I present a finite-difference method for modeling P-SV wave propagation in heterogeneous media This is an extension of the method I previously proposed for modeling SH-wave propagation by using velocity and stress in a discrete grid The two components of the velocity cannot be defined at the same node for a complete staggered grid: the stability condition and the P-wave phase velocity dispersion curve do not depend on the Poisson's ratio, while the S-wave phase velocity dispersion curve behavior is rather insensitive to the Poisson's ratio Therefore, the same code used for elastic media can be used for liquid media, where S-wave velocity goes to zero, and no special treatment is needed for a liquid-solid interface Typical physical phenomena arising with P-SV modeling, such as surface waves, are in agreement with analytical results The weathered-layer and corner-edge models show in seismograms the same converted phases obtained by previous authors This method gives stable results for step discontinuities, as shown for a liquid layer above an elastic half-space The head wave preserves the correct amplitude Finally, the corner-edge model illustrates a more complex geometry for the liquid-solid interface As the Poisson's ratio v increases from 025 to 05, the shear converted phases are removed from seismograms and from the time section of the wave field

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P-SV wave propagation in heterogeneous media: Velocity‐stress finite‐difference method

Dynamics of structures

Statistics and Data Analysis in Geology

We present the spectral element method to simulate elastic-wave propagation in realistic geological structures involving complieated free-surface topography and material interfaces for two- and three-dimensional geometries. The spectral element method introduced here is a high-order variational method for the spatial approximation of elastic-wave equations. The mass matrix is diagonal by construction in this method, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy-momentum conserving scheme that can be put into a classical explicit-implicit predictor/multi-corrector format. Long-term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The associated Courant condition behaves as Δ tC < O ( nel−1/nd N −2), with nel the number of elements, nd the spatial dimension, and N the polynomial order. In practice, a spatial sampling of approximately 5 points per wavelength is found to be very accurate when working with a polynomial degree of N = 8. The accuracy of the method is shown by comparing the spectral element solution to analytical solutions of the classical two-dimensional (2D) problems of Lamb and Garvin. The flexibility of the method is then illustrated by studying more realistic 2D models involving realistic geometries and complex free-boundary conditions. Very accurate modeling of Rayleigh-wave propagation, surface diffraction, and Rayleigh-to-body-wave mode conversion associated with the free-surface curvature are obtained at low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves by three-dimensional (3D) surface topographies and the associated local effects on strong ground motion. Complex amplification patterns, both in space and time, are shown to occur even for a gentle hill topography. Extension to a heterogeneous hill structure is considered. The efficient implementation on parallel distributed memory architectures will allow to perform real-time visualization and interactive physical investigations of 3D amplification phenomena for seismic risk assessment.

The spectral element method: An efficient tool to simulate the seismic response of 2D and 3D geological structures

The spectral ratio between horizontal and vertical components (H/V ra- tio) of microtremors measured at the ground surface has been used to estimate fun- damental periods and amplification factors of a site, although this technique lacks theoretical background. The aim of this article is to formulate the H/V technique in terms of the characteristics of Rayleigh and Love waves, and to contribute to improve the technique. The improvement includes use of not only peaks but also troughs in the H/V ratio for reliable estimation of the period and use of a newly proposed smoothing function for better estimation of the amplification factor. The formulation leads to a simple formula for the amplification factor expressed with the H/V ratio. With microtremor data measured at 546 junior high schools in 23 wards of Tokyo, the improved technique is applied to mapping site periods and amplification factors in the area. mode Rayleigh waves. Mainly based on these facts, in this study, a large proportion of microtremors are thought to con- sist of surface waves, mainly of their fundamental modes. This article first deals with the peak period in the H/V ratio of microtremors and its relation to the fundamental pe- riod of the response to vertically incident S waves. Next, peak values in the H/V ratio are related to the amplification factors for the S waves. In this process, a new smoothing function is proposed for getting a better correlation between them. A simple relation between the amplification factor and the H/V ratio, which is similar to a so-called Nakamura's method, is also formulated. Finally, the periods and the am- plification factors estimated from microtremor measurement at 546 stations in Tokyo are mapped as a demonstration of practicality of the present study.

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Ground-motion characteristics estimated from spectral ratio between horizontal and vertical components of microtremor

Numerical simulation of noise is used to investigate the characteristics of the spectral ratio between horizontal and vertical components (H/V ratio) and its sensitivity to various parameters in order to better appreciate the reliability of the technique proposed by Nakamura (1989) to estimate site amplification effects from single station noise recordings. Noise is simulated as the signal produced at a single site by a set of superficial sources (unidirectional forces or dipoles) disposed all around with random amplitude and time delay. Individual signals from a single source are computed by the discrete wave number technique. Synthetic calculations for 15 soil profiles show that this ratio exhibits a single, clear peak, the location of which is independent of the source excitation function, but strongly correlated with the local geological structure: its frequency is very close to the S wave resonance frequency. This peak appears to be mainly controlled by the polarization curve of the fundamental Rayleigh waves, which in turn exhibits a sharp peak around the fundamental resonance mode of the sedimentary structure. A similar result is found for the H/V ratio obtained for incident plane SV waves. In contrast, the amplitude of this peak exhibits a poor correlation with the ground motion amplification of S waves at resonance frequency. It is shown to be related with a high sensitivity on the value of the Poisson's ratio in the uppermost layer presumed to be the noise source layer, and, though to a much lesser extent, on the mean distance between site and noise sources. It is concluded that Nakamura's method can clearly allow the resonance frequency of a given sedimentary site to be measured very efficiently and very cheaply, but that its use for deriving the amplification at resonance frequency seems still premature from a theoretical point of view.

Numerical and Theoretical Investigations on the Possibilities and Limitations of Nakamura's Technique

The nature of noise wavefield and its applications for site effects studies A literature review

A brief review of experimental and theoretical results about the effect of topography on seismic motion shows that they are consistent only on a qualitative basis. Amplification at mountain tops for wavelengths comparable with mountain widths is predicted and observed, but the numerical simulations often underestimate the actual observed effects. We propose that this disagreement is because current models are not complex enough .

We therefore computed the response to incident SH waves of a set of different complex configurations that include two-dimensional surface topography, with or without periodic ridges and subsurface layering. The results show that: (1) the topographic effect in itself is difficult to isolate from other effects, like surface layering, and therefore the amplification on top of geomorphologically complex sites cannot be predicted by a priori estimations based solely on topography; and (2) the high crest/base amplification ratio observed in the field cannot, usually, be matched even with complex two-dimensional structures with incident plane SH waves, which suggests that more complex models are needed to incorporate more complex wave fields (e.g., SV , surface) and three-dimensional geologic configurations .

The effect of topography on earthquake ground motion: a review and new results

This paper presents a numerical study of the response of relatively embanked sediment-filled valleys to incident plane SH , SV , and P waves, in the two-dimensional case. The Aki-Larner technique used here is shown to be reliable even for steep interface slopes. Numerical results show the existence and the importance in such valleys of specific two-dimensional resonance patterns, which may be classified in three categories: the antiplane shear modes, corresponding to SH motion; the in-plane shear modes ( SV ); and the in-plane bulk modes ( P ). Each of them is characterized by the consistency of the peak frequencies, and the in-phase motion (modulo 180°) across the whole valley. They induce a very large amplification, even in the case of significant damping (up to 4 times the corresponding one-dimensional prediction), a very long duration of motion, and large differential motion. The characteristics of the fundamental modes are in good qualitative agreement with the relevant experimental observations. The existence of this two-dimensional resonance is controlled by the shape ratio (thickness to half-width ratio) and the velocity contrast: a quantitative relationship is proposed in the SH case. The frequency of these resonance modes, for each pattern, is shown to depend only on two parameters: the one-dimensional resonance frequency at the valley center and the shape ratio. A simple model of a soft rectangular inclusion is shown to provide satisfactory quantitative formulas to estimate the fundamental resonance frequencies of any valley. As to the amplitude of this two-dimensional resonance, the general trends of its dependence on the different valley parameters (shape ratio, velocity contrast, Poisson ratio, damping) and on the incident wave field characteristics (wave type, incidence angle) are indicated. An important result, however, for earthquake engineering purposes is that both the two-dimensional resonant frequencies and amplification values differ a lot from their “classical” one-dimensional estimates.

The two-dimensional resonance of sediment-filled valleys

SUMMARY Ambient vibration techniques such as the H/V method may have the potential to significantly contribute to site effect evaluation, particularly in urban areas. Previous studies interpret the so-called Nakamura’s technique in relation to the ellipticity ratio of Rayleigh waves, which, for a high enough impedance contrast, exhibits a pronounced peak close to the fundamental S-wave resonance frequency. Within the European SESAME project (Site EffectS assessment using AMbient Excitations) this interpretation has been tested through noise numerical simulation under well-controlled conditions in terms of source type and distribution and propagation structure. We will present simulations for a simple realistic site (one sedimentary layer over bedrock) characterized by a rather high impedance contrast and low quality factor. Careful H/V and array analysis on these noise synthetics allow an in-depth investigation of the link between H/V ratio peaks and the noise wavefield composition for the soil model considered here: (1) when sources are near (4 to 50 times the layer thickness) and surficial, H/V curves exhibit one single peak, while the array analysis shows that the wavefield is dominated by Rayleigh waves; (2) when sources are distant (more than 50 times the layer thickness) and located inside the sedimentary layer, two peaks show up on the H/V curve, while the array analysis indicates both Rayleigh waves and strong S head waves; the first peak is due to both fundamental Rayleigh waves and resonance of head S waves, the second is only due to the resonance of head S waves; (3) when sources are deep (located inside the bedrock), whatever their distance, H/V ratio exhibit peaks at the fundamental and harmonic resonance frequencies, while array analyses indicate only non-dispersive body waves; the H/V is thus simply due to multiple reflections of S waves within the layer. Therefore, considering that experimental H/V ratio (i.e. derived from actual noise measured in the field) exhibit in most cases only one peak, we conclude that H/V ratio is (1) mainly controlled by local surface sources, (2) mainly due to the ellipticity of the fundamental Rayleigh waves. Then the amplitude of H/V peak is not able to give a good estimate of site amplification factor.

https://hal-insu.archives-ouvertes.fr/insu-00265520/document

Pierre-Yves Bard

Papers

Numerical and Theoretical Investigations on the Possibilities and Limitations of Nakamura's Technique

The nature of noise wavefield and its applications for site effects studies A literature review

The effect of topography on earthquake ground motion: a review and new results

The two-dimensional resonance of sediment-filled valleys

H/V ratio: a tool for site effects evaluation. Results from 1-D noise simulations