Downloaded by guest on September 28, 2021 ntesuc ein.Sc ahmtymyla osltigof splitting to lead first may bathymetry The inclines Such bathymetric Earth. regions. of source presence the or the the strong in within by is supported paths is (PSD) region hypothesis propagation density the in spectral distinct either power along generated pressure be the can where they that envisage waves (18). exceptions few with 17), of (16, predominance a waves show Rayleigh and (15) dependent Love-to-Rayleigh frequency that are found ratios They S1 ). Table Appendix, (SI range frequency quantifying microseism secondary on the recent in focused ratio few data Love-to-Rayleigh A the digital century. high-quality 20th on (14) based middle and studies (13) date early waves the to Love back microseism secondary of pres- microseismic Observations of the records. components horizontal explain the on waves cannot Love of microseisms ence couple secondary directly generation of can the 12), mechanism (11, waves waves Love ocean generate and periods, seafloor the system with longer the at of velocity While phase (10). minimum velocity the phase than ocean. their smaller when becomes waves the Scholte construc- called are of to they due interface, surface seafloor of the the interference below tive at excited are pro- vertical- sources waves are Rayleigh of pressure-like microseisms by content Secondary (9). duced wave records Rayleigh noise component the explains microseisms microseism of component (8). vertical records the and dominate waves, 7). fre- surface waves (6, seismic Hz Rayleigh of interaction 0.3 composed wave–wave predominantly to are ocean 0.1 They nonlinear the by are the in range vibrations imaging excited quency strongest as The microseisms, well (5). secondary Earth as called the (2–4) of climate structure allowing changing internal systems, our Earth probing different for between about exchange information carry energy data the Such (1). Earth recorded to on data due seismic seismometers of signals by majority seismic vast oceans, the represent by storms covered ocean is planet our of surface T noise ambient seismic interaction Earth ocean–solid source realistic for account to distributions. full-wave tomography for requirement ambient-noise a field explains partitioning, that wave model ergodic seismic the an observed for within of evidence present heterogeneity interaction We three-dimensional the Earth. the with bathymetric by field generated at wave is force-splitting seismic majority of boundary the fraction but small by inclines, A generated the waves. is explain Love we those microseism here secondary prop- frequency, century. of wave high a origin seismic unprecedented over global-scale at for world- of wanting agation the simulations been of numerical has two-thirds Using field wave for ambient explanation wide nevertheless justify An waves, not observed. Love does polarized it widely transversely but of type, sec- presence Rayleigh the explains the satisfactorily of the mechanism microseisms in ondary accepted waves The seismic generating Earth. 2020) of 2, solid capable fluctu- July pressure seafloor review produces the for (received at waves 2020 ations surface 6, ocean October of approved and interaction CA, The Berkeley, California, of University Romanowicz, A. Barbara by Edited a waves Love Gualtieri microseism Lucia secondary of origin The www.pnas.org/cgi/doi/10.1073/pnas.2013806117 and eateto epyis tnodUiest,Safr,C 94305-2215; CA Stanford, University, Stanford Geophysics, of Department yohssfrtegnrto fscnaymcoes Love microseism secondary of generation the for Hypotheses secondary for accepted currently mechanism generation The c n ocsta eeaesimcwvs ie ht7%o the of 70% that Given perturb- waves. to seismic subjected generate continuously that is forces ing Earth the of surface he rga nApid&CmuainlMteais rneo nvriy rneo,N 08544 NJ Princeton, University, Princeton Mathematics, Computational & Applied in Program a,1 | ten Bachmann Etienne , oewaves Love P and | opttoa seismology computational SV oywvs tteocean–crust the At waves. body b rdrkJ Simons J. Frederik , | b eateto esine,PictnUiest,Pictn J08544; NJ Princeton, University, Princeton Geosciences, of Department b doi:10.1073/pnas.2013806117/-/DCSupplemental at online information supporting contains article This 1 erpaetefe onaycniina at’ surface—the Earth’s het- at bathymetry. condition speed boundary high-resolution wave free 3D and the replace both Earth We accommodates the is SEM within waves. simulations The erogeneities numerical Love s. our microseism 4 by secondary about resolved period of struc- minimum generation (3D) The the three-dimensional on the and assess ture bathymetry to 22) of (21, (SEM) importance method spectral-element simulations the numerical using high-frequency global-scale, perform We Love Microseism Waves Secondary of have Generation the waves Modeling Love microseism secondary can observed conducted. mechanisms been the which to to compre- no as lead date, To investigations (20). theoretical energy energy wave hensive wave Love Rayleigh to incident converted of be percent can the few a at only that conversion suggest simulations numerical wave early waves the although Rayleigh-to-Love Love boundary, ocean–continent larger hypotheses, from these the originate to waves, addition may Rayleigh In greater energy. the of that wave the propagation noted Love 19 of the from ref. in distance hand, coming originate other the do the waves On waves Rayleigh region. Love source that and concluding Love direction, same effects. observed focusing/defocusing generation and 8 the scattering het- Ref. to to lateral due lead waves of can Love which presence of Earth, the the by within supported erogeneities is The hypothesis waves. per- second a Love for component waves—and inclines—responsible to a Rayleigh tangent component for in inclines—responsible force to pressure pendicular second-order vertical the ulse ne the under Published Submission. y Direct PNAS a is article This interest.y J.T. competing and no F.J.S. the declare from the authors on contributions of The advised with discussion J.T. manuscript the on study; the in work wrote the participated L.G. to on J.T. and code advised and results; F.J.S., J.T. the E.B., and optimized F.J.S. L.G., and frequency; implementation; GPUs high numerical to at code cluster the GPU ported a E.B. analysis; the formed n h optto ftertto ftemto nSPECFEM3D in motion the of rotation sources the noise of ambient the computation implemented the study, the and designed L.G. contributions: Author owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To naymcoes oewvsoiiaeegdclydeto due structure. ergodically Earth in originate heterogeneities waves lateral sec- Love that demonstrating microseism by ondary shed conundrum we 100-y-old propagation, this wave on light seismic global of of simulations beginning ical numer- the high-frequency unprecedented since Using century. archive 20th data the seismic the in of observed waves two-thirds surface polarized microseism horizontally secondary of waves, theo- presence Love the generation the justify to State-of-the-art unable with seafloor. are ries the couples at energy Earth whose solid storms, by ocean global generated wind-driven of are part microseisms major Secondary the data. represent seismographic and Earth seismic the background of strongest vibrations the are microseisms Secondary Significance n eonTromp Jeroen and , NSlicense.y PNAS b,c . y https://www.pnas.org/lookup/suppl/ NSLts Articles Latest PNAS LB,adper- and GLOBE, y | f8 of 1

EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES typical assumption made in seismology—by a realistic distri- where Love waves cannot be generated. We tested this case and bution of pressure point sources in the ocean, accounting observe that indeed no clear transversely polarized arrivals can for space- and frequency-dependent second-order interactions be identified (SI Appendix, Figs. S2A–S4A). between ocean gravity waves (Materials and Methods) in the secondary microseism period band. We show results between Effects of Bathymetry and 3D Earth Structure 4 and 10 s. We use a source configuration in the northern We first investigate the role played by bathymetry and/or 3D hemispheric winter (SI Appendix, Fig. S1), when the strongest Earth structure on the generation of secondary microseism Love sources of secondary microseisms are located in the North waves. Fig. 2 shows the power and arrival direction of Love Atlantic Ocean. Using the same distribution of sources, we per- waves in the synthetics at seismic stations around the North form numerical simulations considering either 1D Earth model Atlantic Ocean, where the strongest sources of secondary micro- PREM (Preliminary Reference Earth Model) (23) or 3D Earth seisms are located (SI Appendix, Fig. S1). Background maps model S40RTS (24) (see Materials and Methods for more details show the median pressure PSD (Pa2/Hz) between about 4 and about the Earth models used in the numerical simulations). To 10 s, while we use period-dependent sources in our simulations assess the effect of bathymetry at the source, we use a high- (SI Appendix, Fig. S1). Simulations presented in Fig. 2A are resolution topography and bathymetry model (25) (Materials performed in the presence of bathymetry in 1D Earth model and Methods). PREM (scenario in Fig. 1B). Simulations in Fig. 2B are per- Fig. 1 shows the four scenarios used in our simulations: a 1D formed in 3D Earth model S40RTS (scenario in Fig. 1D). The Earth model without bathymetry (Fig. 1A) and with bathymetry two scenarios allow us to compare the effects of bathymetry (Fig. 1B) and a 3D Earth model without bathymetry (Fig. 1C) (Fig. 2A) and structure (Fig. 2B) on the generation of Love and with bathymetry (Fig. 1D). In each scenario, we compute waves. The power of the Love waves computed at each sta- the three components of displacement due to a distribution of tion is proportional to the radial length of the polar histograms, sources in the ocean at 199 stations worldwide (Materials and while the histogram is oriented along the direction of the Methods and SI Appendix, Table S2). main arrival. We rotate the north–south and east–west components of dis- Love waves due to bathymetry and force splitting at the placement to the radial and transverse components by computing source (Fig. 2A) are nearly absent in the majority of seismic the rotation rate around the vertical axis at each station. This stations around the North Atlantic Ocean. We observe some enables us to retrieve the azimuth of the main arrival at each sta- larger energy at seismic stations in Europe, possibly due to tion (26, 27) by correlating the rotation rate around the vertical the steep continental slope offshore Spain and France or the axis and the transverse component of the acceleration over mov- North Atlantic Ridge (SI Appendix, Fig. S5). The presence of ing windows for varying azimuths (Materials and Methods and Love waves at all stations dramatically increases when 3D wave SI Appendix, Figs. S2–S4). This allows us to estimate the Love speed heterogeneities are taken into account (Fig. 2B). This wave energy and the Love-to-Rayleigh energy ratio at each sta- indicates that the interaction of the seismic wave field with lat- tion. The first scenario in Fig. 1 represents the null hypothesis, eral variations in Earth structure—likely through scattering and

AB

CD

Fig. 1. The generation of secondary-microseism Love waves under various scenarios. Love waves are not generated in (A) a 1D Earth model without bathymetry. They do exist in the presence of (B) bathymetry and (C) 3D heterogeneity and (D) in a 3D Earth model with bathymetry. In this study we show that only scenario D comes close to explaining the hitherto unexplained observations that have been made worldwide for over a century.

2 of 8 | www.pnas.org/cgi/doi/10.1073/pnas.2013806117 Gualtieri et al. Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 nAeia uoe n fia ecmaerslso simu- al. of et Gualtieri results compare We Africa. stations and seismic Europe, selected shows of America, at 3 ratio Fig. in ratio spectral components. spectral vertical the Love-to-Rayleigh and the as transverse the ratio on Love-to-Rayleigh energy the define We Ratio Energy Love-to-Rayleigh these of colocation the on (28). relying sources methods different conse- imaging important for Earth—with and have quences the seafloor, within generated the sources are at latter wave generated the are Love former strength locations—the the their and source determines not that Rayleigh is alone it interaction generation. that wave–wave his- indication ocean dominant polar the an the the of is from of distance which length the PSD, to radial pressure amount proportional the not The by is (captured scattering. tograms) waves wide-angle Love scat- over low-angle of dominate S40RTS, model may Earth tering the of located. smoothness are Earth—in sources the the strongest Given the within where region heterogeneities geographic waves same Love depth—at to at the Rayleigh from occurs or conversion the seafloor direc- Therefore, 2B the arrival region. Fig. in wave at station every Love splitting at main tion force The boundary. than ocean–continent waves Love Love of of micro- amount seism secondary periods the of between to amount PSD proportional larger pressure a is median focusing/defocusing—yields histogram the polar represents each color of background length The in account. radial names into The station taken S40RTS. and is model Network bathymetry s. Earth scenarios, 10 3D and both (B) 4.5 In and station. PREM each model at Earth waves 1D (A) in computed 2. Fig. mrec fLv ae tsimcsain rudteNrhAlni ca.Mi ria ieto fscnaymcoes oewvsas waves Love microseism secondary of direction arrival Main Ocean. Atlantic North the around stations seismic at waves Love of Emergence B A onstwr h togs source strongest the toward points A ee obt panels. both to refer e fteLv-oRyeg ai ntescnaymicroseism secondary the in val- ratio Observed Love-to-Rayleigh 29). instru- the (18, novel or of motion 16) ues rotational (15, measuring stations for of ments arrays require they because multiscale waves. and Love models of to Earth generation the different needed on by are heterogeneities played studies role Follow-up the Mantle anisotropy. assess crust. account of not into layers does but take three heterogeneities for as large-scale model captures well accounts S40RTS mantle as simu- 2.0 model sediments, 3D Crust our of a model layers in with Crustal two used model Methods ). model crustal and 3D Earth (Materials a 3D combines The heterogeneities, lations (green the curves). 3D power wave magenta in of Love significant vs. values any presence add toward not the (green does ratio bathymetry heterogeneities In the 3D, record. enhance Fig. observational curves) in sec- shown magenta cases the and that a all in observations with indicates at In waves agreement (15–17). only This in Love band, 1 over bands. period microseism exceeds dominate period ondary and mostly narrow cases waves in Rayleigh most This and in heterogeneities. stations 1 3D few below to due remains increase ratio the ratio assess Love-to-Rayleigh (magenta to the models curves), in model (green of Earth Earth bathymetry 3D 3D of a in absence and and the bathymetry curves) of presence (blue the in 1D curves) a in lations bevtoso h oet-alihrtoaeeteeyrare extremely are ratio Love-to-Rayleigh the of Observations NSLts Articles Latest PNAS | f8 of 3

EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES Fig. 3. Computed Love-to-Rayleigh ratio for varying periods at selected stations. Love-to-Rayleigh spectral ratio as a function of period at selected seismic stations in a 1D Earth model with bathymetry (blue curves), a 3D Earth model without bathymetry (green curves), and a 3D Earth model with bathymetry (magenta curves). Love waves are generated by bathymetry and force splitting at the source (blue curves), 3D heterogeneities within the Earth (green curves), and both mechanisms (magenta curves). Insets display the location of the station, while the title of each panel reflects the station name and network. The legend on the top left applies to all of the panels.

period band range from 0.25 to 1.2 (Materials and Methods range (Fig. 3, green and magenta curves). At most stations, the and SI Appendix, Table S2). Our estimated values in the pres- presence of bathymetry alone (blue curves) underpredicts the ence of 3D heterogeneities are compatible with this observed observed ratios.

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EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES where the sources are strongest (SI Appendix, Fig. S1). How- ocean with periods between about 4 and 10 s. Starting from an ever, stations in the Pacific and Indian Oceans—far away from initial condition—pressure sources at the ocean surface—where, the dominant source region—show high Love-to-Rayleigh ratios even with realistic bathymetry, Love waves are not efficiently that increase with increasing period. We also observe stations excited, the wave field evolves to a stochastic stationary state, showing low Love-to-Rayleigh ratios, around 0.1. This occurs i.e., a state containing both Love and Rayleigh waves. Any suf- mostly at short periods (Fig. 4A) and in the southern hemisphere ficiently large time periods used to assess the distribution of (Fig. 4B), mostly far away from strong sources. We interpret this Love/Rayleigh wave power generated by a diffuse wave field yield to be an indication of the seasonal effect of sources on Love-to- similar results. Rayleigh ratios, as observed by ref. 17. Numerical simulations with seasonally varying source configurations will be required Materials and Methods to assess the effect of storm seasonality on the generation of Here we describe the implementation of the 3D numerical simulations of Love waves. seismic wave propagation due to a realistic diffuse secondary microseism source distribution. For figures and tables, we refer the reader to the main Discussion text as well as SI Appendix. The generation of secondary microseism Love waves has been Secondary Microseism Sources. Sources of secondary microseisms are a conundrum for close to a century (13). Attempts to locate extended pressure sources at the surface of the ocean (32, 33). For compu- sources of Love waves and relate them to structural or bathy- tational purposes, they can be discretized on a grid of equivalent pressure metric features through numerical simulations have failed to give point sources (6, 9). Numerical ocean wave models, such as WAVEWATCH an explanation for their generation. For example, pioneering III (34), can be used to estimate the strength of the sources. The out- numerical simulations in the 1970s (20)—focusing on Rayleigh- put of the ocean wave model can be found at ftp://ftp.ifremer.fr/ifremer/ to-Love wave conversion at the ocean continent boundary—were ww3/HINDCAST/SISMO/ (accessed 21 May 2020). Following refs. 7 and 35, the PSD of the pressure field at the surface of limited by computational power and reached 20 s as the min- 2 imum period. Recent efforts toward performing 3D numerical the ocean (Pa / Hz) due to ocean wave–wave interaction can be written simulations in the secondary microseism period band—to assess as a frequency-dependent geographical (at colatitude θ and longitude φ) distribution the effect of 3D structure on Love wave generation—were 2 2 2 2 ρw g fE (fw ) I(fw ) Fp(f, θ, φ) = (2π) , [1] performed at the local scale, with a Cartesian configuration, dS(θ, φ) employing a single source and random-medium properties (30). where ρw is the density of the ocean (assumed constant), g is the gravi- Our numerical simulations of realistic global seismic wave prop- tational acceleration, and f is the frequency of the seismic waves (so that agation in the secondary microseism period band (4 to 10 s) f = 2fw , with fw the frequency of the water waves). Furthermore, E(fw ) 2 reveal that lateral heterogeneities in Earth structure play a dom- represents the PSD of the sea surface elevation (m /Hz), and I(fw ) is the inant role in the ergodic generation of Love waves in the seismic nondimensional ocean gravity wave energy distribution as a function of fre- 2 ambient wave field of the Earth that dominates seismic records quency, integrated over the ocean wave azimuth. The product E (fw )I(fw ) worldwide. Force splitting at the seafloor or the ocean–continent in Eq. 1 can be estimated using WAVEWATCH III. The elementary surface 2 2 boundary plays a smaller role. is dS = R sin θ dθ dφ, where R is the radius of the Earth. The factor (2π) The method used to determine the azimuth of the main enables the conversion of pressure PSD from the wavenumber domain to the spatial domain (9). arrival of secondary microseisms and compute synthetic Love- The ocean wave model is defined on a global scale with a spatial resolu- to-Rayleigh ratios (Materials and Methods) can also be applied to tion of 0.5◦ both in latitude and in longitude. At each grid point, the ocean observed data. We show an example at six seismic stations in SI state is described by 24 azimuths and 22 ocean wave frequencies spaced Appendix, Fig. S6. The data are compared to results of two simu- exponentially between 0.041 Hz (24.37 s) and 0.30 Hz (3.29 s). The corre- lations for two different mantle models, the same crustal model, sponding seismic period ranges from 1.64 to 12.18 s. One key attribute of and the same source configuration. Observed Love-to-Rayleigh this model is that it is the only one to date that takes into account coastal ratios are similar to simulated ones, although we observe small reflection of ocean waves. The amount of ocean wave energy reflected differences as a function of the period. This may be due to sev- from the coast is not well constrained and should be adjusted as a func- eral factors, such as the presence of signals in the data other tion of bathymetry and shape of the coast (36). In this work, since we are ultimately interested in the ratio between Rayleigh and Love waves, than oceanic secondary microseisms, discrepancies of the syn- we set the coastal reflection to 5% as a global average, in agreement thetic source distribution with respect to the actual one, and with ref. 6. small-scale heterogeneities not accounted for by the global-scale models used in the simulations. Interestingly, at some stations Ocean Site Effects. In the global spectral-element solver SPECFEM3D GLOBE we observe a nonnegligible sensitivity of Love-to-Rayleigh ratios (21, 22), the ocean is treated as incompressible, meaning that the entire to the 3D mantle model, which needs to be investigated in ocean moves as a whole as a result of the normal displacement of the future studies. seafloor. The effect of the ocean on seismic wave propagation can be taken The ambient wave field recorded on horizontal-component into account as a load. At long periods (≥ 20 s), the thickness of the ocean seismograms—two-thirds of the seismic record—is often is small compared with the wavelength of the seismic waves, and there- neglected in imaging studies (5). When taken into account, in fore, this is a good approximation (22). At short periods, this assumption no longer holds because the propagation of compressible (P) waves within the the absence of information about the origin of Love waves, water layer has a nonnegligible effect on the seismic wave field. This is par- Rayleigh and Love waves are assumed to share the same source ticularly important in the case of secondary microseisms, whose sources are location, which is considered as a diffuse wave field (31). This is at the ocean surface, and P waves generated at those sources are multiply not the case, as Rayleigh waves arise directly from the pressure reflected between the ocean surface and the seafloor. sources due to nonlinear ocean wave–wave interaction, while The effect of the ocean on the seismic wave field in the source region Love waves predominantly originate within the Earth, as shown (hereafter called source site effect) can be computed analytically (6), and in this study. The main arrival direction of Rayleigh and Love it varies with frequency, ocean depth, and seismic phase (6, 9). Love waves waves is not the same at most of the stations (SI Appendix, are not affected by the presence of the ocean (22). This source site effect Fig. S7). Hence, seismic wave speed variations can be the result accounts for propagation within the water layer and moves the source from of source mislocation, leading to significant errors in models of the top to the bottom of the ocean (9, 37, 38). The PSD of secondary micro- seisms is mostly dominated by surface waves. Therefore, we neglect the the seismic wave speed structure of the Earth. source site effect of body waves, and we multiply the pressure PSD in Eq. Our findings demonstrate that given about 30 min (Materials 1 by the source site effect of Rayleigh waves as computed by (6) and Methods and SI Appendix, Fig. S8), the Earth behaves as an 0 2 ergodic system under the action of heterogeneous sources in the Fp(f, θ, φ) = c (f, θ, φ) Fp(f, θ, φ), [2]

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Data that 0.75. observe of small and threshold a Only the case identified. exceeds this be cross-correlations can tested of arrival percentage We of direction exist. transverse hypoth- not clear no null do indeed waves the Love bathymetry—represents where and esis, topography of absence and velocity: axis ambient vertical for eration transverse the rotation including the the of around 47), amplitude compute the (26, rate to propagation, wave station thus plane Assuming seismic and (27). a noise (46) at arrival recorded main acceleration the of azimuth the amount the station. about a information unknown at trans- get recorded an the waves and in Love compute records of to area, noise straightforward wide deter- of not a component is easily verse it over be consequence, extended a cannot are As location. records sources Component. the noise Transverse since ambient the mined, of Estimating component and transverse Azimuth Arrival Assessing National Ridge Oak at Summit supercomputer Laboratory. the on available (GPUs) units eepoie ytePictnIsiuefrCmuainlSine& from Science OSR-2016-CRG5- funding grant Computational Technology, acknowledge and 2970-01. for and Science supported of Alkhalifah Institute University Facility Abdullah Tariq King Princeton User thank Science the We which resources of Engineering. by Facility, computational Office provided Computing Additional Energy were Leadership DE-AC05-00OR22725. of Ridge contract Department the under Oak research US thank This the a MATLAB. We of is of use model. resources made our wave analysis improved used data greatly ocean and which the comments, Plotting their manuscript. of for the reviewers output SPECFEM3D of anonymous two the implementation in Labo- the available National motion with Ridge making the Oak helping of the for rotation at Schuberth Summit Bernhard supercomputer ratory, and the article on the ulations in included ACKNOWLEDGMENTS. are data study All 2020. Appendix 3, accessed last November were Incorpo- Websites on the (IRIS). of Seismology Center for Management Institutions Data Research rated the from available freely Seismic are ftp://ftp.ifremer.fr/ifremer/ww3/HINDCAST/SISMO/. data at found be can 0.75. ). S6 exceeds azimuth Fig. coefficient the Appendix, of (SI computation S2–S4, condition the state Figs. from nonsteady excluded the are avoid s to 2,000 win- time first 50%-overlapping The azimuth over dows. of computed function are a Cross-correlations as time. components and transverse possible all and axis vertical bathymetry. with S40RTS model PREM Left Earth model 3D 1A, Earth 1D, Fig. 1D Fig. 1B, 1: 1C Fig. Fig. Fig. bathymetry; bathymetry; in without with described PREM scenarios model different Earth 1D the under stations seismic h oainlmto rudtevria xspoie a oestimate to way a provides axis vertical the around motion rotational The n S4C and S3B, S2A, Figs. Appendix, (SI scenario first The three at waves Love of emergence the show S2–S4, Figs. Appendix, SI hwtecreaincefiin ewe h oainrt rudthe around rate rotation the between coefficient correlation the show , a T . h atrrltn hs w uniisi wc h phase the twice is quantities two these relating factor The . Right LB (48). GLOBE hwhsorm faiuh twihtecorrelation the which at azimuths of histograms show , SPECFEM3D etakCnyeCifrrnignmrclsim- numerical running for Cui Congyue thank We DErhmdlS0T ihu ahmty and bathymetry; without S40RTS model Earth 3D , ω z .Teotu fteoenwv model wave ocean the of output The globe/). = LB safel vial oetruhthe through code available freely a is GLOBE ω ˙ 1 2 ω a ˙ z T z  a spootoa otetases accel- transverse the to proportional is = T x ∂ ∂ , rmterotation the from u u − x y y , 2c − LB,adFbieAdunfor Ardhuin Fabrice and GLOBE, u z , ∂ ,adteoedtdntsthe denotes overdot the and ), ∂ u y LB ngahc processing graphics on GLOBE x NSLts Articles Latest PNAS  is S2–S4, Figs. Appendix, SI https://geodynamics. ω —DErhmodel Earth )—1D z optdwith computed IAppendix, SI | f8 of 7 The [5] [4] SI

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