Amplification and Increased Duration of Earthquake Motion on Uneven Stress-free Ground

When a flat stress-free surface (i.e., the ground in seismological applications) separating air from a isotropic, homogeneous or horizontally-layered, solid substratum is solicited by a SH plane body wave incident in the substratum, the response in the substratum is a single specularly-reflected body wave. When the stress-free condition, equivalent to vanishing impedance, is relaxed by the introduction of a spatially-modulated, non-vanishing impedance, the response turns out to take the form of a spectrum of plane body waves and surface waves. It is shown that, in a great variety of situations, resonances are produced at the frequencies of which one or several surface wave amplitudes can become large. Furthermore, at resonance, the amplitude of the motion on the surface is shown to be amplified with respect to the situation in which the surface impedance vanishes. A subsidiary, but all-important, effect of this resonant response is that, when the solicitation is pulse-like, the peak value of the time history of surface motion is larger, and the duration of the signal is considerably longer, for a spatially-modulated impedance surface than for a constant, or vanishing, impedance surface. 1 General introduction An important question in seismology, civil engineering, urban planning, and natural disaster risk assessment is: to what extent does surface topography of different length and height scales (ranging from those of mountains and hills to city blocks and buildings) modify the seismic response on the ground? There exists experimental evidence (Singh and Ordaz, 1993; Davis and West, 1973; Griffiths and Bollinger, 1979) that this modification is real and can attain considerable proportions as regards increases in peak ground motion and signal duration. Some theoretical studies (Wirgin, 1989; Wirgin 1990; Wirgin and Kouoh-Bille, 1993; Groby, 2005) seem to indicate that such effects are indeed possible, but various numerical studies ( Bouchon, 1973; Bard, 1982; Sanchez-Sesma, 1987, Geli et al., 1988; Wirgin and Bard, 1996; Guéguen, 2000; Clouteau and Aubry, 2001; Guéguen et al., 2002; Semblat et al., 2003; Tsogka and Wirgin, 2003; Boutin and Roussillon, 2004; Kham, 2004; Groby and Tsogka, 2005) yield conflicting results in that some of these point to amplification, while others to very weak effects, or even to de-amplification. Contradictory results are also obtained regarding the duration of the earthquakes. The present contribution is devoted to establishing, in a hopefully-decisive manner, whether the aggravation of earthquake effects can or cannot be induced by uneven topography and/or the presence of buildings on the ground. Furthermore, if such aggravated seismic phenomena are produced, we want to know whether they are rare or of systematic nature, and what their underlying causes may be. We provide herein a theoretical and numerical analysis which supports the conclusion that such substantial deleterious effects can, and will, indeed occur repeatedly if the irregularity of the ground is spatially-periodic, as often occurs in portions of modern cities and in various geological formations. 2 Space-time and space-frequency formulations In the following, we shall be concerned with the determination of the vectorial displacement field u on, and underneath, the ground in response to a seismic solicitation. In general, u is a function of the spatial coordinates, incarnated in the vector x and time t, so that u = u(x, t). Since we shall employ the concept of surface impedance (Biot, 1968; Wait, 1971; Guéguen, 2000), and since the latter is defined, stricto sensu, only in the space-frequency domain, we shall carry out our analysis therein, and thus search for u(x, ω), with ω the angular frequency. The time history of response u(x, t) will then be computed via the Fourier transform u(x, t) = ∫ ∞ −∞ u(x, ω) exp(−iωt)dω , (1) wherein u(x, ω) is a generally-complex function, and u(x, t) a real function. 3 Reflection of a SH plane wave from a planar spatially-modulated impedance boundary 3.1 Features of the problem The uneven ground is replaced by the flat, horizontal, planar boundary I {x2 = 0 ; ∀x1 ∈ R ; ∀x3 ∈ R} which separates the lower half-infinite region x2 < 0 from the upper half-infinite region x2 > 0. The medium filling x2 > 0 is air, assumed for the purpose of the analysis, to be the vacumn. The uneveness of the ground is accounted for by a suitably-chosen surface impedance function. The incident plane body seismic wave propagates in x2 < 0 towards I. Attention will be restricted to the displacement field exclusively in x2 ≤ 0 (in fact, this is the purpose of employing the concept of surface impedance). We choose the cartesian coordinate system so that the wavevector associated with the incident shear wave lies in the x1 − x2 plane. This signifies that the displacement associated with this wave is perpendicular to the x1 − x2 plane and therefore lies in a horizontal plane. Thus, the incident wave is a shear-horizontal (SH) wave. Moreover, the motion associated with this wave is, due to the choice of the cartesian reference system, independent of the coordinate x3. This implies that the resultant total motion induced by this incident wave is independent of x3, i.e., the boundary value problem is 2D, so that it is sufficient to look for the displacement field in the x1 − x2 plane. Actually, due to previous comments, we look for the total displacement field (hereafter designated by u(x, ω) := (0, 0, u(x, ω))) only in the lower half of the x1 − x2 plane, i.e., in Ω. The trace of the boundary I in the x1 − x2 plane is designated by Γ . Hereafter, we designate the (real) density and (real) Lamé parameters in Ω by ρ > 0 and λ ≥ 0, μ ≥ 0 respectively.