Preliminary results of the modelling of the Mexico City valley with a two-dimensional boundary element method for the scattering of SH waves

A numerical method is used for calculating the two-dimensional scattering of incident SH waves to try to explain some of the amplification patterns observed from recent data of the Mexico City's accelerometric array. The method is briefly presented and its efficiency is tested against analytical and other numerical solutions for canyons and alluvial valleys.

Spectral ratios computed for transition and lake-bed zones of the Mexico City valley with respect to the average motion at hill-zone sites are also presented. The one-dimensional model is used to explain the amplifications observed at a site where the valley is relatively shallow, while the two-dimensional approach is employed at another site at the centre of the valley where irregular amplification patterns have been observed. Results in the time domain are also shown.     

Site response analysis in the valley of Mexico: Selection of input motion and extent of non-linear soil behaviour

It is well known that seismic waves in the lake-bed zone of Mexico city are dramatically amplified, and that the soft clays in this region exhibit an unusually large elastic range. New field and laboratory data show that clays in certain areas within the Valley of Mexico have lower elastic thresholds and, hence, non-linear soil behaviour would be more likely to take place. In this paper, we analyse geotechnical and strong-motion data to model, using a one-dimensional (1-D) approach, the non-linear soil response at two sites in Mexico City during the September 19, 1985 (M_s = 8.1) Michoacán earthquake. Results suggest that, even for motions as large as those recorded during the 1985 Michoacán event, a linear viscoelastic behaviour must be assumed for the examined area, although, as far as it is presently known, the clays from this area show the greatest modulus reduction among the Mexico City clays. Thus, a linear 1-D propagation analysis continues to be a suitable engineering option in computing expected ground motions at soft sites for the Valley of Mexico. Caution, however, is warranted when selecting input motions, since, as shown in this paper, large differences can result depending on the choice of outcrop motions.     

Computer analysis of regional groundwater flow and boundary conditions in the basin of Mexico

Mexico City is situated in the Valley of Mexico on the extensive lacustrine clays that overlay highly productive aquifers of both volcanic and sedimentary origin. The Valley is closed by volcanic mountains. The natural hydraulic boundary conditions associated withe mountain ranges and their relationship to the important aquifers were studied using a two-dimensional, steady-state finite-element model in cross section. Four cross sections were analysed under hydrologic conditions existing prior to the large scale pumping of the aquifers. Factors such as bulk hydraulic conductivities and regional infiltration rates were obtained from field observations and the literature to assess location of the associated groundwater divides, and the water-table in the mountains. The modeled flow patterns are consistent with the historical hydrologic records piezometric characteristics and observed surface features of the groundwater in the Basin of Mexico. From the modeling results, the groundwater recharge in the mountains is 30–50% of the mean average precipitation. Higher and lower rates result in a flow regime that is not compatible with field observations. In general the location of the divides in the mountains is displaced towards the Valley of Mexico, which influences the groundwater budget of the Valley. The water table in places is several hundred metres below ground surface, in accordance with field observations of a very thick unsaturated zone. Before major aquifer exploitation began about 50 years ago, 40–50% of the total discharge into the Valley was by upward flow through the lacustrine deposits. The best results were obtained using a subsurface distribution of hydrostratigraphic units based on recently published geological interpretations.     

Site effect evaluation at Mexico City: Dominant period and relative amplification from strong motion and microtremor records

Site effects in Mexico City are discussed in terms of simple 1D, one-layer, linear models. The analysis is focussed on two parameters: dominant period and maximum amplification relative to a firm site within the city. The data used is a compilation of strong motion data and microtremor measurements. Strong motion data consist of digital acceleration records for nine events recorded by the Accelerographic Network of Mexico City. The authors analyzed spectral ratios of horizontal components of soft soil sites relative to an average of firm site observations for this data set. Dominant period, maximum relative amplification and an estimate of material damping were computed from the empirical transfer functions thus obtained. Microtremor data were compiled from measurement of different groups during the period 1985–1992. In all, 409 measurement points were analyzed. Values of dominant period obtained from microtremor measurements are in excellent agreement with those obtained from empirical transfer functions for strong motion data. The synthesis of results allows us to draw a detailed and robust map of dominant period for Mexico City. Based on this map, the authors propose some modifications to the current microzonation of Mexico City and evaluate a proposed model to account for site effects in this city.     

Site effects in Mexico City urban zone. A complementary study

Site effects are a major factor affecting seismic risk at Mexico City conurbation and have been carefully studied. However, most of the studies and instrumentation concentrate in Mexico City. We present an evaluation of site effects in the counties of Estado de Mexico, to the North of Mexico City, that have been incorporated into Mexico City conurbation. We measured microtremors at 67 points and operated a temporal digital seismograph network. Our results allow us to extend the dominant period map of Mexico City to the heavily populated and industrialized counties of Estado de Mexico. Additionally, we propose a map of maximum relative amplification. Both maps are consistent with local geology, and could be useful for microzonation of this region.     

Site effects and soil-structure interaction in the valley of Mexico

An engineering approach is proposed for representing both site effects and soil-structure interaction in extended alluvial valleys, by using the one-dimensional model of shear were propagation corrected empirically to account for lateral heterogeneities and generated surface waves. The peak structural response is expressed by means of spectral contours that are a function of the predominant period of the site and the fundamental period of the structure. Variations of the peak spectral ordinates with the prevailing site period can be deduced from these contours. A number of events of firm ground, representative of the most dangerous earthquakes expected in Mexico City, are assumed as design earthquakes. Making use of the resulting spectral contours, the provisions for site effects recommended in the Mexican seismic code are evaluated. Also, considering as control motion the 1985 Michoacan earthquake recorded at a representative firm site, spectral contours with soil-structure interaction are obtained which allow one to identify the significant interaction effects originating in the Valley of Mexico for medium- and long-period structures. The influence and relative importance of the critical parameters involved are examined within practical ranges of interest.     

SH波垂直入射下的斜坡地形放大因子分析

现有大量观测记录表明:斜坡地形对地震波的传播有着非常强烈的影响。为服务于工程抗震设计,基于显式有限元方法,定量分析SH波垂直入射下二维斜坡地形的地震动响应与斜坡角度、土层厚度以及介质阻抗比的关系,总结了位于一维土层基本频率附近斜坡上台面各区域放大因子的变化规律。研究表明:（1）斜坡面对SH波的反射使二维斜坡地形的放大倍数较一维土层存在显著放大,且该现象在缓坡中更加明显。（2）当土层厚度为斜坡高度的1/4,介质阻抗比为0.368,坡度为30°时,放大因子在距坡顶1.67倍斜坡高度处取到最大值1.930。（3）斜坡覆盖土层薄时,放大因子受斜坡角度的影响大,斜坡覆盖土层厚时,阻抗比成为影响放大因子的主要因素。（4）分别考虑土层厚度、斜坡角度、介质阻抗比以及观测点位置对二维斜坡地形地表地震动响应的影响,取1倍斜坡高度作为区域间隔,统计每个区域内各参数对应的放大因子最大值,对比现有规范给出工程抗震设计参考值及放大因子大于1的基频比范围。     

Evaluation of the change in dominant periods in the lake-bed zone of Mexico City produced by ground subsidence through the use of site amplification factors

We compute site amplification functions for several sites in Mexico City using actual accelerograms recorded from 1985 to 2010 and we present field evidence of the change in the dominant period of a given site (T_s) as a consequence of ground subsidence produced by groundwater withdrawal. The changes in T_s are larger in the lake-bed zone where thicker clay deposits exist, although there are sites in the southwest part of the lake-bed zone where T_s has remained constant. With the information obtained from the site amplification functions and available geotechnical soundings we develop an empirical model to estimate the future value of T_s for several sites in Mexico City. Because the practical application of the model requires extrapolation we also present a method to compute the uncertainty of the model when it is used to forecast a future value of T_s at a given site. Our results suggest that significant changes in the dominant period at several sites in Mexico City can be expected in the future.     

The effect of travelling seismic waves on the dynamic response of earth dams

A theoretical investigation of the dynamic response of earth dams to the travelling seismic waves is presented. The earth dam is simplified as a truncated two-dimensional elastic wedge. The dam body consists of an isotropical linear viscoelastic material with homogeneous elastic modulus and density. The seismic waves travel along the longitudinal direction of the earth dam. The numerical calculations show the following. (i) For the longitudinal mode of vibration, the greater the ratio (H/L) of the height to the lenght of the complete wedge, the more the natural transverse period of vibrational of the two-dimensional wedge is less than that of the one-dimensional wedge. Especially for the first two natural transverse period, this influence is large. The decrease of the ratio of the natural transverse period for a two-dimensional wedge to that for a one-dimensional wedge with the ratio H/L is rapid for the higher than for the lower longitudinal modes. (ii) Comparing with the one-dimensional wedge, the natural transverse periods for the two-dimensional case in the complete wedge are lower, and they will increase as the coefficient of truncation, h/H increases. (iii) When the frequency of forced vibration is less than the natural transverse frequency for one-dimensional wedge, the amplification is less for a two-dimensional wedge than for a one-dimensional wedge. (iv) When the phase difference of ground motion at both ends of the dam equals π, the amplification for a two-dimensional wedge is less than that for a one-dimensional wedge, but when the phase difference equals nπ, (n > 1), the situation is reversed. (v) As the coefficient of truncation, h/H, increases, the displacement model partecipations decrease monotically. (vi) In general, the displacement caused by an earthquake is greater for a one-dimensional wedge than for a two-dimensional wedge when considering the seismic waves travelling, but the acceleration response of a two-dimensional wedge with long length of dam to travelling seismic waves with long dominant period is greater than that of a one-dimensional wedge. When the length of the dam is short enough, the response of a two-dimensional wedge without considering the influence of travelling seismic waves always gives the greatest value.     

ENGINEERING PERSPECTIVE FOR THE SEISMIC SITE RESPONSE OF ALLUVIAL VALLEYS

The seismic site response of alluvial valleys with limited width is evaluated using three engineering models. The models are based on the one-dimensional, two-dimensional and the frame model approaches. The objective is to analyse the effects of the main parameters governing surface motions and provide engineering guidance for predicting them. The limitations on the use of the one-dimensional model in site response evaluation in valleys are pointed out. The frame model, which accounts for the limited width of valley, gives response results that are in good agreement with the two-dimensional model results. It is found that the effect of the two-dimensional amplification is significant over a distance from the valley edges beyond which the response may be adequately represented by one-dimensional analysis. The soil amplification varies depending on the soil type, site location relative to the valley and the dominant period and amplitude of input rock record. © 1997 by John Wiley & Sons, Ltd.     

Earthquake ground motion in Mexico City: An analysis of data recorded at Roma array

We analysed in detail three earthquakes recorded in a small-aperture accelerometric array in Mexico City, using the correlation of the records as a function of time along the accelerogram and frequency. Ground response is strongly conditioned by the fundamental period of the soft soils at the site of the array (T₀). Energy at periods longer than 2T₀ is guided by the crustal structure (with a thickness of 45 km). The wave field at periods between T₀ and 2T₀ also consists of surface waves but guided by the upper 2–3 km of volcanic sediments in central Mexico. For periods smaller than T₀, ground motion is uncorrelated among the stations. Our results indicate that seismic response of Mexico City, including its very long duration, results from deeply guided surface waves (between 2 and 45 km depth) interacting with the very local response of the soft surficial clay layer.     

Azimuth dependent wave amplification in alluvial valleys

An extension of the indirect boundary element method (IBEM) to three-dimensional scattering by two-dimensional alluvial valleys is presented. While the valley is two-dimensional, the incident plane waves can arrive outside the 2D plane so the scattering is three-dimensional with coupling of P---SV---SH waves. Such a method makes it possible to take earthquake location into account in the estimation of site effects in alluvial valleys. The method is validated by transparency tests, by comparison with 2D simulations, and by comparison with results of other authors. The advantage of the method is that is combines high accuracy with cost-efficiency in terms of computer-time. It is applied to theoretically estimate site effects across a simplified model of an alluvial valley in the French Alps where azimuth dependence of local amplification has been observed. A parametric study with simulations for a range of azimuths and incidence angles shows that (1) the local amplification depends strongly on both azimuth and incidence of the incoming waves, (2) the global pattern of amplification across the valley is very complex for all azimuths, and (3) it is not possible to predict the 3D response of the valley from 2D modeling. Theoretical spectral ratios are in approximate agreement with observed ones for a station in the center of the valley where the local structure justifies use of a simplified model for the comparison.     

Site effect study in central Mexico using H/V and SSR techniques: Independence of seismic site effects on source characteristics

We study the influence of different source characteristics (depth, distance, type and azimuth) on the site effect in Acapulco and the Valley of Mexico. Site amplification was estimated by means of spectral ratios (both horizontal-to-vertical spectral ratio and standard spectral ratio techniques were applied) from earthquake recordings at soft and hard sites. In Acapulco, 125 Mexican earthquakes covering a hypocentral range of 7–290 km and a depth range (H) of 3–61 km were analyzed in three different groups of hypocentral distances. In the Valley of Mexico, we estimate site effect at five locations using recordings from shallow-coastal interplate (200?Δ?570 km; H?35 km) and normal-faulting, intermediate-depth inslab (132?Δ?738 km; 32?H?178 km) earthquakes, as well as from teleseismic events. Our results seem to point to negligible dependence of site effects on the source location and type.     

Site Response in Las Vegas Valley, Nevada from NTS Explosions and Earthquake Data

We report site response in Las Vegas Valley (LVV) from historical recordings of Nevada Test Site (NTS) nuclear explosions and earthquake recordings from permanent and temporary seismic stations. Our data set significantly improves the spatial coverage of LVV over previous studies, especially in the northern, deeper parts of the basin. Site response at stations in LVV was measured for frequencies in the range 0.2–5.0 Hz using Standard Spectral Ratios (SSR) and Horizontal-Vertical Spectral Ratios (HVR). For the SSR measurements we used a reference site (approximately NEHRP B ``rock' classification) located on Frenchman Mountain outside the basin. Site response at sedimentary sites is variable in LVV with average amplifications approaching a factor of 10 at some frequencies. We observed peaks in the site response curves at frequencies clustered near 0.6, 1.2 and 2.0 Hz, with some sites showing additional lower amplitude peaks at higher frequencies. The spatial pattern of site response is strongly correlated with the reported depth to basement for frequencies between 0.2 and 3.0 Hz, although the frequency of peak amplification does not show a similar correlation. For a few sites where we have geotechnical shear velocities, the amplification shows a correlation with the average upper 30-meter shear velocities, V₃₀. We performed two-dimensional finite difference simulations and reproduced the observed peak site amplifications at 0.6 and 1.2 Hz with a low velocity near-surface layer with shear velocities 600–750 m/s and a thickness of 100–200 m. These modeling results indicate that the amplitude and frequencies of site response peaks in LVV are strongly controlled by shallow velocity structure.     

Site effects in 3D basins using 1D and 2D models: an evaluation of the differences based on simulations of the seismic response of Euroseistest

Site effects are one of the most predictable factors of destructive earthquake ground motion but results depend on the type of model chosen. We compare simulations of ground motion for a 3D model of the Mygdonian basin in northern Greece (Euroseistest) using different approximation for this basin. Site effects predicted using simple 1D models at many points inside the basin are compared to site effects predicted using four different 2D cross sections across the basin and with results for a full 3D simulation. Surface topography was neglected but anelastic attenuation was included in the simulations. We show that lateral heterogeneity may increase ground motion amplification by 100 %. Larger amplification is distributed in a wide frequency range, and amplification may occur at frequencies different from the expected resonant frequencies for the soil column. In contrast, on a different cross section, smaller conversion of incident energy into surface waves and larger dispersion leads to similar amplitudes of ground motion for 2D and 1D models. In general, results from 2D simulations are similar to those from a complete 3D model. 2D models may overestimate local surface wave amplitudes, especially when the boundaries of the basin are oblique to the selected cross section. However, the differences between 2D and 3D site effects are small, especially in regard of the difficulties and uncertainties associated to building a reliable 3D model for a large basin.     

Two-dimensional scattering of SH waves due to a discontinuity in bedrock

The direct boundary integral equation technique is applied to determine the impact on surface amplification caused by an inhomogeneity in a bedrock half-space. The particular soil-rock configuration studied is one in which a soil layer rests on a rock half-space which includes a rock inclusion. The particular rock inclusion considered for this study is a semi-infinite rock layer with its upper boundary bordering the soil layer. Materials are considered viscoelastic except for the section of the rock half-space below the level of the rock inclusion which is considered elastic. A parametric study is performed to determine controlling factors for surface displacement due to a vertically incident shear wave. The study includes varying the stiffness and the thickness of the inclusion for a range of frequencies. Solutions from a one-dimensional analysis are compared with the results of a two-dimensional analysis in order to establish limits inside of which a two-dimensional analysis is required. Copyright © 1999 John Wiley & Sons, Ltd.     

A shallow attenuating anomaly inside the ring fracture of the Valles Caldera, New Mexico

Spectral ratios of teleseismic direct and scattered P waves observed in the Valles Caldera, New Mexico, show a systematic pattern of low amplitudes at sites inside the caldera relative to sites on or outside the ring fracture. Waveforms recorded at caldera stations are considerably more complex than those recorded outside the caldera. The data used in this study were collected during a passive seismic monitoring experiment conducted in 1987. Twenty-four teleseismic events were recorded on two linear arrays spanning the caldera. To first order, the pattern of low amplitudes did not vary with source incidence angle or azimuth of approach, and could not be explained by anomalous amplification at the ring fracture. This observation suggests the presence of a shallow, attenuating zone associated with the caldera fill material inside of the ring fracture. We estimated the general features of the caldera's near-surface structure for the two-dimensional vertical cross section beneath the array, using a modification of the Aki-Larner discrete-wavenumber method to forward model the observed amplitude variations. Our results indicate that the caldera fill material must be subdivided into at least two distinct zones: a strongly attenuating lower zone, extending to depths in excess of 4 km, and a mildly attenuating surface layer. To fit the data we had to assign an unrealistically low value to seismic Q in the deeper attenuating anomaly. We attribute this to the inability of the Aki-Larner method to account for strong re-direction of energy away from the caldera due to local heterogeneity that we could not include within the low-Q anomaly. This interpretation is consistent with the pervasive, fractured hydrothermal system that is known to exist in the caldera fill material.     

An indirect boundary element method applied to simulate the seismic response of alluvial valleys for incident P,S and Rayleigh waves

A boundary integral formulation is presented and applied to model the ground motion on alluvial valleys under incident P, S and Rayleigh waves. It is based on integral representations for the diffracted and the refracted elastic waves using single-layer boundary sources. This approach is called indirect BEM in the literature as the sources' strengths should be obtained as an intermediate step. Boundary conditions lead to a system of integral equations for boundary sources. A discretization scheme based on the numerical and analytical integration of exact Green's functions for displacements and tractions is used. Various examples are given for two-dimensional problems of diffraction of elastic waves by soft elastic inclusion models of alluvial deposits in an elastic half-space. Results are displayed in both frequency and time domains. These results show the significant influence of locally generated surface waves in seismic response and suggest approximations of practical interest. For shallow alluvial valleys the response and its resonant frequencies are controlled by a coupling mechanism that involves both the simple one-dimensional shear beam model and the propagation of surface waves.     

Characterization of site conditions (soil class,V<Subscript>S30</Subscript>, velocity profiles) for 33 stations from the French permanent accelerometric network (RAP) using surface-wave methods

Data provided by accelerometric networks are important for seismic hazard assessment. The correct use of accelerometric signals is conditioned by the station site metadata quality (i.e., soil class, V_S30, velocity profiles, and other relevant information that can help to quantify site effects). In France, the permanent accelerometric network consists of about 150 stations. Thirty-three of these stations in the southern half of France have been characterized, using surface-wave-based methods that allow derivation of velocity profiles from dispersion curves of surface waves. The computation of dispersion curves and their subsequent inversion in terms of shear-wave velocity profiles has allowed estimation of V_S30 values and designation of soil classes, which include the corresponding uncertainties. From a methodological point of view, this survey leads to the following recommendations: (1) perform both active (multi-analysis surface waves) and passive (ambient vibration arrays) measurements to derive dispersion curves in a broadband frequency range; (2) perform active acquisitions for both vertical (Rayleigh wave) and horizontal (Love wave) polarities. Even when the logistic contexts are sometimes difficult, the use of surface-wave-based methods is suitable for station-site characterization, even on rock sites. In comparison with previous studies that have mainly estimated V_S30 indirectly, the new values here are globally lower, but the EC8-A class sites remain numerous. However, even on rock sites, high frequency amplifications may affect accelerometric records, due to the shallow relatively softer layers.     

平面SH波作用下部分充填圆弧形沉积谷的二维土层和地形放大效应

场地效应通常包含土层放大效应和地形放大效应,为了揭示二者的相对贡献,本文构造了平面SH波作用下部分充填沉积谷的解析模型,借助于区域分解策略,在波函数展开法的框架下,提出了超定方程组解法,得到了部分充填圆弧形沉积谷对平面SH波散射的波函数级数解,而且级数解的收敛测试表明了超定方程组解法的必要性.通过与文献结果进行对比,验证了本文方法的正确性.通过调整解析模型中两个子区域内的材料参数,计算了沉积谷引起的场地放大效应和相应的空河谷引起的地形放大效应.对二维土层与地形效应进行对比分析,结果显示,在沉积谷内二维土层放大效应通常强于地形放大效应,而地形放大效应决定了沉积谷外的地面运动放大形态.针对最大地面运动,进行了沉积谷和相应空河谷的参数分析,进一步描述了二维土层放大效应,研究结果表明二维土层放大效应引起的最大地面运动通常远远大于地形放大效应引起的最大地面运动,并且二维土层效应通常随着土层与基岩的阻抗比的增大而增大,但不是一维土层放大效应与二维地形放大效应的简单线性叠加.