Earth's Normal Modes Research Articles

A new formalism for the effect of lateral heterogeneity on normal modes and surface waves--I: isotropic perturbations, perturbations of interfaces and gravitational perturbations

Lateral heterogeneities in the Earth produce a coupling of the normal modes of a laterally homogeneous Earth model, and lead to mode interactions for surface waves. Traditionally, this problem was treated by expanding the heterogeneity in spherical harmonics, and was thereby reduced to a complicated algebraic problem requiring the use of Wigner 3j-symbols and generalized spherical harmonics. However, due to the global character of this theory, the resulting equations are not convenient for obtaining physical insight into the problem, and are cumbersome to manipulate. In this paper, the effects of lateral heterogeneity in density, bulk modulus, shear modulus, interface displacements and gravitation are treated without a global expansion in spherical harmonics. Using a simple-operator formalism, the coupling coefficients between the Earth's normal modes can be expressed by an integral over the horizontal extent of the inhomogeneity. The integrand can be expressed by a set of 17 local frequencies of interaction, and some simple geometrical variables. The mode coupling depends in a simple way on the scattering angle, even for modes with such a long period that the concept of scattering is meaningless. Apart from the restrictions of first-order perturbation theory, there are no other restrictions; specifically, it is not necessary to assume a far-field limit. From the expression for normal-mode coupling, a theory for surface-wave scattering and conversion is derived. This leads to a complete set of local surface-wave interaction coefficients, where the effects of sphericity are fully taken into account. The surface-wave polarization vectors and excitation tensor are derived from the source and receiver operators. The resulting theory for normal-mode interactions and surface-wave scattering leads to an efficient method for generating synthetic seismograms in laterally inhomogeneous media, and is simple enough to allow extensive mathematical manipulation of the resulting equations. The effects of anisotropy are treated in the sequel of this paper.

Multiplet-multiplet coupling due to lateral heterogeneity: asymptotic effects on the amplitude and frequency of the Earth's normal modes

Starting with the first-order formulation of quasi-degenerate splitting theory for the normal modes of a laterally heterogeneous earth, we have obtained an asymptotic expression for the coupling terms corresponding to neighbouring multiplets along the same dispersion branch as the mode considered, valid to order 1/l, where l is the angular order of this mode (l≫ 1). We show that, to order zero, these coupling terms introduce a small shift in epicentral distance into the expression for the long period seismogram obtained by normal mode summation. This shift depends on the difference between the great circle and the minor arc averages of the local frequency. the coupling terms thus permit us to reconcile results obtained by normal-mode summation and by a propagating wave approach, as far as the dependence on structure of the phase of surface waves is concerned. To order 1/l, the coupling terms result in a perturbation in the amplitude of the mode considered, which depends on spatial derivatives of the local frequency and thus on the structure in the vicinity of the source station great circle path. We show that this term is equivalent to that which is found using ray perturbation methods for propagating surface waves. We compare and discuss the assumptions underlying both approaches and illustrate, by an example, the potential of the asymptotic normal-mode formulation for improved modelling of lateral heterogeneity in the earth.

Earth's Third Ocean: The Liquid Core

The history of the discovery of the earth's liquid core and its role in geophysics is briefly reviewed. Among the core's still undetermined properties is the extent to which its density gradient departs from an adiabat, as characterized by the Brunt‐Vaisala frequency. Since this parameter must affect the eigenfrequencies of the hitherto unobserved inertia‐gravity wave segment of the earth's normal mode spectrum, considerable effort has gone into theoretical studies of these core oscillations over the past decade. We relate these studies to those of internal gravity and Rossby waves in oceanography. The deployment of sensitive gravimeters with long‐term recording capability is now begining to yield data in which core modes may already have been observed.

A very broad band analysis of the Michoacan, Mexico, Earthquake of September 19, 1985

Long‐period seismograms from the GDSN, RSTN, and IDA network are used in a centroid‐moment tensor analysis of the September 19, 1985 Michoacan, Mexico, earthquake (Ms=8.1). The long‐period data are also used together with broad band displacement P‐waveforms in a joint very broad band inversion for the moment tensor and the moment rate distribution in time and space. Results are that the earthquake had a total scalar moment of 1.1×1028 dyne‐cm and that the moment release occurred mainly in two subevents, the second of which was located approximately 70 km southeast of the first. Observations of Earth's normal modes at ultra long periods (600 to 2000 sec) are shown to be consistent with the obtained scalar moment and source mechanism.

Variation in apparent attenuation of the Earth's normal modes due to lateral heterogeneity

Study of 3242 full first order synthetic seismograms of isolated normal mode multiplets shows that constructive and destructive interference of the singlets comprising a multiplet can account for the noise‐corrected variance of free oscillation attenuation measurements in the frequency band 3.0‐4.0 mHz. This interference gives rise to a distribution of synthetic Q measurements which is sensitive to the short wavelength velocity features of a model. While individual attenuation measurements depend also on the moment tensor, the overall variance of Q fluctuations is insensitive to the characteristics of the source employed and depends little on the length of the time series processed. The Woodhouse‐Dziewonski model M84A of upper mantle velocity structure causes a scatter in the synthetics large enough to interfere significantly with any signal from lateral Q structure. Correlation of individual synthetics with data is poor, however, which suggests that either there are remaining uncertainties in the velocity model, that mode‐mode coupling occurs in the data in a way that is poorly understood, or that lateral attenuative structure contributes to the observed scatter. Resolution of any such Q structure will be difficult given the present data set.

Excitation of the Earth's eigenvibrations by gravitational radiation from astrophysical sources

The first-order equation of motion for the excitation of a radially heterogeneous rotating self-gravitating Earth model by a plane gravitational wave is explicitly solved in terms of the calculable Earth's normal-mode amplitudes. It is shown that astrophysical sources with quadrupole strength of the orderh=10−20 (e.g. monochromatic waves from binary stars, pulses from supernova explosions or from a black hole in the centre of the Galaxy) can excite the modes n S 2 m to an amplitude level of 10−7 cm (accompanying ground accelerations of 10−12 cm/s2) at favourable locations on Earth. All other toroidal and spheroidal modes are smaller in amplitude by at least a factor of ∼[1000/ n T l m ]·10−4 and are not likely to be observed. Recent recordings of the Earth's eigenvibrations with amplitudes of the order 10−7 cm by means of gravitational-wave antennas give hope that the detection of gravitational signals by means of a quadrupole-quadrupole interaction with the whole Earth is indeed feasible.

Upper mantle anisotropy: evidence from free oscillations

Isotropic earth models are unable to provide uniform fits to the gross Earth normal mode data set or, in many cases, to regional Love-and Rayleigh-wave data. Anisotropic inversion provides a good fit to the data and indicates that the upper 200km of the mantle is anisotropic. The nature and magnitude of the required anisotropy, moreover, is similar to that found in body wave studies and in studies of ultramafic samples from the upper mantle. Pronounced upper mantle low-velocity zones are characteristic of models resulting from isotropic inversion of global or regional data sets. Anisotropic models have more nearly constant velocities in the upper mantle. Normal mode partial (Frediet) derivatives are calculated for a transversely isotropic earth model with a radial axis of symmetry. For this type of anisotropy there are five elastic constant. The two shear-type moduli can be determined from the toroidal modes. Spheroidal and Rayleigh modes are sensitive to all five elastic constants but are mainly controlled by the two compressional-type moduli, one of the shear-type moduli and the remaining, mixed-mode, modulus. The lack of sensitivity of Rayleigh waves to compressional wave velocities is a characteristic only of the isotropic case. The partial derivatives of the horizontal and vertical components of the compressional velocity are nearly equal and opposite in the region of the mantle where the shear velocity sensitivity is the greatest. The net compressional wave partial derivative, at depth, is therefore very small for isotropic perturbations. Compressional wave anisotropy, however, has a significant effect on Rayleigh-wave dispersion. Once it has been established that transverse anisotropy is important it is necessary to invert for all five elastic constants. If the azimuthal effect has not been averaged out a more general anisotropy may have to be allowed for.

Split free oscillation amplitudes for the 1960 Chilean and 1964 Alaskan earthquakes

abstract Splitting of the Earth's normal modes was observed for both the 1960 Chilean and 1964 Alaskan earthquakes. The strong peaks in the observed spectrum of each split multiplet correspond to singlets with much higher amplitudes than the others. Using theoretical results we have derived elsewhere (Stein and Geller, 1977a), we are able to predict this pattern. We show that the source mechanisms inferred for these earthquakes from surface waves are consistent with the observed pattern of relative spectral amplitudes of the split modes. However other mechanisms, such as a slow isotropic volume change, are also consistent with the split-mode amplitudes and are excluded only by additional data.

On Rayleigh's Principle

Summary In the theory of the Earth's normal modes of oscillation it is customary to use Rayleigh's principle in the determination of perturbations in the eigenperiods due to perturbations in the structure of an Earth model. When internal boundaries are perturbed the usual methods have been found to yield incorrect numerical results and this paper is directed towards a correct determination of the perturbations in this case.

On the excitation of the earth's seismic normal modes

The excitation of the earth's normal modes is formulated as an initial value problem. The static state of the earth, stressed from its hydrostatic reference situation, is considered as the initial state. The initial state is relaxed, at the time of the earthquake, by the removal of the forces maintaining the departure from hydrostatic equilibrium. Expressions are derived for the coefficients giving the relative excitation of the individual modes for the cases where these forces are compensating volume forces or compensating tractions on the faces of a dislocation. It is demonstrated that a point slip dislocation has a body force equivalent in the form of a double couple with a deviatoric moment tensor. However, for a source with volume change no moment tensor equivalent can be found. The volume change, apart from an elastic effect which can be represented by an isotropic moment tensor, has a direct gravitational effect on the excitation. This effect is due to a balanced force field consisting of a point force at the source and a continuous distribution of volume forces throughout the earth. The latter distribution, if not taken into account, may give rise to artificial phases in the frequency spectrum of the normal modes.

Re-examination of the earth's free oxcillations excited by the Kamchatka earthquake of November 4, 1952

Benioff's suggestion that the 58-min period sinusoidal oscillation found on a Pasadena strain seismogram after the Kamchatka earthquake of November 4, 1952 may represent the earth's gravest normal mode is re-examined in terms of a slow large-scale post-seismic deformation. The mechanism and the seismic moment of the main shock of the Kamchatka earthquake are determined by using the amplitude and the initial phase of G 2 and R 2 recorded at Pasadena and R 6 recorded at Palisades. By constraining the dip angle and the strike of the fault at 30° (towards NW) and N34°E, respectively, on the basis of the geometry of the Benioff zone, the slip angle is determined as 110° which represents 74% thrust and 26% right-lateral faulting. The direction of the slip angle agrees with the slip direction of the Pacific plate. A seismic moment of 3.5 · 10 29 dyn cm is obtained. If a fault area of 650 · 200 km 2 is assumed, an average dislocation of 5 m is obtained. Spectral analyses of the Pasadena strain records show that the 58-min sinusoidal oscillation in fact consists of a spectral peak near 54 min which is very close to the 0S 2 mode and other high-frequency peaks which can be correlated to the earth's normal modes. The records from two independent recording galvanometers correlate with each other very well, indicating that the recorded oscillation represents a real strain and not instrumental noise. The phase relation between the NS and EW components is consistent with the strain field associated with 0S 2 mode. Although these results provide positive evidence for a slow post-seismic deformation, the cause of the abrupt termination of the oscillation and the excitation mechanism remain unresolved.

A Discrete Model of the Inverse Love Wave Problem

A mathematical model is used in order to understand some unresolved aspects of the uniqueness of the inverse problem for the Earth normal modes. More specifically, the conjecture of Backus & Gilbert regarding the information content of the natural frequencies of modes associated with different angular numbers is examined. This is done for the geophysically artificial case of a vibrating system which is a discrete version of an elastic membrane in the shape of an infinite strip, with a density variation across the width. The question then becomes whether the dispersion relations for the waves propagating along this membrane are sufficient to determine the density uniquely. A definitive answer to this conjecture is still not available, but our results strengthen its validity.

Normal Modes of a Rotating, Self-gravitating Inhomogeneous Earth

This present study focuses on three topics: (i) formulation of the first- order perturbation problem of the Earth's normal modes; (ii) formulation of a variational method based on the solutions of first-order quasi- degenerate perturbation calculation; and (iii) numerical results for both ordinary degenerate and quasi-degenerate multiplets using formulation (i). The Earth considered is a rotating, laterally-inhomogeneous Earth. To zeroth order, however, it is approximated by a spherically symmetric, non-rotating, elastic and isotropic sphere. Lateral perturbations and Earth's rotation are treated as perturbations. Examples of lateral perturbations are Earth's ellipticity of figure and lateral inhomogeneities. Because of the symmetry in the Earth model, unperturbed modes form multiplets with varying degrees of degeneracy. For almost all of these multiplets, the perturbation calculation need be carried out to first order. When one multiplet and another nearly coincide in their frequencies, first-order ordinary degenerate perturbation method must be replaced by first-order quasi-degenerate perturbation method. Since the latter is a more general form of the former, the first-order perturbation problem is completely solved. The perturbations considered in this study consist of Earth's rotation, ellipticity of figure, a continent-ocean discontinuity near the Earth's surface, and an artificially contrived discontinuity about the core-mantle interface. When the effect due to rotation becomes dominant, the first-order calculation becomes insufficient, and a variational method is posed. Such a method can be formulated from quasi-degenerate solutions. The numerical results show that ellipticity corrections are important and should always be included in the computations of the Earth's normal modes, that lateral inhomogeneities in the Earth's deep interior become an important perturbation for high-overtone multiplets, and that Coriolis coupling between a spheroidal and a toroidal multiplet overshadows all other types of coupling.

A Discussion on the measurement and interpretation of changes of strain in the Earth - Ultra sensitive laser interferometers and their application to problems of geophysical interest

A 30 m laser strainmeter is currently being operated in an unworked gold mine near Boulder, Colorado. The strainmeter consists of an evacuated Fabry-Perot interferometer illuminated by a 3.39 pm He-Ne laser. A second 3.39 pm laser is stabilized by means of saturated absorption in methane and its wavelength serves as the reference length for the system. We shall describe the instrument in some detail and present the latest results in our investigation of the Earth tides and the Earth normal modes.

A Review of Tidal, Earth Normal Mode and Seismic Data Obtained with Quartz Torsion Accelerometers

Summary A review of broad-band data obtained with quartz torsion accelerometers is presented. Special emphasis is made on the Earth normal mode and seismic frequency regions. Comparison with more conventional instruments is provided where possible.

Improvements in the wide-band vertical quartz torsion accelerometer

Operational improvements have been made in the wide-band quartz vertical accelerometer. Simultaneous digital data at tidal, earth normal-mode, and seismic frequencies have been taken over many months. Data, power spectra, and ambient noise levels from 1 to 30 cph for two earthquakes (mb = 6.0, 6.3) are presented. Teleseismic data are shown for a series of small earthquakes, and near-field data on the San Fernando earthquake are presented.

A Wide Band Horizontal Accelerometer with Preliminary Earth Normal Mode and Seismic Investigations

A wide band horizontal accelerometer using a quartz torsional pivot and capacitative position sensing with phase-sensitive detection is described. Devices for magnetic damping of unwanted modes of vibration and internal stabilization of temperature and pressure are discussed in detail. The accelerometer is small enough in size and weight to allow deployment in bore holes. Data from the earth normal mode band and the seismic band are shown. Although horizontal noise due to surface tilting is high for periods greater than 50 s, noise in the seismic band can be reduced by as much as an order of magnitude by digital high-pass filtering. The resulting horizontal signal-to-noise ratio for teleseismic events is comparable to that for the vertical signals.

Surface Wave Detection with a Broad Band Accelerometer

WE report preliminary results of a study to determine the capability of a broad band vertical accelerometer to detect surface waves from small earthquakes. The instrument uses a quartz fibre in torsion and capacitive position sensing1. It was designed as a low noise, low drift broad band accelerometer and has produced data on Earth normal mode excitation after earthquakes (5 to 40 cycle h−1, ref. 2). For this study, the output was modified by an active filter to copy the response of Pomeroy et al.3 and Molnar et al.4. Such a response does not use the wide band aspects of this instrument and the shape was dictated primarily by the shape of the ground noise spectrum, which has a minimum in amplitude between periods of 20 and 30 s. This minimum may be inferred from the known decrease in noise amplitude with increasing period at shorter periods and seismic background noise data5. It is also shown in the data of Savino et al.6.

Earth Normal Modes from a 6.5 Magnitude Earthquake

THE Earth normal mode spectrum, between 8 and 39 cycles per hour (c.p.h.), has been obtained with a 20 dB signal-to-noise ratio for a relatively small earthquake (Ms = 6.5), using a low noise broad-band vertical accelerometer of new design. The earthquake occurred in the New Hebrides (18.9° S, 169.3° E) on October 13, 1969, 06 : 56 G.M.T. at a depth of 246 km.

The Normal Modes of a Rotating, Elliptical Earth?11 Near-Resonance Multiplet Coupling

Certain of the Earth's poloidal and toroidal elastic-gravitational normal mode multiplets may be strongly coupled to another multiplet by the Earth's rotation and ellipticity of figure. This paper uses Rayleigh's variational principle to specify the selection rules for strong mode coupling and to examine the nature and effects of this mode coupling. It is found that some of the normal modes of a slowly rotating, slightly elliptical Earth model may consist of about half poloidal type motion and half toroidal type motion. This strong coupling can severely hinder the problem of proper mode identification and measurement of the angular frequencies of oscillation of the Earth's normal modes.