifies the imaginary part of the elastic modulus (Saulson 1990).

- 6.
It should not be forgotten that thermal noise also plays a role in the other two detector designs, but it is a more severe problem for superconducting gravimeters since the mechanical structure supporting the thermal vibrations is much larger. Any method to lower thermal noise, such as cooling of the structure, or lowering its mechanical loss is a greater effort.

- 7.
- 8.
- 9.
Only at really low frequencies, below 10 mHz, where the finite size of the atmosphere starts to matter, pressure oscillations can be isothermal again.

- 10.
Here, we do not consider using seismic data from a gravimeter for Newtonian-noise cancellation.

- 11.
- 12.

## Notes

### Acknowledgements

I was lucky to have been given the opportunity to enter the field of Newtonian noise and terrestrial gravity perturbations at a time when outstanding experimental problems had to be addressed for future GW detector concepts. I took my first steps in this field as part of the group of Prof Vuk Mandic at the UMN, Twin Cities. I have to thank Prof Mandic for his continuous support, and especially for taking the time to return comments on this manuscript. During my years at UMN, I started to collaborate with Prof Giancarlo Cella, who by that time had already written seminal papers on Newtonian-noise modeling and mitigation. I thank Prof Cella for the many discussions on Newtonian noise, and also for pointing out important past work on Newtonian noise missing in an earlier version of the manuscript. While working on the experimental realization of an underground seismic array at the former Homestake mine in the years 2008–2012, I had the privilege to collaborate with and learn from my colleagues Dr. Riccardo DeSalvo and Dr. Mark Beker. Their direct, uncomplicated approach to solving practical problems has inspired me. I thank both of them for comments and contributions to this manuscript. Starting in 2010, I was given the opportunity at Caltech to apply my experience with seismic fields and gravity modeling to investigate Newtonian noise for the LIGO detectors. I have to thank Prof. Rana Adhikari for supporting me not only with my LIGO work, but also for making sure that I keep an open mind and broad view on science. I am especially thankful that I could work with one of Prof. Adhikari’s graduate students, (now Dr.) Jennifer Driggers, with whom I was able to lie the foundation for future work on Newtonian noise at the LIGO sites. I thank Jenne for her dedication and for keeping me focussed on the important problems. Afterwards, much of my Newtonian-noise work was done in a highly creative collaboration with Dr. Michael Coughlin producing the foundation of almost all available codes to model Newtonian noise and its cancellation. I have to thank Michael for his endless patience and productive attitude over so many years. In 2017, I started to work at the Gran Sasso Science Institute (GSSI) in l’Aquila. L’Aquila appeared in the news all over the world on April 6, 2009 when a destructive earthquake hit the city. The opportunity given to this region with the help of European countries and important Italian institutions like INFN to build a center of academic excellence attracting top-level students from all over the world is currently propelling my Newtonian-noise studies, and I am highly thankful for the support of my colleagues at GSSI. As I could hopefully demonstrate in this paper, terrestrial gravity perturbations is a complex problem, which means that observations in the future should be expected to hold surprises for us and unexpected applications may emerge. Last but not least, I want to thank Marica Branchesi who made sure that I never lose motivation to write this article, and whose dedication to science and people is always an inspiration to me. I acknowledge the use of *Mathematica* and *Matlab* for the generation of the plots in this paper, and as a help with some analytical studies.

## Supplementary material

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