First-order P-wave ray synthetic seismograms in inhomogeneous weakly anisotropic media

We propose approximate equations for P -wave ray theory Green's function for smooth inhomogeneous weakly anisotropic media. Equations are based on perturbation theory, in which deviations of anisotropy from isotropy are considered to be the first-order quantities. For evaluation of the approximate Green's function, earlier derived first-order ray tracing equations and in this paper derived first-order dynamic ray tracing equations are used.
The first-order ray theory P -wave Green's function for inhomogeneous, weakly anisotropic media of arbitrary symmetry depends, at most, on 15 weak-anisotropy parameters. For anisotropic media of higher-symmetry than monoclinic, all equations involved differ only slightly from the corresponding equations for isotropic media. For vanishing anisotropy, the equations reduce to equations for computation of standard ray theory Green's function for isotropic media. These properties make the proposed approximate Green's function an easy and natural substitute of traditional Green's function for isotropic media.
Numerical tests for configuration and models used in seismic prospecting indicate negligible dependence of accuracy of the approximate Green's function on inhomogeneity of the medium. Accuracy depends more strongly on strength of anisotropy in general and on angular variation of phase velocity due to anisotropy in particular. For example, for anisotropy of about 8 per cent, considered in the examples presented, the relative errors of the geometrical spreading are usually under 1 per cent; for anisotropy of about 20 per cent, however, they may locally reach as much as 20 per cent.     

Pseudo-bending method for three-dimensional seismic ray tracing in a spherical earth with discontinuities

We present a 'pseudo-bending' approach to 3-D ray tracing in a spherical earth with discontinuities. This method is based on a three-point perturbation associated with a first-order approximation, while Snell's law in curvilinear coordinates is applied at the discontinuities. We demonstrate the computational accuracy and efficiency of the pseudo-bending method in tracing rays for various velocity models by comparing results with analytical solutions and with results from the bending method. The improvement of efficiency is significant, but is reduced as the number of discontinuities increases. Since the bending approach may be computationally unstable in some situations, even though it is exact, the pseudo-bending approach is preferable for automatic calculation of rays.     

Paraxial ray approximations in the computation of seismic wavefields in inhomogeneous media

Summary. Several important applications of the paraxial ray approximation (PRA) to numerical modelling of high-frequency seismic body wavefields are discussed. The PRA can be used to evaluate the displacement vector not only directly on the ray, as in the standard ray method. but also approximately in the vicinity of this ray. The PRA also offers simple ways of approximate evaluation of paraxial rays, situated in the vicinity of the central ray, and of two-point ray tracing. A very important application of the PRA consists in a simple, fast and effective Computation of body-wave synthetic seismograms in general, 3-D, laterally inhomogeneous, layered structures. Examples of synthetic seismograms for 3-D structures, computed using the PRA, are presented.     

Ray tracing and inhomogeneous dynamic ray tracing for anisotropy specified in curvilinear coordinates

Ray tracing has recently been expressed for anisotropy specified in a local Cartesian coordinate system, which may vary continuously in a model specified by elastic parameters. It takes advantage of the fact that anisotropy is often of a simpler nature locally (and is thus specified by a smaller number of elastic parameters) and that the orientation of its symmetry elements may vary. Here we extend this approach by replacing the local Cartesian coordinate system with a curvilinear coordinate system of global extent and by applying the new approach to ray tracing and inhomogeneous dynamic ray tracing. The curvilinear coordinate system is orthogonal and is constructed so that the coordinate axes are consistent with the considered anisotropy of the medium. Our formulation allows for computation of ray attributes (e.g. ray velocity vector and paraxial ray attributes) in the curvilinear coordinate system, while rays are computed in global Cartesian coordinates. Compared to the classic formulation in terms of 21 elastic moduli in global Cartesian coordinates, the main advantages are improved efficiency, lower computer-memory requirements, and conservation of anisotropic symmetry throughout the model.     

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.     

Computation of wave fields in inhomogeneous media — Gaussian beam approach

Summary. An asymptotic procedure for the computation of wave fields in two-dimensional laterally inhomogeneous media is proposed. It is based on the simulation of the wave field by a system of Gaussian beams. Each beam is continued independently through an arbitrary inhomogeneous structure. The complete wave field at a receiver is then obtained as an integral superposition of all Gaussian beams arriving in some neighbourhood of the receiver. The corresponding integral formula is valid even in various singular regions where the ray method fails (the vicinity of caustic, critical point, etc.). Numerical examples are given.     

Energy release in faulting processes in a pre-stressed self-gravitating rotating earth model

Summary. A general expression for the amount of energy released by faulting processes in a rotating, self-gravitating and pre-stressed earth model is given. Previous results in the same subject are obtained as a special case, namely when the configurations before and after the occurrence of the earthquake are assumed static.     

Displacement fields and finite strains in a sandbox model simulating a fold-thrust-belt

A sandbox model consisting of two adjacent mechanically different decollements (frictional and viscous) loosely simulated the southeastern part of the Zagros fold-thrust-belt. Digital images of the model surface are used to coordinate passive markers on the surface and quantify displacement fields and estimate 2-D finite strains. These analyses show that, mapped in a fixed coordinate system, the deformation front propagates at different rates above the two decollements. Strain analysis of the model surface at different stages of deformation also shows that cumulative strain is more heterogeneous above the viscous decollement where strain domains are separated by fault zones. Maps of displacement fields, finite strain ellipses and dilatation also differ in character above the two decollements. Displacements above a viscous decollement decrease gradually towards the foreland, whereas they decrease sharply in front of the frontal thrust above the frictional decollement. Our analyses also show that the estimated finite strain depends not only on the density of the marker points chosen for the analysis, but also their initial distribution relative to the structures. This comparison shows that marker density limits measuring the actual strains in a heterogeneously deforming fold-thrust-belt and marker density and distribution have a strong impact on the strain analysed in nature. The similarity of our model with nature is examined with recent GPS study in the Zagros fold-thrust-belt (SW Iran) and shows, similar to the model results, that a weak salt decollement causes divergent movement in the sedimentary cover in SE Zagros.     

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.