Geophys. J. Int. (2015) ???, 1–11

Atomic as tools to monitor vertical surface motion

Ruxandra Bondarescu1, Andreas Scharer¨ 1, Andrew Lundgren2 Gyorgy¨ Hetenyi´ 3,4, Nicolas Houlie´4,5, Philippe Jetzer1, Mihai Bondarescu6,7 1 Department of Physics, University of Zurich,¨ Switzerland 2 Albert Einstein Institute, Hannover, Germany 3 Swiss Seismological Service, ETH Zurich,¨ Switzerland 4 Department of Earth , ETH Zurich,¨ Switzerland 5 Department of Civil, Environmental and Geomatic Engineering, ETH Zurich,¨ Switzerland 6 Department of Physics and Astronomy, University of Mississippi, Oxford, MS, USA 7 Facultatea de fizica, Universitatea de Vest, Timisoara, Romania

17 September 2018

SUMMARY

According to general relativity, a experiencing a shift in the gravitational potential ∆U will measure a frequency change given by ∆f/f ≈ ∆U/c2. The best clocks are optical clocks. After about 7 of integration they reach stabilities of ∆f/f ∼ 10−18, and can be used to detect changes in the gravitational potential that correspond to vertical displacements of the centimetre level. At this level of performance, ground-based networks emerge as a tool that is complementary to existing technology for monitoring a wide range of geophysical processes by directly measuring changes in the gravitational potential. Vertical changes of the clock’s position due to magmatic, post-seismic or tidal deformations can result in measurable variations in the clock tick rate. We illustrate the geopotential change arising due to an inflat- ing magma chamber using the Mogi model, and apply it to the Etna volcano. Its effect on an observer on the Earth’s surface can be divided into two different terms: one purely due to uplift (free-air gradient) and one due to the redistribution of matter. Thus, with the centimetre-level precision of current clocks it is already possible to monitor volcanoes. The matter redistribution term is estimated to be 3 orders of magnitude smaller than the uplift term. Additionally, clocks can be compared over distances of thousands of kilometres over short periods of , which improves our ability to monitor periodic effects with long-wavelength like the solid Earth tide. Key words: Atomic clocks; Surface deformation; geopotential measurements, Mogi model; Solid Earth tides

arXiv:1506.02457v2 [physics.geo-ph] 9 Jun 2015 1 INTRODUCTION ing (errors due to effects faster than the sampling rate) and from the attenuation of the gravitational field at the location of the satellite. Vertical deformation transients are key to characterising many ge- ological processes such as magmatic or tectonic deformation (Fig. In this paper, we consider dynamic sources that cause both 1). Many of these processes have timescales from hours to years vertical displacement and underground mass redistribution which which are difficult to measure with existing instruments. Atomic produce changes in the local geopotential. Geopotential differ- clocks provide a new tool to resolve vertical displacement, with a ences ∆U are directly measured by the changes in clock tick rate 2 current precision of about 1 cm in equivalent height after an inte- ∆f/f ≈ ∆U/c , where c is the speed of light. To be useful, a gration time of 7 hours (Hinkley et al. 2013; Bloom et al. 2014; clock must always be compared to a reference clock, which could Nicholson et al. 2015). be nearby or thousands of kilometres away. Clocks are connected via ultra-precise fiber links that are capable of disseminating their In the , we argued (Bondarescu et al. 2012) that clocks frequency signals over thousands of kilometers with a stability be- provide the most direct local measurement of the geoid, which yond that of the clock (Droste et al. 2013). As a concrete example is the equipotential surface (constant clock tick rate) that extends we the case of the inflation (or deflation) of an underground the mean sea level to continents. Since a clock network is ground- magma chamber, computed analytically (Mogi 1958), and apply it based, it can provide variable spatial resolution and can be used to to the Etna volcano. We explore whether the magma filling could calibrate and add detail to satellite maps, which suffer from alias- be detected using one or two clocks located on the volcanic edifice. 2 R. Bondarescu et al.

current clock current solid Earth tide has a vertical amplitude that can be as high as 30 cm sensitivity clock integration time (Agnew 2007) with a semi-diurnal period, whose amplitude can be 20 monitored by an atomic clock that is compared with a distant ref- erence clock. We find that geopotential and gravity measurements co- post- dyke seismic seismic intrusion are sensitive to two different combinations of tidal Love , 15 and could be used to calibrate existing measurements of the solid Earth tide. magma chamber 10 2 OVERVIEW OF ATOMIC CLOCKS

5 According to Einstein’s theory of general relativity, time slows

Vertical displacement (cm) Vertical down in the vicinity of massive objects. On a neutron star, clocks tick at about half their rates on Earth. An observer outside the hori- 0 zon of a black hole even sees time stopping all together at the hori- 0 1 2 3 4 5 6 7 8 zon. Similarly, clocks that are closer to Earth tick slightly slower 10 10 10 10 10 10 10 10 10 s Time than clocks that are further away. 1s 1min 1hr 1dy 1mo 1yr Atomic clocks employ atomic transitions that are classified, Figure 1. Phenomena that could be monitored with optical clock networks. depending on the transition frequency, as either microwave or op- The black line shows the lower sensitivity of today’s best optical clocks; see tical clocks. Since the clock frequency depends only on a known equation (1) and ∆ztoday in (7). The vertical dashed lines show current and atomic transition and constants of nature, clocks respond identi- planned clock integration . cally to changes in the gravitational potential and do not require calibration. This is in contrast to relative gravimeters, which suf- fer from instrument drift and have to be calibrated via comparisons with other measurements at the same location. The current defini- The primary tools currently used to monitor vertical dis- tion of the is based on a microwave atomic clock. However, placement are InSAR and GPS. Interferometric Synthetic Aperture optical clocks have the potential for higher stability because they Radar (InSAR) measures millimetre displacements in the line of utilize atomic transitions with resonance line-widths typically 105 sight of radar satellites over wide areas (e.g. Burgmann¨ et al. 2000; narrower than microwave transitions. Since the development of the Biggs et al. 2011), but with limited sampling rates (days to weeks). femtosecond laser frequency comb, optical clocks have been im- GPS is able to measure vertical displacements of 1 cm over short proving extremely rapidly (Poli et al. 2014). Today’s best clocks timescales (∼ an ) only when the displacement is very local- are optical. They are laboratory devices with frequency uncertainty ized in the network and/or the frequency of motion is different from −16 p the frequency of various artefacts that impact GPS accuracy. After ∆f/f ∼ 3 × 10 / τ/sec, (1) surveying areas for more than 10 years, the level of accuracy of where τ is the integration period (Hinkley et al. 2013; Bloom GPS techniques is close to the millimetre level (Blewitt & Lavallee´ et al. 2014; Nicholson et al. 2015), and are likely to continue to 2002; Houlie´ & Stern 2012) or better, enabling us to better char- improve dramatically within the next decade (Poli et al. 2014). A acterize the crustal elastic contrast of plate boundaries (Jolivet et transportable optical clock that monitors and compensates for en- al. 2009; Houlie´ & Romanowicz 2011). Since the primary source vironmental effects (temperature, pressure, electric and magnetic of noise in GPS measurements is due to signal dispersion through fields) and fits within two cubic meters has recently been built (Poli the atmosphere, both differential GPS and post-processed GPS data et al. 2014). perform better if networks are dense (e.g. Khan et al. 2010; Houlie´ In the future, it is expected that clocks will become sensitive et al. 2014) because many artefacts cancel across networks over to surface displacements at the sub-millimetre level. Possible tech- which the ionosphere and troposphere can be assumed to be con- niques include nuclear optical transitions (Campbell et al. 2012), stant. For timescales of , broadband seismometers can be optical transitions in Erbium (Kozlov et al. 2013), and transitions used (Houlie´ & Montagner 2007), but their bandwidth is unsuited in highly charged ions (Derevianko et al. 2012). A stability of for resolving long-term displacements (Boore 2003). −17 p Unlike the GPS or InSAR measurements, local atomic ∆f/f ∼ σtomorrow = 10 / τ/sec (2) clock measurements are insensitive to atmospheric perturbations, should be possible (Hinkley et al. 2013), which would achieve and could resolve ground displacement over shorter integration ∆f/f ∼ 10−20 within one month. timescales (hours to months). Further, clocks in conjunction with In order to take full advantage of the improved stability of gravimeters are also able to resolve density changes in the Earth optical clocks, distant clocks have to be compared reliably to the crust that do not, or just partially, lead to uplift or subsidence. In 10−18 level. This entails a global understanding of vertical dis- the case of a spherical magma chamber, the geopotential term re- placements, the solid Earth tide, and, overall, of the geoid to the sulting from mass redistribution is inversely proportional to the dis- 1-cm level. Any effects that cause perturbations to this level would tance to the source ∼ 1/R, whereas the ground displacement term have to be reliably modelled and understood. Clock comparisons scales with 1/R2. As opposed to this, both terms have the same via satellites are currently limited by the precision of the communi- 1/R2 scaling in gravity surveys. Comparing the measured grav- cation link that passes through a potentially turbulent atmosphere. ity change to the uplift, ∆g/∆h, can reduce this degeneracy to The most precise comparisons of distant clocks currently use un- model processes a volcano is undergoing before eruption (Rymer derground optical fiber links. Optical frequency transfer with stabil- & Williams-Jones 2000)and also more general processes (see Fig. ity better than the clock has been demonstrated over a two-way dis- 1 and De Linage et al. 2007). tance of 1840 km (Droste et al. 2013), with a fiber-link from Braun- Even in areas without active seismic or volcanic processes, the schweig, Germany to Paris, France. A fiber-link network capable Atomic clocks as tools to monitor vertical surface motion 3 of disseminating ultra-stable frequency signals is being planned (0, d) that undergoes a volume change ∆V deforms the half-, throughout Europe (e.g. NEAT-FT collaboration, REFIMEVE+). lifting an observer sitting at (r, 0) by ∆V d |w| = (1 − ν) , (8) π (r2 + d2)3/2 3 METHODS where ν is Poisson’s ratio. For crustal rock the typical value is ν ≈ While clocks are sensitive to changes in the gravitational poten- 0.25. Using Eq. (3) with z = w, this uplift changes the potential by tial, relative gravimeters see changes in the vertical component of the gravitational acceleration, which is a vector (~g = −∇~ U) whose GM⊕ GM⊕ ∆V d ∆U1 ≈ − 2 w = 2 (1 − ν) 2 2 3/2 , (9) amplitude can generally be measured much better than its direction. R⊕ R⊕ π (r + d ) They both provide local measurements of the change in gravity or where higher order terms have been neglected. When we add the potential relative to a reference point where the gravity and poten- contribution of the mass redistribution terms tial are known accurately. Within a large fiber links network, the 1 reference clock could be very far away. ∆Um = Gρm∆V √ . (10) r2 + d2 For a displacement z, the geopotential and gravity changes are The total geopotential change is GM⊕ 2GM⊕ ∆U ≈ − 2 z, ∆g ≈ 3 z. (3) R⊕ R⊕ ∆U = ∆U1 + ∆Um where G is the gravitational constant, and M⊕ and R⊕ the mass GM⊕ ∆V d 1 = (1 − ν) + Gρm∆V √ , 2 2 2 3/2 2 2 and radius of the Earth. The well-known free-air correction ∆g for R⊕ π (r + d ) r + d gravity is the gradient of the potential change ∆U. (11) A vertical displacement of z = 1 cm causes changes in the where ρ is the magma density. Typically, the change due to mass geopotential and in gravity of m redistribuiton ∆Um is several orders of magnitude smaller than that  z  m2 due to uplift ∆U . Measuring the mass redistribution term would ∆U ≈ 0.1 , (4) 1 1 cm sec2 require that we subtract the uplift term which may be obtained in-  z  m  z  ∆g ≈ 3 × 10−8 ∼ 3 µgal. (5) dividually using other techniques like GPS or InSAR. 1 cm sec2 1 cm Similarly, the change in the gravitational acceleration is given Thus, the frequency of a clock changes by by ∆f ∆U  z  ∆g = ∆g + ∆g ≈ ∼ 10−18 . (6) 1 m f c2 1 cm   2GM⊕ 1 − ν d (12) = − 3 + Gρm ∆V 2 2 3/2 . The solid line in Fig. 1 is the vertical displacement ∆ztoday as R⊕ π (r + d ) a function of clock integration time τ, which is obtained by equat- Notice that for both the acceleration and the potential the dif- ing (1) and (6), while ∆z results from (2) and (6). ferent terms of (2), (3) and (4) mostly cancel each other and only −1/2 −1/2  τ   τ  a term ∼ ρm survives (see Appendix A for the complete calcu- ∆ztoday ≈ 300 cm, ∆ztomorrow ≈ 10 cm. sec sec lation). However, this cancellation is a consequence of the elastic (7) half-space assumption of the Mogi model and will be less exact in more realistic scenarios. On the one hand, for the acceleration all terms have the same spatial dependence. On the other hand, for 4 APPLICATIONS the potential they scale differently: while the uplift term scales as d/R3 like all acceleration terms, the mass redistribution term scales We first discuss the geopotential and gravity changes caused by an √ as 1/R, where R = r2 + d2 is the distance to the source. inflating magma chamber, using the Mogi model. This is followed Assuming a clock with stability ∆f/f = σ /pτ/sec and a by a discussion of the measurability of the solid Earth tides. 0 source with a constant volume change rate ∆V/∆τ, the minimum required integration time to measure the uplift at a location (r, d) 4.1 Inflating magma chamber - the Mogi model " #2/3 σ πc2 R3  ∆V −1 τ = 0 sec−1 sec, (13) An inflating or deflating magma chamber can be described by the 1 g(1 − ν) d ∆τ so-called “Mogi model” (Mogi 1958): an isolated point pressure source in an elastic half-space that undergoes a pressure change. 2 with g = GM⊕/R⊕, whereas for the mass redistribution The Mogi model is broadly used in the literature (e.g. Houlie´ et 2/3 al. (2006) and Biggs et al. (2011)). Recently, its predictions were " 2  −1 # σ0Rc ∆V −1 compared to sophisticated simulations, which showed that in many τm = sec sec. (14) Gρm ∆τ situations the discrepancy is quite small (Pascal et al. 2014). Clocks and gravimeters lying above the magma chamber will We focus on the specific example of the Etna volcano. Houlie´ be affected by (1) a ground displacement, (2) the change of mass et al. (2006) used GPS data to investigate its underground magma density within the chamber, (3) the uplifted rock, and (4) the change system and found that a Mogi source located about 9.5 km below in the density of the surrounding material. We refer to effects (2), the summit with a volume change rate of 30 × 106 m3/yr would (3) and (4) as ‘mass redistribution’ (∆U2, ∆U3 and ∆U4 in Ap- yield the observed uplift. In Fig. 2, we plot the ground motion and pendix A). The change of the gravitational potential due to ground the resulting potential change as well as the potential change due −3 displacement (1) arises since the observer is shifted to a position to mass redistribution. We find that ∆Um ∼ 10 ∆U1, which farther away from the centre of the Earth. A Mogi source centred at roughly corresponds to ∆f/f ∼ ∆U/c2 ∼ 10−20. If we assume 4 R. Bondarescu et al.

and δ∆V/∆V as a function of the distance to the summit, assum- (a) 2 −18 3.5 ing a clock sensitive to ∆U/c = 10 . We find that the opti- 3.0 southern slope mal location of a second clock is r ≈ 0.78 d, independent of the 2.5 northern slope clock’s performance. For the performance considered here, this cor- 2.0 1.5 responds to δd/d ≈ 14% and δ∆V/∆V ≈ 30%. 1.0

altitude [km] 0.5 0.0 (b) 4.2 Solid Earth Tides 9 8 Solid Earth tides are the deformation of the Earth by the gravi- 7 vertical (uplift) 6 horizontal tational fields of external bodies, chiefly the Moon and the Sun. 5 In general relativity the Earth is freely falling, so at the centre of 4 3 the Earth the external gravitational force is canceled by the Earth’s 2 1 acceleration toward the external body. Because the external field

displacement [cm] 0 is not uniform, other points experience a position-dependent tidal (c) 9 force (Agnew 2007).

8 8 2 1 ∆U1 /c The tide has three effects of concern on the gravitational po- − 7 0 1 6 103 ∆U /c2 tential. First is the external potential itself, which can be calculated × 5 × m 2 4 directly from the known mass and position of the external bodies. c / i 3 Second is the change in the Earth’s gravitational potential produced U 2 ∆ 1 by the deformation of the Earth. Third is the vertical displacement 0 (d) of the surface, which produces a free-air correction to our measure- 100 ments. The last two effects are proportional to the first, parameter- 80 ized by the Love numbers kn and hn, respectively (Agnew 2007). 60 As shown in Appendix B, clocks (sensitive to geopotential % 40 δ∆V/∆V changes) and gravimeters (sensitive to the downward component 20 δd/d of g) each measure a different linear combination of the Love num- 0 bers. This is because the three effects scale differently with the dis- 0 5 10 15 20 25 30 tance from the centre of the Earth. It is therefore desirable to com- r [km] bine gravimeter and clock measurements to infer the Love numbers with great accuracy. Recall that clock measurements must be made Figure 2. Estimate of vertical ground deformation on the Etna volcano over between a pair of clocks. Since tidal effects are global, both clocks the course of one year with the Mogi model. In (a), the altitude profile of are sensitive to the tidal deformation; one clock cannot simply be Etna is shown. The solid curve shows the southern slope of the mountain, treated as a reference. To measure the tidal amplitude, it is nec- the dashed line shows the mirror image of the northern slope. For the fol- essary to compare instruments over distances of the order of half lowing plots only the former is considered since the latter gives very similar the tidal wavelength on timescales shorter than the period of the results. Assuming a Mogi source located 9.5 km below the summit with a tide. The tidal wavelength is half the circumference of the Earth for 6 3 volume change rate of 30 × 10 m /yr, the vertical and horizontal motion the dominant tidal mode. Both differential GPS and InSAR have over one year is plotted in (b). The change in the gravitational potential due short baseline, so tidal effects are small and are mostly subtracted to this uplift together with the change due to mass redistribution is shown by modelling. A network of clocks where each clock pair is sep- in (c). Notice that the latter is about 3 orders of magnitude smaller. If one clock is positioned at the summit, (d) shows the fractional errors on the arated by hundreds of kilometers could more holistically monitor measurements of the source depth δd/d and the volume change δ∆V/V as solid Earth tides. Current clocks provide measurements of the ver- a function of the horizontal distance to a second clock. tical uplift to within a percent of the maximal tidal amplitude on an hourly basis. Such measurements could be used to monitor stress changes within the crust, and to investigate whether these correlate constant volume change rate, an optical clock on the summit would with triggered seismicity. see the uplift if integrated for about ten days (Eq. (13)), while the Such a network of clocks on the continent scale could accu- mass redistribution signal is out of reach. With a clock stability rately measure the tidal Love numbers. The external tidal potential σtomorrow, such an uplift would be observable within a day, and can be decomposed into a sum of Legendre polynomials the mass redistribution in five months.  n n GMext R⊕ We also give an example of how to best choose the clock loca- Utid(R, α) = − Pn(cos α), (15) tions. We assume that the horizontal position of the magma cham- R R ber is already known from previous surveys, and that the two mea- where R is the distance between the external body of mass Mext surements we wish to make are the depth of the magma chamber d and the Earth’s centre of mass, and α is the angle from the ob- and the volume change ∆V (assuming also that ν is known). With servation point to the line between the centre of the Earth and the one clock directly above the magma chamber, we find the location external body. Since R⊕/R ≈ 1/60 for the Moon, and R⊕/R ≈ of a second clock which minimizes the measurement errors on d 1/23000 for the Sun, it is typically sufficient to just consider the and ∆V ; of course there must also be a distant reference clock. For first few terms of the expansion. By linearity, we treat each n sepa- a flat half-space we find the optimal location to be at a horizontal rately. distance r ≈ 0.78d (note that the minima is broad; see Appendix The potential change measured by a clock, including all three A5). Assuming Etna to be a half-space with chamber depth 9.5 km, effects above, is the fractional errors there are δd/d ≈ 18% and δ∆V/∆V ≈ 38%. n Including the height profile of the mountain, Fig. 2(d) shows δd/d ∆Un = (1 + kn − hn)Utid(R, α) . (16) Atomic clocks as tools to monitor vertical surface motion 5

The change in the vertical gravitational acceleration measured by a Boore, D.M., 2003. Analog-to-Digital Conversion as a Source of Drifts in gravimeter is Displacements Derived from Digital Recordings of Ground Accelera- tion, Bull. Seismol. Soc. Am., 93, 20172024.   n n + 1 2 Utid(R, α) ∆gn = −n 1 − kn + hn . (17) Burgmann,¨ R., Rosen, P.A. & Fielding, E.J., 2000. Synthetic Aperture n n R⊕ Radar Interferometry to Measure Earths Surface Topography and Its n Deformation, Annu. Rev. Earth Planet. Sci., 28, 169209. Combining the two measurements with the known Utid(R⊕, α) we can determine the Love numbers Campbell, C.J. et al., 2012. Single-Ion Nuclear Clock for Metrology at the 19th Place, Phys. Rev. Lett., 108, 120802. n + 2 R ∆g − 2∆U ⊕ n n De Linage, C., Hinderer, J. & Rogister, Y., 2007. A search for the ratio kn = + n , (18) n − 1 (n − 1)Utid(R⊕, α) between gravity variation and vertical displacement due to a surface 2n + 1 R⊕∆gn − (n + 1)∆Un load, Geophys. J. Int., 171, 986-994. hn = + n . (19) n − 1 (n − 1)Utid(R⊕, α) Derevianko, A., Dzuba, V.A., & Flambaum, V.V., 2012. Highly Charged Ions as a Basis of Optical Atomic Clockwork of Exceptional Accuracy, Both Love numbers are believed to be modelled to within a fraction Phys. Rev. Lett., 109, 180801. of a percent (Yuan & Chao 2012). These models will be throughly Droste, S. et al., 2013. Optical-Frequency Transfer over a Single-Span 1840 tested once a global optical clock network becomes available. km Fiber Link, Phys. Rev. Lett., 111, 110801. Gradshteyn, I.S. & Ryzhik, I.M., 2007. Table of Integrals, Series, and Prod- ucts, 7th edn, Elsevier, Burlington, San Diego, London. 5 CONCLUSIONS Hagiwara, Y., 1977. The Mogi model as a possible cause of the crustal uplift in the eastern part of Izu Peninsula and the related gravity change, Bull. We have demonstrated the promise of very precise atomic clocks Earthq. Res. Inst., 52, 301-309. for geophysical measurements with two illustrative examples. In Hinkley, N. et al., 2013. An Atomic Clock with 10−18 Instability, , the inflation or deflation of a spherical magma chamber (the Mogi 341, 1215-1218. model) clocks are primarily sensitive to the local vertical displace- Houlie,´ N. & Montagner, J.P., 2007. Hidden Dykes detected on Ultra Long ment resulting at the Earth’s surface. Such monitoring of local de- Period seismic signals at Piton de la Fournaise volcano?, EPSL, 261, formations can be done using a reference clock anywhere outside 1-8. the zone where the displacement is significant, typically tens of Houlie´ N. & Romanowicz B., 2011. Asymmetric deformation across the kilometres. However, when monitoring solid Earth tides, the best San Francisco Bay Area faults from GPS observations in Northern Cal- accuracy will be obtained with a clock network spanning the globe. ifornia, Phys. Earth Planet. In., 184, 143-153. Beyond the examples given, it should be possible to use clocks Houlie,´ N. & Stern, T., 2012. A comparison of GPS solutions for strain and in conjunction with gravimeters to monitor dyke intrusion, post- SKS fast directions: Implications for modes of shear in the mantle of a seismic deformation, aquifer variations, and other effects causing plate boundary zone, EPSL, 345348, 117125. vertical displacement or subsurface mass change. In contrast to Houlie,´ N., Dreger, D. & Kim, A., 2014. GPS source solution of the 2004 GPS and InSAR measurements, ground clock measurements are Parkfield earthquake, Sci. Rep., 4, 3646. insensitive to the turbulence in the atmosphere. Houlie,´ N., Briole, P., Bonforte, A. & Puglisi, G., 2006. Large scale ground In the future, optical clocks are expected to become part of deformation of Etna observed by GPS between 1994 and 2001, Geo- the global ground-clock network that is used for telecommunica- phys. Res. Lett., 33, L02309. tions increasing the precision with which we can monitor time. Jolivet, R., Burgmann,¨ R. & Houlie,´ N., 2009. Geodetic exploration of the Ground clocks can thus be combined with existing instrumenta- elastic properties across and within the northern San Andreas Fault tions (GPS, InSAR, gravimeters) to track underground mass redis- zone, Earth Planet. Sci. Lett., 288, 26-131. tribution through its effects on the geopotential. Portable optical Khan, S.A. et al., 2010. GPS measurements of crustal uplift near Jakob- clocks have already been developed (Poli et al. 2014), and could shavn Isbræ due to glacial ice mass loss, J. Geophys. Res., 115, B09405. monitor changes in the geopotential of the Earth across fault lines, Kozlov, A., Dzuba, V.A. & Flambaum, V.V., 2013. Prospects of building optical atomic clocks using Er i or Er iii, Phys. Rev. A, 88, 032509. in areas with active volcanoes, and for surveying. Mogi, K., 1958. Relations between eruptions of various volcanoes and the deformation of the ground surfaces around them, Bull. Earthq. Res. Inst., 36, 99-134. ACKNOWLEDGMENTS Nicholson, T. L. et al., 2015. Systematic evaluation of an atomic clock at −18 We acknowledge support from the Swiss National Science Founda- 2×10 total uncertainty, Nat. Commun., 6, 6896. tion, and thank Domenico Giardini for helpful discussions. Pascal, K., Neuberg, J. & Rivalta, E., 2014. On precisely modelling surface deformation due to interacting magma chambers and dykes, Geophys. J. Int., 196, 253278. Poli, N. et al., 2014. A transportable strontium optical lattice clock, Appl. REFERENCES Phys. B, 117, 1107-1116. Agnew, D.C., 2007. Treatise on Geophysics, 3.06 - Earth Tides, Elsevier, Poli, N., Oates, C.W., Gill, P., & Tino, G.M., 2014. Optical atomic clocks, Amsterdam, 163195. arXiv:1401.2378. Biggs, J., Bastow, I.D., Keir, D. & Lewi, E., 2011. Pulses of deformation Rymer, H. & Williams-Jones, G., 2000. Volcanic eruption prediction: reveal frequently recurring shallow magmatic activity beneath the Main Magma chamber physics from gravity and deformation measurements, Ethiopian Rift, Geochem. 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(a) 109 109

d0 w ] 8 r 10

y 8

/ 10 3 r m

[ 7 R u 10 τ τ 1 7 d ∆ 10 /

V 106 ∆

106 a ∆a 105 ]

1 5 9 13 17 21 s [

z (b) i 109 τ 105

Figure A1. Deformation due to inflating cavity. ] r y

/ 4

3 10

APPENDIX A: THE MOGI MODEL IN DETAIL m [ 108 τ A Mogi source is an isolated pressure point source embedded in an τ m ∆ 3 elastic half-space. A pressure change ∆P deforms the surround- / 10 V

ing half-space and uplifts an observer standing on the surface. This ∆ ground displacement, discussed in Sec. 4.1, affects the ticking rate of a clock since its position in the Earth’s gravitational field is 107 102 slightly shifted. Besides that, there are three additional effects af- 1 5 9 13 17 21 fecting both the gravitational potential and acceleration, which are R [km] due to the redistribution of matter. These are discussed in Sec. A2, A3 and A4. We choose coordinates such that the elastic half-space ex- Figure A2. Integration times for current clocks. Considering a clock with −16 tends from minus to plus infinity in both the x and y directions, σ0 = 3 × 10 , the required integration time to resolve uplift (Eq. (13)) and from 0 to +∞ in the z direction. The source is located at is shown in (a) as a function of distance R and (constant) volume change rate ∆V/∆τ. Here, it is assumed that the clock is placed above the magma (0, 0, d) and an observer positioned on the surface has coordinates chamber (r = 0). Analogous, the integration time for seeing the redistribu- (x, y, 0). By symmetry, we typically use coordinates (r, z), where tion of mass (Eq. (14)) is shown in (b). In contrast to the uplift where the p 2 2 r ≡ x + y is the distance from the observer to the place on result depends on both r and d, the integration time for the mass redistribu- the surface directly above the cavity. A scheme is given in Fig. A1. tion term depends on the total distance R only. Even though a point source is considered, the pressure change can be interpreted as the volume change ∆V of a finite size cavity. √ This approximation is valid as long as the radius of the cavity is with ν being the Poisson ratio and R ≡ r2 + d2 being the dis- much smaller than the depth d. Such a volume change can, for ex- tance between the cavity and the observer. Thus, a point on the sur- ample, be caused by the inflation or deflation of a magma chamber. face that was originally at (r, 0) gets shifted to (r + u, w). There- The radius of the cavity changes by (Hagiwara 1977) fore, an observer on the Earth surface is vertically uplifted by a∆P ∆V d ∆a = , (A.1) |w(r)| = (1 − ν) . (A.5) 4µ π (r2 + d2)3/2 where µ is the shear modulus, having the same units as pressure In addition, the density of the elastic body changes at each (note that some authors use G for the shear modulus). Typical val- point by (Hagiwara 1977) 9 9 ues for rock are µ = 10 × 10 Pa to 30 × 10 Pa. Thus, the volume ρa3∆P r2 − 2(z + d)2 ∆ρ(r, z) = changes by λ + µ [r2 + (z + d)2]5/2 4π 4π (A.6) ∆V = (a + ∆a)3 − a3 = 4πa2∆a + O(∆a2). (A.2) ∆V r2 − 2(z + d)2 3 3 = (1 − 2ν)ρ , π [r2 + (z + d)2]5/2 Neglecting terms of order O(∆a2), it can be written as where λ is Lame’s´ constant 3 πa ∆P 3V ∆P 1 2µν ν= ∆V = = . (A.3) λ ≡ =4 µ. (A.7) µ 4µ 1 − 2ν The change in pressure within the cavity induces a volume change Below, we calculate the changes in the clock and gravime- that affects the surrounding elastic medium. An observer on the ter measurements due to ground displacement and the three effects surface at (r, z = 0) is displaced by coming from mass redistribution. Summing the results of Appendix u 1 − ν  r  ∆V  r  A2, A3 and A4 we find = a3∆P /R3 = (1 − ν) /R3, w µ −d π −d 1 ∆Um ≡ ∆U2 + ∆U3 + ∆U4 = −Gρm∆V √ . (A.8) (A.4) r2 + d2 Atomic clocks as tools to monitor vertical surface motion 7

with the expansion (a) 109 109 GM⊕ 2GM⊕ GM⊕ 2 2 g(r, z) ≈ 2 + 3 z − 4 (r − 3z ) . (A.12) R⊕ R⊕ R⊕

] 8 r 10 The linear term in the expansion is known as the free-air cor- y 8 / 10 rection; we can neglect the higher orders. We define the gravita- 3 2 2

m tional acceleration at the surface as g ≡ GM⊕/R⊕ ≈ 9.81m/sec . [ 7 10 τ For a Mogi source z = w (see Eq. A.4). The potential and gravity τ 1 7

∆ 10 changes are /

V 106 ∆V d ∆ ∆U = −g z = g(1 − ν) , (A.13) 1 π (r2 + d2)3/2 106 5 z ∆V d 10 ∆g1 = 2g = −2g(1 − ν) . (A.14) ] 2 2 3/2

1 5 9 13 17 21 s R⊕ π R⊕ (r + d ) [

(b) i 109 τ 105 A2 Direct signal

] If there is inflow or outflow of magma, crustal rock of the vol- r

y ume ∆V will be replaced by magma, or vice versa. This change / 4 3 10 in density gives a direct signal, due to the mass change ∆M = m [ 8 ∆V (ρ − ρ) ρ ρ 10 τ m , where m and are the densities of magma and τ m crustal rock, respectively. The resulting change in the gravitational

∆ 3 / 10 potential from the presence of the density anomaly only is V

∆ G∆M 1 ∆U2 = − = −G(ρm − ρ)∆V √ , (A.15) R d2 + r2 107 102 and the change in the gravitational acceleration is 1 5 9 13 17 21 ∂∆U d R [km] ∆g = 2 = G(ρ − ρ)∆V . (A.16) 2 ∂d m (d2 + r2)3/2

Figure A3. Integration times for future clocks. Analogous to Fig. A2, the integration times to resolve uplift and mass redistribution are shown for A3 The potential of the uplifted rock future clocks with σ = 1 × 10−17. 0 Before the inflation of the cavity, the surface of the half-space is flat. But after the inflation there will be a hat of material peaking

The required integration timescales to resolve uplift (τ1) and mass above the location of the cavity; this is where the clock and the redistribution (τm) as a function of the distance to the magma gravimeter are located. Obviously, this hat of material affects the chamber and the volume change rate ∆V/∆τ is given by Eq. (13) outcome of a measurement. As a simplification, most authors re- −16 and (14), respectively. For a current clock with σ0 = 3 × 10 , place the hat by an infinite disc of height d0. While this gives the the respective integration times are shown in Fig. A2(a) and (b). correct first order result for the gravitational acceleration, the ex- Resolving ∆Um requires either long integration timescales of the pression for the gravitational potential diverges. order of a year or more or better clocks. However, for different ge- In this section we discuss the gravitational potential and accel- ometries (e.g. flatter magma chambers) or more realistic models eration of this additional hat without making the disc approxima- the mass redistribution term could be more significant. For future tion. This allows us to calculate the effect on clock measurements −17 clocks with σ0 = 1 × 10 , the analogous plot is given in Fig. and to compute higher order corrections to the gravitational accel- A3. eration. We denote the uplift directly above the cavity by ∆V 1 A1 Ground displacement d ≡ |w(r = 0)| = (1 − ν) . (A.17) 0 π d2 Treating the Earth as a perfect sphere, its gravitational potential is We shift the coordinates slightly to put the observer at z = 0. Thus, GM z = d now corresponds to the surface of the ground before the U(r, z) = − ⊕ , (A.9) 0 p 2 2 r + (R⊕ − z) uplift. The z coordinate of the new surface at a given distance r is where M⊕ is the mass and R⊕ is the radius of the Earth. z(r) = d0 − |w(r)| We expand the potential in a Taylor series around (r, z) = " −3/2#  r2  (A.18) (0, 0) and neglect all terms of order O(r3), O(z3) and higher to = d 1 − + 1 . 0 d2 obtain

GM⊕ GM⊕ GM⊕ 2 2 This can be inverted to express r as a function of z: U(r, z) ≈ − − 2 z + 3 (r − 2z ) (A.10) R⊕ R 2R ⊕ ⊕  r 2  z −2/3 z = 1 − − 1. (A.19) The gravitational acceleration is given by d d0 GM⊕ The z index is used to emphasize that rz is the distance at which g(r, z) = 2 2 , (A.11) r + (R⊕ − z) the surface has altitude z. 8 R. Bondarescu et al.

In general, the gravitational acceleration and the gravitational A4 Gravitational potential due to density change potential of an observer located at (r, z) = (0, 0) due to the density In response to the inflating magma chamber, the density through- distribution ρ(r, z) are out the surrounding rock changes by ∆ρ(r, z), given by (A.6). The Z ∞ Z ∞ ρ(r, z)z resulting change in the gravitational acceleration g3 = 2πG 2 2 3/2 r dr dz, (A.20) −∞ 0 (r + z ) d ∆g (r) = −G(1 − 2ν)ρ∆V . (A.31) Z ∞ Z ∞ ρ(r, z) 4 (r2 + d2)3/2 U3 = −2πG √ r dr dz. (A.21) r2 + z2 −∞ 0 was calculated by (Hagiwara 1977). Following a similar approach,

Assuming constant density, the acceleration ∆g3 and the potential we determine the change in the gravitational potential due to this ∆U3 due the uplifted mass are density variation. This may be useful for the reader since Hagi- wara’s paper was written in Japanese. d r Z 0 Z z rz First, a new function K is defined ∆g3 = 2πGρ 2 2 3/2 dr dz, (A.22) 0 0 (r + z ) −3/2 K(x, y, z) = x2 + y2 + (z + d)2 , (A.32) Z d0 Z rz r ∆U3 = −2πGρ √ dr dz. (A.23) 2 2 0 0 r + z which allows to rewrite the density change (A.6) as Performing the integrals over r yields ∆V  ∂K(x, y, z)  ∆ρ(x, y, z) = (1 − 2ν)ρ K(x, y, z) + (z + d) . π ∂d Z d0  z  ∆g3 = −2πGρ √ − 1 dz, (A.24) (A.33) 2 2 0 rz + z The gravitational potential due to this change in density is Z d0   p 2 2 ∆U3 = −2πGρ rz + z − z dz, (A.25) Z 0 0 0 0 0 0 0 G∆ρ(x , y , z )dx dy dz ∆U4(x, y, z) = − p 3 (x − x0)2 + (y − y0)2 + (z − z0)2 and defining a new variable ζ ≡ z/d0, we obtain R Z  ∂    = κ 1 + (z0 + d) Φ (x, y, z, z0)dz0, ∂d K Z 1 R  ζ  (A.34) ∆g3 = −2πGρd0  − 1 dζ r 2 0  rz 2  2 + ζ where we defined d0   Z 0 0 0 0 0 1 0 K(x , y , z )dx dy Z ΦK (x, y, z, z ) ≡ , d0 ζ p 0 2 0 2 0 2 = −2πGρd0 − 1 dζ, 2 (x − x ) + (y − y ) + (z − z )  q 2  R 0 d −2/3 d0 2 (1 − ζ) + d2 ζ − 1 (A.35) (A.26) ∆V κ ≡ −G(1 − 2ν)ρ . (A.36) Z d0   π p 2 2 ∆U3 = −2πGρ rz + z − z dz 0 Further, we assume that before the inflation of the magma chamber r ! the density ρ is constant. Z d0 d2 d = −2πGρd d (1 − ζ)−2/3 + 0 ζ2 − 1 − 0 ζ dζ. The 2D Fourier transform, assuming z = 0, is 0 d2 d 0 Z ∗ 1 0 −i(kxx+ky y) (A.27) ΦK (kx, ky) = ΦK (x, y, 0, z )e dx dy 2π 2 R Because d0/d  1, we can expand around d0/d = 0 and drop all Z 0 0  0 0 0 −i(kxx +ky y ) 0 0 O(d2/d2) = K(x , y , z )e dx dy terms that are of order 0 , then we can integrate and find 2 R      Z −i(kxu+ky v)  15π d0 1 e ∆g3 ≈ 2πGρd0 1 − 2 − , × √ du dv 32 d 2π 2 u2 + v2 + z02 (A.28) R√ √   0 2 2 0 2 2 1 d0 e−|z +d| kx+ky e−|z | kx+ky ∆U3 ≈ −2πGρd0d 1 − . = 2π 2 d 0 p 2 2 z + d kx + ky √ 0 2 2 To combine this with the other mass redistribution terms, we need e−(2z +d) kx+ky the result as a function of r. In the infinite disk model in the litera- = 2π . (z0 + d)pk2 + k2 ture, the gravitational acceleration at r is taken to be that produced x y (A.37) by a disk with a thickness equal to |w(r)|, the local value of the uplift, We substituted u = x−x0 and v = y−y0 and used dx dy dx0 dy0 = du dv dx0 dy0. Further, we used the Fourier integrals ∆g = 2πGρ|w(r)| 3 √ 2 2 d (A.29) 1 Z e−i(kxx+ky y) e−|z| kx+ky = 2Gρ(1 − ν)∆V . dx dy = , (A.38) (r2 + d2)3/2 2 2 2 1/2 p 2 2 2π 2 (x + y + z ) kx + ky R √ 2 2 Integrating with respect to d we obtain the corresponding potential 1 Z e−i(kxx+ky y) e−|z| kx+ky 2 2 2 3/2 dx dy = , (A.39) 2π 2 (x + y + z ) z 1 R ∆U3 = −2Gρ(1 − ν)∆V √ . (A.30) r2 + d2 and we used that the density vanishes for z0 < 0. Taking the Fourier Atomic clocks as tools to monitor vertical surface motion 9 transform of ∆U4 where d is the depth of the magma chamber, ri the horizontal dis- 1 Z tance of clock i from the magma chamber. We assume that ν is ∆U ∗(k , k ) = ∆U (x, y)e−i(kxx+ky y)dx dy 4 x y 4 known, thus C0 is a constant. 2π 2 R We measure two quantities, the differences between each of Z  ∂  = κ 1 + (z0 + d) the clocks and the reference clock. These are ∂d R  Z  1 1 −i(kxx+ky y) 0 ∆U1 = C0∆V , (A.47) × ΦK (x, y)e dx dy dz d2 2π 2 R d Z  ∂  ∆U2 = C0∆V . (A.48) = κ 1 + (z0 + d) Φ∗ (k , k )dz0, (r2 + d2)3/2 ∂d K x y R (A.40) The measurements of each of these quantities is affected by ∗ clock noise. The three clocks will have uncorrelated noise, but the and using the expression for ΦK above, we obtain √ ∆Ui will be correlated because they each depend on the reference 0 2 2 Z   −(2z +d) kx+ky ∗ 0 ∂ e 0 clock. We write the variance of the two clocks near the magma ∆U4 (kx, ky) = 2πκ 1 + (z + d) dz 2 2 ∂d 0 p 2 2 chamber as σM , and the variance of the reference clock as σR. R (z + d) kx + ky ∞ √ Then the variances and covariance of ∆U1 and ∆U2 are Z 0 2 2 = −2πκ e−(2z +d) kx+ky dz0 2 2 0√ Var(∆U1) = σM + σR, (A.49) 2 2 e−d kx+ky 2 2 = −πκ , Var(∆U2) = σM + σR, (A.50) p 2 2 kx + ky 2 Cov(∆U1, ∆U2) = σR . (A.51) (A.41) where we used that outside the half-space (i.e. for z0 < 0) the The noise variance in the measurement of d is density (contained in κ) vanishes. The inverse Fourier transform is 2 2  ∂d   ∂d  Z Var(d) = Var(∆U1) + Var(∆U2) 1 ∗ i(kxx+ky y) ∂∆U1 ∂∆U2 ∆U4(x, y) = ∆U4 (kx, ky)e dkxdky 2π 2 R √ ∂d ∂d −d k2 +k2 +2 Cov(∆U1, ∆U2) Z x y ∂∆U1 ∂∆U2 1 e i(kxx+kyy ) = − κ e dkxdky, (A.52) 2 2 p 2 2 R kx + ky (A.42) and the same holds replacing d with ∆V . It is easiest to calculate and introducing polar coordinates kx = k cos θ, ky = k sin θ (such the partial derivatives by calculating first the partial derivatives of that dkxdky = k dk dθ) and orientating the coordinate axes such the ∆Ui with respect to d and α, and inverting. They are that (x, y) = (r, 0), the integral can be written as ∂∆V d2(r2 − 2d2) Z ∞ Z 2π −dk = , (A.53) 1 e ikr cos θ 2 ∆U (x, y) = − κ e k dθ dk ∂∆U1 3C0 r 4 2 k 0 0 ∂∆V 2(r2 + d2)5/2 ∞  2π  = , 1 Z Z 2 (A.54) = − κ e−dk eikr cos θ dθ dk (A.43) ∂∆U2 3C0d r 2 0 0 ∂d d3(r2 + d2) π = − 2 , (A.55) = −κ √ . ∂∆U1 3C0∆V r r2 + d2 ∂d (r2 + d2)5/2 Here, we used the definition of the Bessel functions and the integral = 2 . (A.56) ∂∆U2 3C0∆V r 6.751(3) in (Gradshteyn & Ryzhik 2007). Finally, we find 1 We find the optimal location for the second clock by minimiz- ∆U4(r) = G(1 − 2ν)ρ∆V √ . (A.44) ing the variance of either ∆V or d, as a function of r, the horizontal r2 + d2 distance to the second clock. Fortunately, we find that d and ∆V are optimized for similar values of r, and the minima are fairly A5 Optimizing the placement of a second clock broad. A reasonable compromise, in the case where the reference We now consider the question of the optimal location for measur- clock has the same performance as the other clocks, is r ≈ 0.78 d. ing the depth and volume change of a magma chamber. We will With this clock placement, we have assume that one clock is directly over the magma chamber. What is  2 2 d 2 the optimal location of a second clock? Recall that there must also Var(∆V ) ≈ 4.6 σM , (A.57) be a distant reference clock. For this example, we assume that the C0  3 2 horizontal location of the magma chamber is known from a previ- d 2 Var(d) ≈ 2.2 σM . (A.58) ous survey. If this were not known, it could also be measured, but C0∆V more clock locations would be required. The shift in the gravitational potential at clock i can be written We can summarize this more succinctly in terms of fractional d errors. We define δh ≡ σM /g, which is the standard devia- ∆Ui(r) = C0∆V , (A.45) tion of the measurement of a single clock, in terms of equiva- (r2 + d2)3/2 i lent height. The maximum value of the uplift (at the summit) is GM⊕ 1 − ν 2 C0 ≡ , (A.46) h = C0∆V/(gd ). The standard deviation of one clock is then R2 π 2 ⊕ σM = (C0∆V/d )δh/h. Given this, the fractional accuracies of 10 R. Bondarescu et al. our measurements are Just taking the first (n = 2) term, this is δ∆V δh ≈ 4.6 , (A.59) U (2)(R , α) ∆V h ∆g(α) = −2 tid ⊕ δd δh R⊕ (B.4) ≈ 2.2 . (A.60) GM d h = ext R (3 cos2 α − 1). R3 ⊕ A frequency stability of 10−18 corresponds to 1 cm, which is achieved with current clocks after an integration of 7 hours. If each Thus, there is an outward pull, ∆g > 0, at the sides of the Earth clock has this performance and the maximum uplift is 10 cm, this facing (α = 0) and opposing (α = π) the external body. Halfway is a fractional accuracy of 10%, giving a fractional accuracy on d between, there is an inward pull (α = π/2, 3π/2). Since the Earth of 22% and on ∆V of 46%. If instead, we integrate for a month, is not a rigid body, it is deformed by tidal forces. Its tidal response the fractional accuracies improve by an order of magnitude leading is quantified by the Love numbers kn and hn; there is also a Love to fractional accuracies of 2.2% and 4.6%, respectively. ln which we do not use here. The larger the Love numbers, In the case of a reference clock that is much better than the the stronger the deformation. others (σR  σM ), the numbers become r ≈ 0.87d, δ∆V/∆V ≈ The gravitational acceleration arising from the tidal interac- 3.7 δh/h, and δd/d ≈ 2.0 δh/h. tion has three components. First, there is the direct effect from the external body (B.3). Second, the deformation of the Earth gives rise (n) of an additional gravitational potential knUtid , resulting in an ad- ditional acceleration. Third, due to the deformation, the altitude of APPENDIX B: SOLID EARTH TIDES (n) the observer changes by −hnUtid /g compared to a non-deformed Throughout this section we follow (Agnew 2007) and we use stan- Earth. Thus, the effective tidal potential measured by a clock on the dard spherical coordinates. The gravitational potential of an exter- surface, for a given n, is nal body can be written as a multipole expansion (n) (n) U = (1 + kn − hn)U . (B.5) GMext tid,eff tid Uext = − ρ The potential of the deformed Earth is knU(n). It can be writ- ∞ n (B.1) ten as an expansion around its value U (R , α) at the undistorted GMext X  r  tid ⊕ = − P (cos α). R R n surface. Since the gravitational field satisfies Poisson’s equation, it n=0 is a harmonic function outside the Earth and therefore, we can write Here, ρ is the distance between the external body and the location it as an expansion of the form of the clock, R is the distance between the body and the Earth’s ∞  n+1 X R⊕ centre of mass, r is the distance between the Earth’s centre and the U (r, α) = F (α), (B.6) Earth def r n clock and α is the angle between the two vectors pointing from n=0 the centre of the Earth to the external mass and the clock. The Pn denote the Legendre polynomials. In the sum, the n = 0 term can where the functions Fn(α) have to be chosen such that they match be neglected since it is constant and therefore does not contribute to Utid(R⊕, α) at the Earth’s surface, giving the force since it will drop out when taking the gradient. The n = 1 ∞  n  n+1 2 GMext X R⊕ R⊕ (U = −GM /R r cos α) UEarth def(r, α) = − Pn(cos α). term ext,n=1 ext causes the orbital accelera- R R r tion and therefore, by the definition of tides, the tidal potential is n=2 ∞ (B.7) n GMext X  r  U (r, α) = − P (cos α). (B.2) tid R R n The tidal acceleration is now given by n=2 n ∂(Utid + knUEarth def) By Utid we will denote the n-th order component of the expansion. − ∂r The tidal deformation is mainly caused by the Moon and the r=R⊕ Sun; the other planets cause very minor effects. Since typically r/R ∞  n GMext X 1 R⊕ (B.8) is small (for the Moon: r/R ≈ 1/60, Sun: r/R ≈ 1/23000), it is = (n − kn(n + 1)) R R⊕ R sufficient to just consider the first few orders of the expansion. n=2 If the Earth was a completely rigid spherical body with no × Pn(cos α), external mass acting tidally, the gravitational acceleration at the 2 which can be written as surface would be a constant g ≡ GM⊕/R⊕. Changing the ra- ∞   n dial distance by a small amount ∆r would change g by ∆g = X n + 1 Utid(R⊕, α) 3 − n 1 − kn . (B.9) −2GM⊕/R ∆r. Thus, going farther away (∆r > 0) weakens n R⊕ ⊕ n=2 the gravitational acceleration. First, we consider just the effect of an external body, letting the The change in the gravitational acceleration due to a small radial Earth be perfectly rigid. This corresponds to the Love numbers (in- displacement ∆r close to the surface of the Earth is troduced below) being zero. The value of the tidal potential on the 2GM GM ∆r ∆r − ⊕ ∆r = −2 ⊕ = −2g . (B.10) surface is Utid(R⊕, α). This induces a change in the gravitational 3 2 R⊕ R⊕ R⊕ R⊕ acceleration at the surface of The final effective tidal acceleration on the surface is ∂Utid(r, α) ∆g(α) = − n ∂r (n) Utid(R⊕, α) r=R⊕ ∆gtid,eff = (n − (n + 1)kn + 2hn) . (B.11) R⊕ ∞ (n) (B.3) X U (R⊕, α) = − n tid . From (B.5) and (B.11), we see a key feature of clocks relative R⊕ n=2 to gravimeters. Gravimeters are relatively more sensitive to effects Atomic clocks as tools to monitor vertical surface motion 11 at higher n, i.e. shorter wavelengths, than are clocks. This is be- cause they measure a spatial derivative of the potential rather than the potential itself. In contrast, the response to uplift only depends on motion in the entire gravitational field of the Earth, and so the coefficients of hn do not depend on n. Note that clocks are best at constraining the lower order mul- tipoles. Instruments that measure higher order derivatives of the geopotential (e.g. gradiometers) are more sensitive to the higher or- der multipoles. Clocks will only be useful for measuring the n = 2 and possibly n = 3 tides. However, a detailed analysis of the mul- tipole structure based on clock data is beyond the subject of this paper.