# Terrestrial Gravity Fluctuations

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## Abstract

Different forms of fluctuations of the terrestrial gravity field are observed by gravity experiments. For example, atmospheric pressure fluctuations generate a gravity-noise foreground in measurements with super-conducting gravimeters. Gravity changes caused by high-magnitude earthquakes have been detected with the satellite gravity experiment GRACE, and we expect high-frequency terrestrial gravity fluctuations produced by ambient seismic fields to limit the sensitivity of ground-based gravitational-wave (GW) detectors. Accordingly, terrestrial gravity fluctuations are considered noise and signal depending on the experiment. Here, we will focus on ground-based gravimetry. This field is rapidly progressing through the development of GW detectors. The technology is pushed to its current limits in the advanced generation of the LIGO and Virgo detectors, targeting gravity strain sensitivities better than 10^{−23} Hz^{−1/2} above a few tens of a Hz. Alternative designs for GW detectors evolving from traditional gravity gradiometers such as torsion bars, atom interferometers, and superconducting gradiometers are currently being developed to extend the detection band to frequencies below 1 Hz. The goal of this article is to provide the analytical framework to describe terrestrial gravity perturbations in these experiments. Models of terrestrial gravity perturbations related to seismic fields, atmospheric disturbances, and vibrating, rotating or moving objects, are derived and analyzed. The models are then used to evaluate passive and active gravity noise mitigation strategies in GW detectors, or alternatively, to describe their potential use in geophysics. The article reviews the current state of the field, and also presents new analyses especially with respect to the impact of seismic scattering on gravity perturbations, active gravity noise cancellation, and time-domain models of gravity perturbations from atmospheric and seismic point sources. Our understanding of terrestrial gravity fluctuations will have great impact on the future development of GW detectors and high-precision gravimetry in general, and many open questions need to be answered still as emphasized in this article.

### Keywords

Terrestrial gravity Newtonian noise Wiener filter Mitigation### Notation

*c*= 299792458 m/sspeed of light

*G*= 6.674 × 10^{−11}Nm^{2}/kg^{2}gravitational constant

- \(\vec{r}, \, {\vec{e}_r}\)
position vector, and corresponding unit vector

*x*,*y*,*z*Cartesian coordinates

*r*,*θ*,*Φ*spherical coordinates

*ϱ*,*Φ, z*cylindrical coordinates

- dΩ ≡ d
*Φ dθ*sin(*θ*) solid angle

- δ
_{ij} Kronecker delta

*δ*(·)Dirac

*δ*distribution- ℜ
real part of a complex number

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

*∂*_{x},*∂*_{y},*∂*_{z})- \(\vec{\xi} (\vec{r}, \, t)\)
displacement field

- \({\phi _{\rm{s}}}(\vec{r}, \, t)\)
potential of seismic compressional waves

- \({\vec{\psi}_{\rm{s}}}(\vec{r}, \, t)\)
potential of seismic shear waves

*ρ*_{0}time-averaged mass density

- α, β
compressional-wave and shear-wave speed

- μ
shear modulus

- ⊗
dyadic product

**M**, \(\vec{\upsilon}\)*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

- \(h_n^{(2)}(x)\)
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 coming years, we will see a transition in the field of high-precision gravimetry from observations of slow lasting changes of the gravity field to the experimental study of fast gravity fluctuations. The latter will be realized by the advanced generation of the US-based LIGO [1] and Europe-based Virgo [7] gravitational-wave (GW) detectors. Their goal is to directly observe for the first time GWs that are produced by astrophysical sources such as inspiraling and merging neutron-star or black-hole binaries. Feasibility of the laser-interferometric detector concept has been demonstrated successfully with the first generation of detectors, which, in addition to the initial LIGO and Virgo detectors, also includes the GEO600 [119] and TAMA300 [161] detectors, and several prototypes around the world. The impact of these projects onto the field is two-fold. First of all, the direct detection of GWs will be a milestone in science opening a new window to our universe, and marking the beginning of a new era in observational astronomy. Second, several groups around the world have already started to adapt the technology to novel interferometer concepts [60, 155], with potential applications not only in GW science, but also geophysics. The basic measurement scheme is always the same: the relative displacement of test masses is monitored by using ultra-stable lasers. Progress in this field is strongly dependent on how well the motion of the test masses can be shielded from the environment. Test masses are placed in vacuum and are either freely falling (e.g., atom clouds [137]), or suspended and seismically isolated (e.g., high-quality glass or crystal mirrors as used in all of the detectors listed above). The best seismic isolations realized so far are effective above a few Hz, which limits the frequency range of detectable gravity fluctuations. Nonetheless, low-frequency concepts are continuously improving, and it is conceivable that future detectors will be sufficiently sensitive to detect GWs well below a Hz [88].

Terrestrial gravity perturbations were identified as a potential noise source already in the first concept laid out for a laser-interferometric GW detector [171]. Today, this form of noise is known as “terrestrial gravitational noise”, “Newtonian noise”, or “gravity-gradient noise”. It has never been observed in GW detectors, but it is predicted to limit the sensitivity of the advanced GW detectors at low frequencies. The most important source of gravity noise comes from fluctuating seismic fields [151]. Gravity perturbations from atmospheric disturbances such as pressure and temperature fluctuations can become significant at lower frequencies [51]. Anthropogenic sources of gravity perturbations are easier to avoid, but could also be relevant at lower frequencies [163]. Today, we only have one example of a direct observation of gravity fluctuations, i.e., from pressure fluctuations of the atmosphere in high-precision gravimeters [128]. Therefore, almost our entire understanding of gravity fluctuations is based on models. Nonetheless, potential sensitivity limits of future large-scale GW detectors need to be identified and characterized well in advance, and so there is a need to continuously improve our understanding of terrestrial gravity noise. Based on our current understanding, the preferred option is to construct future GW detectors underground to avoid the most dominant Newtonian-noise contributions. This choice was made for the next-generation Japanese GW detector KAGRA, which is currently being constructed underground at the Kamioka site [17], and also as part of a design study for the Einstein Telescope in Europe [140, 139]. While the benefit from underground construction with respect to gravity noise is expected to be substantial in GW detectors sensitive above a few Hz [27], it can be argued that it is less effective at lower frequencies [88].

Alternative mitigation strategies includes coherent noise cancellation [42]. 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. This technique is successfully applied in gravimeters to reduce the foreground of atmospheric gravity noise using collocated pressure sensors [128]. It is also noteworthy that the models of the atmospheric gravity noise are consistent with observations. This should give us some confidence at least that coherent Newtonian-noise cancellation can also be achieved in GW detectors. It is evident though that a model-based prediction of the performance of coherent noise cancellation schemes is prone to systematic errors as long as the properties of the sources are not fully understood. Ongoing experiments at the Sanford Underground Research Facility with the goal to characterize seismic fields in three dimensions are expected to deliver first data from an underground seismometer array in 2015 (see [89] for results from an initial stage of the experiment). While most people would argue that constructing GW detectors underground is always advantageous, it is still necessary to estimate how much is gained and whether the science case strongly profits from it. This is a complicated problem that needs to be answered as part of a site selection process.

More recently, high-precision gravity strainmeters have been considered as monitors of geophysical signals [83]. Analytical models have been calculated, which allow us to predict gravity transients from seismic sources such as earthquakes. It was suggested to implement gravity strainmeters in existing earthquake-early warning systems to increase warning times. It is also conceivable that an alternative method to estimate source parameters using gravity signals will improve our understanding of seismic sources. Potential applications must still be investigated in greater detail, but the study already demonstrates that the idea to use GW technology to realize new geophysical sensors seems feasible. As explained in [49], gravitational forces start to dominate the dynamics of seismic phenomena below about 1 mHz (which coincides approximately with a similar transition in atmospheric dynamics where gravity waves start to dominate over other forms of oscillations [164]). Seismic isolation would be ineffective below 1 mHz since the gravitational acceleration of a test mass produced by seismic displacement becomes comparable to the seismic acceleration itself. Therefore, we claim that 10 mHz is about the lowest frequency at which ground-based gravity strainmeters will ever be able to detect GWs, and consequently, modelling terrestrial gravity perturbations in these detectors can focus on frequencies above 10 mHz.

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 Section 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 Section 4. A theoretical framework to calculate gravity perturbations from objects is given in Section 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, Section 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 new analyses especially with respect to the impact of seismic scattering on gravity perturbations (Sections 3.3.2 and 3.3.3), active gravity noise cancellation (Section 7.1.3), and timedomain models of gravity perturbations from atmospheric and seismic point sources (Sections 4.1, 4.5, and 5.3).

*m*

_{1},

*m*

_{2}is given by

*G*= 6.672 × 10

^{−11}N m

^{2}/kg

^{2}is the gravitational constant. Eq. (1) gives rise to many complex phenomena on Earth such as inner-core oscillations [156], atmospheric gravity waves [157], ocean waves [94, 177], and co-seismic gravity changes [122]. Due to its importance, we will honor the eponym by referring to gravity noise as Newtonian noise in the following. It is thereby clarified that the gravity noise models considered in this article are non-relativistic, and propagation effects of gravity changes are neglected. While there could be interesting scenarios where this approximation is not fully justified (e.g., whenever a gravity perturbation can be sensed by several sensors and differences in arrival times can be resolved), it certainly holds in any of the problems discussed in this article. We now invite the reader to enjoy the rest of the article, and hope that it proves to be useful.

## 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 [53, 181], the gravity gradiometer [125], and the gravity strainmeter. The last category includes the large-scale GW detectors Virgo [6], LIGO [91], GEO600 [119], KAGRA [17], and a new generation of torsion-bar antennas currently under development [13]. Also atom interferometers can potentially be used as gravity strainmeters in the future [62]. 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. Later on in this article, we almost exclusively discuss gravity models relevant to gravity strainmeters since the focus lies on gravity fluctuations above 10 mHz. Today, the sensitivity near 10 mHz of gravimeters towards gravity fluctuations is still competitive to or exceeds the sensitivity of gravity strainmeters, but this is likely going to change in the future so that we can expect strainmeters to become the technology of choice for gravity observations above 10 mHz [88]. The following sections provide further details on this statement. Space-borne gravity experiments such as GRACE [167] will not be included in this overview. The measurement principle of GRACE is similar to that of gravity strainmeters, but only very slow changes of Earth gravity field can be observed, and for this reason it is beyond the scope of this article.

The different response mechanisms to terrestrial gravity perturbations are summarized in Section 2.1. While we will identify the tidal forces acting on the test masses as dominant coupling mechanism, other couplings may well be relevant depending on the experiment. The Shapiro time delay will be discussed as the only relativistic effect. Higher-order relativistic effects are neglected. All other coupling mechanisms can be calculated using Newtonian theory including tidal forces, coupling in static non-uniform gravity fields, and coupling through ground displacement induced by gravity fluctuations. In Sections 2.2 to 2.4, the different measurement schemes are explained including a brief summary of the sensitivity limitations (choosing one of a few possible experimental realizations in each case). As mentioned before, we will mostly develop gravity models relevant to gravity strainmeters in the remainder of the article. Therefore, the detailed discussion of alternative gravimetry concepts mostly serves to highlight important differences between these concepts, and to develop a deeper understanding of the instruments and their role in gravity measurements.

### 2.1 Gravity response mechanisms

#### 2.1.1 Gravity acceleration and tidal forces

*δg*(ω) and induced test mass acceleration

*δa*(

*ω*) assumes the form

*γ*, and

*ω*

_{0}is the resonance frequency. Well below resonance, the response is proportional to

*ω*

^{2}, while it is constant well above resonance. Above resonance, the supported test mass responds like a freely falling mass, at least with respect to “soft” directions of the support. The test-mass response to vibrations

*δα*(

*ω*) of the support is given by

*ω*

^{−2}, while no vibration isolation is provided below resonance. The situation is somewhat more complicated in realistic models of the support especially due to internal modes of the mechanical system (see for example [76]), or due to coupling of degrees of freedom [121]. Large mechanical support structures can feature internal resonances at relatively low frequencies, which can interfere to some extent with the desired performance of the mechanical support [173]. While Eqs. (2) and (3) summarize the properties of isolation and response relevant for this paper, details of the readout method can fundamentally impact an instrument’s response to gravity fluctuations and its susceptibility to seismic noise, as explained in Sections 2.2 to 2.4.

*δg*

_{12}(

*ω*) between two freely falling test masses according to

*r*

_{12}between the test masses is sufficiently small, which also depends on the frequency. The term \(\nabla \otimes \nabla \psi (\vec{r}, \, t)\) is called gravity-gradient tensor. In Newtonian approximation, the second time integral of this tensor corresponds to gravity strain \({\bf{h}}(\vec{r}, \, t)\), which is discussed in more detail in Section 2.4. Its trace needs to vanish in empty space since the gravity potential fulfills the Poisson equation. Tidal forces produce the dominant signals in gravity gradiometers and gravity strainmeters, which measure the differential acceleration or associated relative displacement between two test masses (see Sections 2.3 and 2.4). If the test masses used for a tidal measurement are supported, then typically the supports are designed to be as similar as possible, so that the response in Eq. (2) holds for both test masses approximately with the same parameter values for the resonance frequencies (and to a lesser extent also for the damping). For the purpose of response calibration, it is less important to know the parameter values exactly if the signal is meant to be observed well above the resonance frequency where the response is approximately equal to 1 independent of the resonance frequency and damping (here, “well above” resonance also depends on the damping parameter, and in realistic models, the signal frequency also needs to be “well below” internal resonances of the mechanical support).

#### 2.1.2 Shapiro time delay

*c*is the speed of light, d

*s*is the so-called line element of a path in spacetime, and \(\psi (\vec{r}, \, t)/{c^2} \ll 1\). 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 d

*s*

^{2}= 0. For the spacetime metric in Eq. (5), we can immediately write

*t*, \(\vec{r}\) in terms of a parameter

*λ*. The weak-field geodesic equation is obtained from the metric in Eq. (5):

*λ*. Since the deviation of a straight path is due to a weak gravity potential, we can solve these equations by perturbation theory introducing expansions \(\vec{r} = {\vec{r}^{(0)}} + {\vec{r}^{(1)}} + \ldots\) and

*t*=

*t*

^{(0)}+

*t*

^{(1)}+ …. The superscript indicates the order in

*ψ*/

*c*

^{2}. The unperturbed path has the simple parametrization

*t*

^{(0)}and parameter

*λ*can be used interchangeably (apart from a shift by

*t*

_{0}). Inserting these expressions into the right-hand side of Eq. (7), we obtain

^{1}. In other words, we can integrate the time delay along a straight line as defined in Eq. (8), and so the total phase integrated over a travel distance

*L*is given by

*t*/d

*λ*.

#### 2.1.3 Gravity induced ground motion

As we will learn in Section 3, seismic fields produce gravity perturbations either through density fluctuations of the ground, or by displacing interfaces between two materials of different density. It is also well-known in seismology that seismic fields can be affected significantly by self-gravity. Self-gravity means that the gravity perturbation produced by a seismic field acts back on the seismic field. The effect is most significant at low frequency where gravity induced acceleration competes against acceleration from elastic forces. In seismology, low-frequency seismic fields are best described in terms of Earth’s normal modes [55]. Normal modes exist as toroidal modes and spheroidal modes. Spheroidal modes are influenced by self-gravity, toroidal modes are not. For example, predictions of frequencies and shapes of spheroidal modes based on Earth models such as PREM (Preliminary Reference Earth Model) [68] are inaccurate if self-gravity effects are excluded. What this practically means is that in addition to displacement amplitudes, gravity becomes a dynamical variable in the elastodynamic equations that determine the normal-mode properties. Therefore, seismic displacement and gravity perturbation cannot be separated in normal-mode formalism (although self-gravity can be neglected in calculations of spheroidal modes at sufficiently high frequency).

In certain situations, it is necessary or at least more intuitive to separate gravity from seismic fields. An exotic example is Earth’s response to GWs [67, 49, 47, 30, 48]. Another example is the seismic response to gravity perturbations produced by strong seismic events at large distance to the source as described in Section 4. It is more challenging to analyze this scenario using normal-mode formalism. The sum over all normal modes excited by the seismic event (each of which describing a global displacement field) must lead to destructive interference of seismic displacement at large distances (where seismic waves have not yet arrived), but not of the gravity amplitudes since gravity is immediately perturbed everywhere. It can be easier to first calculate the gravity perturbation from the seismic perturbation, and then to calculate the response of the seismic field to the gravity perturbation at larger distance. This method will be adopted in this section. Gravity fields will be represented as arbitrary force or tidal fields (detailed models are presented in later sections), and we simply calculate the response of the seismic field. Normal-mode formalism can be avoided only at sufficiently high frequencies where the curvature of Earth does not significantly influence the response (i.e., well above 10 mHz). In this section, we will model the ground as homogeneous half space, but also more complex geologies can in principle be assumed.

*ψ*[146, 169], or as tidal strain The latter method was described first by Dyson to calculate Earth’s response to GWs [67]. The approach also works for Newtonian gravity, with the difference that the tidal field produced by a GW is necessarily a quadrupole field with only two degrees of freedom (polarizations), while tidal fields produced by terrestrial sources are less constrained. Certainly, GWs can only be fully described in the framework of general relativity, which means that their representation as a Newtonian tidal field cannot be used to explain all possible observations [124]. Nonetheless, important here is that Dyson’s method can be extended to Newtonian tidal fields. Without gravity, the elastodynamic equations for small seismic displacement can be written as

*λ*,

*μ*are the Lamé constants (see Section 3.1). Its trace is equal to the divergence of the displacement field. Dyson introduced the tidal field from first principles using Lagrangian mechanics, but we can follow a simpler approach. Eq. (12) means that a stress field builds up in response to a seismic strain field, and the divergence of the stress field acts as a force producing seismic displacement. The same happens in response to a tidal field, which we represent as gravity strain \({\bf{h}}(\vec{r}, \, t)\). A strain field changes the distance between two freely falling test masses separated by \(\vec{L}\) by \(\delta \vec{L}(\vec{r}, \, t) = {\bf{h}}(\vec{r}, \, t) \cdot \vec{L}\)

^{2}. For sufficiently small distances

*L*, the strain field can be substituted by the second time integral of the gravity-gradient tensor \(\nabla \otimes \nabla \psi (\vec{r}, \, t)\). If the masses are not freely falling, then the strain field acts as an additional force. The corresponding contribution to the material’s stress tensor can be written

*z*= 0 with unit normal vector \({\vec{e}_n}\). The response to gravity strain fields is obtained applying the boundary condition of vanishing surface traction, \({\vec{e}_n} \cdot \sigma (\vec{r},t) = 0\):

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

*λ*, over which gravity acceleration varies significantly. Hence, we can rewrite the last equation in terms of the associated test-mass displacement

*ζ*

*z*-axis) seismic noise

*ξ*

_{z}(

*t*) coupling into the horizontal (

*x*-axis) motion of the test mass through the term

*∂*

_{x}

*g*

_{z}=

*∂*

_{z}

*g*

_{x}dominates over the gravity response term in Eq. (2). Due to additional coupling mechanisms between vertical and horizontal motion in real seismic-isolation systems, test masses especially in GW detectors are also isolated in vertical direction, but without achieving the same noise suppression as in horizontal direction. For example, the requirements on vertical test-mass displacement for Advanced LIGO are a factor 1000 less stringent than on the horizontal displacement [22]. Requirements can be set on the vertical isolation by estimating the coupling of vertical motion into horizontal motion, which needs to take the gravity-gradient coupling of Eq. (16) into account. Although, because of the frequency dependence, gravity-gradient effects are more significant in low-frequency detectors, such as the space-borne GW detector LISA [154].

Next, we calculate an estimate of gravity gradients in the vicinity of test masses in large-scale GW detectors, and see if the gravity-gradient coupling matters compared to mechanical vertical-to-horizontal coupling.

*∂*

_{z}

*g*

_{x}vanish if the test mass is located on the symmetry axis or at height

*L*/2. There are also two additional

*∂*

_{z}

*g*

_{x}= 0 contour lines starting at the symmetry axis at heights ∼ 0.24 and ∼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 kg, and the cylinder height is

*L*= 4 m. In this case, the gravity-gradient induced vertical-to-horizontal coupling factor at 20 Hz is

^{3}). Even though the vacuum chamber was modelled 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 of course to have a 3D model of the near test-mass infrastructure available and to use it for a precise calculation of the gravity-gradient field.

### 2.2 Gravimeters

Let us take a closer look at the basic measurement scheme of a superconducting gravimeter shown in Figure 2. The central part is formed by a spherical superconducting shell that is levitated by superconducting coils. Superconductivity provides stability of the measurement, and also avoids some forms of noise (see [96] for details). In this gravimeter design, the lower coil is responsible mostly to balance the mean gravitational force acting on the sphere, while the upper coil modifies the magnetic gradient such that a certain “spring constant” of the magnetic levitation is realized. In other words, the current in the upper coil determines the resonance frequency in Eq. (2).

*ξ*(

*t*) of a levitated sphere against a static gravity gradient (see Section 2.1.4). As explained above, feedback control suppresses relative motion between sphere and gravimeter frame, which causes the sphere to move as if attached to the frame or ground. In the presence of a static gravity gradient

*∂*

_{z}

*g*

_{z}, the motion of the sphere against this gradient leads to a change in gravity, which alters the feedback force (and therefore the recorded signal). The full contribution from gravitational,

*δa*(

*t*), and seismic, \(\ddot{\xi}(t) = \delta \alpha (t)\), accelerations can therefore be written

*additional*noise due to non-linearities and cross-coupling. As is explained further in Section 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.

Gravimeters target signals that typically lie well below 1 mHz. Mechanical or magnetic supports of test masses have resonance frequencies at best slightly below 10 mHz along horizontal directions, and typically above 0.1 Hz in the vertical direction [23, 174]^{4}. 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 was 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).

^{−12}m/s

^{2}. A detailed study of noise in superconducting gravimeters over a larger frequency range can be found in [145]. 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. (19) have to be understood only to accurately translate data into normal-mode amplitudes [55].

### 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 [102, 125]. The test masses have to be located close to each other so that the approximation in Eq. (4) 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 Section 2.4), which denotes any type of instrument that measures relative gravitational acceleration (including the even more general concept of measuring space-time strain).

Gravity gradiometers can be realized in two versions. First, one can read out the position of two test masses with respect to the same rigid, non-inertial reference. The two channels, each of which can be considered a gravimeter, are subsequently subtracted. This scheme is for example realized in dual-sphere designs of superconducting gravity gradiometers [90] or in atom-interferometric gravity gradiometers [159].

*δα*(

*ω*) of the optics or laser source limits the sensitivity of a tidal measurement according to

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

*ωL/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. (20) 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:

*ω*

_{1},

*γ*

_{1}are the resonance frequency and damping of the gravimeter support, while

*ω*

_{2},

*γ*

_{2}are the resonance frequency and damping of the test-mass support. The response and isolation functions

*R*(·),

*S*(·) are defined in Eqs. (2) and (3). Remember that Eq. (21) is obtained as a differential measurement of test-mass acceleration versus acceleration of the reference frame. Therefore,

*δg*

_{1}(

*ω*) denotes the gravity fluctuation at the center-of-mass of the gravimeter frame, and

*δg*

_{2}(

*ω*) at the test mass. An infinitely stiff gravimeter suspension,

*ω*

_{1}→ ∞, yields R(

*ω*;

*ω*

_{1},

*γ*

_{1}) = 0, and the response turns into the form of the non-isolated gravimeter. The seismic isolation is determined by

*ω*

_{1}, the seismically isolated gravimeter responds like a gravity gradiometer, and seismic noise is strongly suppressed. The deviation from the pure gradiometer response ∼

*δg*

_{2}(

*ω*) −

*δg*

_{1}(

*ω*) is determined by the same function

*S*(

*ω*;

*ω*

_{1},

*γ*

_{1}) that describes the seismic isolation. In other words, if the gravity gradient was negligible, then we ended up with the conventional gravimeter response, with signals suppressed by the seismic isolation function. Well below

*ω*

_{1}, the seismically isolated gravimeter responds like a conventional gravimeter without seismic-noise reduction. If the centers of the masses

*m*(test mass) and

*M*(reference frame) coincide, and therefore

*δg*

_{1}(

*ω*) =

*δg*

_{2}(

*ω*), then the response is again like a conventional gravimeter, but this time suppressed by the isolation function S(

*ω*;

*ω*

_{1},

*γ*

_{1}).

*δg*

_{1}(

*ω*) −

*δα*(

*ω*), i.e., using a single test mass for acceleration measurements, seismic and gravity perturbations contribute in the same way. A transfer function needs to be multiplied to the acceleration signals, which accounts for the mechanical support and possibly also electronic circuits involved in the seismometer readout. To cancel the seismic noise of the platform that carries the gravimeter, the effect of all transfer functions needs to be reversed by a matched feed-forward filter. The output of the filter is then equal to

*δg*

_{1}(

*ω*) −

*δα*(

*ω*) and is added to the motion of the platform using actuators cancelling the seismic noise and adding the seismometer’s gravity signal. In this case, the seismometer’s gravity signal takes the place of the seismic noise in Eq. (3). The complete gravity response of the actively isolated gravimeter then reads

*ω*

_{2},

*γ*

_{2}are the resonance frequency and damping of the gravimeter’s test-mass support. In reality, instrumental noise of the seismometer will limit the isolation performance and introduce additional noise into Eq. (23). Nonetheless, Eqs. (21) and (23) show that any form of seismic isolation turns a gravimeter into a gravity gradiometer at frequencies where seismic isolation is effective. For the passive seismic isolation, this means that the gravimeter responds like a gradiometer at frequencies well above the resonance frequency

*ω*

_{1}of the gravimeter support, while it behaves like a conventional gravimeter below

*ω*

_{1}. From these results it is clear that the design of seismic isolations and the gravity response can in general not be treated independently. As we will see in Section 2.4 though, tidal measurements can profit strongly from seismic isolation especially when common-mode suppression of seismic noise like in gradiometers is insufficient or completely absent.

### 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 [106]. Fundamentally, gravity strain corresponds to a perturbation of the metric that determines the geometrical properties of spacetime [124]. We will briefly discuss GWs, before returning to a Newtonian description of gravity strain.

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

^{5}. Note that the strain amplitude

**h**in Eq. (24) 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*= Δ

*L/L, while the effect of a GW with amplitude a*

_{GW}

*on the separation of two test mass is determined by a*

_{GW}=

*2*Δ

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

*x*,

*y*-axes are oriented along two perpendicular baselines (arms) of the detector. The two schemes are shown in Figure 6. The most sensitive GW detectors are based on the conventional method, and distance between test masses is measured by means of laser interferometry. The LIGO and Virgo detectors have achieved strain sensitivities of better than 10

^{−22}Hz

^{−1/2}between about 50 Hz and 1000 Hz in past science runs and are currently being commissioned in their advanced configurations [91, 7]. The rotational scheme is realized in torsion-bar antennas, which are considered as possible technology for sub-Hz GW detection [155, 69]. However, with achieved strain sensitivity of about 10

^{−8}Hz

^{−1/2}near 0.1 Hz, the torsion-bar detectors are far from the sensitivity we expect to be necessary for GW detection [88].

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 Section 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 Section 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. (4) does not apply. Certainly, the distinction between gravity gradiometers and strainmeters remains somewhat arbitrary since at any frequency the approximation in Eq. (4) 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. Section 2.1.1 introduced a simplified response and isolation model based on a harmonic oscillator characterized by a resonance frequency *ω*_{0} and viscous damping *γ*^{6}. In a multi-stage isolation and suspension system as realized in GW detectors (see for example [37, 121]), 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. (2) and (3). 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} 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 Section 2.1.3, and from the Shapiro time delay as described in Section 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. (2) and (3), 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. (4). 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 [155], 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 [43]. 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 *ξ*(*ω*), then the corresponding strain noise in atom-interferometric strainmeters is of order *ωξ*(*ω*)/*c*, where *c* is the speed of light, and *ω* the signal frequency [18]. While this noise is lower than the corresponding term *ξ*(*ω*)/*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 *ωL*/*c* compared to the contribution from distance changes between atom clouds. Here, 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 *ωξ*(*ω*)/*c* ∼ 10^{−17} Hz^{−1/2} at 0.1 Hz without seismic isolation (too high at least for GW detection [88]), but also to suppress seismic-noise contributions through additional channels (e.g., tilting optics in combination with laser-wavefront aberrations [97]). The additional channels dominate in current experiments, which are already seismic-noise limited with strain noise many orders of magnitude higher than 10^{−17} Hz^{−1/2} [60]. 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 Section 2.2). Subtracting local readouts of test-mass displacement from each other constitutes the basic strain-meter scheme [135]. 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^{7} (similar to the situation with torsion-bar antennas [88]). 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 Section 2.3). Another solution could be to substitute the mechanical structure by an optically rigid body as suggested in [88] 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 [16, 54]. 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 [125].

### 2.5 Summary

Summary of gravity sensors

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

Gravimeter | Differential displacement between ground and test mass | Superconducting gravimeters: \(\sim 1\;{\rm{nm/}}{{\rm{s}}^2}/\sqrt {{\rm{Hz}}}\) 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: \(\sim {10^{- 9}}{\rm{/}}{{\rm{s}}^2}/\sqrt {{\rm{Hz}}}\) between 0 Hz 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): \(\sim {10^{- 23}}{\rm{/}}\sqrt {{\rm{Hz}}}\) at 100 Hz |

Superconducting strainmeters: \(\sim {10^{- 10}}{\rm{/}}\sqrt {{\rm{Hz}}}\) 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 [171]. 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 [151]. 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 [99, 24]. 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 Section 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 Sections 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 [85, 26]. We will give a brief summary in Section 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 [83]. Since this work also inspired potential applications in geophysics and seismology, we devote Section 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 Section 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, for example Aki & Richards [9]. 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 [55]. 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

*ω*and a wave vector \({\vec{k}^{\rm{P}}}\). While one typically assumes

*ω*=

*k*

^{P}

*α*with compressional wave speed

*α*, 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

*λ*,

*μ*:

*μ*is also known as shear modulus, and

*ρ*is the density of the medium. Shear waves produce transversal displacement and do not exist in media with vanishing shear modulus. They are also known as “S-waves”, where S stands for

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

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

*ν*is the Poisson’s ratio of the medium. It should be mentioned that there are situations when a wave field cannot be described as a superposition of compressional and shear waves. This is for example the case in the near field of a seismic source. In the remainder of this section, we will calculate gravity perturbations for cases where the distinction between compressional and shear waves is meaningful. The more complicated case of gravity perturbations from seismic fields near their sources is considered in Section 4.

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

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

*k*

_{ϱ}to be larger than

*k*

^{P}and

*k*

^{S}. In this case, the vertical wavenumbers have imaginary values. Hence, in the case of Rayleigh waves, it is convenient to define new wave parameters as:

*k*

_{ϱ}is the horizontal wavenumber of the Rayleigh wave. Note that the order of terms in the square-roots are reversed with respect to the case of body waves as in Eq. (34). Rewriting the equations in [92] in terms of the horizontal and vertical wavenumbers, the horizontal and vertical amplitudes of the three-dimensional displacement field of a Rayleigh wave reads

*c*

_{R}=

*k*

_{ϱ}/

*ω*of the fundamental Rayleigh wave obeys the equation

*higher-order Rayleigh waves*that can exist in these media [99]. For this reason, we will occasionally refer to Rayleigh waves as Rf-waves. According to Eqs. (31) & (37), given a shear-wave speed

*β*, the compressional-wave speed

*α*and Rayleigh-wave speed

*c*

_{R}are functions of the Poisson’s ratio only. Figure 8 shows the values of the wave speeds in units of

*β*. 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

*ν*= 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 \(\vec \xi (\vec{r},t)\), then in terms of seismic potentials \({\phi _{\rm{s}}}(\vec{r}, \, t), \, {\vec \psi _{\rm{s}}}(\vec{r}, \, t)\), and this section concludes with a discussion of gravity perturbations in transform domain.

#### 3.2.1 Gravity perturbations from seismic displacement

_{0}denotes the gradient operation with respect to \({\vec{r}_0}\). In this form, it is straight-forward to implement gravity perturbations in finite-element simulations (see Section 3.5), where each finite element is given a mass \(\rho (\vec{r})\delta V\). This equation is valid whenever the continuity Eq. (39) holds, and describes gravity perturbations inside infinite media as well as media with surfaces.

*ρ*

_{0}, the density of the entire space can be written as

#### 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 \(\phi_{\rm{s}}(\vec r,t)\), which cannot be measured or inferred in general from measurements. The shear-wave potential enters as \(\nabla\times\vec\psi_{\rm{s}}(\vec r,t)\), 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 \(\vec\xi(\vec k_\varrho,z,t)\), and in infinite space in terms of \(\vec\xi(\vec{k},t)\). As shown in Section 4.5, 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*= 0, are obtained by calculating the Fourier transforms of Eqs. (43) and (44) with respect to

*x*

_{0},

*y*

_{0}. This yields the bulk term

*z*is straight-forward to calculate.

*Φ*

_{s}(

*z*> 0) = 0, and dependence of the potentials on \(\vec k_\varrho\) and

*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.

*x*

_{0},

*y*

_{0}and

*z*

_{0}. Since there are no surface terms, the result is simply

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

Test masses of underground detectors, as for example KAGRA [17], 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. [84]. 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 Section 3.3.1, and present for the first time a calculation of gravity perturbations from seismic waves scattered from a spherical cavity in Sections 3.3.2 and 3.3.3.

#### 3.3.1 Gravity perturbations without scattering

*j*

_{n}(·) is the spherical Bessel function. In the case that the length of the seismic wave is much larger than the cavity radius, the ratio can be approximated according to

*ka*)

^{2}is made in the modelling of the gravity perturbation. The numerical factor in this equation is smaller by −4

*π*/3 compared to the factor in Eq. (52). This means that the bulk gravity perturbation is partially cancelled by cavity-surface contributions, which can be verified by directly evaluating the surface term:

*ka*→ 0 of the expression in brackets is 1/3, which is consistent with Eqs. (60) and (52). If the seismic field consisted only of pressure waves propagating in a homogeneous medium, then Eq. (60) would mean that a seismometer placed at the test mass monitors all information required to estimate the corresponding gravity perturbations.

If the cavity has a radius of about 0.4*λ*, 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

*k*

_{P},

*k*

_{S}are the wave numbers of compressional and shear waves respectively,

*θ*is the angle between the direction of propagation of the scattered wave with respect to the direction of the incident compressional wave,

*ξ*

_{0}is the displacement amplitude of the incoming compressional wave, and the origin of the coordinate system lies at the center of the cavity. The spherical Hankel functions of the second kind \(h_n^{(2)}(\cdot)\) are defined in terms of the spherical Bessel functions of the first and second kind as:

*A*

_{l},

*B*

_{l}; need to be determined from boundary conditions at the cavity surface, which was presented in detail in [176]. Here we just mention that for small cavities, i.e., in the Rayleigh-scattering regime with {

*k*

_{P},

*k*

_{S}} ·

*a*≪ 1, the dependence of the scattering coefficients on the cavity radius

*a*is (

*k*

_{P}

*a*)

^{3}or higher order.

*u*≡ cos(

*θ*). If the gravity perturbations are to be calculated at the center \(\vec r_0=\vec 0\) of the spherical cavity, then the integrals are easily evaluated by substituting powers of cos(

*θ*) according to the right-hand-side of Table 3, and making use of the orthogonality relation in Eq. (248). We first outline the calculation for the bulk term. Identifying the

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

*x, y*vanish. Also the surface contribution is readily obtained with integration by parts:

*x, y*vanish. Adding the bulk and surface contributions, we finally obtain

*k*

_{P},

*k*

_{S}} ·

*a*≪ 1 using the following approximations of the scattering coefficients

*A*

_{1}(

*a*),

*B*

_{1}(

*a*) given in [176]:

*a*→ 0, which may seem intuitive, but notice that the surface contribution of the incoming wave does not vanish in the same limit. Instead, it is a consequence of perfect cancellation of leading order terms from scattered shear and compressional waves. Therefore, this result shows explicitly that neglecting contributions from scattered waves has no influence on leading order terms of the full gravity perturbation, at least if the incident wave is of compressional type.

#### 3.3.3 Incident shear wave

*y*

_{lm},

*s*

_{lm}in Eq. (C.2) of [111], but it should also be clear from symmetry considerations that gravity perturbation can be non-zero only along the displacement direction of the incident wave. The term under the last integral takes the form

*l*= 1 scattered

*P*-wave to the

*l*= 1 amplitude of the incident S-wave. It can be calculated using equations from [111] (note that the explicit solutions given in the appendix are wrong). Inserting this expression into the last equation, we finally obtain

*k*

^{P}

*a*≪ 1 and

*k*

^{S}

*a*≪ 1, the scattering coefficients can be expanded according to

*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 Section 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 \(\vec r\) above surface, \(\vec\varrho\) being the projection of \(\vec r\) onto the surface, and \(\vec k_\varrho\) 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*= 0), then the total half-space gravity perturbation is simply half of the infinite-space perturbation.

*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 \(\vec k_\varrho\) couple at reflection from a flat surface [9]. Therefore, the total gravity perturbation above surface in the case of an incident compressional wave can be written

*k*

_{ϱ}), PS(

*k*

_{ϱ}), as functions of the horizontal wave number. Their explicit form can for example be found in [9], which leads to the gravity perturbation

*k*

_{ϱ}) vanishes because of interference of the incident and reflected P-wave, while there is no conversion PS(

*k*

_{ϱ}) from P to S-waves for these two angles. The gravity amplitude

*δ*(·) depends on the Poisson’s ratio, and the angle of incidence of the P-wave. Its absolute value is plotted in Figure 10 for three different angles of incidence 10°, 45°, 80° of the P-wave with respect to the surface normal. Important to note is that above surface, the gravity perturbation produced by shear and body waves assumes the form of a surface density perturbation with exponential suppression as a function of height above ground, determined by the horizontal wavenumber. The expression for an incident S-wave can be constructed analogously.

#### 3.4.2 Gravity perturbations from Rayleigh waves

*H*

_{i},

*V*

_{i},

*h*

_{i},

*v*

_{i}are real numbers, see Eq. (36), and so there is a constant 90° phase difference between horizontal and vertical displacement leading to elliptical particle motion. The surface displacement and the density change inside the medium caused by the Rayleigh wave lead to gravity perturbations. The surface contribution valid below and above ground is given by

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

*H*(

*z*),

*V*(

*z*) from Eq. (88) into the last equation, we finally obtain

*H*

_{i},

*V*

_{i},

*h*

_{i},

*v*

_{i}have been substituted by the expressions in Eq. (36) for fundamental Rayleigh waves. The gravity perturbation underground contains an additional contribution from the compressional-wave content of the Rayleigh 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 Section 3.3.1 and amplitudes of shear and compressional waves dependent on depth as given in Eq. (36).

*A*by vertical surface displacement:

*ν*of the half-space medium (see Section 3.1). Therefore,

*γ*itself is fully determined by

*ν*. A plot of

*γ*(

*ν*) is shown in Figure 10. The maximum value of

*γ*(

*ν*) is equal to 1, which also corresponds to the case of gravity perturbations from pure surface displacement. This means that the density perturbations generated by the Rayleigh wave inside the medium partially cancel the surface contribution for

*ν*< 0.5.

### 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. (42). 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 Figure 11.

The results were presented in [26]. 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 [142]. 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*_{P}, *t*_{S} and *t*_{R} respectively. The gravity perturbations are also divided into contributions from density perturbations inside the medium according to Eq. (43) and surface contributions according to Eq. (44).

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 [109, 108]. Recently, Eq. (42) has been implemented for gravity calculations [83]. 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 Section A.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

*Φ*is the angle of propagation with respect to the

*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 Section 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

*λ*

_{r}≡

*2π*/

*k*

_{ϱ}, the response functions, i.e., the square roots of the components of the vector in Eq. (98), divided by

*L/λ*

_{R}are shown in Figure 13.

*x*= {0,

*L*},

*y*=

*z*= 0 are uncorrelated for sufficiently large distances, and therefore the strain response decreases with increasing

*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 [88]. Eq. (98) 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

*ν*= 0.27 should be a good approximation in general [179]). In GW detectors, the relevant noise component is along the

*x*-axis. Taking the square-root of the expression in Eq. (98), and using a measured spectrum of vertical seismic motion, we obtain the Newtonian-noise estimate shown in Figure 14. Virgo’s arm length is

*L*= 3000 m, and the test masses are suspended at a height of about

*h*= 1 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 \(\rm{m/s/\sqrt{Hz}}\) within the displayed frequency range, which according to Eq. (98) 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 Figure 13 lies well below the frequency range of the spectral plots, and therefore does not influence the frequency dependence of the Newtonian-noise spectrum.

Since seismic noise is non-stationary in general, and therefore can show relatively large variations in spectra, it is a wise idea to plot the seismic spectra as histograms instead of averaging over spectra. The plots can then be used to say for example that Newtonian noise stays below some level 90% of the time (the corresponding level curve being called 90th percentile). In the shown example, a seismic spectrum was calculated each 128 s for 7 days. Red colors mean that noise spectra often assume these values, blue colors mean that seismic spectra are rarely observed with these values. No colors mean that a seismic spectrum has never assumed these values within the 7 days of observation. Interesting information can be obtained in this way. For example, it can be seen that between about 11 Hz–12 Hz a persistent seismic disturbance increases the spectral variation, which causes the distribution to be wider and therefore the maximum value of the histogram to be smaller.

#### 3.6.2 Corrections from anisotropy measurements

Anisotropy of the seismic field can be an important factor in Newtonian-noise modelling. According to Eq. (93), Rayleigh waves that propagate perpendicularly to the relevant displacement direction of a test mass (which is along the arm of a GW detector), do not produce Newtonian noise. The chances of the Rayleigh-wave field to show significant anisotropy at Newtonian-noise frequencies are high since the dominant contribution to the field comes from nearby sources, probably part of the detector infrastructure. At one of the end stations of the LIGO Hanford detector, an array of 44 vertical seismometers was used to show that indeed the main seismic source of waves around 10 Hz lies in the direction of an exhaust fan [82]. Coincidentally, this direction is almost perpendicular to the direction of the detector arm.

*ϕ*of the Rayleigh waves. An azimuth of 90½ 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 Figure 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

*ϱ*,

*h*. For vanishing test-mass height

*h*, the kernel is to be substituted by the Delta-distribution

*δ*(

*ϱ*). 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*(

*ξ*

_{z};

*ω*) =

*C*(

*ξ*

_{z}; 0,

*ω*).

*ϱ*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 Figure 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. (101) is a function of the variables

*k*

_{ϱ},

*h*with maximum at

*k*

_{ϱ}= 1

*/*(2

*h*). It is displayed in Figure 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

*ϱ*and smaller values of

*k*

_{ϱ}. 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.

*L*→ ∞ of Eq. (98) is twice as high). As mentioned already, in the form given here with the numerical factor

*γ*(

*ν*), the results are strictly only valid for Rayleigh waves. Contributions from other types of waves to \(C(\xi_z;\vec \varrho,\omega)\) could potentially be integrated separately with different numerical factors, but then one needs some prior information helping to distinguish wave types in these spectra (e.g., based on estimated seismic speeds).

*γ*(

*ν*) < 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

*γ*(

*ν*) 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 Section 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* ≪ *λ*/(2*π*). This should typically be the case below about 1 Hz.

*L*:

*L*

^{2}, and we recall that the test masses are separated by

*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 Figure 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. (110) is valid only above surface. As we have seen in Eq. (84), 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. (110) 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 [88]), 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 [166]. 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 [61]. 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 Section 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 [129, 116], 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. (103) is prone to aliasing. A review on this problem is given in [112]. 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 Section 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, it is conceivable to obtain information about a seismic source based on observations of gravity perturbations. For example, it was suggested to promptly detect and characterize fault ruptures leading to earthquakes using low-frequency gravity strain meters [83]. In this case, the analysis of gravity data from high-precision gravity strain meters can be understood as a new development in the field of terrestrial gravimetry. Until today, observation of only very slow changes in the terrestrial gravity field (slower than about 1 mHz) were possible using networks of ground-based gravimeters [52] or the satellite mission GRACE [167] with applications in hydrology, seismology and climate research. Also co-seismic gravity changes, i.e., changes following large earthquakes, were observed with gravimeters [100] as well as with GRACE (see for example [168, 40]). These observations were predicted based on a theory of static gravity perturbations from fault rupture first developed by Okubo in [132, 133, 134]. Only lasting gravity changes can be detected with these instruments, and it is to be expected that high-precision gravity strain meters will contribute significantly to this field by opening a window to gravity changes at higher frequencies. New models need to be constructed that describe time-varying gravity changes from seismic fields produced by various seismic sources. The first steps thereof are outlined in the following. We also want to point out that the same formalism can be applied to point sources of sound waves as shown in Section 5.3. We emphasize that all known time-domain models of gravity perturbations from seismic sources are for infinite media. The inclusion of surface effects, which is not always necessary, is one of the important calculations that needs to be done still. We will give some ideas how to approach this problem in Section 4.6. According to the title of this section, the models presented here are for point sources only. It is, however, numerically trivial to combine point source solutions to represent an extended source. Also certain analytical calculations of gravity perturbations from extended sources should be feasible.

### 4.1 Gravity perturbations from a point force

*F*(

*t*) = 0 for

*t*< 0, and \({\vec{e}_f}\) being the normal vector pointing along the direction of the force. Such a force generates a complicated seismic field that is composed of a near field, and shear and compressional waves propagating in the intermediate and far field [9], all components with different radiation patterns (explicit expressions for a point shear dislocation are given in Section 4.2). However, Eq. (52) can be applied here, which means that we only need to know the potential of compressional waves in infinite media to simply write down the corresponding gravity perturbation. It is not too difficult to calculate the seismic potential, but one can also find the solution in standard text books [9]. The solution for the perturbed gravity acceleration reads

*t*<

*r*

_{0}/

*α*, i.e., when the seismic waves produced by the source have not yet reached the location \({\vec r_0}\), the second term vanishes while the integral can be rewritten as double time integral

*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 [84], 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}/

*α*, where

*r*

_{0}is the distance between the source and the test mass. The situation is illustrated in Figure 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. Eq. (114) 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*(

*ω*), 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*,

*θ*,

*ϕ*will be used in the following that are related to the Cartesian coordinates via

*x*=

*r*sin(

*θ*) cos(

*ϕ*),

*y*=

*r*sin(

*θ*) sin(

*ϕ*),

*z*=

*r*cos(

*θ*), with 0 <

*θ*<

*π*, and 0 <

*ϕ*< 2

*π*. The double couple drives a displacement field that obeys conservation of linear and angular momenta. Its explicit form is given in Aki & Richards [9]. It consists of a near-field component:

*M*

_{0}(

*t*) of the double couple is called moment function. As for the point force, we assume again that the source function is zero for

*t*< 0. If a double couple is used to represent fault ruptures, than the source function increases continuously as long as the fault rupture lasts.

*t*→ ∞. According to Eq. (39), density perturbations in infinite, homogeneous media can only be associated with compressional waves, since the divergence of the shear field is zero. This is confirmed by inserting the total displacement field into Eq. (39). One obtains the density change

*M*

_{0}(

*t*→ ∞) of the shear dislocation. In Figure 21, the gravity perturbation is shown for

*θ*=

*π*/4,

*ϕ*= 0. The source function is

*M*

_{0}(

*t*) =

*M*

_{0}tanh(

*t*/

*τ*) for

*t*> 0 and zero otherwise. A log-modulus transform is applied to the density field since its value varies over many orders of magnitude [103]. This transform preserves the sign of the function it is applied to. A transient perturbation carried by compressional waves propagates parallel to the line

*t*=

*r/α*. A lasting density change, which quickly decreases with distance to the source, forms after the transient has passed.

### 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 [55]: 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 [83]. 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 second, more elegant approach to solve Equation (119) is again based on Eq. (52). Given the known solution for seismic potentials from a point force in infinite media [9], one can derive the corresponding expression for a double-couple by applying derivatives to the gravity potential with respect to the source coordinates along two orthogonal directions, and rescale it according to Eq. (119) to obtain the expression of the perturbed gravity potential given in Eq. (124).

*t*<

*r*

_{0}/

*α*, i.e. before the arrival of P waves at \({\vec{r}_{0}}\):

#### 4.3.1 Gravity-gradient tensor

*t*, the result is a symmetric tensor that can be divided into four distinct parts. The first part is proportional to the density perturbation at \(\vec r_0\):

*αt*<

*r*

_{0}since

*M*

_{0}(

*t*) = 0 for

*t*< 0, and the integral of the second contribution can be rewritten into

*t*→ ∞. Instead the time derivatives of the moment function go to zero, and the moment function itself can be substituted by its final value

*M*

_{0}(

*t*→ ∞). The result is a gravity-gradient tensor whose components decrease with \(1/r_0^3\). In addition, the gravity gradient for

*t*→ ∞ is identical to the static gravity perturbation found by [132] for shear dislocations in a half space, provided that his result is evaluated for an event far from the surface (so that surface effects are suppressed).

For small times *αt* < *r*_{0}, the gravity-gradient perturbation is not delayed by *r*_{0}*/α*. This delay only emerges once the P waves have reached the point \(\vec r_0\). In other words, the point-shear dislocation behaves as a point source of gravity perturbations for *αt* < *r*_{0} even though the actual source is an expanding wavefront of seismic compressional waves. In this case, the effective (point) source function of gravity-gradient perturbations is the fourth time integral of the moment function, which also entails that contributions to the gravity gradient from higher-frequency components of the moment spectrum are strongly suppressed.

### 4.4 Gravity perturbation from the Tohoku earthquake

In this section, the specific example of the 2011 Tohoku earthquake will be used to estimate gravity perturbations. The Tohoku event had a magnitude of 9.0, and ruptured a fault of width and length of several hundred kilometers [12]. The hypocenter was located at latitude N37.52 and longitude E143.05. The estimate of the gravity perturbation will be based on the early-time approximation given in Eq. (125). The result should not be expected to be accurate, since the source cannot be approximated as a point source, and the influence of the surface on the gravity perturbation may be substantial, but nonetheless, it serves as an order-of-magnitude estimate.

^{8}, and translate them into the fault-slip oriented coordinate system of Section 4.3. The focal mechanisms can be specified by three angles: the strike angle

*γ*

_{S}, the dip angle

*γ*

_{d}, and the rake angle

*γ*

_{r}. The strike angle is subtended by the intersection of the fault plane with the horizontal plane, and the North cardinal direction. The dip is the angle between the fault plane and the horizontal plane. Finally, the rake is subtended by the slip vector and the horizontal direction on the fault plane. The fault geometry is displayed in Figure 22. For the Tohoku earthquake, the angles are

*γ*

_{S}= 3.54, the dip angle

*γ*

_{d}= 0.17, and the rake angle

*γ*

_{r}= 1.54. In the coordinate system shown in Figure 20, the normal vector of the fault defines the direction of the

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

*z*-axis. In this coordinate system, the gravity perturbation is given by:

*α*. Sensors designed to monitor changes in gravity acceleration are called gravimeters. For example, networks of gravimeters have been used in the past to detect co-seismic gravity changes following large earthquakes [100]. However, these were pre-post event comparisons of DC gravity changes. A prompt detection of a co-seismic gravity perturbation using gravimeters has not been achieved yet.

^{9}, which had a total rupture duration of about 300 s, with almost all of the total seismic moment, 5 × 10

^{22}Nm, already released after 120 s. After 68 s, the first seismic waves reach the gravimeter, which makes gravity measurements impossible for more than a day. A signal of about −6 × 10

^{−9}m/s

^{2}is substantial. The rms of the data between 2 mHz and 0.5 Hz is about 5 × 10

^{−9}m/s

^{2}during relatively quiet times, and gravimeter data are highly non-stationary (mostly due to direct seismic perturbation of the instrument). It may be possible to detect this signal before arrival of the seismic waves, based on a fit to the predicted gravity perturbation and integrating over the available 68 s of data.

### 4.5 Seismic sources in a homogeneous half space

*z*-axis is chosen as surface normal: \(\vec n=(0,0,1)\). The goal is to calculate a gravity perturbation directly above the free surface at

*z*= 0. In general, the gravity potential above surface takes the form

*z*= 0, and gravity above surface so that

*z*

_{0}> 0, the inverse distance can be expanded according to

*p*can be interpreted as horizontal wavenumber of the harmonics that constitute the seismic field, while the vertical wavenumbers \(k_z^{\rm{P}}(p), \, k_z^{\rm{S}}(p)\) have the form in Eq. (34). Explicit expressions of the amplitudes \(a_m^x(p), \, a_m^y(p), \, b_m(p)\),

*b*

_{m}(

*p*) depend on the nature of the seismic source, and can be found for a few important cases in [107]. They also depend on the depth

*z*

_{S}of the seismic source. The evaluation of the surface integrals is analogous for the three potentials. We outline the calculation for the P-wave potential

*ϕ*

_{S}:

*k*. We have seen this already happening in the explicit solutions for Rayleigh and plane body waves, where the seismic amplitude changes with depth

*z*in terms of a vertical wavenumber, but the gravity perturbation changes with the horizontal wavenumber (see Sections 3.4.1 and 3.4.2). For this reason, it is unfortunately impossible to express the P-wave contribution to the gravity potential directly in terms of the P-wave potential. However, the solution simplifies if the gravity field is to be calculated directly above surface at

*z*

_{0}= 0:

### 4.6 Summary and open problems

We have shown how to calculate 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. (52). We have seen demonstrations of this principle in Section 4.1 for point forces, and in Section 4.3 for point shear dislocations.

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 due to the strong low-pass filtering effect of gravity perturbations from distant seismic wavefronts, seismic sources need to have very peculiar properties to produce significant, instantaneous gravity perturbations at the test mass. Consequently, 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 Sections 4.2 and 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. Especially a thorough analysis of surface effects is essential for future developments. In Section 4.5, a first calculation of gravity perturbations from point sources in half spaces was outlined. The full solution still needs to be analyzed in detail. Open questions are how the Rayleigh waves generated in half spaces affect gravity perturbations at larger distances, and also how the contribution of body waves is altered by reflection from the surface. In light of the possible applications of low-frequency GW detectors in geophysics, further development of the theory may significantly influence future directions in this field.

## 5 Atmospheric Gravity Perturbations

The properties of the atmosphere give rise to many possible mechanisms to produce gravity perturbations. Sound fields are one of the major sources of gravity perturbations. Typically, sound is produced at boundaries between air and solid materials, but in general, one also needs to consider the *internal* production of sound via the Lighthill process. The models of gravity perturbations from sound fields are very similar to perturbations from seismic compressional waves as given in Section 3. The main difference in the models is related to the fact that the two fields are observed by different types of sensors. Additional mechanisms of producing atmospheric gravity noise are related to the fact that air can flow. This can lead to the formation of vortices or convection, and turbulence can always play a role in these phenomena. The Navier-Stokes equations directly predict density perturbations in these phenomena [57]. Also static density perturbations produced by non-uniform temperature fields can be transported past a gravity sensor and cause gravity noise. One goal of Newtonian-noise modelling 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.

Atmospheric gravity perturbations have been known since long to produce noise in gravimeter data [128], where they can be observed below about 1 mHz. At these frequencies, they are modelled accurately as a consequence of pressure fluctuations and loading of Earth’s surface. Atmospheric gravity perturbations are generally expected to be the dominant contribution to ambient Newtonian noise below 1 Hz [88]. In contrast, Creighton showed that atmospheric Newtonian noise can likely be neglected above 10 Hz in large-scale GW detectors [51]. His paper is until today the only detailed study of atmospheric Newtonian noise at frequencies above the sensitive band of gravimeters, and includes noise models for infrasound waves, quasi-static temperature fluctuations advected in various modes past test masses, and shockwaves. His results will be reviewed in the following with the exception that a new solution is given for gravity perturbations from shockwaves based on the point-source formalism of Section 4. Preliminary work on modelling gravity perturbations from turbulence was first published in [39], and is reviewed and improved in Section 5.4.

### 5.1 Gravity perturbation from atmospheric sound waves

*p*

_{0}. The pressure change can be translated into perturbation of the mean density

*ρ*

_{0}. The relation between pressure and density fluctuations depends on the adiabatic index

*γ*≈ 1.4 of air [175]

*γ*> 1 is that the temperature increases when the sound wave compresses the gas sufficiently slowly, and this temperature increase causes an increase of the gas pressure beyond what is expected from compression at constant temperature. Note that in systems whose size is much smaller than the length of a sound wave, the statement needs to be reversed, i.e., fast pressure fluctuations describe an adiabatic process, not slow changes. An explanation of this counter-intuitive statement in terms of classical thermodynamics is given in [71]. It can also be explained in terms of the degrees of freedom of gas molecules [93]. At very high frequencies (several kHz or MHz depending on the gas molecule), vibrations and also rotations of the molecules cannot follow the fast sound oscillation, and their contribution to the specific heat freezes out (thereby lowering the adiabatic index). At low audio frequencies, sound propagation in air is adiabatic

^{10}.

*z*> 0:

*z*

_{0}= 0. The gravity potential and acceleration are continuous across the surface. We neglect the surface term here, but this is mostly to simplify the calculation and not fully justified. Part of the energy of a sound wave is transmitted into the ground in the form of seismic waves. Intuitively, one might be tempted to say that only a negligible amount of the energy is transmitted into the ground, but at the same time the density of the ground is higher, which amplifies the gravity perturbations. Let us analyze the case for a sound wave incident at a normal angle to the surface. In this case, the sound wave is transmitted as pure compressional wave into the ground. We denote the air medium by the index “1” and the ground medium by “2”. Multiplying the seismic transmission coefficient (see [9]) by

*ρ*

_{2}/

*ρ*

_{1}, the relative amplitude of gravity perturbations is

*α*

_{1}is the speed of sound, and

*α*

_{2}the speed of compressional waves. The sum in the denominator can be approximated by

*ρ*

_{2}

*α*

_{2}, which leaves 2

*α*

_{1}/

*α*

_{2}as gravity ratio. The ratio of wave speeds does not necessarily have to be small at the surface. We know that the Rayleigh-wave speed at the LIGO sites is about 250 m/s [87], which we can use to estimate the compressional-wave speed to be around 600 m/s (by making a guess about the Poisson’s ratio of the ground medium). This means that the effective transmissivity with respect to gravity perturbations could even exceed a value of 1! Therefore, it is clear that the physics of infrasound gravity perturbations is likely more complicated than outlined in this section. Nonetheless, we will keep this for future work and proceed with the simplified analysis assuming that sound waves are fully reflected by the ground.

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

*T*

_{0}and density

*ρ*

_{0}of the atmosphere, and according to the ideal gas law at constant pressure, small fluctuations in the temperature field cause perturbations of the density:

*temperature structure function D*(

*δT*;

*r*):

*r*is sufficiently small. This relation also breaks down at distances similar to and smaller than the Kolmogorov length scale, which is about 0.4 mm for atmospheric surface layers [14]. Turbulent mixing enforces power laws with

*p*∼ 2/3 [14]. Applying Taylor’s hypothesis, the distance Δ

*r*can be substituted by the product of wind speed

*v*with time

*τ*, and Eq. (148) can be reformulated as

*c*

_{T}depends on the dissipation rate of turbulent kinetic energy and the temperature diffusion rate, and

*σ*

_{T}is the standard deviation of temperature fluctuations. Since Taylor’s hypothesis is essential for the following calculations, we should make sure to understand it. Qualitatively it states that turbulence is transported as frozen pattern with the mean wind speed. More technically, it links measurements in Eulerian coordinates, i.e., at points fixed in space, with measurements in Lagrangian coordinates, i.e., that are connected to fluid particles. The practical importance is that two-point spatial correlation functions such as Eq. (148) can be estimated based on a measurement at a single location when carried out over some duration

*τ*as in Eq. (150). In either case, the hypothesis can be expected to fail over sufficiently long periods

*τ*or distances Δ

*r*, which are linked to the maximal scale of turbulent structures [56]. In any case, we assume that Taylor’s hypothesis is sufficiently accurate for our purposes. The Fourier transform of Eq. (150) yields the spectral density of temperature fluctuations

*τ*. This means that the spectral density given here only holds at sufficiently high frequencies (at the same time not exceeding the Kolmogorov limit defined by the size

*l*of the smallest turbulence structures,

*ω*<

*υ*/

*l*, which is of the order kHz). Technically, the Fourier transform can be calculated by multiplying an exponential term exp(−

*ϵτ*) to the integrand, and subsequently taking the limit

*ϵ*→ 0.

*v/ω*(which is shown in the following). Consider the scenario displayed in Figure 25. Two air pockets are shown at locations \(\vec r, \, \vec r\,{\prime}\) and times

*t*,

*t*′ on two steam lines that we denote by

*S*and

*S*′.

*τ*=

*t*−

*t*′ is sufficiently small, then the separation of the two pockets can be written (

*s*

^{2}+ (

*vτ*)

^{2})

^{1/2}, where the distance

*s*of the two streamlines

*S*,

*S*′ and

*v*are evaluated at \(\vec r\). Together with Taylor’s hypothesis, temperature fluctuations between the two pockets are significant if

*τ*is sufficiently close to the time

*τ*

_{0}it takes for the pocket at \(\vec r{\prime}\) to reach the reference plane, and also

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

*τ*in Eq. (147):

*K*

_{n}(·) is the modified Bessel function of the second kind, and

*v*,

*s*are functions of \(\vec r\). Again, the integral can only be evaluated if an exponential upper cutoff on the variable

*τ*is multiplied to the integrand, which means that we neglect contributions from large-scale temperature perturbations. The correlation spectrum is plotted in Figure 26.

*v*/

*s*, the spectrum falls exponentially since \(K_\nu(x)\rightarrow \sqrt{\pi/(2x)}\exp(-x)\) for

*x*≫ ∣

*ν*

^{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. (147) can be turned into an integral over streamlines

*S*′ that lie within a bundle

*s*≲

*υ*/

*ω*of streamline

*S*, which allows us to approximate the volume element as cylindrical bundle \({\rm{d}}V{\prime} = 2\pi s{\rm{d}}s\,{\rm{d}}{\tau _0}v(\vec r)\). 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 \(\vec r\). With this notation, the integral can be carried out over 0 <

*s*<∞ since the modified Bessel function automatically enforces the long-distance cutoff necessary for our approximations, which yields

*τ*

_{0}. This result can be interpreted as follows. We have two streamlines

*S*,

*S*′, whose contributions to this integral are evaluated in terms of the duration

*τ*

_{0}it takes for the pocket at \(\vec r{\prime}\) to reach the reference plane that goes through all streamlines, and contains the test mass at \({\vec{r}_{0}}\) and location \(\vec{r}\) (as indicated in Figure 25). Since we consider the pocket on streamline

*S*to be at the reference plane at time

*t*, we can set

*τ*

_{0}=

*t*−

*t*′, and integrating contributions from all streamlines over the reference plane with area element d

*A*, with wind speed \(\upsilon(\vec\varrho)\), and \(\vec{\varrho}\) pointing from the test mass to streamlines on the reference plane, we can finally write

*v*= const, and the remaining integrals can be solved with the results given in Section 6.2. Other examples have been calculated by Creighton [51].

### 5.3 Gravity perturbations from shock waves

*M*=

*Vρ*

_{0}determined by the source volume

*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

*α*is the speed of sound,

*γ*is the adiabatic coefficient of air,

*p*

_{0}the mean air pressure, Δ

*p*(

*t*) the pressure change, and 1 the unit matrix. Since shock-wave generation is typically non-linear [172], the source volume should be chosen sufficiently large so that wave propagation is linear beyond its boundary. This entails that the pressure change Δ

*p*is also to be evaluated on the boundary of the source volume. Note that in comparison to solitons, shock waves always show significant dissipation, which means that there should not be a fundamental problem with this definition of the source volume. Alternatively, if nonlinear wave propagation is significant over long distances, then one can attempt to linearize the shock-wave propagation by introducing a new nonlinear wave speed, which needs to be used instead of the speed of sound [172]. In general, a sudden increase of atmospheric pressure by an explosive source must relax again in some way, which means that Δ

*p*(

*t*→ ∞) = 0.

*r*

_{0}as distance at closest approach of the air craft to the test mass. The source volume is replaced by the rate

*V*→

*Av*(

*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.

*p*at times

*t*= 0 and

*t*= ∇

*t*= 0.1r

_{0}/

*α*, and a linear pressure fall between these two times. The aircraft trajectory is assumed to be horizontal and passing directly above the test mass. Time

*t*= 0 corresponds to the moment when the aircraft reaches the point of closest approach. If

*v*<

*α*, then sound waves reach the test mass well before the aircraft reaches the closest point of approach. In the case of supersonic flight,

*α*<

*v*, the first sound waves reach the test mass at

*t*=

*r*

_{0}/

*α*. Inserting the pressure change into Eq. (160), we see that the far-field gravity perturbation is characterized by two

*δ*-peaks. The derivative of the linear pressure change between the peaks cancels with a contribution of the near-field term. As can be understood from the left plot in Figure 27, the gravity perturbation falls gradually after the initial peak since a test mass inside the cone still responds to pressure changes associated with two propagating wavefronts.

### 5.4 Gravity perturbations in turbulent flow

*c*

_{s}is the speed of sound in the uniform medium. The terms in the effective stress tensor are the fluctuating Reynolds stress

*ρv*

_{i}

*v*

_{j}, the compressional stress tensor

*σ*

_{ij}, and the stress \(c_{\rm{s}}^2\rho\delta_{ij}\) of a uniform acoustic medium at rest. In other words, the effective stress tensor acting as a source term of sound is the difference between the stresses in the real flow and the stress of a uniform medium at rest. Eq. (162) is exact.

*σ*

_{ij}unimportant (we neglect viscous damping in sound propagation), and therefore the temperature field can be assumed to be approximately uniform. This means that the difference \(\sigma_{ij}-c_{\rm{s}}^2\rho\delta_{ij}\) is negligible with respect to the fluctuating Reynolds stress. Furthermore, we will assume that the root mean square of the velocities

*v*

_{i}are much smaller than the speed of sound

*c*

_{s}(i.e., the turbulence has a small Mach number), and consequently the relative pressure fluctuations \(\delta p(\vec r,t)/p_0\) produced by the Reynolds stress is much smaller than 1. In this case, we can rewrite the Lighthill equation into the approximate form

*i*,

*j*). This equation serves as a starting point for the calculation of the pressure field. It describes the production of sound in turbulent flow through conversion of shear motion into longitudinal motion. The Reynolds stress represents a quadrupole source, which means that sound production is less efficient in turbulent flow than for example at vibrating boundaries where the source has dipole form. The remaining task is to characterize the velocity fluctuations in terms of spatial correlation functions, translate these into a two-point correlation function of the pressure field using Eq. (163), and finally obtain the spectrum of gravity fluctuations from these correlations. The last step is analogous to the calculation carried out in Section 5.2, specifically Eq. (147), for the perturbed temperature field. The calculation of gravity perturbations will be further simplified by assuming that the velocity field is stationary, isotropic, and homogeneous. These conditions can certainly be contested, but they are necessary to obtain an explicit solution to the problem (at least, solutions for a more general velocity field are unknown to the author).

*π*)

^{4}is different from the convention used elsewhere in this article, where the inverse Fourier transform obtains this factor. The fact that noise amplitudes at different wave vectors and frequencies do not couple is a consequence of homogeneity and stationarity of the velocity field. Once the spectral density of pressure fluctuations is known, we can use it to calculate the gravity perturbation according to

*ϵ*the total (specific) energy dissipated by viscous forces

*ν*is the fluid’s viscosity. The Kolmogorov energy spectrum holds for the inertial regime \(\mathcal{I}\) (viscous forces are negligible), i.e., for wavenumbers between \(k_0=2\pi/\mathcal R\) and

*k*

_{ν}= (

*ϵ*/

*ν*

^{3})

^{1/4}, where \(\mathcal R\) is the linear dimension of the largest eddy in the turbulent flow. In Eq. (167), we have only written the equal-time correlations (the first following from the second equation). The velocities in the second equation should however be evaluated at two different times

*t*,

*t*+

*τ*. In [105], we find that for

*k*≫

*k*

_{0}

*v*

_{i}is any of the components of the velocity vector. The first term in Eq. (166) is independent of time for a stationary velocity field (both expectation values are equal-time). Therefore, its energy only contributes to frequency

*ω*= 0, and we can neglect it. The Fourier transform in Eq. (164) of the second and third terms in Eq. (166) with respect to

*τ*can be carried out easily using Eq. (169). Also integrating over the angular coordinates of the spatial Fourier transform in Eq. (164), the gravity spectrum can be written

*r*. The integrands are products of three spherical Bessel functions. An analytic solution for this type of integral was presented in [123] where we find that the integral is non-zero only if the three wavenumbers fulfill the triangular relation ∣

*k*′ −

*k*″∣ ≤

*k*≤

*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*″∣ ≤

*n*≤

*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 [39] is a valuable option. The square-roots of the noise spectra normalized to units of GW amplitude, \(2S(\delta\vec a;\vec k,\omega)/(L\omega^2)^2\), are shown in Figure 28 for

*k*= 0.1 m

^{−1},

*k*= 0.67 m

^{−1},

*k*= 1.58 m

^{−1},

*k*= 3.0 m

^{−1}, where

*L*= 3000 m is the length of an interferometer arm.

Each spectrum is exponentially suppressed above the corner frequency 1/*τ*_{0}(*k*) with *τ*_{0} = 3.5s, 0.52s, 0.22s, 0.12s. Below the corner frequency, the spectrum is proportional to 1/*ω*^{2}. In order to calculate the dissipation rate *ϵ*, a measured spectrum was used [10], which has a value of about 1 m^{3} s^{−2} at *k* = 1m^{−1}, and wavenumber dependence approximately equal to the Kolmogorov spectrum. In this way, we avoid the implicit relation of the dissipation rate in Eq. (168), since *ϵ* also determines the Kolmogorov energy spectrum. Solving the implicit relation for *ϵ* 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 \(\mathcal{R}\) (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, …). 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 calculate the surface noise spectra here. Newtonian noise from advected temperature perturbations decreases strongly with depth and should not play a role in underground detectors. Suppression of infrasound gravity noise with depth depends strongly on the isotropy of the infrasound field. Using Eq. (143), it is straight-forward to modify the results of this section to include noise suppression with depth once the infrasound field is characterized.

*z*

_{0}= 0 due to an infrasound wave is given by

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

*λ*

_{IS}= 2

*π*/

*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, while the small-distance response corresponds to the regime where the two test masses sense gravity-gradient fluctuations.

*r*

_{min}around the test mass. The excision enforces a minimum distance between stream lines and test masses, for example because of buildings hosting the test masses. Due to the exponential suppression, this noise contribution can be expected to be insignificant deep underground. The integration includes an average over streamline directions. If the expression is to be converted into a GW strain sensitivity of a two-arm interferometer, then it is not fully accurate to simply multiply the strain noise by 2 due to gravity correlations between the two inner test masses of the two arms. Nonetheless, for the noise budget presented in this section, we will use the factor 2 conversion.

The Newtonian noise from advected temperature fields is evaluated using a wind speed of *v* = 15 m/s, and a minimum distance of 5 m to the test masses. With respect to the advanced GW detectors LIGO/Virgo, atmospheric Newtonian noise will be insignificant according to these results.

The slope of infrasound Newtonian noise is steeper than of seismic Newtonian noise (see Figure 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 [88]. 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. From Section 7.1.5, we also know 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 the noise prediction cannot be extended to much below a few Hz without modifying the model. 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 [113]. Also, the part of the model shown in Figure 30 is characterized by an exponential suppression (effective above 3 Hz).

### 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. We have seen that atmospheric Newtonian noise will very likely be insignificant in GW detectors of the advanced generation. For surface detectors, atmospheric Newtonian noise starts to be significant below 10 Hz according to these models.

According to Eqs. (172) and (173), and comparing with seismic Newtonian noise (see Figure 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 [88], including modelling of a half-space atmosphere, neglecting wind, etc. Also the noise model of advected temperature fluctuations likely does not hold at low frequencies since it is based on the assumption that the temperature field is quasi-stationary. In addition, at low frequencies, near-surface temperature spectra can be affected by variations of ground temperature in addition to turbulent mixing.

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 modelled forms of atmospheric noise are likely insignificant in GW detectors sensitive above 10 Hz, they may become important in low-frequency detectors. Another open problem is to study systematically the decrease in atmospheric Newtonian noise with depth in the case of underground GW detectors. Especially, it is unclear how much atmospheric noise is suppressed in sub-Hz underground detectors. We have argued that seismic Newtonian noise does not vary significantly with detector depth in low-frequency GW detectors, but some forms of atmospheric Newtonian noise depend strongly on the minimal distance between source and test mass. So the conclusion might be different for atmospheric noise. 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 Section 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 [136]. A first analytical study of gravity perturbations from objects was performed by Thorne and Winstein who investigated disturbances of anthropogenic origin [163]. The paper of Creighton has a section on gravity perturbations from moving tumbleweeds, which was considered potentially relevant to the LIGO Hanford detector [51]. Interesting results were also presented by Lockerbie [117], 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 Sections 6.5 and 6.6.

### 6.1 Rules of thumb for gravity perturbations

*ξ*, and ground density

*ρ*:

*V*and density

*ρ*

_{0}at distance

*r*to the test mass that oscillates with amplitude

*ξ*(

*t*) ≪

*r*. We can use the dipole form in Eq. (42) to calculate the gravity perturbation at \(\vec r_0=\vec 0\):

*δV*/

*r*

^{3}. In numbers, a solid object with 1 m diameter at a distance of 5 m oscillating with amplitude equal to seismic amplitudes, and equal density to the ground would produce Newtonian noise, which is about a factor 100 weaker than seismic Newtonian noise. Infrastructure at GW detectors near test masses include neighboring chambers, which can have diameters of several meters, but the effective density is low since the mass is concentrated in the chamber walls.

If the distance *r* is decreased to its minimum when the test mass and the perturbing mass almost touch, then the factor *δV*/*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 [117] is presented in Section 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 Section 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 \(\vec r_1, \, \vec r_2\) pointing from the test mass to the points of closest approach are perpendicular to the velocity \(\vec v\). The closest approach to the first test mass occurs at time *t*_{1}, and at *t*_{2} to the second test mass.

*K*

_{n}(

*x*) being the modified Bessel function of the second kind. This equation already captures the most important properties of the perturbation in frequency domain. The ratio

*v*/

*r*

_{1}marks a threshold frequency. Above this frequency, the argument of the modified Bessel functions is large and we can apply the approximation

*x*≫ ∣

*n*

^{2}− 1/4∣. We see that the Fourier amplitudes are exponentially suppressed above

*v*/

*r*

_{1}. The expression in Eq. (177) has the same form for the second test mass. We can however eliminate

*t*

_{2}in this equation since the distance travelled by the object between

*t*

_{1}and

*t*

_{2}is \(L({\vec e_{12}}\, \cdot \, \vec \upsilon)/\upsilon\), where

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

*D*, and a car is driving directly above the test masses with \(\vec v\) parallel \(\vec e_{12}\) and perpendicular to \(\vec r_1\). Therefore, \(\vec r_1=\vec r_2\), and

*t*

_{2}−

*t*

_{1}=

*L*/

*v*. The corresponding strain amplitude is

*L*at frequencies

*ω*≪

*v*/

*L*. The plots in Figure 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 m, in the latter case, the speed is kept constant at

*v*= 20 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 Section 5.2. For uniform airflow, the remaining integrals in Eq. (155) are the Fourier transform of Eq. (176), 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 Section 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 Section 6.4.

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

*L*to each other separated along the direction of the unit vector \(\vec e_{12}\).

*ξ*(

*ω*) keeping in mind that these only have symbolic meaning and need to be translated into spectral densities. We only allow for small oscillations, i.e., with

*ξ*being much smaller than the distance of the object to the two test masses. The acceleration of the first test mass has the well-known dipole form

*r*

_{1}is the distance between them. The acceleration of the second test mass has the same form, and we can substitute \(\vec e_{r_2}=(\vec e_{r_1}+\lambda\vec e_{12})/\delta\) and

*r*

_{2}=

*r*

_{1}

*δ*with \(\delta\equiv(1+\lambda^2-2\lambda(\vec e_{r_1}\cdot\vec e_{12}))^{1/2}\) and

*λ*≡

*L*/

*r*

_{1}.

*h*

_{∥}(

*ω*) becomes arbitrarily small with decreasing λ,

*h*

_{⊥}(

*ω*) approaches a constant value. Towards high frequencies,

*h*

_{⊥}falls rapidly since there is no force along \(\vec e_{12}\) on the first test mass, and the distance of the object to the second test mass increases with growing λ, and also the projection of the gravity perturbation at the second test mass onto \(\vec e_{12}\) becomes smaller.

*ϕ*=

*π*/2. The response grows to infinity for λ = 1 and polar angle

*ϕ*= 0 since the object is collocated with the second test mass. Note that

*λ*> 1 and

*ϕ*= 0 means that the object lies between the two test masses.

### 6.4 Interaction between mass distributions

*R*

_{AB}between the two centers of mass is greater than the object diameters at largest extent. The so-called bipolar expansion allows us to express the gravitational force in terms of mass multipole moments. The idea is to split the problem into three separate terms. One term depends on the vector \(\vec R_{\rm{AB}}\) that points from the center of mass

*A*to the center of mass

*B*. Each individual mass is expanded into its multipoles according to Eq. (274) calculated in identically oriented coordinate systems, but with their origins corresponding to the two centers of mass. The situation is depicted in Figure 34.

*L*≡

*l*

_{1}+

*l*

_{2},

*M*≡

*m*

_{1}+

*m*

_{2}. It is not very difficult to generalize this equation for arbitrary mass distributions (one object inside another hollow object, etc), but we will leave this for the reader. The method is essentially an exchange of irregular and regular solid spherical harmonics in Eq. (185) together with Eqs. (274) and (275). Also, in general it may be necessary to divide the multipole integral in Eq. (274) into several integrals over regular and irregular harmonics. A practical method to calculate the Clebsch-Gordan coefficients 〈

*l*

_{1},

*m*

_{1};

*l*

_{2},

*m*

_{2}∣

*L*,

*M*〉 is outlined in Section A.4.

*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

*θ*in \(I_2^0(\vec R_{AB})\) represents the angle between the symmetry axis and the separation vector \(\vec R_{AB}\). 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. (266)

### 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.

*ξ*is assumed to be small, we only keep terms up to linear order in

*ξ*:

*ξ*/

*R*

_{AB}(the vibration amplitude is at most a few millimeters). Therefore it is clear that corrections from higher-order moments only matter if gravitational interaction is measured very precisely, or the vibrating point mass is very close to the cylinder.

### 6.6 Rotating objects

*l*of spherical harmonics.

*α*,

*β*,

*γ*around three axes derived from the body-fixed system. The first rotation is by

*α*around the

*z*-axis of the body-fixed system, then by

*β*around the

*y*′-axis of the once rotated coordinate system (following the convention in [160]), and finally by

*γ*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 \(D_{m,m{\prime}}^{(l)^{\ast}}(\alpha,\beta,\gamma)\) of the rotation matrix are given by[160]:

*k*that give non-negative factorials in the two binomial coefficients: max(0,

*m*−

*m*′) ≤

*k*≤ 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 Section A.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. (186) as starting point. The rotation transforms the exterior multipole moments \(X_l^{m,\rm{B}}\). We have seen that multipole moments of the ring vanish unless

*m*= 0,

*N*, 2

*N*,… and

*l*+

*m*must be even. Only the first (or last) Euler rotation by an angle

*α*=

*ωt*is required, which yields

*ϕ*

_{k}that determines the position of a point mass on the ring appears in the spherical harmonics as phase factor exp(i

*m*(i

*mϕ*

_{k}). When the ring rotates, all azimuthal angles change according to

*ϕ*

_{k}(

*t*) =

*ϕ*

_{k}(0) −

*ϕt*.

Since \(X_l^m\) vanishes unless *m* = 0, *N*, 2*N*, …, only specific multiples of the rotation frequency *ω* 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 [136, 66]. So far, none of the potential sources turned out to be relevant. In Section 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. (97), the sound Newtonian noise using Eq. (172), 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 Section 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 [180]). 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 & Thorne [99]. 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 seismic channels serve as input to a linear filter, whose output is then subtracted from the target channel (i.e., the data of a GW detector). The output of an optimal filter can be interpreted as the best possible linear estimation of gravity perturbations based on seismic data. This method will be discussed in Section 7.1. The second idea was to construct narrow moats around the test masses that reflect incoming Rf 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 Section 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.

The idea of coherent cancellation of seismic Newtonian noise has gained popularity in the GW community, probably because it is based on techniques that have already been implemented successfully in GW detectors to mitigate other forms of noise [74, 65, 59]. These techniques are known as *active* noise mitigation. It is mostly considered as a means to reduce seismic Newtonian noise, but the same scheme may also be applied to atmospheric Newtonian noise (see especially Section 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 cancelled in the data of the GW detector. Coherent cancellation comes without (known) ultimate limitations, which means that in principle any level of noise reduction can be achieved provided that the environmental sensors are sufficiently sensitive, and one can deploy as many senors as required. The prediction by Hughes & 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 [42]. 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 solution to reduce variance in a target channel as explained in Section 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 Section 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 [27] 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 Section 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 Section 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 also signal and other noise in the target channel can limit the performance. Furthermore, the filter that represents the noise model, which in the context of Newtonian-noise cancellation is a multiple-input-single-output (MISO) filter with reference channels providing the inputs, and the noise model being the output, also maps sensor noise into the noise model, which means that sensor noise is 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* [165]. Wiener filters are linear filters calculated from correlation measurements between reference and target channels. They are introduced in Section 7.1.1. In the context of seismic or atmospheric Newtonian-noise cancellation, the auxiliary sensors monitor a field of density perturbations, which means that correlation between auxiliary sensors is to be expected. In this case, if the field is wide-sense stationary (defined in Section 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 [141]. In Sections 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 density fields. Noise cancellation from finite-size sources is mostly a practical problem, and trivial from the theory perspective. The optimization of array configurations for noise cancellation is a separate problem, which is discussed in Section 7.1.6.

#### 7.1.1 Wiener filtering

A linear, time-invariant filter that produces an estimate of a random stationary (target) process minimizing the deviation between target and estimation is known as Wiener filter [32]. 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*∇

*t*, where ∇

*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

**w**is a (

*N*+ 1) ×

*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 〈(

*y*

_{n}−

*ŷ*

_{n})

^{2}〉 between the target channel and filter output is minimized, which directly leads to the Wiener-Hopf equations:

*N M*-dimensional vector that is obtained by concatenating the

*M*columns of the matrix

**w**. The (

*N*+ 1)

*M*× (

*N*+ 1)

*M*matrix

**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*∇

*t*. It contains the autocorrelations of each reference channel as (

*N*+ 1) × (

*N*+ 1) blocks on its diagonal:

*c*

_{k}≡ 〈

*x*

_{n}

*x*

_{n+k}〉 for each of the

*M*reference channels. In this form it is a symmetric Toeplitz matrix. The (

*N*+ 1)

*M*-dimensional vector \(\vec C_{xy}\) is a concatenation of correlations between each reference channel and the target channel. The components contributed by a single reference channel are

*s*

_{k}≡ 〈

*x*

_{n}

*y*

_{n+k}〉. Note that we do not assume independence of noise between different reference channels. This is important since there can be forms of noise correlated between reference channels, but uncorrelated with the target channel (e.g., shear waves in Newtonian-noise cancellation, see Section 7.1.3). In general, the correlations that determine the Wiener-Hopf equations are unknown and need to be estimated from measurements using data from reference and target channels. An elegant implementation of the code that provides these estimates and solves the Wiener-Hopf equations can be found in [136].

*y*

_{n}is given by

*y*

_{n}corresponds to the GW strain signal contaminated by Newtonian noise, and

*y*

_{n}is the estimate of Newtonian noise provided by the Wiener filter using reference data from seismometers or other sensors. Time-domain Wiener filters were successfully implemented in GW detectors for the purpose of noise reduction [65, 59]. Results from a time-domain simulation of Newtonian-noise cancellation using Wiener filters was presented in [66]. Not all coherent cancellation schemes are necessarily implemented as Wiener filters. For example in [74], noise cancellation was optimized by solving a system-identification problem.

A frequency-domain version of the Wiener filter can be obtained straight-forwardly by dividing the data into segments and calculating their Fourier transforms. Eq. (206) 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 (keep in mind that amplitudes of stationary, random processes do not exist as Fourier amplitudes, and therefore this statement needs a suitable definition of these amplitudes, see Section A.5 and [144]). 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. (203) is reduced from *NM* to *M* (for *N* different frequency bins). In contrast, time-domain correlations *c*_{k}, *s*_{k}. can be large for small values of *k*. This can cause significant numerical problems to solve the Wiener-Hopf equations, and as observed in [50], FIR filters of lower order can be more effective (even though theoretically, increasing the filter order should not make the cancellation performance worse).

*s*(

*ω*) as a function of frequency in the case of a single reference channel is related to the reference-target coherence

*c*(

*ω*) via

*M*collocated, identical sensors leads at least to a \(\sqrt{M}\) reduction of the sensor noise limit. If the

*M*sensors monitor a field whose values at nearby points are dynamically correlated (i.e., the two-point spatial correlation is not just a

*δ*-peak), then further gain is to be expected for example by being able to distinguish between modes of the field that produce correlation with the target channel, and modes that do not. This will be discussed in detail in Section 7.1.3 and Section 7.1.6.

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

As we have seen in Section 7.1.1, the correlations between reference channels and the target channel determine the Wiener filter. For seismic fields, correlations between reference channels (seismometers) can be measured, but we still need a model consistent with the seismic correlations that provides the correlation with the gravity channel. Obviously, as long as 2D arrays are used for the characterization of the seismic field, the predicted correlation with the gravity channel can be subject to systematic errors. For example, we will have to guess the types of seismic waves that contribute to the seismic correlations. In this section, the problem will be solved assuming that all seismic waves are Rayleigh waves. Here, we will also discuss the cancellation problem in low-frequency detectors explicitly since it is qualitatively different.

*x*-axis measured at height

*h*above surface, the correlation is given by (see Section 3.6.3 for spectral representation of noise)

*y*′ =

*y*. Next, we will consider the explicit example of an isotropic Rayleigh wave field. The easiest way to obtain the result is to insert the known solution of the wavenumber spectrum, Eq. (108), into the first line in Eq. (209), which gives:

*ϱ*= 0. This is a consequence of the fact that any elastic perturbation of the ground must fulfill the wave equation. If instead the ground were considered as a collection of infinitely many point masses without causal link, then the correlation of displacement of point masses nearest to the test mass with the gravity perturbation would be strongest.

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 Section 7.1.3.

*x*′ =

*x*,

*y*′ =

*y*independent of test-mass height. Now, for the homogeneous and isotropic field, the solution with respect to the strain acceleration reads

*ϱ*→ 0. Also notice that the result is independent of the distance

*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 Section 3.6.4.

*σ*(

*ω*) with respect to measurements of seismic displacement. According to Eq. (107), the correlation between two seismometers at locations \(\vec\varrho_i, \, \vec\varrho_j\) can then be written

**C**

_{xx}in Eq. (203). The correlation of each seismometer with the gravity perturbation will be denoted as

*C*

_{NN}(

*ω*) =

*S*(

*δa*

_{x};

*ω*) is [42]

*ϕ*

_{1}= 0 or

*π*, and

*ϱ*

_{1}is chosen to maximize the value of the Bessel function. In the presence of

*N*> 1 seismometers, the optimization problem is non-trivial. The optimal array configuration fulfills the relation

^{N}contains 2

*N*derivatives with respect to the two horizontal coordinates of

*N*seismometers. Already with a few seismometers, it becomes very challenging to find numerical solutions to this equation (see Section 7.1.6). An easier procedure that we want to illustrate now is to perform a step-wise optimal placement of seismometers. In other words, one after the other, seismometers are added at the best locations, with all previous seismometers having fixed positions. The procedure can be seen in Figure 37. The first seismometer must be placed at

*x*

_{1}= ±0.3

*λ*

^{R}and

*y*

_{1}= 0. We choose the side with positive

*x*-coordinate. Assuming a signal-to-noise ratio of

*σ*= 10, the single seismometer residual would be 0.38. The second seismometer needs to be placed at

*x*

_{2}= − 0.28λ

^{R}and

*y*

_{2}= 0, with residual 0.09. The third seismometer at

*x*

_{3}= 0.75

*λ*

^{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 λ

^{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 Figure 38 for a sub-optimal spiral array, and seismometers with frequency-independent

*σ*= 100. Rayleigh-wave speed is constant

*c*

_{R}= 250 m/s. 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 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 Figure 38 after subtraction of gravity-gradient noise (i.e., the low-frequency case). The sensor signal-to-noise ratio is the same as before. As we have seen in Section 3.6.4, Newtonian noise is suppressed at low frequencies in gravity strainmeters due to common-mode rejection of correlated gravity perturbations between two test masses. However, as soon as coherent cancellation is required, one has to pay the price for this gain. Each seismometer measures seismic displacement that is similarly correlated with gravity perturbations at both test masses. This means that the dominant part of the seismic data is useless since the corresponding gravity perturbations are rejected as common mode. Therefore, the data provided by the seismic array must make it possible to distinguish between the common-mode and differential noise. The Wiener filter needs to cancel the common-mode noise in the seismic data by combining data from different seismometers. An underlying weak correlation with the differential gravity signal then needs to be sufficient to optimize the Wiener filter for noise cancellation. It can be seen that the common-mode rejection causes the residuals to be higher, but only if the number of seismometers lies below a critical value. With the *N* = 20 arrays it is possible to distinguish the common-mode noise from differential noise, and subtraction residuals are similar to the standard Newtonian-noise cancellation. However, in all cases, suppression of common-mode noise becomes less efficient at long wavelengths. For this reason, the low-frequency slope of the residual spectra has an additional 1/*ω*, which causes the cancellation to be less broadband. Further results from this analysis can be found in [82]. In the future, it should be analyzed if an inherently differential seismic sensor, such as a seismic strainmeter, naturally provides the required common-mode rejection of seismic data, leading to more efficient noise subtraction.

#### 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 Sections 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 Section 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 this section, we will investigate the properties of coherent noise cancellation of the bulk contribution. Therefore, this section is without purpose to low-frequency GW detectors. The reason is that a low-frequency detector (i.e., sub-Hz detector) can always be considered to be located at the surface with respect to seismic Newtonian noise, since feasible detector depths are only a small fraction of the length of seismic waves. In other words, surface perturbations will always vastly dominate bulk contributions. Since we consider the high-frequency case, we can assume here that Newtonian noise is uncorrelated between test masses. In order to simplify the analysis, only homogeneous and isotropic body-wave fields are considered, without contributions from surface waves.

The evaluation of Wiener-filter performance requires the calculation of two-point spatial correlation functions between seismic measurements and gravity measurements. Since gravity perturbations are assumed to be uncorrelated between test masses, we can focus on gravity perturbations at a single test mass. The test mass is assumed to be located underground inside a cavity. We know from Section 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. From the theory perspective, cancellation of noise from cavity walls is straight-forward and will not be discussed here. More interesting is the cancellation of noise from density perturbations in the medium. A seismic measurement is represented by the projection \(\vec e_n \, \cdot \, \vec \xi(\vec r,\omega)\) where \(\vec e_n\) is the direction of the axis of the seismometer, and the gravity measurement by a similar projection \(\vec e_n \, \cdot \, \delta\vec a(\vec r,\omega)\). Therefore, the general two-point correlation function depends on the directions \(\vec e_1, \, \vec e_2\) of two measurements, and the unit vector \(\vec e_{12}\) that points from one measurement location at \(\vec r_1\) to the other at \(\vec r_2\).

*θ*,

*ϕ*by choosing the

*z*-axis parallel to \(\vec e_{12}\) so that \(\vec e_k \, \cdot \, \vec e_{12}=\cos(\theta)\) Instead of writing down the explicit expression of \(\vec e_k\otimes\vec e_k\) 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 1 and \(\vec e_{12}\otimes \vec e_{12}\) For symmetry reasons, it cannot depend explicitly on any other combination of the coordinate basis vectors \({\vec e_x} \otimes {\vec e_x},{\vec e_x} \otimes {\vec e_y}, \ldots\) Expressing the integral as linear combination of basis vectors, \(P_1(\Phi_{12})\mathbf 1+P_2(\Phi_{12})(\vec e_{12}\otimes \vec e_{12})\) with \(\Phi_{12}\equiv k^{\rm{P}}\vert\vec r_2-\vec r_1\vert\) solutions for

*P*

_{1}(Φ

_{12}),

*P*

_{2}(Φ

_{12}) can be calculated as outlined in [70, 11], and the correlation function finally reads

*p*needs to be introduced that parameterizes the ratio of energy in the P-wave field over the total energy in P- and S-waves. The correlation between seismometers depends on

*p*:

_{12}= 0, i.e., when the seismometer is placed at the test mass. The residual is solely limited by the mixing ratio and signal-to-noise ratio. The case was different for Rayleigh waves, see Eq. (216), where a limitation was also enforced by the correlation pattern of the seismic field. This is a great advantage of underground detectors. In fact, if the mixing ratio is

*p*= 1 (only P-waves), then it can be shown that the optimal placement of all seismometers would be at the test mass. With a single seismometer, a residual of ≈ 1/

*σ*

^{2}would be achieved over all frequencies (assuming that

*σ*is constant). However, the case is different for mixing ratios smaller than 1. Assuming a conservative mixing ratio of

*p*= 1/3 (P-waves are one out of three possible body-wave polarizations), the single-seismometer residual is about 2/3 provided that

*σ*≫ 1.

*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 theses 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}= α0.33

*λ*

^{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

*λ*

^{P}. With three seismometers, a residual of 0. 44 can be achieved.

The left plot in Figure 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 *σ* = 100.

*p*. The seismic strain field \(\mathbf{h}(\vec r, \, t)\) produced by a compressional wave can be written as

*T*

_{1}(Φ

_{12}),

*T*

_{2}(Φ

_{12}),

*T*

_{3}(Φ

_{12}). The result is the following:

*θ*,

*ϕ*. For each value of these two angles, the resulting correlation between the two strainmeters corresponds to the radial coordinate of the plotted surfaces. Since the focus lies on the angular pattern of the correlation function, each surface is scaled to the same maximal radius. It can be seen that there is a rich variety of angular correlation patterns, which even includes near spherically symmetric patterns (which means that the orientation of the second strainmeter weakly affects correlation).

#### 7.1.4 Cancellation of Newtonian noise from infrasound

*z*

_{0}= 0 that sound waves are reflected from the surface (apart from a doubling of the amplitude). This also means that the direction average can be carried out over the full solid angle. For

*z*

_{0}> 0, one has to be more careful, explicitly include the reflection of sound waves, and only average over propagation directions incident “from the sky” (assuming also that there are no sources of infrasound on the surface).

*x*-coordinate can be technically obtained by calculating the derivative

*∂*

_{x}, with

*x*≡

*x*

_{2}−

*x*

_{1}and Eq. (228), we find

As a final remark, infrasound waves have properties that are very similar to compressional seismic waves, and the result of Section 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. (227), 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 [128]. 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 [20]. In light of the results presented in Section 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 [78]. Nonetheless, from a practical point of view, the full result is more similar to the coherent relations such as Eq. (143), 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 \(C(\delta\rho;\vec r,\omega)\). We start with a general discussion and later present results for the isotropic Rayleigh-wave field.

*R*defined in Eq. (215) as a function of sensor locations \(\vec r_i\) Accordingly, the optimal sensor locations fulfill the equation

*M*sensors, i.e.,

*k*∈ 1,…,

*M*. In homogeneous fields, the Newtonian-noise spectrum and seismic spectrum are independent of sensor location,

*k*of the vector \(\vec C(\vec s;n)\) and the

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

*k*. This means that the derivative ∇

_{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. (233) 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. (233).

*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. (233) either for the derivatives *∂*_{x}, *∂*_{y} or *∂*_{x1}, *∂*_{x2}. 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.

In order to find optimal arrays with many sensors, it is recommended to build these solutions gradually from optimal solutions with one less sensor. In other words, for the initial placement, one should use locations of the *M* − 1 optimal array, and then add another sensor randomly nearby the test mass. The search relocates all sensors, but it turns out that sensors of an optimal array with a total of *M* − 1 sensors only move by a bit to take their optimal positions in an optimal array with *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.

*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. (207), is given by 1/SNR.

In many situations, it will not be possible to model the correlations **C**_{ss} and \(\vec C_{\rm{SN}}\). In this case, observations of seismic correlations **C**_{ss} can be used to calculate \(\vec C_{\rm{SN}}\), see Eq. (209), and also *C*_{NN}, see Eq. (104). 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 [112].

*N*= 7 array is the first optimal array that requires two seismometers placed on top of each other. Consequently, the broadband performance of the

*N*= 6 array is similar to the

*N*= 7 array. Residuals of optimal arrays can be compared with the stepwise optimized arrays as discussed in Section 7.1.2, taking into account that SNR= 10 was used in Section 7.1.2.

Cancellation of Newtonian noise from isotropic Rayleigh-wave fields at wavelength *λ*. Shown are the optimal arrays for 1 to 6 sensors with SNR = 100.

Sensor coordinates [ | Noise residual \(\sqrt R\) |
---|---|

(0.293,0) | 0.568 |

(0.087,0), (−0.087,0) | 2.28 × 10 |

(0.152,−0.103), (0.152,0.103), (−0.120,0) | 1.24 × 10 |

(0.194,0.112), (0.194,−0.112), (−0.194,0.112), (−0.194,−0.112) | 7.90 × 10 |

(0.191,0.215), (0.299,0), (0.191,−0.215), (−0.226,0.116), (−0.226,−0.116) | 6.69 × 10 |

(0.206,0.196), (0.295,0), (0.206,−0.196), (−0.206,0.196), (−0.295,0), (−0.206,−0.196) | 6.04 × 10 |

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* ≥ 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.

*k*

_{ϱ}is the wavenumber of a Rayleigh wave. This result turns into Eq. (109) for

*ϱ*→ 0.

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^{11}. As we have discussed in Section 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. (234). 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 Δ*L*, where Δ*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 Figure 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*_{ϱ}*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.

*δ*

_{z}

*δa*

_{x}=

*δ*

_{x}

*δa*

_{z}, where \(\delta\vec a\) are the fluctuations of gravity acceleration, and

*x*points along the arm of the large-scale detector, are perfectly correlated with

*δa*

_{x}. This can be seen from Eq. (97), since derivatives of the acceleration

*δa*

_{x}with respect to

*z*, i.e., the vertical direction, does not change the dependence on directions

*ϕ*. The coherence (normalized correlation) between

*δa*

_{x}and

*δ*

_{z}

*δa*

_{x}is shown in the left of Figure 46 making use of \(\langle\delta a_x(\vec 0),\partial_z\delta a_x(\vec \varrho)\rangle_{\rm{norm}}=\langle\delta a_x(\vec 0),\delta a_x(\vec \varrho)\rangle_{\rm{norm}}\).

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. (207), the maximal noise reduction can be calculated as a function of the coherence. In Figure 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.

*r*≪

*λ*is the distance between test masses of the large-scale detector and the gradiometer, which one can also interpret as maximal size of the gradiometer to achieve a suppression

*s*. A numerical factor of order unity is omitted. Given a Newtonian strain noise

*h*

_{NN}of the large-scale detector with arm length

*L*, the gradiometer observes

*ξ*

_{NN}denotes the relative displacement noise in the large-scale detector. Now, the relative displacement noise in the gradiometer is

*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 Sections 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 [27, 28], 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

^{12}(archiving global seismic data), and the Japanese seismic broadband network F-Net

^{13}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 Figure 47 for a seismic station in the US.

The colors of the markers on the map signify the median of the spectral histograms at a specific frequency. The frequency can be changed with a video slider. Clicking on a marker pops up additional site-specific information. 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 [45]. 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 [33, 45]. 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. To be specific here, ambient seismicity should be understood as the quasi-stationary noise background, which excludes for example the occasional strong earthquake. Larger distances to seismically active zones may be necessary for 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. (36), 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 ground. The last means that amplitude decreases over relatively short distances to the surface. Seismic measurements have been carried out in boreholes [63, 153], and specifically in the context of site characterization for future GW detectors at former or still active underground mines [89, 27, 127, 28]. There are however hardly any underground array measurements to characterize the seismic field in terms of mode composition. 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,…). Currently, a larger seismic array is being deployed for this purpose as part of the DUGL (Deep Underground Gravity Laboratory) project at the former Homestake mine, now known as the Sanford Underground Research Facility, equipped with broadband seismometers, state-of-the-art data acquisition, and auxiliary sensors such as infrasound microphones. As a consequence of the high cost, the effort could only be realized as collaboration between several groups involving seismologists and GW scientists.

The underground sites have similar seismic spectra above about 1 Hz, which are all lower by orders of magnitude compared to the surface spectrum measured inside one of the Virgo buildings. The Virgo spectrum however shows strong excess noise even for a surface site. This can be seen immediately since the spectrum exceeds the global high-noise model drawn as dashed curve between 1 Hz and 3 Hz, which means that there is likely no natural cause for the seismic energy in this range. The Virgo infrastructure may have enhanced response to ambient noise at these frequencies, or the seismic sources may be part of the infrastructure. The Netherland spectrum is closer to spectra from typical surface locations, with somewhat decreased noise level though above a few Hz since the measurement was taken 10 m underground. Nevertheless, the reduction of seismic Newtonian noise to be expected by building a GW detector underground relative to typical surface sites is about 2 orders of magnitude, which is substantial. Whether the reduction is sufficient to meet the requirements set by the ET sensitivity goal is not clear. It depends strongly on the noise models. While results presented in [25] indicate that the reduction is sufficient, results in [81] show that further reduction of seismic Newtonian noise would still be necessary.

#### 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 Section 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 [75, 5, 98, 131]. 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 Section 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

Hughes and Thorne suggested that one way to reduce Newtonian noise at a surface site may be to dig moats at some distance around the test masses [99]. The purpose is to reflect incident Rayleigh waves and thereby create a region near the test masses that is seismically quieter. The reflection coefficient depends on the depth of the moat [120, 73, 35]. If the moat depth is half the length of a Rayleigh wave, then the wave amplitude behind the moat is weakened by more than a factor 5. Only if the moat depth exceeds a full length of a Rayleigh wave, then substantially better reduction can be achieved. If the distance of the moat to the test mass is sufficiently large, then the reduction factor in wave amplitude should translate approximately into the same reduction of Newtonian noise from Rayleigh waves. There are two practical problems with this idea. First, the length of Rayleigh waves at 10 Hz is about 20 m (at the LIGO sites), which means that the moat needs to be very deep to be effective. It may also be necessary to fill moats of this depth with a light material, which can slightly degrade the isolation performance. The second problem is that the scheme requires that Rayleigh waves are predominantly produced outside the protected area. This seems unlikely for the existing detector sites, but it may be possible to design the infrastructure of a new surface site such that sources near the test masses can be avoided. For example, fans, pumps, building walls set into vibration by wind, and the chambers being connected to the arm vacuum pipes are potential sources of seismicity in the vicinity of the test masses. The advantage is that the moats do not have to be wide, and therefore the site infrastructure is not strongly affected after construction of the moats. Another potential advantage, which also holds for the recess structures discussed below, is that the moat can host seismometers, which may facilitate coherent cancellation schemes since 3D information of seismic fields is obtained. This idea certainly needs to be studied quantitatively since seismic scattering from the moats could undo this advantage.

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 Section 3.5 for details). The Newtonian-noise spectrum calculated from the reduction curve is shown in the right of Figure 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 Figure 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. Especially the coherent cancellation of Newtonian noise from seismic body waves depends on many factors, and in this section we could only develop the tools to address this problem systematically. The role of S-waves as coherent noise contribution among seismic sensors serving as reference channels in Wiener filtering has been described in Section 7.1.3. Since the cancellation performance presented in Figure 41 is relatively poor and possibly insufficient for future GW detectors that rely on substantial reduction of Newtonian noise, it can be said that developing an effective scheme is one of the top priorities of future investigations in this field. Possible solutions may be to combine seismometers and strainmeters in sensor arrays, and to use multi-axes sensors instead of the single-axis sensors modelled here. Nonetheless, it is remarkable that a simple approach does not lead to satisfactory results as we have seen for the cancellation of Rayleigh-wave and infrasound Newtonian noise in Figures 38 and 42. However, we have also been conservative with the body-wave modelling in the sense that we assumed isotropic fields and relatively low P-wave content. Since P-waves experience weaker damping compared to S-waves, it may well be possible that P-wave content is higher in seismic fields. We have also reviewed our current understanding of site-dependent effects on coherent noise cancellation in Section 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 a 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 at surface sites only as explained in Section 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 problems with this approach.

It should be emphasized that in general, the null constraint given by Eq. (6) 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. (9) is that we used the null constraint together with the geodesic equation to obtain Eq. (9), which is therefore valid only for massless particles.

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

Winterflood explains in his thesis why vertical resonance frequencies are higher than horizontal, and why this does not necessarily have to be so [173].

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 [124].

In reality, the dominant damping mechanism in suspension systems is not viscous damping, but structural damping characterized by the so-called loss angle *Φ*, which quantifies the imaginary part of the elastic modulus [152].

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.

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

## 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 University of Minnesota, Twin Cities. I have to thank Prof Mandic for his continuous support, and especially for taking the time to return comments on this manuscript. With his DUGL project currently proceeding at a steep rise, his time is very precious. During these first two years, I started to collaborate with Prof Giancarlo Cella, who by that time had already written seminal papers on Newtonian-noise modelling 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 as part of the DUGL project, I had the privilege to collaborate with and learn from my colleagues Dr Riccardo DeSalvo and Dr Mark Beker (at that time graduate student). Their motivation to do science in its best way without hesitation in complicated situations has inspired me since then. 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 modelling 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, 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. I am currently supported by a Marie-Curie Fellowship (FP7-PEOPLE-2013-IIF) at the Universit à di Urbino, which gives me the freedom to contribute to the development of the field in any possible direction. Therefore, I want to thank the committee of the European Commission who evaluated my past accomplishments in a favorable way. I want to thank Prof Flavio Vetrano and my colleagues at the INFN Firenze, who involved me in exciting experimental developments in Europe on low-frequency gravity sensing, especially atom interferometry. 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.

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