Earth's Normal Modes Research Articles

Atmosphere-solid earth coupling signals generated by the 15 January 2022 Hunga-Tonga eruption

The January 15th 2022 eruption of the Hunga-Tonga volcano was associated to one of the highest energy volcanic explosions of the past few decades. Here we present a compilation of data from global seismic networks to explore three main topics: the time evolution of the eruption, the propagation of the atmospheric waves around the Earth and the low-frequency, long-lasting seismic signals worldwide after the main event. We find that the eruption started around 04:00, included two large explosions at 05:30 and 08:25 and produced atmospheric waves which circled the Earth more than two times during a time span of 3.5 days and were detected seismically. We also identify very low frequency signals, detected over several hours, which we interpret as resulting from the excitation of Earth normal modes. To our knowledge, there are no previous examples of atmospheric-solid Earth coupling over such a long time interval and only two examples of normal mode excitation following volcanic eruptions.

Tilt Corrections for Normal Mode Observations on Ocean Bottom Seismic Data, an example from the PI-LAB experiment

Earth's normal modes are fundamental observations used in global seismic tomography to understand Earth structure. Land seismic station coverage is sufficient to constrain the broadest scale Earth structures. However, 70% of Earth's surface is covered by the oceans, hampering our ability to observe variations in local mode frequencies that contribute to imaging small-scale structures. Broadband ocean bottom seismometers can record spheroidal modes to fill in gaps in global data coverage. Ocean bottom recordings are contaminated by signals from complex interactions between ocean and solid Earth dynamics at normal mode frequencies. We present a method for correcting tilt on broadband ocean bottom seismometers by rotation. The correction improves the ability of some instruments to observe spheroidal modes down to 0S4. We demonstrate this method using 15 broadband ocean bottom seismometers from the PI-LAB array. We measure normal mode peak frequency shifts and compare with 1-D reference mode frequencies and predictions from 3-D global models. Our measurements agree with the 3-D models for modes between 0S14 - 0S37 with small but significant differences. These differences likely reflect real Earth structure. This suggests incorporating ocean bottom normal mode measurements into global inversions will improve models of global seismic velocity structure.

The global seismographic network reveals atmospherically coupled normal modes excited by the 2022 Hunga Tonga eruption

SUMMARY The eruption of the submarine Hunga Tonga-Hunga Haʻapai (Hunga Tonga) volcano on 15 January 2022, was one of the largest volcanic explosions recorded by modern geophysical instrumentation. The eruption was notable for the broad range of atmospheric wave phenomena it generated and for their unusual coupling with the oceans and solid Earth. The event was recorded worldwide across the Global Seismographic Network (GSN) by seismometers, microbarographs and infrasound sensors. The broad-band instrumentation in the GSN allows us to make high fidelity observations of spheroidal solid Earth normal modes from this event at frequencies near 3.7 and 4.4 mHz. Similar normal mode excitations were reported following the 1991 Pinatubo (Volcanic Explosivity Index of 6) eruption and were predicted, by theory, to arise from the excitation of mesosphere-scale acoustic modes of the atmosphere coupling with the solid Earth. Here, we compare observations for the Hunga Tonga and Pinatubo eruptions and find that both strongly excited the solid Earth normal mode 0S29 (3.72 mHz). However, the mean modal amplitude was roughly 11 times larger for the 2022 Hunga Tonga eruption. Estimates of attenuation (Q) for 0S29 across the GSN from temporal modal decay give Q = 332 ± 101, which is higher than estimates of Q for this mode using earthquake data (Q = 186.9 ± 5). Two microbarographs located at regional distances (<1000 km) to the volcano provide direct observations of the fundamental acoustic mode of the atmosphere. These pressure oscillations, first observed approximately 40 min after the onset of the eruption, are in phase with the seismic Rayleigh wave excitation and are recorded only by microbarographs in proximity (<1500 km) to the eruption. We infer that excitation of fundamental atmospheric modes occurs within a limited area close to the site of the eruption, where they excite select solid Earth fundamental spheroidal modes of similar frequencies that are globally recorded and have a higher apparent Q due to the extended duration of atmospheric oscillations.

Uniformly asymptotic eigensolutions of the Earth's toroidal modes

SUMMARYA novel approach to computing the Earth's normal modes is presented. With the Langer approximation incorporated into the framework of the theory of the generalized reflection/transmission coefficients, our method, when applied to calculate the toroidal modes, produces results that agree extremely well with the exact results from Mineos and successfully overcomes the limitations of the WKBJ analysis. Given any 1-D earth model, regardless of the number of discontinuities, our method can perform the calculations reliably with satisfactory accuracy at high frequencies. The success achieved with the toroidal modes encourages us to tackle in a future study the asymptotic computation of the spheroidal modes, especially those high-frequency trapped modes for which the accuracy of Mineos is demonstrably inadequate.

Probabilistic estimation of structure coefficients and their uncertainties, for inner-core sensitive modes, using matrix autoregression

SUMMARY Normal-mode structure coefficients are crucial observations to infer the velocity, density and attenuation structure of the deep Earth interior, but estimating these coefficients from Earth's normal mode spectra is a non-linear inverse problem. Additionally, complete source information is typically unknown for large earthquakes, and there is a trade-off between the earthquake source and attenuation. Therefore, proper estimation of elastic and anelastic structure coefficients with their uncertainties becomes challenging. Here, we combine a matrix autoregression and a fully non-linear probabilistic sampling to address existing limitations. After successful feasibility experiments using synthetic data with noise, we apply this combined approach to the data for 19 inner-core sensitive spheroidal (S) modes measured for earthquakes from 1994 to 2016. We further implement a model selection criterion to assess whether anelastic structure is significant. Our model selection criterion indicates that anelastic structure coefficients are required only for modes with strong shear-wave energy in the inner core. Inversion results also show a strong correlation between elastic and anelastic splitting functions for these modes. This indicates that the seismic waves travel faster and strongly attenuate along polar paths such that the m = 0 singlet remains poorly observed for these modes.

Earth’s spheroidal motion induced by a gravitational wave in flat spacetime

We revisit the theoretical modeling of astrophysical gravitational wave (GW) interaction with Earth's normal modes in the millihertz frequency band. We consider a force term associated with plane waves in a flat spacetime acting on a nonrotating anelastic Earth model. Our study confirms that the leading spheroidal induced displacement consists of an angular order-2 term, while the induced azimuthal order terms depend on the incident angles of the GW. We show that some of the previous estimates of the Earth's acceleration due to incoming gravitational waves were too optimistic. Current ground-based detectors are embedded in seismic and environmental noise that is several orders of magnitude larger than the expected GW signal. However, with the prospect of new technologies and improved noise reduction techniques, detection may become feasible.

Anelasticity from seismic to tidal timescales: Theory and observations

We examine strain energy dissipation of Earth's normal modes and body tides in order to compare these observations with predictions from an experimental model of intrinsic dissipation. For this comparison we employ a recently developed self-consistent treatment of modes and tides, that includes the separation of dynamical processes (self-gravity and inertia) from intrinsic attenuation. We select a set of normal modes and tides with similar depth sampling centered on the lower mantle, without contributions from the core. This dataset ranges from 7 min to 18.6 yr, or about 6 decades in frequency. The results show that the attenuation of the higher-frequency modes defines a broad plateau in dissipation (inferred experimentally to occur at the transition from elastic to anelastic behavior) whereas the attenuation of lower-frequency tides falls on a frequency-dependent absorption band. This reconciles published, seemingly contradictory models of dissipation for modes versus tides: the absorption band for modes terminates at periods where tidal models infer its onset. Microphysically, the experimental data can be explained by elastically accommodated grain boundary sliding, producing a broad plateau at short periods, transitioning to diffusionally assisted grain boundary sliding at longer periods, corresponding to an absorption band. Extrapolation of the fit to the experimental data with parameters adopted for the lower mantle can reconcile the seismological and geodetic observations.

Estimation of splitting functions from Earth's normal mode spectra using the neighbourhood algorithm

Estimation of splitting functions from Earth's normal mode spectra using the neighbourhood algorithm

Unexpected, high-Q, low-frequency peaks in seismic spectra

It was established over a decade ago that the normal modes of the Earth are continuously excited at times without large earthquakes, but the sources of the ‘seismic hum’ have remained unresolved. In addition to the normal modes of the Earth, we show spectral lines in seismic data with frequencies which correspond closely to normal modes of the Sun. Moreover, the widths of the low-frequency lines in the seismic spectra are similar to those of solar modes and much narrower than those of the Earth's normal mode peaks. These seismic lines are highly coherent with magnetic fields measured on both the Geostationary Operations Environmental Satellite (GOES)–10 satellite and the Advanced Composition Explorer (ACE) spacecraft located at L1, 1.5 million km sunward of Earth suggesting that the solar modes are transmitted to the Earth by the interplanetary magnetic field and solar wind. The solar modes are split by multiples of a cycle/day and, surprisingly, by the ‘quasi two-day’ mode and other frequencies. Both the phase of the coherences and slight frequency offsets between seismic and geomagnetic data at observatories exclude the possibility that these effects are simply spurious responses of the seismometers to the geomagnetic field. We emphasize data from low-noise seismic observatories: Black Forest (BFO), Piñon Flat (PFO), Eskdalemuir (ESK) and Obninsk (OBN). Horizontal components of seismic velocity show higher coherences with the external (ACE) magnetic field than do the vertical components. This effect appears to be larger near the seismic torsional, or T-mode, frequencies.

Anisotropy of Earth's inner core intrinsic attenuation from seismic normal mode models

The Earth's inner core, the slowly growing sphere of solid iron alloy at the centre of our planet, is known to exhibit seismic anisotropy. Both normal mode and body wave studies have established that, when the global average is taken, compressional waves propagate faster in the North–South direction than in the equatorial plane. Recent body wave studies also indicate that this fast direction may be more attenuating, and interpret this anisotropic attenuation in terms of anisotropic scattering due to inner core texturing. Here we use the Earth's normal modes to study the attenuation anisotropy of both compressional and shear waves in the inner core. As normal modes have wavelengths several orders of magnitude longer than estimates of inner core grain size, any attenuation anisotropy quantified using normal modes must reflect the anisotropy of intrinsic (viscoelastic) attenuation of the crystalline inner core alloy. By inverting zonal anelastic and elastic normal mode splitting function coefficients of twenty inner core sensitive modes, we construct models of inner core intrinsic attenuation and velocity anisotropy. We find that, for compressional waves, the North–South direction is both fast and more strongly attenuating. The existence of intrinsic inner core attenuation anisotropy can be interpreted in terms of anisotropic Zener relaxation in the metallic alloy comprising the inner core. Such anisotropic Zener relaxation has only been observed in the presence of solute atoms, and is thus entirely consistent with the presence of a few atomic per cent of light elements in the Earth's inner core.

Constraining the gravitational wave energy density of the Universe using Earth’s ring

The search for gravitational waves is one of today's major scientific endeavors. A gravitational wave can interact with matter by exciting vibrations of elastic bodies. Earth itself is a large elastic body whose so-called normal-mode oscillations ring up when a gravitational wave passes. Therefore, precise measurement of vibration amplitudes can be used to search for the elusive gravitational-wave signals. Earth's free oscillations that can be observed after high-magnitude earthquakes have been studied extensively with gravimeters and low-frequency seismometers over many decades leading to invaluable insight into Earth's structure. Making use of our detailed understanding of Earth's normal modes, numerical models are employed for the first time to accurately calculate Earth's gravitational-wave response, and thereby turn a network of sensors that so far has served to improve our understanding of Earth, into an astrophysical observatory exploring our Universe. In this paper, we constrain the energy density of gravitational waves to values in the range 0.035--0.15 normalized by the critical energy density of the Universe at frequencies between 0.3 and 5 mHz, using ten years of data from the gravimeter network of the Global Geodynamics Project that continuously monitors Earth's oscillations. This work is the first step towards a systematic investigation of the sensitivity of gravimeter networks to gravitational waves. Further advances in gravimeter technology could improve sensitivity of these networks and possibly lead to gravitational-wave detection.

Spherical harmonic stacking for the singlets of Earth's normal modes of free oscillation

AbstractWe extend the spherical harmonic stacking (SHS) method of Buland et al. (1979) for the radial (vertical) component in the seismogram to the transverse (horizontal) components of the displacement field. Taking advantage of the orthogonality of the spherical harmonic functions (scalar and vectorial), SHS isolates and accentuates the signals of individual singlets of the Earth's normal modes of free oscillation. We apply the SHS on the broadband Incorporated Research Institutions for Seismology (IRIS) seismograms from up to 97 IRIS seismic stations for the 2004 Sumatra‐Andaman earthquake, in experiments targeted to spheroidal as well as toroidal modes—2S1, 0S3, 2S2, 3S1, 1S3, 0T2, and 0T3. We report the complete resolution of the singlet frequencies of these multiplets, some for the first time, and estimate the singlets' complex frequencies using the frequency domain autoregressive method of Chao and Gilbert (1980). The latter contain useful information to be used in inversions for the 3‐D structure of the Earth's interior.

Constraints on core‐mantle boundary topography from normal mode splitting

The frequencies of Earth's normal modes are split by rotation, ellipticity, and internal structure of the Earth. Thus, models of mantle heterogeneity and discontinuity topography generate splitting that may be tested against observations. We insert maps of core‐mantle boundary (CMB) topography, which are derived via either a purely seismic or a joint tomographic/geodynamic inversion of body waves data, on top of tomographic model S20RTS. We then calculate synthetic splitting functions for normal modes that have been shown to be sensitive to CMB topography and compare these to observed normal mode splitting data. The CMB topography maps obtained via geodynamically constrained tomography fit normal mode data better than purely seismic maps, in particular when the geodynamic constraint also accounts for the presence of post‐perovskite in the D″ region. We test the significance of the reduction in misfit using the concept of observability which suggests that normal modes are able to observe the difference between the different CMB topography maps. In addition, we find that the statistical significance, assessed by checking what fraction of 1000 randomly generated CMB models achieve a comparatively good fit as the desired model, is higher than 92% for degree 2 and 98% for all degrees. In summary, we have identified a model of CMB topography that fits body wave data and improves, at least to some extent, the fit to normal mode data, and is coherent with the large‐scale pattern of deep mantle heterogeneity expected on the basis of convection modeling.

Earth's free oscillations recorded by free-fall OBS ocean-bottom seismometers at the Lesser Antilles subduction zone

Three unburied ocean bottom seismometers (OBS) equipped with Trillium 240 s broad-band seismometers recorded spheroidal free oscillations of the Earth out to periods over 1000 s period, for the M = 8.1, April 1, 2007 Solomon Islands earthquake. In contrast to broadband observatories of the global network that operate in quiet continental locations, these instruments were dropped on the several-km thick layer of sediments of the forearc and accretionary wedge of the Lesser Antilles subduction zone. Furthermore, a high ambient noise level due to the ocean surface infragravity waves is expected to cover the frequency band of Earth's normal modes band when recorded at these sites. In spite of these hostile environmental conditions, the frequency of clearly defined peaks of the Earth's normal modes were measured after the earthquake. This suggests that the recording of normal modes and long period waves can be extended to parts of the hitherto inaccessible ocean with currently available OBS technology.

Synthetic seismograms by normal-mode summation: a new derivation and numerical examples

SUMMARY We present a new derivation of the explicit expression of displacement in response to a point dislocation source in terms of the summation of the Earth's normal modes. Starting from the well-known formal expression of normal-mode summation, we obtain the explicit expressions of displacements for the three conventional components in seismology (vertical, radial and transverse). Aided by the generalized spherical harmonics and coordinate transformation, our derivation is more comprehensive and concise and easier to handle. Numerical examples of normal-mode synthetics in different but realistic Earth models for a variety of source–receiver geometries are presented to illustrate the influence of self-gravitation, attenuation and radial anisotropy on waveforms. We also examine how waveforms of Love and Rayleigh waves are affected by the density and P-wave velocity structures in the upper mantle. Results show that the shear-wave structure obtained from surface wave tomography may be biased if the sensitivities of surface waves to the variation of density and compressional wave speed are not properly taken into account in the inversion.

Wide-band coupling of Earth's normal modes due to anisotropic inner core structure

SUMMARY We investigate the importance of wide-band coupling of normal modes due to inner core anisotropy. We compare four different seismic models of inner core anisotropy, which were obtained by others using the splitting of Earth’s normal modes. These models have been developed using a self-coupling (SC) approximation, which assumes that coupling between nearby modes through anisotropic inner core structure is negligible. We test the SC approximation by comparing the frequencies and quality factors of 90 inner core sensitive modes, computed for these models using either the SC approximation or full-coupling (FC) among large groups of modes. We find significant shifts in the quality factors and frequencies for some modes. Groups of modes which significantly couple together are constructed for six target modes. These groups are model dependent and in some cases contain large numbers of modes. Synthetic seismograms are calculated to show that the difference between SC and FC is observable on the scale of seismograms and of the same order of magnitude as the difference between synthetic and observed seismograms. Thus, future models of inner core anisotropy should take cross-coupling between large groups of modes into account.

The Earth's hum: the excitation of Earth normal modes by ocean waves

SUMMARY The recent discovery that the seismic normal modes of the Earth are excited to a nearly constant level during seismically quiet days (‘the hum of the Earth’) has lead to much speculation as to what drives the observed background in the absence of large earthquakes. Other authors have shown that the hum cannot be explained by the many small earthquakes occurring each day and have also precluded excitation by ‘slow’ earthquakes as the source. The non-linear interaction of low frequency ocean waves (infragravity waves) generates high phase velocity components that excite the seismic normal modes of the Earth. I show the forcing of the normal modes of the Earth by infragravity waves interacting over the continental shelves and over the deep ocean basins is sufficient to explain the background seismic acceleration spectrum observed during seismically quiet days. Atmospheric turbulence is shown to be a negligible source of the Earth’s hum. Observations suggesting enhanced spectral levels for two seismic modes that couple strongly with atmospheric modes (0S27 and 0S39) have been cited as evidence that the source of the hum lies in the atmosphere. However, the wave interaction mechanism also couples energy into infrasound and thus may explain these observations. A calculation of the coupling between ocean waves and normal modes expected from ocean waves interacting over the shelves reproduces the vertical acceleration spectrum observed at quiet seismic sites from 2 to 40 mHz including the small ‘hump’ between 5 and 15 mHz and the observed rise at higher frequencies. The model diverges above 40 mHz because the ‘single frequency’ microseism peak is generated by a different mechanism. The shape of the hum spectrum is controlled primarily by the elastic properties of the Earth with the attenuation structure controlling mode amplitudes. A second calculation shows the excitation of Earth’s normal modes by infragravity waves interacting over the deep ocean basins can also explain the observed spectrum of the Earth’s hum. Infragravity waves are less energetic and the forcing mechanism weaker over deep water, but the integrative effect of forcing over a large area offsets the weaker forcing per unit area. It is likely that both pelagic and shelf ocean wave sources contribute significantly to the forcing of the Earth’s hum and both coastal and pelagic source regions have been identified using seismic techniques. Infragravity wave spectra in deep water over the Pacific basin vary little, thus explaining the small variation in the hum that is observed. However, time varying source regions are also observed in the hum, and may be best explained by intense near shore sources. The great similarity between the spectra predicted from continental shelf forcing and from forcing over the deep ocean basins precludes discriminating between coastal and pelagic sources based on the shape of the spectrum. A more complete knowledge of the climatology of infragravity waves coupled with better seismic observations are needed to resolve this question.

Preliminary result of the Earth's free oscillations by Galerkin method

AbstractWe use a Galerkin method to compute the eigenfunctions and eigenperiods of some of the Earths spheroidal and toroidal modes. The boundary conditions are treated using a Tau method. We show that for a realistic Earth model the difference between the computed and observed periods is less than 1.4%. We conclude that a Galerkin method may be an effective tool for the studies of the Earth's normal modes.

An energy-duration procedure for rapid determination of earthquake magnitude and tsunamigenic potential

We introduce a rapid and robust, energy-duration procedure, based on the Haskell, extended-source model, to obtain an earthquake moment and a moment magnitude, MED. Using seismograms at teleseismic distances (30°–90°), this procedure combines radiated seismic energy measures on the P to S interval of broadband signals and source duration measures on high-frequency, P-wave signals. The MED energy-duration magnitude is scaled to correspond to the Global Centroid-Moment Tensor (CMT) moment-magnitude, MCMTw, and can be calculated within about 20 min or less after origin time (OT). The measured energy and duration values also provide the energy-to-moment ratio, Θ, used for identification of tsunami earthquakes. The MED magnitudes for a set of recent, large earthquakes match closely MCMTw, even for the largest, great earthquakes; these results imply that the MED measure is accurate and does not saturate. After the 2004 December 26 Sumatra-Andaman mega-thrust earthquake, magnitude estimates available within 1 hr of OT ranged from M = 8.0 to 8.5, the CMT magnitude, available about 3 hr after OT, was MCMTw = 9.0, and, several months after the event, Mw = 9.1–9.3 was obtained from analysis of the earth normal modes. The energy-duration magnitude for this event is MED = 9.2, a measure that is potentially available within 20 min after OT. After the 2006 July 17, Java earthquake, the magnitude was evaluated at M = 7.2 at 17 min after OT, the CMT magnitude, available about 1 hr after OT, was MCMTw = 7.7; the energy-duration results for this event give MED = 7.8, with a very long source duration of about 160 s, and a very low Θ value, indicating a possible tsunami earthquake.

MTSU : Recovering Seismic Moments from Tsunameter Records

We define a new magnitude scale, M TSU, allowing the quantification of the seismic moment M0 of an earthquake based on recordings of its tsunami in the far field by ocean-bottom pressure sensors (“tsunameters”) deployed in ocean basins, far from continental or island shores which are known to affect profoundly and in a nonlinear fashion the amplitude of the tsunami wave. The formula for M TSU, M TSU=log10 M O‒20=log10 X (ω) + C D TSU + C S TSU +C O, where X(ω) is the spectral amplitude of the tsunami, C D TSU a distance correction and C S TSU a source correction, is directly adapted from the mantle magnitude M m introduced for seismic surface waves by Okal and Talandier. Like M m, its corrections are fully justified theoretically based on the representation of a tsunami wave as a branch of the Earth's normal modes. Even the locking constant C 0, which may depend on the nature of the recording (surface amplitude of the tsunami or overpressure at the ocean floor) and its units, is predicted theoretically. M TSU combines the power of a theoretically developed algorithm, with the robustness of a magnitude measurement that does not take into account such parameters as focal geometry and exact depth, which may not be available under operational conditions in the framework of tsunami warning. We verify the performance of the concept on simulations of the great 1946 Aleutian tsunami at two virtual gauges, and then apply the algorithm to 24 records of 7 tsunamis at DART tsunameters during the years 1994–2003. We find that M TSU generally recovers the seismic moment M 0 within 0.2 logarithmic units, even under unfavorable conditions such as excessive focal depth and refraction of the tsunami wave around continental masses. Finally, we apply the algorithm to the JASON satellite trace obtained over the Bay of Bengal during the 2004 Sumatra tsunami, after transforming the trace into a time series through a simple ad hoc procedure. Results are surprisingly good, with most estimates of the moment being over 1029 dyn-cm, and thus identifying the source as an exceptionally large earthquake.