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Transient gravity perturbations induced by earthquake

rupture

J. Harms1,2,7, J.-P. Ampuero3, M. Barsuglia4, E. Chassande-Mottin4,

J.-P. Montagner5, S. N. Somala3and B. F. Whiting4,6

1Universit`

a degli Studi di Urbino “Carlo Bo”, Urbino, 61029, Italy

2INFN Sezione di Firenze, Sesto Fiorentino, 50019, Italy

3Seismological Laboratory, California Institute of Technology, Pasadena, CA 91125, USA

4AstroParticule et Cosmologie (APC), CNRS: UMR7164-IN2P3-Observatoire de Paris-

Universit´

e Denis Diderot-Paris 7, CEA: DSM/IRFU, France

5D´

epartement de Sismologie, Institut de Physique du Globe, 75005 Paris, France

6University of Florida, Gainesville, FL 32611, USA

7After the ﬁrst name, the author list is in alphabetic order.

SUMMARY

The static and transient deformations produced by earthquakes cause density perturba-

tions which, in turn, generate immediate, long-range perturbations of the Earth’s grav-

ity ﬁeld. Here, an analytical solution is derived for gravity perturbations produced by a

point double-couple source in homogeneous, inﬁnite, non-self-gravitating elastic media.

The solution features transient gravity perturbations that occur at any distance from the

source between the rupture onset time and the arrival time of seismic P waves, which

are of potential interest for real-time earthquake source studies and early warning. An

analytical solution for such prompt gravity perturbations is presented in compact form.

We show that it approximates adequately the prompt gravity perturbations generated by

2J. Harms et al

strike-slip and dip-slip ﬁnite fault ruptures in a half-space obtained by numerical simula-

tions based on the spectral element method. Based on the analytical solution, we estimate

that the observability of prompt gravity perturbations within 10 s after rupture onset by

current instruments is severely challenged by the background microseism noise but may

be achieved by high-precision gravity strainmeters currently under development. Our an-

alytical results facilitate parametric studies of the expected prompt gravity signals that

could be recorded by gravity strainmeters.

Key words: gravity perturbation – double couple – time domain – seismic potentials.

1 INTRODUCTION

Earthquakes generate transient and static deformation, including volumetric deformation and displace-

ment of material interfaces, which modify the spatial distribution of material density. This mass re-

distribution induces changes in the Earth’s gravitational ﬁeld. Permanent gravity changes generated

by static deformation induced by the co-seismic and post-seismic slip of large earthquakes have been

observed by superconducting gravimeters and gravity ﬁeld satellite missions on multiple occasions

(Imanishi et al. 2004; Wang et al. 2012a; Fuchs et al. 2013; Cambiotti & Sabadini 2013). Theoreti-

cal models of these static gravity perturbations have been developed (Okubo 1992; Sun et al. 2009)

and compared to observations (Matsuo & Heki 2011; Wang et al. 2012b). The transient effects of

elasto-gravitational coupling on the Earth’s gravest normal modes are well established and routinely

included in normal mode computations for long-period seismology (Dahlen & Tromp 1998). Transient

perturbations of the gravity ﬁeld concurrent with the passage of seismic waves have been examined

as a broad-band source of so-called Newtonian noise for gravitational wave detectors (Beker et al.

2011; Driggers et al. 2012). Here we address, for the ﬁrst time, the problem of modelling the gravity

perturbations generated by earthquakes at short time scales, including those generated by transient

deformation induced by seismic waves and those occurring during the fault rupture.

Our focus here is on a particular short-term elasto-gravitational effect that has received no at-

tention in the literature. Owing to the long-range and virtually instantaneous (speed-of-light) effect

of gravitational forces, density perturbations lead immediately to global perturbations of the Earth’s

gravity ﬁeld. In particular, the volumetric deformation carried by P waves induces remote gravity per-

turbations even at distances beyond the P wave front. Hence, at any distance from an earthquake source

and its dynamically deforming region, gravity perturbations are expected to be induced even between

the onset time of the rupture and the arrival time of seismic waves. We refer to these as prompt gravity

Transient gravity perturbations induced by earthquake rupture 3

perturbations. If practically measurable, these signals would add a new dimension to real-time seis-

mology that may enhance rapid earthquake source detection and parameter estimation capabilities and

may contribute to earthquake and tsunami early warning. Some analogies may be drawn to prompt

electromagnetic perturbations caused, in principle, by earthquakes via the electrokinetic effect in sat-

urated porous media (Gao et al. 2013) or via the motional induction effect of the Earth’s conductive

crust and magnetic ﬁeld (Gao et al. 2014). Here we develop building blocks for the quantitative anal-

ysis of prompt gravity perturbations, as needed to assess their detectability and information content.

An analysis of their potential contribution to earthquake early warning will be reported elsewhere, as

well as results of a search for prompt gravity signals in superconducting gravimeter recordings of a

recent mega-earthquake.

While normal mode theory in global seismology accounts for gravitational effects (Takeuchi &

Saito 1972; Dahlen & Tromp 1998), the structure of the prompt gravity perturbation ﬁeld is not im-

mediately evident from it. To facilitate systematic, parametric studies of transient gravity signals, we

develop in section 2 an analytical model of transient Newtonian gravity perturbations from point shear

dislocations in unbounded, uniform, non-self-gravitating elastic media. The model accounts for the ef-

fects of density perturbations induced by both static and transient deformation and provides a complete

gravity time series, before and after the passage of seismic waves. The key result is a compact expres-

sion for prompt gravity perturbations. Its relevance in more realistic settings is assessed in section 3

by comparing the analytical results with results from numerical simulations that include ﬁnite fault

and free surface effects. In section 4, after realizing that prompt gravity perturbations are generally

too weak to be detected by conventional gravimeters within 10 s after rupture onset, we derive ex-

pressions for transient gravity gradients and assess the theoretical potential of high-sensitivity gravity

strainmeters to observe prompt gravity signals.

2 GRAVITY PERTURBATIONS FROM POINT-SHEAR DISLOCATIONS IN INFINITE,

HOMOGENEOUS, NON-SELF-GRAVITATING ELASTIC MEDIA

2.1 Model assumptions and density perturbations

We consider a point shear dislocation in an inﬁnite, elastic and homogeneous medium. The defor-

mation is related to the dislocation source by the seismic wave equation and the Newtonian gravity

potential perturbation is related to the deformation-induced density perturbations by Poisson’s equa-

tion. We ignore the effects of self-gravitation, i.e. we assume that gravity perturbations induced by

deformation do not act back on the deformation and their contribution to the seismic wave equation

is ignored. Self-gravitation effects are signiﬁcant only at periods much longer than 100 s (see p. 142

4J. Harms et al

of Dahlen & Tromp (1998)), whereas our primary interest here is in the short time-scales over which

signiﬁcant prompt gravity perturbations may develop (the P wave travel time from the hypocenter to

distances of up to a few 100 km). We adopt a source-based coordinate system with origin at the loca-

tion of the shear dislocation, z-axis parallel to the slip direction and x-axis perpendicular to the fault

plane. Spherical coordinates r, θ, φ are related to the Cartesian coordinates via x=rsin(θ) cos(φ),

y=rsin(θ) sin(φ),z=rcos(θ), with 0< θ < π, and 0< φ < 2π. The point-shear dislocation

source is represented by a double-couple moment tensor point-source, with the following equivalent

body-force:

f(r, t) = −M0(t)∂δ(r)

∂z ex+∂δ(r)

∂x ez,(1)

where exand ezare the unit vectors pointing along the corresponding coordinate axes. We assume a

ﬁxed focal mechanism but allow for an arbitrary seismic moment time function M0(t).

The displacement ﬁeld induced by seismic waves is denoted by ξ(r, t). The resulting density

perturbation δρ(r, t)is given by the linearized continuity equation (e. g. equation 3.46 of Dahlen &

Tromp (1998)):

δρ(r, t) = −ρ0∇·ξ(r, t)(2)

where ρ0is the unperturbed mass density. It depends on the divergence of the displacement ﬁeld,

which describes volumetric deformations of the medium. From the known expression of the seismic

wave ﬁeld generated by a point double-couple source (Aki & Richards 2009):

δρ(r, t) = RP(θ, φ) ∆(r, t),(3)

where

RP(θ, φ) = cos(φ) sin(2θ)

= 2(ex·er)(ez·er)

(4)

is the quadrupolar P-wave radiation pattern, er≡r/r the radial unit vector, and

∆(r, t)≡3

4πr3α2M0(t−r/α)

+r

α˙

M0(t−r/α) + r2

3α2¨

M0(t−r/α)(5)

where αis the P-wave speed. Because S waves produce no volumetric deformation, density perturba-

tions in inﬁnite, uniform, elastic media are carried exclusively by P waves and hence depend on the

moment time function delayed by the P-wave travel time, M0(t−r/α). The density perturbation is

plotted in Figure 1 as a function of distance and time, for φ= 0 and θ=π/4. The plot shows that in

addition to a lasting density change in the vicinity of the source, a propagating density perturbation is

transported by the compressional waves.

Transient gravity perturbations induced by earthquake rupture 5

Figure 1. Density perturbation ﬁeld as a function of distance and time, for φ= 0,θ=π/4and seismic moment

time function M0tanh(t/τ)Θ(t), where Θ(·)denotes the unit step function. Since the positive and negative

values vary over many orders of magnitude, the color scale is based on a log-modulus transformation (John &

Draper 1980).

2.2 Prompt gravity perturbation

Prompt gravity perturbations are essentially the exterior gravitational ﬁeld induced by the density per-

turbation outside its spatial support, that is, beyond the P-wave front. Their structure can be determined

by exploiting classical results in potential theory (Jekeli 2007) as follows. The density perturbation

ﬁeld inherits the symmetries of a quadrupole from the P-wave radiation pattern, which features four

lobes of alternating signs. A quadrupole density perturbation generates an exterior gravitational poten-

tial, which has also a quadrupole structure and decays as 1/r3

0, where r0is the distance to the centroid

of the quadrupole.

In this section, we use seismic potentials (of the seismic ﬁelds and alternatively of the seismic

sources) to obtain an explicit expression of the gravity perturbation in homogeneous space valid for

all times. In the appendix, a second method is described, which uses the spherical multipole expansion.

While the method based on potentials is not very intuitive and does not give a direct interpretation of

the effective source of gravity perturbations, it is more elegant and should prove useful also to solving

more complicated calculations in the future, for example concerning gravity perturbations in a half

space.

The perturbation of the Newtonian gravity potential, δψ(r0, t), is determined by the density ﬂuc-

6J. Harms et al

tuations according to Poisson’s equation

∇2δψ = 4πGδρ (6)

where Gis the gravitational constant. In a uniform medium, the density perturbation is governed by

the seismic displacement ﬁeld via equation (2). We represent the seismic ﬁeld in terms of its Lam´

e

potentials:

ξ(r, t) = ∇φs(r, t) + ∇×ψs(r, t)(7)

The scalar potential φsgives rise to compressional waves, whereas the vector potential ψsgives rise

to shear waves and also obeys the condition ∇·ψs= 0. Inserting equation (7) into equation (2) and

noting that ∇·∇×ψs= 0, one ﬁnds

δρ =−ρ0∇2φs(8)

and, from equation (6), ∇2δψ =−4πGρ0∇2φs. Hence, considering the anticipated fast (1/r3

0) decay

of the gravity potential away from the density perturbation region, one ﬁnds for an inﬁnite medium

δψ(r0, t) = −4πGρ0φs(r0, t).(9)

This equivalence between a potential giving rise to a seismic compressional ﬁeld and the correspond-

ing gravity potential perturbation is remarkable. Given the known solution for seismic potentials from

a point force in inﬁnite media (Aki & Richards 2009), one can derive the expression for a double-

couple and rescale it according to equation (9) to obtain the following expression of the perturbed

gravity potential:

δψ(r0, t) = GRP(θ0, φ0)

·1

r0α2M0(t−r0/α)−3

r3

0

r0/α

Z

0

du uM0(t−u)(10)

Note that this result does not depend explicitly on the reference density of the medium. Before the

arrival of P waves, when t<r0/α, the ﬁrst term in brackets vanishes, the upper limit of integration of

the second term can be substituted by t, and

t

Z

0

du uM0(t−u) =

t

Z

0

dt0

t0

Z

0

dt00M0(t00 )≡I2[M0](t)(11)

if M0(t)=0for t < 0. Therefore, the early gravity potential assumes a remarkably simple form

independent of the speed αof compressional waves:

δψ(r0, t) = −RP(θ0, φ0)3G

r3

0

I2[M0](t)(12)

Transient gravity perturbations induced by earthquake rupture 7

As anticipated, the prompt gravity potential inherits the quadrupole distribution of the P wave radiation

pattern and decays as 1/r3

0. Its relation to the seismic moment is remarkably simple: it is proportional

to the second integral of the seismic moment time function.

The derivation, in particular equation (9), has the unintuitive feature that the prompt gravity po-

tential perturbation appears to emerge from the “acausal” component of the P-wave potential. This

component has actually no physical signiﬁcance in the seismic-wave equation, where its contribution

to the near-ﬁeld seismic waveﬁeld is cancelled out by a similar contribution from the S-wave potential.

We can provide an alternative derivation of the prompt gravity potential perturbation based on

source potentials, which allows further insight into the origin of its simple structure. Taking the second

time derivative of equation (9) and considering the wave equation for the P-wave potential,

¨

φs=α2∇2φs+ Φ/ρ0,(13)

where Φ(r, t)is the scalar Helmholtz potential of the equivalent body force representation of a dislo-

cation (e.g. equation 4.9 of Aki & Richards (2009)), we ﬁnd

δ¨

ψ=−4πGρ0(α2∇2φs+ Φ/ρ0).(14)

Making use of equation (8), we get

δ¨

ψ=−4πG(−α2δρ + Φ).(15)

The ﬁrst term in brackets is a propagative contribution carried by P waves. The second term is long-

ranged. Before the arrival of P waves to the detector, only the second term is non zero and hence the

prompt gravity potential perturbation satisﬁes

δ¨

ψ=−4πGΦ.(16)

Combining spatial derivatives of the point-force potential given e.g. in equation (4.17) of Aki &

Richards (2009), it can be shown that the scalar Helmholtz potential of the double-couple source

given in equation (1) is

Φ(r, t) = M0(t)

2π

∂21/r

∂x∂z .(17)

Porting this into equation (16) and integrating twice leads to equation (12). This derivation, in par-

ticular equation (16), provides a more elegant proof that the prompt gravity potential perturbation is

proportional to the second integral of M0(t). Also, equation (15) is consistent with equation (10) but

shows more clearly the separation between propagative and long-ranged components of the gravity

potential ﬁeld.

Finally, we provide an explicit expression of the prompt perturbation of gravity acceleration, δa=

8J. Harms et al

−∇δψ, which takes the form

δa(r0, t) = 6G

r4

0

((ez·er)ex+ (ex·er)ez−5(ex·er)(ez·er)er)I2[M0](t).(18)

Reﬂecting the long-range and instantaneous nature of gravity changes, these expressions show that the

prompt gravity perturbation extends over arbitrary distances and involves the seismic moment time-

function without P-wave time delay. Its proportionality to the second integral of seismic moment has

important implications. Compared to far-ﬁeld seismic ground displacements, which are proportional to

seismic moment rate, the high-frequency content of prompt gravity potential perturbations is damped.

For self-similar ruptures, whose moment function has an onset proportional to t3, the prompt gravity

potential signal is predicted to start as t5. After the rupture ends and before the P wave arrives, it grows

as t2. At the arrival of P waves, t=r0/α and δa∝t2/r4

0∝1/r2

0, consistent with the distance decay

of gravity perturbations produced by static earthquake deformation (Okubo 1991). In the calculations

leading to equation (12), it is found that the near-, intermediate- and far-ﬁeld terms of the density

perturbation, represented by the three terms in equation (5), have comparable contributions to the

prompt gravity perturbation. In particular, the contribution of the static deformation (the near-ﬁeld

term) is not dominant at any distance.

3 COMPARISON WITH SPECTRAL ELEMENT SIMULATIONS

The theory developed in section 2 does not capture all effects that may be present in nature. Even if

for some purposes the Earth can be sufﬁciently approximated by a homogenous medium, it remains

to be determined under which conditions this theory adequately describes ﬁnite fault sources in a half

space.

First, the theoretical results are derived for a point source. Additional ﬁnite-source effects are

expected close to a ﬁnite-size rupture. We do not consider this a major drawback, since the problem

is linear and ﬁnite-sized sources of arbitrary shape and with complicated rupture propagation can

be represented by superposition of point sources. A suitable approximation to describe ﬁnite-size

ruptures on a single planar fault is proposed here. The prompt gravity perturbation in that scenario is a

superposition of quadrupole gravity ﬁelds, each of the form given by equation (12). The leading term

of the multipole expansion of a superposition of quadrupoles is also a quadrupole, whose quadrupole

moment is simply the sum of the individual ones. The higher order multipole terms (octupole, etc) are

not necessarily zero but they decay faster with distance. Hence, at a distance from the P-wave front, the

prompt gravity perturbation produced by a ﬁnite earthquake source is approximately given by equation

(12) if, as conventionally in seismology, M0(t)is understood as the seismic moment integrated over

the whole ﬁnite rupture surface. To minimize the octupole contribution, r0should be understood as the

Transient gravity perturbations induced by earthquake rupture 9

distance to the instantaneous earthquake centroid (or, more generally, to the instantaneous centroid of

the density perturbation ﬁeld). The quality of this approximation is expected to degrade as the P-wave

front approaches.

Second, the analytical solution was derived for inﬁnite media. If an event occurs at a depth h, then

we can expect deviations from our results as soon as P waves reach the surface at time h/α, which is

typically a few seconds only. Therefore, simulating many tens of seconds of gravity time series using

the analytical results of section 2 would in general lead to inaccurate predictions. However, for early

warning applications of prompt gravity signals the important question is if the inﬂuence of the surface

is signiﬁcant within a few seconds of the rupture onset.

Finite-source and free-surface effects can be assessed by comparing the analytical prediction

with results obtained from a numerical simulation on a half-space. The simulation tool used here

is SPECFEM3D, a seismic wave propagation code based on the spectral element method (Komatitsch

& Vilotte 1998; Komatitsch & Tromp 1999) which can simulate ﬁnite earthquake sources with pre-

scribed slip rate (S. N. Somala, J.-P. Ampuero and N. Lapusta, Finite-fault source inversion using ad-

joint methods in 3D heterogeneous media, submitted manuscript). We implemented in SPECFEM3D

the computation of gravity acceleration perturbations above the half-space surface produced by the

seismic displacement ﬁeld ξ(r, t)as a function of time, based on the following equation (Harms et al.

(2009); equations 3.100-3.101 of Dahlen & Tromp (1998)),

δa(r0, t) = Gρ0ZdV1

|r−r0|3(ξ(r, t)−3(er·ξ(r, t)) ·er).(19)

This equation has the form of a dipole perturbation associated with displaced point masses ρ0dV. It is

valid as long as the displacement is much smaller than the distance |r−r0|. The integral is performed

over a ﬁxed domain but it accounts for the gravity perturbations induced by deformation of the Earth’s

surface. Equation (19) gives the Eulerian gravity acceleration measurable by a seismically isolated

sensor, whereas the Lagrangian quantity measured by a non-isolated sensor involves an additional term

representing advection through the initial gravity gradient (Dahlen & Tromp 1998). This distinction is,

however, irrelevant before seismic waves arrive to the sensor. In this section, we compare the simulated

half-space gravity perturbation with the full-space analytical solution of prompt gravity acceleration

perturbations, equation (18).

The assumed kinematic source model is speciﬁed in appendix B. It represents a circular pulse-

like rupture of ﬁnite duration τwith cosine slip rate function, uniform ﬁnal slip δ, rise time Tand

constant rupture speed vrup. From the rupture model, an explicit expression of the moment function

M0(t)is obtained that is inserted into equation (18) to calculate an analytical prediction of the gravity

perturbation. We present results of three simulations, a very deep strike-slip event, a shallow one and

10 J. Harms et al

Parameter Symbol Value

Rupture speed vrup 3 km/s

P-wave speed α6.4 km/s

Mass density ρ02670 kg/m3

Shear modulus µ27 GPa

Slip δ0.5 m

Strike-slip events

Total ruptured area A110 km2

Total seismic moment M01.5×1018 Nm

Moment magnitude Mw6.1

Dip-slip event

Total ruptured area A400 km2

Total seismic moment M05.4×1018 Nm

Moment magnitude Mw6.5

Table 1. Parameter values used for the comparison between the numerical simulation and theory.

a shallow dip-slip event, and compare them with the analytical predictions. The duration of the strike-

slip events is τ= 2 s, while the duration of the dip-slip event is 4 s. The rise time in all cases is

T= 1 s. We simulated about 5 s of time series, a short time scale relevant to anticipated applications

in early warning. It is computationally expensive to increase the simulation duration and at the same

time increase the size of the simulation domain to encompass the complete P wave front and preserve

accuracy at high frequencies. Further parameter values of the simulations are listed in Table 1. For the

strike-slip scenarios, the total ruptured area is A=π(vrupτ)2and the seismic moment M0=µδA. In

the dip-slip scenario the rupture reaches the surface.

Figure 2 shows the comparison between theory and simulation for the deep strike-slip event with

hypocenter at 50 km depth. We compare the gravity acceleration in the horizontal direction normal to

the fault evaluated at a horizontal distance of 1km to the epicenter and to the fault (to avoid a null of

gravity acceleration) and 2 km above ground (to avoid exponentially decaying artifacts of the gravity

calculation related to the ﬁnite grid density). The relative deviation (shown as dashed line) grows with

time but stays smaller than 7% over the entire 5 s. This serves as veriﬁcation of our SPECFEM3D

implementation and to show that ﬁnite source effects are not signiﬁcant at this distance.

Figure 3-left shows the comparison for the shallow strike-slip event with hypocenter depth 7.5 km.

Gravity is evaluated at 50 km horizontal distance to the epicenter and 2 km above ground. The hori-

Transient gravity perturbations induced by earthquake rupture 11

Figure 2. Comparison between analytical predictions gth and numerical simulation gnum of prompt gravity

acceleration perturbations for a deep strike-slip earthquake. The solid curve is plotted against the right y-axis,

while the dashed and dash-dotted curves are plotted against the left y-axis. The hypocenter is 50 km deep so that

surface effects are non-existent during the ﬁrst 5 s. The detector is located at a horizontal distance of 1 km from

the epicenter to avoid a null of gravity acceleration. The result shows that approximating the fault rupture as a

point source leads to relative deviations smaller than 0.07 within the ﬁrst 5s.

zontal distance was chosen larger than the propagation distance of seismic waves during the ﬁrst 5 s.

The plot in the right of Figure 3 shows the comparison for the 7.5 km deep dip-slip event with 45◦dip

angle, and 4 s duration. In these two examples, the relative deviation is smaller than 1.5% and 6%, re-

spectively. This is even smaller than in the previous example, which may seem surprising given that at

t= 5 s seismic waves have already been reﬂected from the surface or have been converted into surface

waves, which was not considered in the theoretical analysis. Especially concerning the dip-slip event,

one might have expected more signiﬁcant deviations since the surface experiences a differential lift

across the fault trace when the rupture reaches the surface at about t= 3.5s. The agreement between

the theoretical and numerical results even in this case may be taken as a hint of a more fundamental

principle that is to be discovered in the full half-space solution.

Strong deviations between theory and numerical simulations were only observed for shallow

events in cases where gravity was evaluated close to the epicenter so that the contribution from passing

Rayleigh waves was dominant. An open question that should be addressed in future studies is whether

12 J. Harms et al

Figure 3. Comparison between analytical predictions gth and numerical simulation gnum of prompt gravity

acceleration perturbations for a shallow strike-slip earthquake and a surface-breaking dip-slip earthquake. Same

representation as in Figure 2. Left: strike-slip event with 2 s duration. Right: dip-slip event with dip angle of 45◦

and duration of 4 s. In both cases, the event hypocenter is 7.5 km deep so that surface effects can be expected

shortly after 1 s, and the detector is located at 50 km distance from the epicenter so that seismic waves do not

reach it within the 5 s of simulation time. For the dip-slip scenario, the rupture reaches the surface at about 3.5 s.

The impact of the surface onto gravity perturbations stays weak within the ﬁrst 5 s, leading to relative deviations

smaller than 0.015 for the strike-slip event, and 0.06 for the dip-slip event.

results still agree well even long after 5 s, at least as long as contributions from passing Rayleigh waves

can be excluded. The effect of heterogeneity of the crust also deserves further scrutiny.

4 TRANSIENT GRAVITY GRADIENTS

4.1 Motivations

Prompt gravity perturbations produced by earthquakes have not been observed with existing gravime-

ters or seismometers. An order-of-magnitude estimate of the amplitude of the prompt gravity accelera-

tion signal can be obtained from equation (18), neglecting the dependence on directions. We assume a

self-similar rupture model, whose seismic moment function starts as M0(t)∼t3. Based on empirical

relations between seismic moment and rupture duration (Houston 2001) we set M0(1 s) = 1017 Nm.

We ﬁnd that

δa =O(106m5s−7)t5

r4

0

∼O(10 nm/s2)(t/10 s)5

(r0/50 km)4(20)

with t < r0/α (before the P-wave arrival). Note that 50 km is the typical distance travelled by a P wave

in 10 s. This estimate is only valid before the half-duration of the rupture; after that it over-estimates

the signal amplitude. At late times, if the rupture duration and distance are large enough, prompt

gravity perturbations above 10 nm/s2are predicted. However, gravity acceleration changes forming

within the ﬁrst few seconds at local distances to the source (50 km or less) lie well below 10 nm/s2.

Transient gravity perturbations induced by earthquake rupture 13

This is also illustrated in the ﬁgures in Section 3. Measurements of such weak gravity transients are

masked by a foreground of seismic noise (Berger et al. 2004; Brown et al. 2014), which is of order

100 nm/s2between 0.1 Hz and 1 Hz. A novel class of instruments, for example seismically isolated

gravity gradiometers (also referred to as gravity strainmeters; (Harms et al. 2013)), is required to

detect these early transients.

The past two decades have seen rapid progress in the development of ultra-sensitive gravity strain-

meters, mainly driven by scientiﬁc communities working on gravitational-wave (GW) detection from

astrophysical sources. Gravity strain is the relative change in physical distance between two test

masses in response to a changing gravitational ﬁeld. This change can be produced by a GW of as-

trophysical origin as well as by gravity perturbations from terrestrial sources. Laser-interferometric

GW detectors such as LIGO (LIGO Scientiﬁc Collaboration 2009), Virgo (Accadia, T. et. al. 2011),

GEO600 (L¨

uck et al. 2010), and KAGRA (Aso et al. 2013) measure gravity strain as differential dis-

placement between seismically isolated test masses using high-power, in-vacuum lasers. Strain sen-

sitivities of better than 10−22Hz−1/2have been demonstrated in the frequency range between about

50 Hz and 1000 Hz. The LIGO and Virgo detectors are currently being upgraded to advanced con-

ﬁgurations with design strain sensitivity better than 10−23Hz−1/2between about 30 Hz and 2000 Hz.

Furthermore, future ground-based detectors have been proposed, such as the Einstein Telescope (ET

Science Team 2011) with even more enhanced strain sensitivity, and a detection band extending down

to a few Hz. In parallel to these kilometer-scale detectors, groups are developing gravity strainmeters

targeting signals below 1 Hz, which are better suited to detect gravity perturbations changing over

timescales of a few seconds (Ando et al. 2010; Hohensee et al. 2011; Dickerson et al. 2013; Shoda

et al. 2014).

In the next section, we derive expressions for gravity strain perturbations (expressed as gravity gra-

dients, i. e. the second time derivative of gravity strain) useful to assess the capability of future sensors

to observe prompt gravity signals. Current concepts and sensitivity goals of these ”low-frequency”

instruments are brieﬂy reviewed in section 4.4.

4.2 Gravity-gradient tensor

The gravity-gradient tensor is given by

D(r0, t) = ∇δa(r0, t) = −(∇⊗∇)δψ(r0, t)(21)

where ’⊗’ denotes the Kronecker product (also known as dyadic or tensor product). It is a symmetric

tensor. Substituting equation (10) for the gravity potential, one obtains a tensor that can be divided

into four parts, distinguished by their dependence on directions. The ﬁrst part is proportional to the

14 J. Harms et al

local density perturbation:

D1(r0, t) = −4πG δρ(r0, t)er⊗er(22)

It is the only contribution with non-vanishing trace. Using Tr(a⊗b) = a·b, one obtains

Tr(D1(r0, t)) = −4πG δρ(r0, t),(23)

consistent with Poisson’s equation. The second part can be cast into the form

D2(r0, t) = −6G

r5

0

S(θ, φ)

r0/α

Z

0

du uM0(t−u)(24)

where

S(θ, φ) = 5(ex·er)(ez·er)(31 −7er⊗er)

+ 4(ex⊗ez)sym + 5((ex×er)⊗(ez×er))sym.

(25)

Here (a⊗b)sym ≡a⊗b+b⊗a. The third part is given by

D3(r0, t) = 2G

5r3

0α26M0(t−r0/α) + r0

α˙

M0(t−r0/α)

·(S(θ, φ)+(ex⊗ez)sym ),

(26)

and the last part is proportional to the delayed moment function

D4(r0, t) = −2G

α2r3

0

M0(t−r0/α)·(ex⊗ez)sym,(27)

Note that the unit vectors exand ezare not arbitrary coordinate axes, but correspond to the fault normal

and slip direction. The full gravity-gradient tensor is simply the sum of these four contributions. For

small times relevant to prompt perturbations, t<r0/α, the ﬁrst and last two contributions vanish

since M0(t)=0for t < 0, and the integral of the second contribution can be rewritten as

D(r0, t) = −6G

r5

0

S(θ, φ)I2[M0](t).(28)

None of the four contributions vanishes for t→ ∞. Instead the time derivatives of the moment

function go to zero, and the moment function itself can be substituted by its ﬁnal value M0(t→ ∞).

The result is a gravity-gradient tensor whose components decrease with 1/r3

0, and is consistent with

the static gravity perturbation found by Okubo (1991) for shear dislocations in a half space, provided

that his result is evaluated for an event far from the surface.

4.3 Gravity-strain response

The instruments anticipated to observe weak gravity transients of the form given in equation (28)

measure changes in distance between test masses or relative rotations between suspended bars. These

Transient gravity perturbations induced by earthquake rupture 15

instruments measure the so-called gravity-strain tensor h≡I2[D](t), rather than the gravity-gradient

tensor D. The concept of gravity strain is similar to that of elastic strain. While elastic strain deter-

mines the relative change in distance between test masses bolted to an elastic medium, gravity strain

causes a relative change in distance between freely-falling test masses. The concept of gravity strain is

predominantly used in the gravitational-wave community, and since it emerged from studies of gravity

in the framework of general relativity, it should also be emphasized that some of the predicted effects

on instruments cannot be explained by Newtonian gravity theory. Nonetheless, in the context of this

paper, it is sufﬁcient to understand gravity strain as if produced by a tidal force and therefore it can be

calculated from the gravity-gradient tensor by double integration as stated above.

There are two types of gravity strain measurements: (1) the conventional scheme, which is a

measurement of the relative displacement of test masses typically carried out along two perpendicular

baselines (arms); and (2) measurement of the relative rotation between two suspended bars. In either

case, the response of a sensor is obtained by projecting the gravity strain tensor onto a combination of

two unit vectors, e1and e2, that characterize the orientation of the detector, such as the directions of

the two bars in a rotational gravity strainmeter, or the two arms of a conventional gravity strainmeter.

This requires us to deﬁne two different gravity strain projections. The projection for the rotational

strain measurement is given by

h×(r0, t)=(e>

1·h(r0, t)·er

1−e>

2·h(r0, t)·er

2)/2,(29)

where the subscript ×indicates that the response to gravity ﬁelds is based on a relative rotation of the

bars. The vectors er

1and er

2are rotated counter-clockwise by 90◦with respect to e1and e2. In the

case of perpendicular bars er

1=e2and er

2=−e1. The corresponding projection for the conventional

gravity strainmeter reads

h+(r0, t)=(e>

1·h(r0, t)·e1−e>

2·h(r0, t)·e2)/2(30)

The subscript +indicates that the response is based on a distance change between test masses. It is

straight-forward to evaluate the projections of the strain tensor in the form given in equations (22),

(24), (26) and (27) using the relation e>

1·(a⊗b)·e2= (e1·a)(e2·b).

Transformations between coordinate systems governed by a rotation matrix Rcan be applied to

the strain tensor in the usual way

h0(r0

0, t) = R>·h(r0, t)·R, (31)

and the projection is now carried out expressing the unit vectors in the transformed coordinates. This

transformation can also be interpreted as a real rotation of the detector frame representing a change in

detector orientation. For the case of perpendicular bars or arms of the gravity strainmeter, we calculate

16 J. Harms et al

the detector response to gravity strain before the arrival of P waves, as given in equation (28). Denoting

the square-root of the average of the squared strain amplitude over detector orientations (represented

by the frame rotations R), fault orientation, and slip direction by h·i, one obtains:

hh+(r0, t)i=hh×(r0, t)i=6p14/5G

r5

0

I4[M0](t),(32)

where I4is the forth time integral. This expression allows us to make simple evaluations of expected

gravity strain signal amplitudes without having to specify directions and orientations. Adopting similar

assumptions as in section 4.1, we derive the following order-of-magnitude estimate of the amplitude

of prompt gravity strain signals:

hhi ∼ 6 104m5s−7t7

r5

0

∼10−12 (t/10 s)7

(r0/50 km)5(33)

As explained in the following section, there are at least three detector concepts that can reach these

sensitivities.

4.4 Sensitivity of low-frequency gravity strainmeters

We conclude with a brief discussion about instruments and instrumental concepts potentially able

to detect the gravity transients from earthquakes discussed in this work. Several concepts have been

proposed for gravity strainmeters that target signals between 10 mHz – 10 Hz. These include atom-

interferometric, laser-interferometric, torsion-bar, and superconducting gravity strainmeters (Moody

et al. 2002; Harms et al. 2013). While none of the concepts has reached the sensitivity yet required

for the detection of earthquake transients, sensitivities of 10−15 Hz−1/2in the region 0.1 – 1Hz seem

within reach. Correspondingly and according to equation (33), transient gravity signals from earth-

quakes at ∼10 s after rupture onset and at epicentral distances ∼50 km can be expected to have

high signal-to-noise ratio. We point out that these low-frequency strainmeters are much smaller scale

(of order 1 m to 10 m) than the km-scale GW detectors LIGO and Virgo, which operate at higher fre-

quencies. In fact, some of the modern concepts of low-frequency gravity strainmeters evolved from

well-known gravity gradiometer technology. We also want to emphasize that the sensitivity of low-

frequency gravity strainmeters required for the detection of earthquake transients lies well below the

sensitivity required for GW detection at the same frequencies, which is about 10−19 Hz−1/2at 0.1 Hz

(Harms et al. 2013). So it is conceivable that these instruments can be either early prototypes of future

GW detectors, or instruments speciﬁcally built for geophysical observations.

Groups working on the realization of these new concepts rely on modern understanding of ter-

restrial gravity noise, seismic isolation, and thermal noise acquired over the past decade. For seismic-

noise suppression, each concept follows a different strategy. Atom-interferometric detectors use freely

Transient gravity perturbations induced by earthquake rupture 17

falling ultracold atom clouds, and seismic noise couples strongly suppressed into the system through

the laser interacting with the atoms (Baker & Thorpe 2012). Superconducting gravity strainmeters

achieve seismic-noise reduction by common-mode suppression by differential readout of test-mass

displacements versus a common rigid reference (Moody et al. 2002). Laser-interferometric concepts

rely on active and passive seismic isolation of suspended test masses (Harms et al. 2013). Torsion-bar

detectors proﬁt from the mechanical ﬁltering of seismic noise through a potentially very low-frequency

torsion resonance (M. Ando et al. 2001). In reality, isolation performance is impeded by cross-coupling

between degrees of freedom allowing seismic noise to couple into the strainmeter output through un-

wanted channels. The goal is to improve the mechanical designs to suppress these couplings.

As explained by Harms et al. (2013), terrestrial gravity noise is expected to pose sensitivity lim-

itations at a level 10−15 Hz−1/2at 0.1 Hz mainly due to atmospheric infrasound, and possibly also

seismic ﬁelds (depending on the instrument site). It was proposed to use microphone or seismome-

ter arrays to coherently cancel associated gravity ﬂuctuations from the strainmeter data. However, this

technique is mostly unexplored and therefore its performance difﬁcult to predict. Some experience has

been gained with the cancellation of atmospheric gravity noise in gravimeter data (Neumeyer 2010),

but gravity-noise cancellation in gravity gradiometers is not equivalent to cancellation in gravimeters.

The best strain sensitivity at 0.1 Hz so far has been demonstrated with superconducting grav-

ity strainmeters reaching 10−10 Hz−1/2(Moody et al. 2002). Torsion-bar sensitivities have surpassed

10−7Hz−1/2at 0.1 Hz (Shoda et al. 2014). Naturally, work on detector designs needs to be accom-

panied by careful selection of instrument sites, which can have a big impact on ambient seismic or

infrasound ﬁelds (Berger et al. 2004; Brown et al. 2014), and also on the associated gravity noise.

5 CONCLUSION

In this paper we addressed for the ﬁrst time the problem of modelling transient gravity perturbations

produced by an earthquake during fault rupture. Since gravity changes propagate essentially instanta-

neously in comparison with seismic waves, gravity perturbations are generated at any distance as soon

as the rupture starts. In particular, a prompt gravity perturbation is expected before the arrival of P

waves, and has been computed here by an original analytical model, validated with numerical simula-

tions. The model indicates that it would be challenging to observe prompt gravity signals within 10 s

of the rupture onset with current instruments, but that they may be within the reach of high-precision

gravity strainmeters currently under development.

This study sets building blocks to evaluate the detectability and information content of prompt

gravity perturbations. The analysis of a potential application to earthquake early warning will be re-

ported elsewhere, in which a few seconds of gravity signal would enable warning signiﬁcantly faster

18 J. Harms et al

than conventional systems based on seismic data. The detection of prompt gravity perturbations may

also open new directions in earthquake seismology, since it consists in a direct measurement of the

mass redistribution during fault rupture. How complementary is the information about the earthquake

source contained in the transient gravity ﬁeld and in seismic waves remains to be assessed.

Further analysis is needed to extend this study to cases where the Earth’s surface affects more

signiﬁcantly the gravity perturbation ﬁeld. Surface effects can be studied numerically, as was done

here, but an analytical model of gravity perturbations in the presence of a surface would provide

further insight.

APPENDIX A: CALCULATING GRAVITY PERTURBATIONS USING SPHERICAL

HARMONICS

A straight-forward and possibly more intuitive, but slightly more involved method to calculate gravity

perturbations is to directly integrate over the ﬁeld of density perturbations produced by a double-

couple source according to

δψ(r0, t) = −GZdVδρ(r, t)

|r−r0|.(A.1)

The integrand can be expanded into its spherical multipoles. The integration over the radial coordinate

ris divided into two intervals: 0< r < r0and r0< r. Over the ﬁrst interval, one obtains the exterior

spherical multipole expansion:

δψext (r0, t) = G

∞

X

l=0

l

X

m=−l

Im

l(r0)∗·

r0

Z

0

dr r2ZdΩ δρ(r, t)Rm

l(r),(A.2)

where

Rm

l(r)≡r4π

2l+ 1rlYm

l(θ, φ)

Im

l(r)≡r4π

2l+ 1

1

rl+1 Ym

l(θ, φ)

(A.3)

are the regular and irregular solid spherical harmonics (in Racah’s normalization), respectively, Ym

l(·)

the spherical harmonics, and dΩ = dφdθsin(θ). The corresponding expression for the interior spher-

ical multipole expansion is

δψint(r0, t) = G

∞

X

l=0

l

X

m=−l

Rm

l(r0)∗·

∞

Z

r0

dr r2ZdΩ δρ(r, t)Im

l(r).(A.4)

Transient gravity perturbations induced by earthquake rupture 19

Inserting the density perturbation of equation (3) into the exterior multipole expansion of the gravity

potential we have

δψext (r0, t) = G

∞

X

l=0

l

X

m=−l

Im

l(r0)∗·

r0

Z

0

dr r2∆(r, t)

·ZdΩ RP(θ, φ)Rm

l(r)

(A.5)

The integral over angles can be carried out by expressing the radiation pattern of the density ﬁeld in

equation (4) in terms of the spherical harmonics Y1

2and Y−1

2,

RP(θ, φ)=2p2π/15 Y−1

2(θ, φ)∗−Y1

2(θ, φ)∗,(A.6)

and making use of the orthogonality relation

ZdΩ Ym

l(θ, φ)Ym0

l0(θ, φ)∗=δll0δmm0.(A.7)

This leads to

δψext (r0, t) = G4π

5

1

r3

0

r0

Z

0

dr r4∆(r, t)

·X

m=−1,1

Ym

2(θ0, φ0)∗ZdΩ RP(θ, φ)Ym

2(θ, φ)

(A.8)

Further simpliﬁcation shows that the angular dependence is given by the quadrupole radiation pat-

tern RP(θ0, φ0). The integral over the radius can be simpliﬁed considerably by integration by parts.

An expression for the interior multipole expansion is obtained by a similar procedure. The solution

δψ(r0, t) = δψext(r0, t) + δψint(r0, t)for the gravity potential perturbation can then be written in

the form given in equation (10).

The form of the early gravity perturbation, equation (12), suggests that one can obtain this re-

sult from a simple effective source term. Due to the quadrupolar form of the radiation pattern (i.e. it

involves only spherical harmonics of degree 2), we can represent the source by a mass quadrupole

moment

Qij =ZdV δρ(3xixj−δij r2)(A.9)

and zero monopole and dipole moments. Such a quadrupolar density distribution generates the follow-

ing exterior gravity potential:

δψ(r0, t) = −G

r3

0X

i,j

Qij(t)(ei·er0)(ej·er0)(A.10)

The integral in equation (A.9) can be solved again by expanding the integrand into its multipoles. For

early times, t<r0/α, the same result is obtained for the gravity perturbation with the only non-zero

20 J. Harms et al

components of the quadrupole moment tensor

Qxz(t) = Qzx(t) = 3I2[M0](t)(A.11)

Thereby, we have established an effective source model for prompt gravity perturbations generated by

a double-couple.

APPENDIX B: SOURCE MODEL OF NUMERICAL SIMULATION

We consider a circular pulse-like rupture of ﬁnite duration with cosine slip rate function and constant

rupture speed:

˙s(r, t) = 2δ

Tsin π

T(t−r/vrup)2

·Θ(t−r/vrup)Θ(T−t+r/vrup)Θ(vrup τ−r)

(B.1)

where Θ(·)denotes the unit step function. In words, the fault slip δat distance rfrom the hypocenter

builds up over a rise time T. The rupture propagates radially outwards with velocity vrup , until it

reaches the distance vrupτ. Therefore, τcan be regarded as the duration of the rupture, but the ﬁnal

fault slip occurs at time τ+T.

The moment function for a circular fault rupture can be calculated as

M0(t)=2πµ

t

Z

0

dt0

∞

Z

0

dr r ˙s(r, t0),(B.2)

where µis the shear modulus. Using the slip velocity model of equation (B.1), the moment function

is found to be

M0(t) = µ δ v2

rupT2

12π2·

−6πu + 4π3u3+ 3 sin(2πu), t ≤T

2π(−3 + π2(2 + 6(u−1)u)), T < t ≤τ

−2π3−3u+ 2π2((u−1)3−3uv2+ 2v3)

+ 3vcos(2π(u−v))−3 sin(2π(u−v)) τ < t ≤τ+T

12π3v2, τ +T < t

(B.3)

with u≡t/T , and v≡τ /T . After the rupture ends at t=τ+T, the moment function stays at a

constant value of µ δ π(vrup τ)2, and according to equation (18) the gravity acceleration grows as ∼t2

until P waves reach r0.

Transient gravity perturbations induced by earthquake rupture 21

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ACKNOWLEDGMENTS

We thank Eric Cl´

ev´

ed´

e for discussions during the initial phase of this study and Mauricio Fuentes for

assistance with veriﬁcation and simpliﬁcation of analytical derivations. This work was supported by

NSF Grants PHY 0855313 and PHY 1205512 to UF. BFW acknowledges sabbatical support from the

Universit´

e Paris Diderot and the CNRS through the APC, where part of this work was carried out.

JPA acknowledges support by a grant from the Gordon and Betty Moore Foundation to Caltech. We

acknowledge the ﬁnancial support from the UnivEarthS Labex program at Sorbonne Paris Cit´

e (ANR-

10-LABX-0023 and ANR-11-IDEX-0005-02) and the ﬁnancial support of the Agence Nationale de la

Recherche through the grant ANR-14-CE03-0014-01.

SPECFEM3D is available at http://www.geodynamics.org/cig/software/specfem3d.