Excitation of the normal modes of a rotating earth model by an earthquake fault

Summary. The motion excited in a rotating earth model by a kinematically prescribed earthquake fault is solved for in closed form. In addition, expressions for the total energy released and the energy dissipated by bodily friction subsequent to faulting are obtained in terms of the normal-mode excitation amplitudes.     

A normal mode expansion for the forced response of a rotating earth

Summary. It is well known that the forced motion of a conservative physical system can be represented as a sum of the normal modes of that system. Although normal mode expansions have been successfully developed and applied to examine the excitation of a non-rotating earth, these techniques have been less successful when the Earth's rotation is included. We develop here an expansion formalism for the rotating Earth which completely decouples all non-trivial normal modes. We consider the response of the Earth both to applied body forces and surface tractions. We consider forces which are purely sinusoidal in time and forces which have power distribution over a continuous frequency interval.     

Surface waves and free oscillations in a regionalized earth model

Summary. The linearized equation is derived which relates observed long-period seismic waveforms to the aspherical perturbations of a spherically symmetric earth model. This is accomplished by formulating the theory of spectral splitting in the time domain. It is shown to be possible greatly to simplify the resulting equations in a way which makes it apparent that for each modal multiplet the 'scattered' field depends only upon three local functional of earth structure. The effect of regional structural variations may then be quantified in a manner analogous to that assumed in the 'pure path technique', but without making the usual asymptotic approximations. These results are used to investigate the validity of the asymptotic result for the locations of the centroids of spectral peaks in individual recordings, for a regionalized model of the Earth. A technique is suggested for retrieving information about geographical structural variations from low-frequency waveform data.     

A new nutation series for a more realistic model earth

The frequency-dependent correction coefficients with respect to the forced nutations of a rigid earth are computed using the complex scalar gravitational-motion equations for an earth model with an anelastic mantle. Oceanic loads and tidal currents enter the model via outer boundary conditions. The ellipticity of the core-mantle boundary and the dynamical ellipticity are adjusted to observations. This requires the behaviour inside the model earth to be regarded as non-hydrostatic. Some relevant equations for the evaluation of boundary conditions and some terms in the equations of motion are expanded to second order in ellipticity. The computation of the equipotential-surface ellipticity profile is carried to second order as well. These second-order expansions lead to increased accuracy of the results in general. Moreover, one achieves a better reliability for the integration at frequencies close to a resonance. This allows the integration of the equations of motion at any relevant nutation period without the need for a normal-mode expansion. A complete new nutation series for a realistic model earth is presented.     

Implementing a new data model for simulating processes

The paper describes the development of a new methodological approach for simulating geographic processes through the development of a data model that represents a process. This methodology complements existing approaches to dynamic modelling, which focus on the states of the system at each time step, by storing and representing the processes that are implicit in the model. The data model, called nen, focuses existing modelling approaches on representing and storing process information, which provides advantages for querying and analysing processes. The flux simulation framework was created utilizing the nen data model to represent processes. This simulator includes basic classes for developing a domain specific simulation and a set of query tools for inquiring after the results of a simulation. The methodology is prototyped with a watershed runoff simulation.     

Modes to replace transitional asymptotic ray fields in a vertically inhomogeneous earth model

Summary. Asymptotic ray theory (ART) fails in transition regions near critically reflected, bottom glancing or caustic-forming rays in a vertically inhomogeneous layered earth. These deficiencies are repaired here by replacing the transitional ray fields with guided modes plus truncation remainders. Exact ray-mode equivalences and their high-frequency asymptotic approximations are formulated, and their validity and efficiency are verified by numerical comparisons for SH motion in a two-layer earth model comprised of an inhomogeneous sediment above an homogeneous semi-infinite bedrock.     

Dynamic nucleation process of shallow earthquake faulting in a fault zone

Two distinct phases are commonly observed at the initial part of seismograms of large shallow earthquakes: low-frequency and low-amplitude waves following the onset of a P wave ( P ₁) are interrupted by the arrival of the second impulsive phase P₂ enriched with high-frequency components. This observation suggests that a large shallow earthquake involves two qualitatively different stages of rupture at its nucleation.
We propose a theoretical model that can naturally explain the above nucleation behaviour. The model is 2-D and the deformation is assumed to be anti-plane. A key clement in our model is the assumption of a zone in which numbers of pre-existing cracks are densely distributed; this cracked zone is a model for the fault zone. Dynamic crack growth nucleated in such a zone is intensely affected by the crack interactions, which exert two conflicting effects: one tends to accelerate the crack growth, and the other tends to decelerate it. The accelerating and decelerating effects are generally ascribable to coplanar and non-coplanar crack interactions, respectively. We rigorously treat the multiple interactions among the cracks, using the boundary integral equation method (BIEM), and assume the critical stress fracture criterion for the analysis of spontaneous crack propagation.
Our analysis shows that a dynamic rupture nucleated in the cracked zone begins to grow slowly due to the relative predominance of non-coplanar interactions. This process radiates the P₁ phase. If the crack continues to grow, coalescence with adjacent coplanar cracks occurs after a short time. Then, coplanar interactions suddenly begin to prevail and crack growth is accelerated; the P₂ phase is emitted in this process. It is interpreted that the two distinct phases appear in the process of the transition from non-coplanar to coplanar interaction predominance.     

A solution of the elastodynamic equation in an anelastic earth model

A formal solution to the elastodynamic equation in an anelastic earth model is presented. The derivation also incorporates the effects of aspherical structure, rotation, self-gravitation, and pre-stress. It is found that the solution can be expressed as a sum of the normal modes of the earth model along with additional terms accounting for anelastic relaxation processes. However, the derivation does not assume that such an eigenfunction expansion is possible, and so avoids difficulties previously encountered due to the non self-adjointness of the problem.