Effect of three‐dimensional topography on seismic motion

Abstract

We present a semianalytical, seminumerical method to calculate the diffraction of elastic waves by an irregular topography of arbitrary shape. The method is a straightforward extension to three dimensions of the approach originally developed to study the diffraction of S.H waves [Bouchon, 1985] and P-SV waves [Gaffer and Bouchon, 1989] by two-dimensional topographies. It relies on a boundary integral equation scheme formulated in •he frequency domain where the Green functions are evaluated by the discrete wavenumber method. The principle of the method is simple. The diffracted wave field is represented as the integral over the topographic surface of an unknown source density function times the medium Green functions. The Green functions are expressed as integrals over the horizontal wavenumbers. The introduction of a spatial periodicity of the t0pogr,aphy combined with the discretization of the surface at equal intervals results in a discretization of the wavenumber integrals and in a periodicity in the horizontal wavenumber space. As a result, the Green functions are expressed as finite sums of analytical terms. The writing of the boundary conditions of free stress at the surface yields a linear system of equations where the unknowns are the source density functions representing the diffracted wave field. Finally, this system is solved iteratively using the conjugate gradient approach. We use this method to investigate the effect of a hill on the ground motion produced during an earthquake. The hill considered is 120 rn high and has an elliptical base and ratios of height-to-half-width of 0.2 and 0.4 along its major and minor axes. The results obtained show that amplification occurs at and near the top of the hill over a broad range of frequencies. For incident shear waves polarized along the short dimension of the hill the amplification at the top reaches 100% around 10 Hz and stays above 50% for frequencies between 1.5 Hz and 20 Hz. For incident shear waves polarized along the direction of elongation of the topography, the maximum amplification Occurs between 2 Hz and 5 Hz with values ranging from 50% to 75%. The results also' show that the geometry of the topography exe..rts a very strong directivity on the wave field diffracted away from the hill and thai' at some distance from the hill this diffracted wave field consists mostly of Rayleigh waves.

Parametric Analysis of Horizontal Acceleration Response of Rock Slope to Seismic Waves in a Shaking Table Test

“…Frequency-dependent amplification was observed at the crest of a ridge and a valley (Bouchon 1973, Rogers et al 1974, Wong 1982. A systematic review on topographic effects was made by Geli et al (1988), who compared theoretical and experimental results, and found qualitative agreement about the amplification at mountain tops.…”

“…Fourier spectral ratios of accelerations are widely used to obtain frequency characteristics dominating the seismic responses (Murphy et al 1971, Bouchon, 1973, Pedersen et al 1994, Fiore 2010. A dimensionless frequency, Df/V = D/λ, is additionally calculated to present the relation between frequency content (f) of incident wave and dimension (D) of topography.…”

Frequency‐Scaled Curvature as a Proxy for Topographic Site‐Effect Amplification and Ground‐Motion Variability

“…The topographic site effect is known to produce an amplification of the seismic waves that is strongly source dependent (Trifunac, 1972;Bouchon, 1973;Wong et al, 1977). This can be seen in the work by Maufroy et al (2012) for our database.…”

Ground-Motion Observations at Hotel Montana during the M 7.0 2010 Haiti Earthquake: Topography or Soil Amplification?

“…The problem of scattering and diffraction of seismic waves by topographic irregularities has been studied by many researchers (e.g., Boore, 1972;Bouchon, 1973;Smith, 1975;Bard, 1982;Sanchez-Sesma et al, 1982;Tucker et al, 1984) who studied topographic effects using numerical techniques such as finite differences, finite elements, boundary elements, and discrete-wavenumber methods. A limited number of examples, which involve more complex simulations such as numerical models with soil layering and/or 3D effects, can be found in Zhenpeng et al (1980), Sanchez-Sesma (1983), Bard and Tucker (1985), Geli et al (1988), , Ashford et al (1997), and Paolucci et al (1999).…”