Normal Modes of the Earth and Planets

doi:10.1016/S0074-6142(02)80213-3

International Geophysics

Volume 81, Part A, 2002, Pages 125-147, cp1

https://doi.org/10.1016/S0074-6142(02)80213-3 Get rights and content

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Introduction and Historical Key Dates

During almost a century, and up to 1960, normal modes were only theoretical concepts. Normal modes of the Earth were first described by Lord Kelvin (Thomson, 1863a, b) with a computation of the lowest fundamental spheroidal mode ₀S₂ frequency for a homogeneous Earth model, either fluid or rigid. This early attempt was followed by a complete description of the normal modes of a spherical nonrotating elastic isotropic (SNREI) Earth in the early works of Rayleigh (1894, 1906) and Love (1911). The

General Properties

Let us consider a general model of the Earth or another planet, including if necessary an ocean and an atmosphere, and let us suppose that all equations are expressed in the inertial frame, rotating with the angular velocity Ω. In the state of equilibrium, the prestress (or static pressure for fluid parts) is compensated by the gravity, the centrifugal force, as well as the Reynolds stress and thermal stress associated with convection and temperature gradient. We will focus on a hydrostatic

Normal Modes of a Spherical NonRotating Elastic Isotropic Earth Model

Normal modes of a spherical nonrotating elastic isotropic (SNREI) Earth constitute for many purposes the basis of normal mode theory, even in an aspherical Earth. They indeed constitute a complete basis of all functions verifying the same boundary and continuity conditions on the spherical surface of the Earth and internal discontinuities, respectively, as shown by Valette (1987) in the elastic case. Normal modes of a spherical Earth can therefore be used to describe the normal modes of all

Normal Modes of a Spherical Nonrotating Anelastic Isotropic Earth Model

Attenuation or a radiating surface affect the normal modes due to the nonconservation of energy. For intrinsic attenuation and at a given frequency, this implies an imaginary part of the stiffness tensor, as shown by Liu et al. (1976) and Anderson and Given (1982). For causality reasons originally demonstrated for similar problems by Kramers and Kronig, this also implies a frequency dependence of the stiffness tensor (Anderson, 1989; Anderson and Minster, 1979), which in seismology is necessary

Normal Mode Orthonormality Relations

In the general case, normal modes cannot be described by a single spherical harmonic and have a more complicated shape on the sphere. Many general properties remain, however. In the general case, following Lognonné (1991), let us first consider the right eigenmodes | u_k〉 and associated eigenfrequencies σ_k satisfying $- σ_{k}^{2} | u_{k} 〉 + σ_{k} B | u_{k} 〉 + A (σ_{k}) | u_{k} 〉 = 0$ and the left eigenmodes, with left eigenfunctions 〈v_k | and their associated eigenfrequencies σ_k verifying $- σ_{k}^{2} 〈 v_{k} | + σ_{k} 〈 v_{k} | B^{†} + 〈 v_{k} | A^{†} (σ_{k}) = 0$ Here

Mode Summation and Seismograms

In the general case, as well as in the different simplified cases (e.g., Gilbert, 1970, in the elastic case), normal modes can be used for computing the response of the Earth produced by an equivalent body force | f (t)〉. In the frequency domain, the solution has to satisfy $[- σ^{2} + σ B + A] | u (σ) 〉 = | f (σ) 〉$ and can moreover be expressed as a projection in the complete basis of eigenfunctions such as $| u (σ) 〉 = \underset{k^{'}}{Σ} c_{k^{'}} (σ) | u_{k^{'}} 〉$ As noted above, we will restrict the summation to the seismic modes (e.g.,

Normal Modes of a Rotating, Aspherical, Anelastic, and Anisotropic (RA₃) Earth Model: Perturbation Theory

As noted in the introduction, the early observations almost immediately pointed out the limits of the spherical Earth theory and showed that the departure of sphericity and rotation resulted in an observable degeneracy of the singlets. For each normal mode multiplet, the frequencies of the 2l + 1 singlets are therefore slightly different, and each has excitation and observation amplitude depending on the position of both the source and station and on the mechanism of the earthquake. Therefore,

Higher-Order Perturbation Theory

Let us now consider a singlet k and note its eigenfrequency σ_k. The associated primal eigenfunction | u_k〉 and dual eigenfunction | v_k〉 verify $\begin{array}{l} λ_{k} | u_{k} 〉 = H (λ_{k}) | u_{k} 〉 \\ λ_{k} | v_{k} 〉 = \hat{H} (λ_{k}) | v_{k} 〉 \end{array}$ where λ_k = σ²_k is the squared frequency and $\hat{H}$ is the dual operator with reversed rotation. They have to follow the bi-orthogonality relation (35) and to be normalized with (36). All the singlets of the multiplet K characterized by l and n in the reference model will define a subspace of the spectrum noted S_K and

Kernels and Computation of Splitting Matrices

All previous expressions need the computation of the coupling matrices expressing the coupling between two singlets of the reference model. As an example, this is the case of the first-order perturbation, which can be rewritten as $\begin{array}{l} | u_{k}^{(1)} 〉 = Δ δ H | u_{k}^{(0)} 〉 \\ = \underset{K^{'} \neq K}{Σ} \frac{1}{λ_{K} - λ_{K^{'}}} \underset{m^{'}, m}{Σ} | l^{'}, m^{'}, n^{'} 〉〈 l^{'}, m^{'}, n^{'} | δ H | l, m, n 〉〈 l, m, n | u_{k}^{(0)} 〉 \end{array}$ where the multiplets K = (n, l) and K′ = (n′, l′) have dimensions 2l′ + 1 and 2l + 1 and where | l, m, n〉 are the SNREI or SNRAI singlets, associated with a symmetric eigenproduct and

Lateral Variation: Normal Mode Synthetics, Observables and Seismograms

The first effect of lateral variations is to split the degenerated eigenfrequencies. The splitting for the model SAW12D of Li and Romanowicz (1996) is shown in Figure 6 for the spheroidal and toroidal fundamental branches. The splitting of normal modes increases with the angular order, as the consequence of both the increased sensitivity of modes to the shallow structure and the stronger lateral variations in the shallow structures. A shift of the mean frequencies of the singlets is also

Conclusion and Perspectives

Almost 40 years after their first observations, the results published on normal modes have clearly demonstrated that observing and inverting the normal modes of a planet is one of the most powerful methods for recovering its internal structure. It is also the only way to get from seismic data a direct sensitivity to density, including lateral variations (Tanimoto, 1991), as all other waves are sensitive mostly to the seismic velocities. The theory is now very mature, and almost completely

Acknowledgments

This work was sponsored by several CNRS/INSU grants, including ASP Tomographie, Programme Intérieur de la Terre, and Programme National des Risques Naturels. We thank J. Artru for her contribution, as well as B. Romanowicz, F. Pollitz, and an anonymous reviever for their comments and corrections. This is IPGP contribution 1708.

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Source models of great earthquakes from ultra low-frequency normal mode data
2014, Physics of the Earth and Planetary Interiors
We present a new earthquake source inversion technique based on normal mode data for the simultaneous determination of the rupture duration, length and moment tensor of large earthquakes with unilateral rupture. We use ultra low-frequency (f <1 mHz) mode singlets and multiplets which are modelled using Higher Order Perturbation Theory (HOPT), taking into account the Earth’s rotation, ellipticity and lateral heterogeneities. A Monte Carlo exploration of the model space is carried out, enabling the assessment of source parameter tradeoffs and uncertainties. We carry out synthetic tests to investigate errors in the source inversions due to: (i) unmodelled 3-D Earth structure; (ii) noise in the data; (iii) uncertainties in spatio-temporal earthquake location; and, (iv) neglecting the source finiteness in point source inversions. We find that unmodelled 3-D structure is the most serious source of errors for rupture duration and length determinations especially for the lowest magnitude events. The errors in moment magnitude and fault mechanism are generally small, with the rake angle showing systematically larger errors (up to 20°). We then investigate five real thrust earthquakes ( $M_{w} ⩾ 8.5$ ): (i) Sumatra-Andaman (26th December 2004); (ii) Nias, Sumatra (28th March 2005); (iii) Bengkulu (12th September 2007); (iv) Tohoku, Japan (11th March 2011); (v) Maule, Chile (27th February 2010); and, (vi) the 24 May 2013 $M_{w} 8.3$ Okhotsk Sea, Russia, deep (607 km) event. While finite source inversions for rupture length, duration, magnitude and fault mechanism are possible for the Sumatra-Andaman and Tohoku events, for all the other events their lower magnitudes only allow stable point source inversions of mode multiplets. We obtain the first normal mode finite source model for the 2011 Tohoku earthquake, which yields a fault length of 461 km, a rupture duration of 151 s, and hence an average rupture velocity of 3.05 km/s, giving an independent confirmation of the compact nature of this event. For all the other earthquakes studied, our new source models agree well with previous studies. We do not find any unexplained systematic differences between our results and those in the literature, suggesting that for the wave frequencies considered, the moment magnitude and the fault mechanism of the earthquakes studied do not show a strong frequency dependence.
Large impacts detected by the Apollo seismometers: Impactor mass and source cutoff frequency estimations
2011, Icarus
Citation Excerpt :
We do a very crude approximation for seismogram modeling and limit ourselves to radially symmetric models. We compute the seismograms for such a model by using a normal mode summation technique (Lognonné and Clévédé, 2002). Using the code MINOS (Woodhouse, 1988), spheroidal frequencies up to 400 mHz (2.5 s in period) have been calculated, which includes not only the normal modes associated with surface waves (with low radial orders), but also those associated with body waves, including converted ones (with high radial orders).
Meteoroid impacts are important seismic sources on the Moon. As they continuously impact the Moon, they are a significant contribution to the lunar micro-seismic background noise. They also were associated with the most powerful seismic sources recorded by the Apollo seismic network. We study in this paper the largest impacts. We show that their masses can be estimated with a rather simple modeling technique and that high frequency seismic signals have reduced amplitudes due to a relatively low (about 1 s) corner frequency resulting from the duration of the impact process and the crater formation. If synthetic seismograms computed for a spherical model of the Moon are unable to match the waveforms of the observations, they nevertheless provide an approximate measure of the energy of seismic waves in the coda. The latter can then be used for an estimation of the mass of the impactors, when the velocity of the impactor is known. This method, for the artificial impacts of the LM and SIVB Apollo upper stages, allows us to retrieve the mass within 20% of relative error. The estimated mass of the largest impacts observed during the 7 years of activity of the Apollo seismic network provides an explanation for the non-detection of surface waves on the seismograms. The specifications of future Moon seismometers, in order to provide the detection of surface waves, are given in conclusion.
Planetary Seismology
2007, Treatise on Geophysics
In-situ extraterrestrial seismic measurements have, to date, been limited to data collected during the lunar Apollo missions. These data have provided significant contributions to our understanding of the interior structure of the Moon, and by inference, permissible thermal and mineralogical models. In addition, the seismic activity recorded by the Apollo stations provides information on both internal (moonquakes) and external (meteoroid) sources.
The seismic activity of other bodies, notably Mars and Venus, has never been measured. It can however be estimated, subject to approximations regarding the thermal structure of the lithospheres of these bodies, strain rates, and plausible present-day tectonic activity. For both Mars and Venus, events detectable by surface seismometers are likely: Martian estimates are 50(10) quakes annually with moment magnitude M_w greater than 3.8(4.5). In addition, impacts large enough for seismology may occur at a rate comparable to the Moon. Venusian estimates are 100(25) quakes annually with M_w greater than 5(6). Such seismic activities, much larger than the lunar ones, justify the deployment of seismic monitoring systems on these planets.
In the foreseeable future, Mars is the new body most likely to be explored through surface landers: current understanding of the interior structure, seismic noise levels, and scattering on this body indicate that body-wave and regional surface-wave investigations could yield enormous insight into the structure and evolution of this planet. Other targets, like asteroids, Europa will also benefit similarly from passive seismic experiments, as well as a return to the Moon with more sensitive seismometers.
The practical deployment and operation of surface landers on Venus is difficult due to the high surface temperature. Acoustic coupling of a planet’s atmosphere with its internal body provides, however, opportunities for seismic investigations, and can be used either on Venus, a planet with a dense atmosphere, or on giant planets. On Venus, atmospheric signals can have amplitudes 600 times their terrestrial counter parts for a given quake seismic moment and atmospheric height. The possibility of detection of Venusian quakes from orbit exists through a combination of the near-source thermal and albedo signatures and ionospheric perturbations associated to Rayleigh waves. Detection of atmospheric normal modes of the giant planets with ground or space-based techniques and could provide opportunities for future exploration of the interiors of these bodies.
10.03 - Planetary Seismology
2007, Treatise on Geophysics: Volume 1-10
In-situ extraterrestrial seismic measurements have, to date, been limited to data collected during the lunar Apollo missions. These data have provided significant contributions to our understanding of the interior structure of the Moon, and by inference, permissible thermal and mineralogical models. In addition, the seismic activity recorded by the Apollo stations provides information on both internal (moonquakes) and external (meteoroid) sources.
The seismic activity of other bodies, notably Mars and Venus, has never been measured. It can however be estimated, subject to approximations regarding the thermal structure of the lithospheres of these bodies, strain rates, and plausible present-day tectonic activity. For both Mars and Venus, events detectable by surface seismometers are likely: Martian estimates are 50(10) quakes annually with moment magnitude M_w greater than 3.8(4.5). In addition, impacts large enough for seismology may occur at a rate comparable to the Moon. Venusian estimates are 100(25) quakes annually with M_w greater than 5(6). Such seismic activities, much larger than the lunar ones, justify the deployment of seismic monitoring systems on these planets.
In the foreseeable future, Mars is the new body most likely to be explored through surface landers: current understanding of the interior structure, seismic noise levels, and scattering on this body indicate that body-wave and regional surface-wave investigations could yield enormous insight into the structure and evolution of this planet. Other targets, like asteroids, Europa will also benefit similarly from passive seismic experiments, as well as a return to the Moon with more sensitive seismometers.
The practical deployment and operation of surface landers on Venus is difficult due to the high surface temperature. Acoustic coupling of a planet’s atmosphere with its internal body provides, however, opportunities for seismic investigations, and can be used either on Venus, a planet with a dense atmosphere, or on giant planets. On Venus, atmospheric signals can have amplitudes 600 times their terrestrial counter parts for a given quake seismic moment and atmospheric height. The possibility of detection of Venusian quakes from orbit exists through a combination of the near-source thermal and albedo signatures and ionospheric perturbations associated to Rayleigh waves. Detection of atmospheric normal modes of the giant planets with ground or space-based techniques and could provide opportunities for future exploration of the interiors of these bodies.
Introduction to mode coupling methods for surface waves
2007, Advances in Geophysics
Citation Excerpt :
Reduced to its simpler expression, it gives the “location-parameter” defined by Jordan (1978). A recent review of mode coupling methods in the framework of free oscillation theory is given in Lognonné and Clévédé (2002). Considerable progress has been done concerning the modelisation of surface waves in laterally heterogeneous structures in the last two decades.
In laterally homogeneous models, free oscillations and surface waves are composed of modes propagating independently of each other. In laterally heterogeneous structures, the seismic wavefield is more complex, but can still be expressed up to a certain degree of approximation by modes, which are then coupled to each other by the lateral heterogeneities. We review here the different types of mode coupling methods which have been developed to model surface waves in laterally heterogeneous structures.
In the surface wave formalism, that is for a single wavetrain propagating in a flat structure, two classes of mode coupling methods exist. Those based on coupling of local modes are most appropriate for analysing the gradual evolution of the surface wavefield from one type of structure to another, and are related to surface wave ray-tracing. For models with localised heterogeneities, the modelling methods are based on coupling of modes calculated in a laterally homogeneous reference structure. In that case, the modes are coupled by the heterogeneities, which act as scatterers. Single- as well as multiple-scattering algorithms have been developed. We review here principally the methods for direct problems, but mention also their applicability in inverse problems.
New determinations of Q quality factors and eigenfrequencies for the whole set of singlets of the Earth's normal modes <inf>0</inf> S<inf>0</inf>, <inf>0</inf>S<inf>2</inf>, <inf>0</inf>S<inf>3</inf> and <inf>2</inf>S<inf>1</inf> using superconducting gravimeter data from the GGP network
2006, Journal of Geodynamics
We present new modal Q measurements of the ₀S₀ and ₀S₂ modes, and first modal frequencies and Q measurements of ₂S₁ and ₀S₃ modes.
The high quality of the GGP (Global Geodynamics Project) superconducting gravimeters (SGs) contributes to the clear observation of seismic normal modes at frequencies lower than 1 mHz and offers a good opportunity for studying the behaviour of these modes.
The interest of scientists in the gravest normal modes is due to the fact that they do contribute to a better knowledge of the density profile in the Earth, helping to constrain Earth's models.
These modes have been clearly identified after some large recent events recorded with superconducting gravimeters, particularly well-suited for low-frequency investigations. The Ms = 8.1 (Mw = 8.4) Peruvian earthquake of June 2001 and the Ms = 9.0 (Mw = 9.3) Sumatra-Andaman earthquake of December 2004 provide us with individual spectra which exhibit a clear splitting of the spheroidal modes ₀S₂, ₀S₃ and ₂S₁, and a clear identification of each of the individual singlets, with a resolution never obtained from broad-band seismometers records.
The Q quality factors have been determined from the apparent decrease of the amplitude of each singlet with time, according to a well-suited technique [Roult, G., Clévédé, E., 2000. New refinements in attenuation measurements from free-oscillation and surface-wave observations. Phys. Earth Planet. Inter. 121, 1–37]. The results are compared to the theoretical frequencies and Q quality factors computed for the PREM and 1066A models, taking into account both rotation and ellipticity effects of the Earth. The two observed datasets (frequencies and Q quality factors) exhibit a splitting on the observed values different from the predicted one. That seems to point out that some parameters as density or attenuation values used in the theoretical models do not explain the observations.
A new dataset of frequencies and Q quality factors of the whole set of singlets of the gravest spheroidal modes is thus under construction. That dataset includes the five individual singlets of the ₀S₂ mode clearly identified on the SG records, the three singlets of the ₂S₁ mode recently observed for the first time by [Rosat, S., Hinderer, J., Rivera, L., 2003b. First observation of ₂S₁ and study of the splitting of the football mode ₀S₂ after the June 2001 Peru earthquake of magnitude 8.4. Geophys. Res. Lett. 30, 21211, doi:10-1029/2003L018304], the ₀S₀ radial mode, and the seven individual singlets of the ₀S₃ mode.

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10 - Normal Modes of the Earth and Planets