Existence of transverse waves in anisotropic poroelastic media

Out of the four waves in an anisotropic poroelastic medium, two are termed as quasi-transverse waves. The prefix 'quasi' refers to their polarizations being nearly, but not exactly, perpendicular to direction of propagation. In this composite medium, unlike perfectly elastic medium, the propagation of a longitudinal wave along a phase direction may not be accompanied by transverse waves. The existence of a transverse wave in anisotropic poroelastic media is ensured by the two equations restricting the choice of elastic coefficients of porous aggregate as well as fluid–solid coupling. Necessary and sufficient conditions for the existence of transverse waves along the coordinate axes and in the coordinate planes for general anisotropy are discussed. The discussion is extended to the case of orthotropic materials and existence for few specific phase directions is also explored. The conditions for the transverse waves decided on the basis of their apparent polarizations, that is, particle motion being perpendicular to ray direction, are also discussed. For a particular numerical model, the existence of these apparent transverse waves is solved numerically for phase directions in coordinate planes. For general directions of phase propagation, the existence of these transverse waves is checked graphically for the chosen numerical model.     

Propagation of harmonic plane waves in a general anisotropic porous solid

Wave propagation is studied in a general anisotropic poroelastic solid. The presence of dissipation due to fluid-viscosity as well as hydraulic anisotropy of pore permeability are also considered. Biot's theory is used to derive a system of modified Christoffel equations for the propagation of plane harmonic waves in porous media. A non-trivial solution of this system is ensured by a determinantal equation. This equation is separated into two different polynomial equations. One is the quartic equation whose roots represent the complex velocities of four attenuating waves in the medium. The other is a eighth-degree polynomial whose roots represent the vertical slowness values for the four waves propagating upward and downward in a finite porous medium. Procedure is explained to associate the numerically obtained roots with the waves propagating in the medium. The slowness surfaces of waves reflected at the boundary of the medium are computed for a realistic numerical model. The behaviours of phase velocity surfaces are analysed with the help of numerical examples.     

Shear wave anisotropy in the crust around the San Andreas fault near Parkfield: spatial and temporal analysis

We systematically analysed shear wave splitting (SWS) for seismic data observed at a temporary array and two permanent networks around the San Andreas Fault (SAF) Observatory at Depth. The purpose was to investigate the spatial distribution of crustal shear wave anisotropy around the SAF in this segment and its temporal behaviour in relation to the occurrence of the 2004 Parkfield M 6.0 earthquake. The dense coverage of the networks, the accurate locations of earthquakes and the high-resolution velocity model provide a unique opportunity to investigate anisotropy in detail around the SAF zone. The results show that the primary fast polarization directions (PDs) in the region including the SAF zone and the northeast side of the fault are NW–SE, nearly parallel or subparallel to the SAF strike. Some measurements on the southwest side of the fault are oriented to the NNE–SSW direction, approximately parallel to the direction of local maximum horizontal compressive stress. There are also a few areas in which the observed fast PDs do not fit into this general pattern. The strong spatial variations in both the measured fast PDs and time delays reveal the extreme complexity of shear wave anisotropy in the area. The top 2–3 km of the crust appears to contribute the most to the observed time delays; however substantial anisotropy could extend to as deep as 7–8 km in the region. The average time delay in the region is about 0.06 s. We also analysed temporal patterns of SWS parameters in a nearly 4-yr period around the 2004 Parkfield main shock based on similar events. The results show that there are no appreciable precursory, coseismic, or post-seismic temporal changes of SWS in a region near the rupture of an M 6.0 earthquake, about 15 km away from its epicentre.     

Surface wave tomography for azimuthal anisotropy in a strongly reduced parameter space

Large scale seismic anisotropy in the Earth's mantle is likely dynamically supported by the mantle's deformation; therefore, tomographic imaging of 3-D anisotropic mantle seismic velocity structure is an important tool to understand the dynamics of the mantle. While many previous studies have focused on special cases of symmetry of the elastic properties, it would be desirable for evaluation of dynamic models to allow more general axis orientation. In this study, we derive 3-D finite-frequency surface wave sensitivity kernels based on the Born approximation using a general expression for a hexagonal medium with an arbitrarily oriented symmetry axis. This results in kernels for two isotropic elastic coefficients, three coefficients that define the strength of anisotropy, and two angles that define the symmetry axis. The particular parametrization is chosen to allow for a physically meaningful method for reducing the number of parameters considered in an inversion, while allowing for straightforward integration with existing approaches for modelling body wave splitting intensity measurements. Example kernels calculated with this method reveal physical interpretations of how surface waveforms are affected by 3-D velocity perturbations, while also demonstrating the non-linearity of the problem as a function of symmetry axis orientation. The expressions are numerically validated using the spectral element method. While challenges remain in determining the best inversion scheme to appropriately handle the non-linearity, the approach derived here has great promise in allowing large scale models with resolution of both the strength and orientation of anisotropy.     

Simulating three-dimensional seismograms in 2.5-dimensional structures by combining two-dimensional finite difference modelling and ray tracing

Finite difference (FD) simulation of elastic wave propagation is an important tool in geophysical research. As large-scale 3-D simulations are only feasible on supercomputers or clusters, and even then the simulations are limited to long periods compared to the model size, 2-D FD simulations are widespread. Whereas in generally 3-D heterogeneous structures it is not possible to infer the correct amplitude and waveform from 2-D simulations, in 2.5-D heterogeneous structures some inferences are possible. In particular, Vidale & Helmberger developed an approach that simulates 3-D waveforms using 2-D FD experiments only. However, their method requires a special FD source implementation technique that is based on a source definition which is not any longer used in nowadays FD codes. In this paper, we derive a conversion between 2-D and 3-D Green tensors that allows us to simulate 3-D displacement seismograms using 2-D FD simulations and the actual ray path determined in the geometrical optic limit. We give the conversion for a source of a certain seismic moment that is implemented by incrementing the components of the stress tensor.
Therefore, we present a hybrid modelling procedure involving 2-D FD and kinematic ray-tracing techniques. The applicability is demonstrated by numerical experiments of elastic wave propagation for models of different complexity.     

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.