... theoretical eigenfunctions and eigenfre- quencies for laterally heterogeneous models of the Earth are: first-order degenerate perturbation theory (Saito, 1971; Madariaga, 1972; Luh, 1974), quasi-degenerate perturbation theory (Luh, 1973, 1974), and the Rayleigh-Ritz variational technique (Usami, 1971; Johnson...

... with azimuthal orders − l ≤ m ≤ l and degenerate, complex eigenfrequencies. The singlets as defined in (3) form a complete vector space of a dissipative, spherical, nonrotating, and isotropic earth model. With a perturbative rescaling into the (approximately) elliptical coordinates of the real Earth...

... the normal modes of a laterally heterogeneous earth model has been considered by Madariaga (1972), Saito (1971), and Luh (1974) using first- order degenerate perturbation theory. Luh (1973) considered the effect of quasi- degeneracy. Usami (1971) used variational methods to find some very low-order tor...

... of the Christoffel tensor (associated with the quasi S-waves) degenerate into one and polarization directions of the S-waves, which lay in the plane orthogonal to the polarization of the quasi longitudinal wave, are not uniquely defined. In the vicinity of these points, slowness sheets of the S-waves have...

... what mathematicians call “singular or degenerate behavior.” This may mean either that the coefficient becomes equal to zero (the degenerate case) or that the coefficient tends to infinity and so becomes undefined (the singular case). Moreover, the treatment of the saturated–unsaturated interface...

... ′ = − sin α and u 2 ′ = cos α . In the language of degenerate perturbation theory ( Landau and Lifshitz, 1977 ), these results give us the zero-order approximation for the unit polarization vectors corresponding to the faster and slower qS-waves: (25) u S 0 + = cos α...

... isotropy; the latter corresponds to the two crack systems degenerating to one. The zero shear-wave splitting corresponds to a singularity, at which the vertical velocities of the two quasi-shear waves converge to a single value that is both predicted theoretically and illustrated numerically...

..., a different analysis, by Brown and Korringa (1975) (here “B&K”), produced a formula that differed from Gassmann’s formula, containing an additional physical parameter. B&K argued that in the special case considered by Gassmann (homogeneous isotropic solid), the additional parameter degenerated...

... on coronitic structures. Therefore, the modern theory of metamorphic reactions can be named after the three scientists who have contributed so much to the knowledge of metamorphism kinetics ( Sheplev, 1998 ). The Korzhinskii–Fisher–Joesten model stems from the assumption of local thermodynamic equilibrium...

... by the effect of the small H mass. The quasi-harmonic behavior of OH-stretching modes explains the successful harmonic modeling of vibrational properties of hydrous minerals using density functional perturbation theory (e.g., Balan et al. 2001 ; Winkler et al. 2008b ). We thank Thierry Allard (IMPMC...

... spectroscopic features of this mineral phase ( Fleet et al. 2004 ; Khan et al. 2013 ; Yoder et al. 2019 ). Type-AB carbonated apatite thermodynamic properties elastic properties density functional theory quasi-harmonic approximation Since the pioneering work of Albee in the first half...

... ): Phonons and related crystal properties from density-functional perturbation theory . Rev. Modern Phys ., 73 , 515 – 561 . Belousov , M.V. , Pogarev , D.E. , Shultin , A.A. ( 1970 ): Band intensity of extraplanar n 2 vibration of CO 3 2− and NO 3 − ions in crystals having...

..., compared with the Jin (2019) model that neglects viscoelastic theory. It is found that the new model and the Jin (2019) model satisfy the quasi P-wave velocity increases with the increase of water saturation. However, in the Jin (2019) model, the quasi P-wave velocity increases with increasing angle...

... relies on the l-BFGS quasi-Newton method, and we regularize the inversion by smoothing both the gradient and the resulting model on the hexahedral mesh ( Trinh et al., 2017 ). Wavefields and EFWI gradients are computed in fully anisotropic triclinic media to be able to evolve toward more complex...

.... The prototypical model manifests uniformity with a seismic wave velocity of 3500 m/s, along with a velocity perturbation ∼±23% relative to the background velocity. We apply a Ricker wavelet with a dominant frequency of 10 Hz. Thirty‐nine evenly distributed seismic events are arranged 1.95 km deep at the model’s...

... , Tectonophysics 612 , 56 – 62 , doi: 10.1016/j.tecto.2013.11.033 . Kilic B. , and Madenci E. 2010 . An adaptive dynamic relaxation method for quasi‐static simulations using the peridynamic theory , Theor. Appl. Fract. Mech. 53 , no.  3 , 194 – 204 . Lapusta N. , and Liu Y...

... five years ago ( Iversen, 2001b ). In this paper, I provide examples of velocity rays corresponding to heterogeneity of the velocity model. The methodology part of the paper starts with a section reviewing elements of the perturbation theory for conventional rays. The succeeding section addresses...

...) . Free oscillations of the laterally inhomogeneous earth: quasi-degenerate multiplet coupling , Geophys. J. 32 , 187 - 202 . Luh P. C. (1974) . Normal modes of a rotating self-gravitating inhomogeneous earth , Geophys. J. 38...

.... Seismol. Soc. Am. 99 , no.  2B , 1132 - 1136 . Teisseyre R. Białecki M. Górski M. ( 2006 ). Degenerated asymmetric continuum theory , in Earthquake Source Asymmetry, Structural Media and Rotation Effects , Teisseyre R. Takeo M. Majewski E. (Editors), Springer...

..., whereas the mode at 3750 cm −1 corresponds to their in-phase motion. The OH2 and OH4 groups, which are quasi parallel to the c * axis, lead to four vibrational modes. Two modes degenerated at 3607 cm −1 correspond to the intralayer out-of-phase motion of OH2 and OH4. The modes corresponding...