3-D linearized scattering of surface waves and a formalism for surface wave holography

Summary. Scattering of surface waves by lateral heterogeneities is analysed in the Born approximation. It is assumed that the background medium is either laterally homogeneous, or smoothly varying in the horizontal direction. A dyadic representation of the Green's function simplifies the theory tremendously. Several examples of the theory are presented. The scattering and mode conversion coefficients are shown for scattering of surface waves by the root of an Alpine-like crustal structure. Furthermore a 'great circle theorem'in a plane geometry is derived. A new proof of Snell's law is given for surface wave scattering by a quarter-space. It is shown how a stationary phase approximation can be used to simplify the Fourier synthesis of the scattered wave in the time domain. Finally a procedure is suggested to do 'surface wave holography'.     

Effects of slight anisotropy on surface waves

We present a complete ray theory for the calculation of surface-wave observables from anisotropic phase-velocity maps. Starting with the surface-wave dispersion relation in an anisotropic earth model, we derive practical dynamical ray-tracing equations. These equations allow calculation of the observables phase, arrival-angle and amplitude in a ray theoretical framework. Using perturbation theory, we also obtain approximate expressions for these observables. We assess the accuracy of the first-order approximations by using both theories to make predictions on a sample anisotropic phase-velocity map. A comparison of the two methods illustrates the size and type of errors which are introduced by perturbation theory. Perturbation theory phase and arrival-angle predictions agree well with the exact calculation, but amplitude predictions are poor. Many previous studies have modelled surface-wave propagation using only isotropic structure, not allowing for anisotropy. We present hypothetical examples to simulate isotropic modelling of surface waves which pass through anisotropic material. Synthetic data sets of phase and arrival angle are produced by ray tracing with exact ray theory on anisotropic phase-velocity maps. The isotropic models obtained by inverting synthetic anisotropic phase data sets produce deceptively high variance reductions because the effects of anisotropy are mapped into short-wavelength isotropic structure. Inversion of synthetic arrival-angle data sets for isotropic models results in poor variance reductions and poor recovery of the isotropic part of the anisotropic input map. Therefore, successful anisotropic phase-velocity inversions of real data require the inclusion of both phase and arrival-angle measurements.     

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.     

Global crustal thickness from neural network inversion of surface wave data

We present a neural network approach to invert surface wave data for a global model of crustal thickness with corresponding uncertainties. We model the a posteriori probability distribution of Moho depth as a mixture of Gaussians and let the various parameters of the mixture model be given by the outputs of a conventional neural network. We show how such a network can be trained on a set of random samples to give a continuous approximation to the inverse relation in a compact and computationally efficient form. The trained networks are applied to real data consisting of fundamental mode Love and Rayleigh phase and group velocity maps. For each inversion, performed on a 2°× 2° grid globally, we obtain the a posteriori probability distribution of Moho depth. From this distribution any desired statistic such as mean and variance can be computed. The obtained results are compared with current knowledge of crustal structure. Generally our results are in good agreement with other crustal models. However in certain regions such as central Africa and the backarc of the Rocky Mountains we observe a thinner crust than the other models propose. We also see evidence for thickening of oceanic crust with increasing age. In applications, characterized by repeated inversion of similar data, the neural network approach proves to be very efficient. In particular, the speed of the individual inversions and the possibility of modelling the whole a posteriori probability distribution of the model parameters make neural networks a promising tool in seismic tomography.