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.     

Phase–velocity dispersion curves and small-scale geophysics using noise correlation slantstack technique

It has been demonstrated both theoretically and experimentally that the Green's function between two receivers can be retrieved from the cross-correlation of isotropic noise records. Since surface waves dominate noise records in geophysics, tomographic inversion using noise correlation techniques have been performed from Rayleigh waves so far. However, very few numerical studies implying surface waves have been conducted to confirm the extraction of the true dispersion curves from noise correlation in a complicated soil structure. In this paper, synthetic noise has been generated in a small-scale (<1 km) numerical realistic environment and classical processing techniques are applied to retrieve the phase velocity dispersion curves, first step toward an inversion. We compare results obtained from spatial autocorrelation method (SPAC), high-resolution frequency-wavenumber method (HRFK) and noise correlation slantstack techniques on a 10-sensor array. Two cases are presented in the (1–20 Hz) frequency band that corresponds to an isotropic or a directional noise wavefield. Results show that noise correlation slantstack provides very accurate phase velocity estimates of Rayleigh waves within a wider frequency band than classical techniques and is also suitable for accurately retrieving Love waves dispersion curves.     

An alternative method for calculation of Rayleigh and Love wave phase velocities by using three-component records on a single circular array without a central station

A method for the computation of phase velocities of surface waves from microtremor waveforms is shown. The technique starts from simultaneous three-component records obtained in a circular array without a central station. Then, Fourier spectra of vertical, radial and tangential components of motion are calculated for each station and considered as complex-valuated functions of the azimuthal coordinate. A couple of intermediate real physical quantities, B and C , can be defined from the 0- and ±1-order coefficients of the Fourier series expansion of such functions. Finally, phase velocities of Rayleigh and Love waves can be retrieved from B and C by solving respective one-unknown equations. The basic assumption is the possibility of expanding the wavefield as a sum of plane surface waves with Rayleigh and Love wavenumbers being univocal functions of the circular frequency. The method is tested in synthetic ambient noise wavefields confirming its reliability and robustness for passive seismic surveying.     

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.     

Cross-dependence of finite-frequency compressional waveforms to shear seismic wave speeds

Seismic tomography has been one of the primary tools to image the interior of the earth and other elastic structures. To date the inversions of compressional ( P ) and shear ( S ) wave speeds have been carried out separately under the assumption that P traveltimes are affected only by the P wave speed of the elastic media and S traveltimes by the S wave speed. Using numerical and analytical solutions, we show that for finite-frequency seismic waves, S wave speed perturbations may have significant effects on P waveforms. This suggests that when waveform-derived traveltime and amplitude anomalies are used in tomographic inversions, the P -wave measurements should be related to not only P wave speed perturbations but also S wave speed perturbations.     

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.     

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.     

An analytical approach for the scattering of SH waves by a symmetrical V-shaped canyon: shallow case

Based on the application of the region-matching technique, an analytical approach is presented for the scattering of plane SH waves from a shallow symmetrical V-shaped canyon, and then a series solution is derived. The analysed region is divided into an enclosed and an open region by introducing a semi-circular auxiliary boundary. In each region, the displacement field can be expressed as infinite sum of appropriate wavefunctions satisfying partial boundary conditions, respectively. The unknown coefficients can be determined by enforcing the continuity conditions in connection with the Graf's addition formula. The frequency- and time-domain responses are both evaluated and displayed for several physical parameters. From graphical results, the effects of the canyon depth on surface ground motion are conspicuous. The proposed series solutions can serve as benchmark for numerical methods, in particular for those at much higher frequencies.     

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.     

A 2-D spectral-element method for computing spherical-earth seismograms—II. Waves in solid–fluid media

We portray a dedicated spectral-element method to solve the elastodynamic wave equation upon spherically symmetric earth models at the expense of a 2-D domain. Using this method, 3-D wavefields of arbitrary resolution may be computed to obtain Fréchet sensitivity kernels, especially for diffracted arrivals. The meshing process is presented for varying frequencies in terms of its efficiency as measured by the total number of elements, their spacing variations and stability criteria. We assess the mesh quantitatively by defining these numerical parameters in a general non-dimensionalized form such that comparisons to other grid-based methods are straightforward. Efficient-mesh generation for the PREM example and a minimum-messaging domain decomposition and parallelization strategy lay foundations for waveforms up to frequencies of 1 Hz on moderate PC clusters. The discretization of fluid, solid and respective boundary regions is similar to previous spectral-element implementations, save for a fluid potential formulation that incorporates the density, thereby yielding identical boundary terms on fluid and solid sides. We compare the second-order Newmark time extrapolation scheme with a newly implemented fourth-order symplectic scheme and argue in favour of the latter in cases of propagation over many wavelengths due to drastic accuracy improvements. Various validation examples such as full moment-tensor seismograms, wavefield snapshots, and energy conservation illustrate the favourable behaviour and potential of the method.