Scattering of surface waves modelled by the integral equation method

The integral equation method is used to model the propagation of surface waves in 3-D structures. The wavefield is represented by the Fredholm integral equation, and the scattered surface waves are calculated by solving the integral equation numerically. The integration of the Green's function elements is given analytically by treating the singularity of the Hankel function at   R = 0  , based on the proper expression of the Green's function and the addition theorem of the Hankel function. No far-field and Born approximation is made. We investigate the scattering of surface waves propagating in layered reference models imbedding a heterogeneity with different density, as well as Lamé constant contrasts, both in frequency and time domains, for incident plane waves and point sources.     

Resolution potential of surface wave phase velocity measurements at small arrays

The deployment of temporary arrays of broadband seismological stations over dedicated targets is common practice. Measurement of surface wave phase velocity across a small array and its depth-inversion gives us information about the structure below the array which is complementary to the information obtained from body-wave analysis. The question is however: what do we actually measure when the array is much smaller than the wave length, and how does the measured phase velocity relates to the real structure below the array? We quantify this relationship by performing a series of numerical simulations of surface wave propagation in 3-D structures and by measuring the apparent phase velocity across the array on the synthetics. A principal conclusion is that heterogeneities located outside the array can map in a complex way onto the phase velocities measured by the array. In order to minimize this effect, it is necessary to have a large number of events and to average measurements from events well-distributed in backazimuth. A second observation is that the period of the wave has a remarkably small influence on the lateral resolution of the measurement, which is dominantly controlled by the size of the array. We analyse if the artefacts created by heterogeneities can be mistaken for azimuthal variations caused by anisotropy. We also show that if the amplitude of the surface waves can be measured precisely enough, phase velocities can be corrected and the artefacts which occur due to reflections and diffractions in 3-D structures greatly reduced.     

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.     

On crustal corrections in surface wave tomography

Mantle models from surface waves rely on good crustal corrections. We investigated how far ray theoretical and finite frequency approximations can predict crustal corrections for fundamental mode surface waves. Using a spectral element method, we calculated synthetic seismograms in transversely isotropic PREM and in the 3-D crustal model Crust2.0 on top of PREM, and measured the corresponding time-shifts as a function of period. We then applied phase corrections to the PREM seismograms using ray theory and finite frequency theory with exact local phase velocity perturbations from Crust2.0 and looked at the residual time-shifts. After crustal corrections, residuals fall within the uncertainty of measured phase velocities for periods longer than 60 and 80 s for Rayleigh and Love waves, respectively. Rayleigh and Love waves are affected in a highly non-linear way by the crustal type. Oceanic crust affects Love waves stronger, while Rayleigh waves change most in continental crust. As a consequence, we find that the imperfect crustal corrections could have a large impact on our inferences of radial anisotropy. If we want to map anisotropy correctly, we should invert simultaneously for mantle and crust. The latter can only be achieved by using perturbation theory from a good 3-D starting model, or implementing full non-linearity from a 1-D starting model.     

Shallow layer correction for Spectral Element like methods

Today's numerical methods like the Spectral Element Method (SEM) allow accurate simulation of the whole seismic field in complex 3-D geological media. However, the accuracy of such a method requires physical discontinuities to be matched by mesh interfaces. In many realistic earth models, the design of such a mesh is difficult and quite ineffective in terms of numerical cost. In this paper, we address a limited aspect of this problem: an earth model with a thin shallow layer below the free surface in which the elastic and density properties are different from the rest of the medium and in which rapid vertical variations are allowed. We only consider here smooth lateral variations of the thickness and elastic properties of the shallow layer. In the limit of a shallow layer thickness very small compared to the smallest wavelength of the wavefield, by resorting to a second order matching asymptotic approximation, the thin layer can be replaced by a vertically smooth effective medium without discontinuities together with a specific Dirichlet to Neumann (DtN) surface boundary condition. Such a formulation allows to accurately take into account complex thin shallow structures within the SEM without the classical mesh design and time step constraints. Corrections at receivers and source—when the source is located within the thin shallow layer—have been also derived. Accuracy and efficiency of this formulation are assessed on academic tests. The stability and limitations of this formulation are also discussed.     

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.     

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.     

Seismic evidence for a sharp lithospheric base persisting to the lowermost mantle beneath the Caribbean

Broad-band data from South American earthquakes recorded by Californian seismic networks are analysed using a newly developed seismic wave migration method—the slowness backazimuth weighted migration (SBWM). Using the SBWM, out-of-plane seismic P -wave reflections have been observed. The reflection locations extend throughout the Earth's lower mantle, down to the core–mantle boundary (CMB) and coincide with the edges of tomographically mapped high seismic velocities. Modelling using synthetic seismograms suggests that a narrow (10–15 km) low- or high-velocity lamella with about 2 per cent velocity contrast can reproduce the observed reflected waveforms, but other explanations may exist. Considering the reflection locations and synthetic modelling, the observed out-of-plane energy is well explained by underside reflections off a sharp reflector at the base of the subducted lithosphere. We also detect weaker reflections corresponding to the tomographically mapped top of the slab, which may arise from the boundary between the Nazca plate and the overlying former basaltic oceanic crust. The joint interpretation of the waveform modelling and geodynamic considerations indicate mass flux of the former oceanic lithosphere and basaltic crust across the 660 km discontinuity, linking processes and structure at the top and bottom of the Earth's mantle, supporting the idea of whole mantle convection.     

Formulation of the multiple anisotropic scattering process in two dimensions for anisotropic source radiation

The radiative transfer theory (RTT) describes the energy transport through a random heterogeneous medium, neglecting phase information. It provides an adequate framework for modelling high-frequency seismogram envelopes. For isotropic scattering and sources, the radiative transfer equation (RTE) has been formulated analytically and numerically simulated using Monte Carlo methods for acoustic and elastic media. Here, we derive an exact analytical solution of the RTE in 2-D space for the acoustic case, including anisotropic scattering for a anisotropic point-like impulsive source. For this purpose, we generalize the path integral method, which has been used before in the isotropic case, to take into account the anisotropy of both the source radiation pattern and scattering processes, simultaneously. Then we obtain a general solution, which is written in a closed form in the Fourier space. To illustrate the theoretical results, we compute the full space and time evolution of the specific intensity for an arbitrary case. We also compare the time traces computed from our general solution with cases in which the source and/or the scattering process are isotropic. The importance of taking into account both anisotropies simultaneously becomes obvious in our examples. We also show that at long lapse time, our example approaches the solution of the diffusion equation.     

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.