Many situations of practical interest involving seismic wave modelling require curved interfaces and free-surface topography to be taken into account. Collocation methods, for instance pseudospectral or finite-difference algorithms, are attractive approaches for modelling wave propagation through these complex realistic models, particularly in view of their ease of implementation. Nonetheless, these methods formulated in Cartesian coordinates are not well suited to such models because the sharp interfaces and free surface do not coincide with grid lines. This leads to a slow convergence rate, resulting in visible artefacts such as diffractions from staircase discretizations of interfaces and the free surface. Such problems can be overcome through the use of curved grids whose lines follow sharp interfaces and whose density increases in the vicinity of these interfaces. One approach is to solve the wave equation in Cartesian coordinates by using the chain rule to express the Cartesian partial derivatives in terms of derivatives computed in the new coordinate system. However, it is more natural to solve the tensorial form of the wave equation directly in the desired curvilinear coordinate system, making use of a transformation of a square grid onto the curved grid. The tensorial approach, which is independent of the coordinate system, requires the same number of derivatives to be computed as in the Cartesian case, whereas the chain rule approach requires 25 per cent more in 2-D and 50 per cent more in 3-D. While the tensorial approach is less computationally expensive than the chain rule method, it requires more memory. Numerical tests validate the tensorial approach by comparing the results with the analytical solution of the tilted Lamb problem. Other numerical experiments demonstrate the ability of the tensorial formulation to model wave propagation in the presence of free-surface topography. Mode conversions between Rayleigh and body waves are observed when bumps on the free surface are encountered.
Recent earthquakes have triggered renewed interest to understand better earthquake site response. Most of the studies comparing various techniques for estimating site response were based on real data (from earthquakes, nuclear blasts, and seismic noise). A theoretical approach, using synthetic data generated with the pseudospectral method, is used to compare four site-response estimation techniques. The limits of applicability of each method were determined by modeling microtremors and incoming SV waves (with different incidence angles) and analyzing the site amplifications. The first two techniques investigated consist of dividing the spectrum of the horizontal motion at a site by that of a reference site using either incident S waves or microtremors. The latter was unable to reveal either the resonant frequencies or peak amplitudes in any cases. The two other techniques are based on the horizontal-to-vertical (H/V) spectral ratio using S waves or microtremors. These techniques were found to reveal at least the fundamental resonant frequency and amplitude (former method only) within 10% error, in the case of simple geology (flat layers). However, the results show that these techniques are unable to take into account 2D effects such as focusing effects and basin-edge effects and yield unreliable or incorrect results in such cases.
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