Teleseismic waveform modelling with a one-way wave equation

It is now widely accepted that elastic properties of the continental lithosphere and the underlying sublithospheric mantle are both anisotropic and laterally heterogeneous at a range of scales. To fully exploit modern three-component broad-band array data sets requires the use of comprehensive modelling tools. In this work, we investigate the use of a wide-angle, one-way wave equation to model variations in teleseismic 3-D waveforms due to 2-D elastic heterogeneity and anisotropy. The one-way operators are derived based on a high-frequency approximation of the square-root operator and include the effects of wave propagation as well as multiple scattering. Computational cost is reduced through a number of physically motivated approximations. We present synthetic results from simple 1-D (layer over a half-space) and 2-D (subduction zone) models that are compared with reference solutions. The algorithm is then used to model data from an array of broad-band seismograph stations deployed in northwestern Canada as part of the IRIS-PASSCAL/LITHOPROBE CANOE experiment. In this region radial-component receiver functions show a clear continental Moho and the presence of crustal material dipping into the mantle at the suture of two Palaeo-Proterozoic terranes. The geometry of the suture is better defined on the transverse component where subduction is associated with a ∼10 km thick layer exhibiting strong elastic anisotropy. The modelling reproduces the main features of the receiver functions, including the effects of anisotropy, heterogeneity and finite-frequency scattering.     

Controls on Rayleigh wave amplitudes: attenuation and focusing

A large data set of amplitude measurements of minor and major arc Rayleigh waves in the period range 73–171 s is collected. By comparing these amplitudes with the amplitudes of synthetic waveforms calculated by mode summation, maps of lateral variations in the apparent attenuation structure of the Earth are constructed. An existing formalism for predicting the effects of focusing is employed to calculate amplitude perturbations for the same data set. These perturbations are used to construct 'pseudo‐attenuation' maps and these results are compared with the apparent attenuation maps calculated from the data. It is shown that variations in Rayleigh wave amplitude perturbations in the Earth are dominated by attenuation at long wavelengths (below about degree 8) and by elastic structure at shorter wavelengths. It is also shown that the linear approximation for focusing is successful at predicting Rayleigh wave amplitudes using existing phase velocity maps. These results indicate that future attempts to model the velocity structure of the Earth would be assisted by incorporating amplitude data and by jointly inverting for Q structure.     

Geometrical structure of shear wave surfaces near singularity directions in anisotropic media

There are three types of surfaces which are used for studying wave propagation in anisotropic media: normal surfaces, slowness surfaces and wave surfaces. Normal surfaces and slowness surfaces have been researched in detail. Wave surfaces are the most complicated and comparatively poorly known compared with the other two. Areas of complicated geometrical structure of the wave surfaces are located in the vicinity of conical acoustic axes. There is an elliptical hole on the quick shear wave surface and complicated folds and cusps on the slow shear wave surface. Decomposition of the slow shear wave surface into smooth sheets is used for the study of its geometrical structure. Complexity of shear wave surfaces can be expressed by the number of waves corresponding to a fixed ray. An original approach to the calculation of wave normals depending on ray direction is presented.     

Reciprocity theorems for one-way wavefields

Acoustic reciprocity theorems have proved their usefulness in the study of forward and inverse scattering problems. The reciprocity theorems in the literature apply to the two-way (i.e. total) wavefield, and are thus not compatible with one-way wave theory, which is often applied in seismic exploration. By transforming the two-way wave equation into a coupled system of one-way wave equations for downgoing and upgoing waves it appears to be possible to derive 'one-way reciprocity theorems" along the same lines as the usual derivation of the 'two-way reciprocity theorems'. However, for the one-way reciprocity theorems it is not directly obvious that the 'contrast term' vanishes when the medium parameters in the two different states are identical. By introducing a modal expansion of the Helraholtz operator, its square root can be derived, which appears to have a symmetric kernel. This symmetry property appears to be sufficient to let the contrast term vanish in the above-mentioned situation.
The one-way reciprocity theorem of the convolution type is exact, whereas the one-way reciprocity theorem of the correlation type ignores evanescent wave modes. The extension to the elastodynamic situation is not trivial, but it can be shown relatively easily that similar reciprocity theorems apply if the (non-unique) decomposition of the elastodynamic two-way operator is done in such a way that the elastodynamic one-way operators satisfy similar symmetry properties to the acoustic one-way operators.