Coupled free oscillations of an aspherical, dissipative, rotating Earth: Galerkin theory

Abstract

Variational theory based on self-adjoint equations of motion cannot fully represent the interaction of the earth's seismic free oscillations in the presence of lateral structure, attenuation, and rotation. The more general Galerkin procedure can model correctly the frequencies and attenuation rates of hybrid oscillations. Implementation of either algorithm leads to a generalized matrix eigenvalue problem in which the potential and kinetic energy interactions are separated into distinct matrices. The interaction of the earth's seismic free oscillations due to aspherical structure, attenuation, and rotation is best treated as a matrix eigenvalue problem. The presence of attenuation causes the matrices to be non-Hermitian and requires the use of a general Galerkin procedure. Physical dispersion, represented as a logarithmic function in frequency, must be represented by a truncated Taylor series about a fiducial frequency in order to be incorporated in the Galerkin formalism in a numerically tractable manner. The earth's rotation introduces an interaction matrix distinct from the potential and kinetic energy matrices, leading to a quadratic eigenvalue problem. A simple approximation leads to an eigenvalue problem linear in squared frequency. Tests show that this approximation is accurate for calculations using modes of frequencies f •_. 1 mHz, unless interaction across a wide frequency band is modeled. Hybrid oscillation particle motions are represented by matrix eigenvectors that can be significantly nonorthogonal. The degrees of freedom in the low-frequency seismic system remain distinct, since source excitation is calculated by using dual eigenvectors. Synthetic seismograms that are constructed from Galerkin coupling calculations without reference to this eigenvector nonorthogonality can be disastrously noncausal. Paper number 5B5475. 0148-0227/86/005 B-5475505.00 tiplets due to Jordan [1978], that at bottom, also relies on a tomographic (i.e., great circle) approximation to the equations of motion. Spheroidal overtone splitting observations [Masters and Gilbert, 1981; M. Ritzwoller et al. , unpublished manuscript, 1986] have often appeared to be characterized by simple spherical harmonic surface dependence. Although this behavior is a blessing so far as observing isolated singlets is concerned, it restricts our inferential capability to only axisymmetric models (i.e., spherical harmonic expansions with azimuthal order t= 0).Although a bootstrap-style iterative improvement in lateral structure models using successively refined representations of the full equations of motion is virtually inevitable, it has become both possible and feasible to model complete low-frequency seismograms using sums of coupled free oscillations. The approximations that must be made in such calculations in return for numerical tractability are different in nature from those associated with the high-frequency, geometric ray approximation. A firstorder splitting calculation, in which no coupling is assumed between distinct free oscillation ...

Equipartitioning is not sufficient for Green’s function extraction

“…The normal mode analysis of section 2.2 is not applicable in the presence of attenuation because in that case the normal mode frequencies are complex, and one needs to use adjoint modes in the orthogonality relation (Park and Gilbert, 1986). In the time domain, attenuation can be described by a timedependent compressibility κ that accounts for the relaxation of the acoustic medium (Dahlen and Tromp, 1998).…”

The Earth's free oscillations and the differential rotation of the inner core

“…As can be seem from Figure 8, the rotation angle is largely underestimated using the complete splitting functions, so an accurate mantle correction is indeed essential for that approach. To test the forward approach laid out in this section, we calculate synthetic seismograms for our model using the coupled-mode code of Park and Gilbert (1986). Only self-coupling and 1D attenuation is considered but the Earth's rotation and hydrostatic ellipticity is included in the calculations and the same steps are performed in this test as with real data.…”

Coseismic Slip and Afterslip of the GreatMw 9.15 Sumatra–Andaman Earthquake of 2004

“…To compute the verylong-period normal modes, modal coupling caused by Earth's rotation, ellipticity and heterogeneities of earth structure are taken into account (Park and Gilbert, 1986;Dahlen and Tromp, 1998;Park et al, 2005). Because inversion of geodetic data only provides static information, that is, the final slip distribution, we assume constant rise time, taken to be 20 sec, and constant rupture velocity between 2.2 and Figure 9.…”