An Atlas of ShakeMaps and population exposure catalog for earthquake loss modeling

We present an Atlas of ShakeMaps and a catalog of human population exposures to moderate-to-strong ground shaking (EXPO-CAT) for recent historical earthquakes (1973–2007). The common purpose of the Atlas and exposure catalog is to calibrate earthquake loss models to be used in the US Geological Survey’s Prompt Assessment of Global Earthquakes for Response (PAGER). The full ShakeMap Atlas currently comprises over 5,600 earthquakes from January 1973 through December 2007, with almost 500 of these maps constrained—to varying degrees—by instrumental ground motions, macroseismic intensity data, community internet intensity observations, and published earthquake rupture models. The catalog of human exposures is derived using current PAGER methodologies. Exposure to discrete levels of shaking intensity is obtained by correlating Atlas ShakeMaps with a global population database. Combining this population exposure dataset with historical earthquake loss data, such as PAGER-CAT, provides a useful resource for calibrating loss methodologies against a systematically-derived set of ShakeMap hazard outputs. We illustrate two example uses for EXPO-CAT; (1) simple objective ranking of country vulnerability to earthquakes, and; (2) the influence of time-of-day on earthquake mortality. In general, we observe that countries in similar geographic regions with similar construction practices tend to cluster spatially in terms of relative vulnerability. We also find little quantitative evidence to suggest that time-of-day is a significant factor in earthquake mortality. Moreover, earthquake mortality appears to be more systematically linked to the population exposed to severe ground shaking (Modified Mercalli Intensity VIII+). Finally, equipped with the full Atlas of ShakeMaps, we merge each of these maps and find the maximum estimated peak ground acceleration at any grid point in the world for the past 35 years. We subsequently compare this “composite ShakeMap” with existing global hazard models, calculating the spatial area of the existing hazard maps exceeded by the combined ShakeMap ground motions. In general, these analyses suggest that existing global, and regional, hazard maps tend to overestimate hazard. Both the Atlas of ShakeMaps and EXPO-CAT have many potential uses for examining earthquake risk and epidemiology. All of the datasets discussed herein are available for download on the PAGER Web page (). T. I. Allen and M. G. Hearne—contracted through Synergetics Incorporated.     

Improving ShakeMap performance by integrating real with synthetic data: tests on the 2009 M w?=?6.3 L’Aquila earthquake (Italy)

The ShakeMap software automatically generates maps of the peak ground motion parameters (shakemaps) and of instrumental intensity soon after an earthquake. Recorded data are fundamental to obtaining accurate results. In case observations are not available, ShakeMap relies on ground motion predictive equations, but due to unmodelled site conditions or finite fault effects, large uncertainties may appear, mainly in the near-source area where damage is relevant. In this paper, we aim to account for source effects in ShakeMap by computing synthetics to be used for integrating observations and ground motion predictive equations when near-source data are not available. To be effective, the computation of synthetics, as well as of the finite fault, should be done in near real time. Therefore, we computed rapid synthetic seismograms, by a stochastic approach, including the main fault features that were obtained through inversion of regional and teleseismic data. The rapidity of calculation is linked to a number of assumptions, and simplifications that need testing before the procedure can run in automatic mode. To assess the performance of our procedure, we performed a retrospective validation analysis considered as case study of the M _w = 6.3 earthquake, which occurred in central Italy on April 6, 2009. In that case, the first shakemaps, generated a few minutes after the earthquake, suffered large uncertainties on ground motion estimates in an area closer to the epicenter due to the lack of near-field data. To verify our approach, we recomputed shakemaps for the L’Aquila earthquake, integrating data available soon after the earthquake at different elapse times with synthetics, and we compared our shaking map with the final shakemap, obtained when all the data were available. Our analysis evidences that (1) when near-source data are missing, the integration of real data with synthetics reduces discrepancies between computed and actual ground shaking maps, mainly in the near-field zone where the damage is relevant and (2) the approach that we adopted is promising in trying to reduce such discrepancies and could be easily implemented in ShakeMap, but some a priori calibration is necessary before running in an automatic mode.     

震后近断层震动图的快速产出研究—以2022年1月8日青海门源地震为例

震后近断层震动图的快速产出对于政府相关部门快速确定地震影响区、评估震害及其导致的经济损失和人员伤亡、科学决策应急救援方案和措施以减轻人员伤亡和财产损失具有重要意义。以2022年门源地震为例,利用滑动分布、应力降均不同的两个震源模型（即,王卫民等反演的震源模型和本研究生成的随机滑动震源模型）以及相同的路径、场地模型和其它输入参数,使用动力学拐角频率的随机有限断层方法和本研究建立的峰值地面速度（PGV）、水平向最大谱烈度（SI）和中国仪器地震烈度（I_I）的经验模型研究了快速产出近断层震动图（包括峰值地面加速度图、PGV图、SI图、I_I图和中国地震烈度图）的实效性。结果表明:（1）上述方法和经验模型可用于震后震动图的快速产出,其实效性主要取决于震源、路径和场地模型的可靠性;（2）基于上述两种震源模型产出的地震烈度图与中国地震局发布的该次地震的烈度图在总体上具有高度一致性,均可用于确定地震影响区,但基于反演震源模型产出的地震烈度图可以给出极震区,而基于随机滑动震源模型产出的地震烈度图则需要根据其最大等震线和发震断层的位置大致估计极震区的位置。     

Numerical simulation of strong ground motion for the <Emphasis Type="Italic">M</Emphasis><Subscript>s</Subscript>8.0 Wenchuan earthquake of 12 May 2008

The Wenchuan earthquake of 12 May 2008 is the most destructive earthquake in China in the past 30 years in terms of property damage and human losses. In order to understand the earthquake process and the geo-morphological factors affecting the seismic hazard, we simulated the strong ground motion caused by the earthquake, incorporating three-dimensional (3D) earth structure, finite-fault rupture, and realistic surface topography. The simulated ground motions reveal that the fault rupture and basin structure control the overall pattern of the peak ground shaking. Large peak ground velocity (PGV) is distributed in two narrow areas: one with the largest PGV values is above the hanging wall of the fault and attributed to the locations of fault asperities and rupture directivity; the other is along the northwestern margin of the Sichuan Basin and caused by both the directivity of fault rupture and the amplification in the thick sediment basin. Rough topography above the rupture fault causes wave scattering, resulting in significantly larger peak ground motion on the apex of topographic relief than in the valley. Topography and scattering also reduce the wave energy in the forward direction of fault rupture but increase the PGV in other parts of the basin. These results suggest the need for a localized hazard assessment in places of rough topography that takes the topographic effects into account. Finally, had the earthquake started at the northeast end of the fault zone and ruptured to the southwest, Chengdu would have suffered a much stronger shaking than it experienced on 12 May, 2008. Supported by the U.S. National Science Foundation (Grant Nos. EAR 0738779 and OCE 0727919), the National Basic Research Program of China (Grant No. 2004CB418404), and partially by the National Nature Science Foundation of China (Grant No. 40521002)     

Three-dimensional simulations of strong ground motion in the Shidian basin based upon the spectral-element method

The strong motion of a small long and narrow basin caused by a moderate scenario earthquake is simulated by using the spectral-element method and the parallel computing technique.A total of five different geometrical profiles within the basin are used to analyze the generation and propagation of surface waves and their relation to the basin structures in both the time and frequency domain.The amplification effects are analyzed by the distribution of peak ground velocity(PGV)and cumulative kinetic energy(Ek) in the basin.The results show that in the 3D basin,the excitation of the fundamental and higher surface wave modes are similar to that of the 2D model.Small bowls in the basin have great influence on the amplification and distribution of strong ground motion,due to their lateral resonances when the wavelengths of the lateral surface waves are comparable to the size of the bowls.Obvious basin edge effects can be seen at the basin edge closer to the source for constructive interference between direct body waves and the basin-induced surface waves.The Ek distribution maps show very large values in small bowls and some corners in the basin due to the interference of waves propagating in different directions.A high impedance contrast model can excite more surface wave modes,resulting in longer shaking durations as well as more complex seismograms and PGV and Ek distributions.     

Earthquake shaking scenarios for the metropolitan area of Lisbon

In this study, we simulate and compare ground motion shaking in the city of Lisbon and surrounding counties (metropolitan area of Lisbon (MAL)), using two possible earthquake models: the onshore source area of Lower Tagus Valley, M5.7 and M4.7 and the offshore source area, Marques de Pombal Fault, M7.6, one of the possible source of the 1755 Lisbon earthquake. The stochastic and a new hybrid stochastic-deterministic approach (DSM) are used in order to evaluate the ground shaking and to characterize its spatial variability. Results are presented in terms of response acceleration spectra (PSA) and peak ground acceleration (PGA) with respect to bedrock and surface. Site effects are evaluated by means of equivalent stochastic non-linear one-dimensional ground responses analysis, performed for a set of stratified soil profile units properly designed to cope with the soil site conditions of MAL region. A sensitive study is carried out using different input parameters and different approaches in order to give the basic information to evaluate the range of uncertainty in seismic scenarios.     

A macroseismic intensity prediction equation for intermediate depth earthquakes in the Vrancea region,Romania

The use of shake maps in terms of macroseismic intensity in earthquake early warning systems as well as intensity based seismic hazard assessments provides a valuable supplement to typical studies based on recorded ground motion parameters. A requirement for such applications is ground motion prediction equations (GMPE) in terms of macroseismic intensity, which have the advantages of good data availability and the direct relation of intensity to earthquake damage. In the current study, we derive intensity prediction equations for the Vrancea region in Romania, which is characterized by the frequent occurrence of large intermediate depth earthquakes giving rise to a peculiar anisotropic ground shaking distribution. The GMPE have a physical basis and take the anisotropic intensity distribution into account through an empirical regional correction function. Furthermore, the relations are easy to implement for the user. Relations are derived in terms of epicentral, rupture and Joyner–Boore distance and the obtained relations all provide a new intensity estimate with an uncertainty of ca. 0.6 intensity units.     

Inferred ground motions on Guadeloupe during the 2004 Les Saintes earthquake

Accurate estimates of the ground motions that occurred during damaging earthquakes are a vital part of many aspects of earthquake engineering, such as the study of the size and cause of the uncertainties within earthquake risk assessments. This article compares a number of methods to estimate the ground shaking that occurred on Guadeloupe (French Antilles) during the 21st November 2004 (M _w 6.3) Les Saintes earthquake, with the aim of providing more accurate shaking estimates for the investigation of the sources of uncertainties within loss evaluations, based on damage data from this event. The various techniques make differing use of the available ground-motion recordings of this earthquake and by consequence the estimates obtained by the different approaches are associated with differing uncertainties. Ground motions on the French Antilles are affected by strong local site effects, which have been extensively investigated in previous studies. In this article, use is made of these studies in order to improve the shaking estimates. It is shown that the simple methods neglecting the spatial correlation of earthquake shaking lead to uncertainties similar to those predicted by empirical ground-motion models and that these are uniform across the whole of Guadeloupe. In contrast, methods (such as the ShakeMap approach) that take account of the spatial correlation in motions demonstrate that shaking within roughly 10 km of a recording station (covering a significant portion of the investigated area) can be defined with reasonable accuracy but that motions at more distant points are not well constrained.     

粤西沿海震动图的生成和研究

震动图是描绘地震产生的地面运动和可能破坏情况的有效工具,主要包括峰值地面速度(PGV)等值线图、峰值地面加速度(PGA)等值线图和仪器烈度分布图等.基于华南数字地震台网记录到的2010年7月9日MS3.7级阳江地震的数据资料,利用SAC和GMT软件从记录到的地震波形数据提取地面运动参数,生成了PGV和PGA等值线图,并...     

浙江省文泰震区地震滑坡危险性评估

地震滑坡灾害的震前预测与震后快速评估已成为减轻地震次生灾害的重要手段之一。本文使用简化Newmark模型,设定地震震级(M_S5.0),利用区域地质图、数字高程模型等基础数据,考虑地形对地震动的放大效应,对文泰震区潜在同震滑坡区域开展评估工作。研究表明,干燥与饱和状态下,设定地震作用下研究区内地震滑坡高危险区均主要分布在距设定震中15km以内的范围内,其分布与区内岩土体处于临界稳定状态的分布趋势相同。区内水库坝址与水库库体未受到潜在同震滑坡的影响,划定的重点关注区内位于潜在滑坡体下方的千秋门村、驮加村、高西村、杜山村、南峤村、包坑村、龙前村、新厂村以及各级公路易受到同震滑坡的影响,应提升重点关注区内承灾体的风险防范能力,尽可能减少潜在同震滑坡对区内生命财产安全造成的威胁。     

High-frequency maximum observable shaking map of Italy from fault sources

We present a strategy for obtaining fault-based maximum observable shaking (MOS) maps, which represent an innovative concept for assessing deterministic seismic ground motion at a regional scale. Our approach uses the fault sources supplied for Italy by the Database of Individual Seismogenic Sources, and particularly by its composite seismogenic sources (CSS), a spatially continuous simplified 3-D representation of a fault system. For each CSS, we consider the associated Typical Fault, i.e., the portion of the corresponding CSS that can generate the maximum credible earthquake. We then compute the high-frequency (1–50?Hz) ground shaking for a rupture model derived from its associated maximum credible earthquake. As the Typical Fault floats within its CSS to occupy all possible positions of the rupture, the high-frequency shaking is updated in the area surrounding the fault, and the maximum from that scenario is extracted and displayed on a map. The final high-frequency MOS map of Italy is then obtained by merging 8,859 individual scenario-simulations, from which the ground shaking parameters have been extracted. To explore the internal consistency of our calculations and validate the results of the procedure we compare our results (1) with predictions based on the Next Generation Attenuation ground-motion equations for an earthquake of M_w 7.1, (2) with the predictions of the official Italian seismic hazard map, and (3) with macroseismic intensities included in the DBMI04 Italian database. We then examine the uncertainties and analyse the variability of ground motion for different fault geometries and slip distributions.     

Seismic Observations and Seismic Hazard Studies in the Philippines

It has taken more than a hundred years for seismic observations in the Philippines to evolve to a modern observation system.The responsibility of seismic observations was likewise transfeered from one agency to another during this same period of time.At present,the mandate of conducting seismic observatins in the Philippines rests with the Philippine Institute of Volcanology and Seismology(PHIVOLCS),In 2000,through a grant aid from the Japan International Cooperation Agency(JICA),the Philippine Seismic netowrk was upgraded to a digital system.As a result,a new set of seismic monitoring equipments was installed in all of the 34 PHIVOLCS seismic stations all over the country,Digital waveforms are now available for high level seismic data processing.and data acquisition and processing are now automated.Included in the upgrade is the provision of strong motion accelerographs in all stations whose data can now be used for studying ground motion and intensity attenuation relations,The new setup is now producing high-resolution data that can now be used for conducting basic seismological researches,Earthquake locations have now improved allowing for the modeling and delineation of earthquake source regions necessary for earthquake hazard studies.Current seismic hazard studies in the Philippines involve the estimation of ground motion using both probabilitstic and deterministic approaches,seismic microzonation studies of key cities using microtremor observations,paleoseismology and active faults mapping ,and identification of liquefaction-prone,landslide-prone nd tsunami-affected areas.The earthquake database is now being reviewed and completed with the addition of historical events and from data from regional databases,While studies of seismic hazards were primarily concentrated on a regional level ,PHIVOLCS is now focusing on doing these seismic hazard studies on a micriolevel.For Metro Manila,first generation hazard maps showing ground rupture,ground shaking and liquefaction hazards have recently been completed.Other large cities that are also at risk from large earthquakes are the next targets.The elements at risk such as population,lifelines,and vertical and horizontal structures for each of these urban centers are also being incorporated in the hazard maps for immediate use of planners,civil defense officials,policy-makers and engineers.The maps can also now be used to describe possible scenarios during times of strong events and how appropriate socio-economic and engineering responses could be designed.In addition,a rapid earthquake damage assessment system has been started which will attempt to produce immediate or rapid assessments identification of elements at risk durin times of strong earthquakes     

Development of Long-period Ground Motions from the Nankai Trough, Japan, Earthquakes: Observations and Computer Simulation of the 1944 Tonankai (Mw 8.1) and the 2004 SE Off-Kii Peninsula (Mw 7.4) Earthquakes

Strong ground motions recorded in central Tokyo during the 1944 Tonankai Mw8.1 earthquake occurring in the Nankai Trough demonstrate significant developments of very large (>10 cm) and prolonged (>10 min) shaking of long-period (T > 10–12 s) ground motions in the basin of Tokyo located over 400 km from the epicenter. In order to understand the process by which such long-period ground motions developed in central Tokyo and to mitigate possible future disasters arising from large earthquakes in the Nankai Trough, we analyzed waveform data from a dense nation wide strong-motion network (K-NET and KiK-net) deployed across Japan for the recent SE Off-Kii Peninsula (Mw 7.4) earthquake of 5 September 2004 that occurred in the Nankai Trough. The observational data and a corresponding computer simulation for the earthquake clearly demonstrate that such long-period ground motion is primarily developed as the wave propagating along the Nankai Trough due to the amplification and directional guidance of long-period surface waves within a thick sedimentary layer overlaid upon the shallowly descending Philippine Sea Plate below the Japanese Island. Then the significant resonance of the seismic waves within the thick cover of sedimentary rocks of the Kanto Basin developed large and prolonged long-period motions in the center of Tokyo. The simulation results and observed seismograms are in good agreement in terms of the main features of the long-period ground motions. Accordingly, we consider that the simulation model is capable of predicting the long-period ground motions that are expected to occur during future Nankai Trough M 8 earthquakes.     

Impacts of simulated M9 Cascadia Subduction Zone earthquakes considering amplifications due to the Georgia sedimentary basin on reinforced concrete shear wall buildings

Southwest British Columbia has the potential to experience large‐magnitude earthquakes generated by the Cascadia Subduction Zone (CSZ). Buildings in Metro Vancouver are particularly vulnerable to these earthquakes because the region lies above the Georgia sedimentary basin, which can amplify the intensity of ground motions, particularly at medium‐to‐long periods. Earthquake design provisions in Canada neglect basin amplification and the consequences of accounting for these effects are uncertain. By leveraging a suite of physics‐based simulations of M9 CSZ earthquakes, we develop site‐specific and period‐dependent spectral acceleration basin amplification factors throughout Metro Vancouver. The M9 simulations, which explicitly account for basin amplification for periods greater than 1s, are benchmarked against the 2016 BC Hydro ground motion model (GMM), which neglects such effects. Outside the basin, empirical and simulated seismic hazard estimates are consistent. However, for sites within the basin and periods in the 1‐5 s range, GMMs significantly underestimate the hazard. The proposed basin amplification factors vary as a function of basin depth, reaching a geometric mean value as high as 4.5 at a 2‐s period, with respect to a reference site located just outside the basin. We evaluate the impact of the M9 simulations on tall reinforced concrete shear wall buildings, which are predominant in the region, by developing a suite of idealized structural systems that capture the strength and ductility intended by historical seismic design provisions in Canada. Ductility demands and collapse risk conditioned on the occurrence of the M9 simulations were found to exceed those associated with ground motion shaking intensities corresponding to the 975 and 2475‐year return periods, far exceeding the ～500‐year return period of M9 CSZ earthquakes.     

Northridge, California, earthquake of 1994: density of pipe breaks and surface strains

Empirical scaling equations are presented which relate the average number of water pipe breaks per km², , with the peak strain in the soil or intensity of shaking at the site. These equations are based on contour maps of peak surface strain evaluated from strong motion recordings, and observations of intensity of ground shaking and damage following the Northridge, California, earthquake of 17 January 1994. Histograms for the number of pipe breaks per km², n, are presented for several ranges of values of the horizontal peak strain and for several values of the site intensity. The observed distribution of pipe breaks is also used to speculate on possible more detailed geographical distribution of near surface strains in the San Fernando Valley and in central Los Angeles. The results can be used to predict number of pipe breaks in the San Fernando Valley and in Los Angeles, for a scenario earthquake or in probabilistic seismic hazard calculations, considering all possible scenarios that contribute to the hazard and the likelihood of their occurrence during specified exposure. Such predictions will be useful for emergency response planning (as the functioning of the water supply is critical for sanitation and in fighting fires caused by earthquakes), to estimate strains during future and past earthquakes in areas where no strong motion was recorded and in defining design guidelines for pipelines and other structures and structural systems sensitive to strains in the ground.     

Seismic rotational displacement of gravity walls by pseudo-dynamic method: Passive case

Prediction of the seismic rotational displacements of retaining wall under passive condition is an important aspect of design in earthquake prone region. In this paper, the pseudo-dynamic method is used to compute the rotational displacements of rigid retaining wall supporting cohesionless backfill under seismic loading for the passive earth pressure condition. The proposed method considers time, phase difference and effect of amplification in shear and primary waves propagating through both the backfill and the retaining wall. The influence of ground motion characteristics on rotational displacement of the wall is evaluated. Also the effects of variation of parameters like wall friction angle, soil friction angle, amplification factor, shear wave velocity, primary wave velocity, period of lateral shaking, horizontal and vertical seismic accelerations on the rotational displacements are studied. The rotational displacement of the wall increases substantially with increase in amplification of both shear and primary waves, time of input motion, period of lateral shaking and decreases with increase in soil friction angle, wall friction angle. The rotational displacements of the wall also increase when the effect of wall inertia is taken into account. Results are provided in graphical form.     

震动图快速生成系统研究

震动图（ShakeMap）是描绘地震发生之后地震动分布情况的一种工具,它显示的是地震产生的地面运动和可能的烈度破坏情况.本文的研究重点是如何实现用计算机自动绘制震动图,主要选择了峰值地面加速度等值图、仪器烈度分布图为研究对象.并对如何确定强震动质心、如何在台站稀疏地区估计地面运动加速度值、如何进行场地校正、如何将PGA/PGV转换为仪器烈度值做了深入的研究.     

2010年新西兰MW7.0主震与2011年MW6.1余震近场强地面运动特征比较分析

2010年9月3日16时35分46秒新西兰南岛Greendale附近发生了M_W7.0地震, 震源深度约10.0 km. 2011年2月21日新西兰南岛又发生了M_W6.1地震, 为2010年M_W7.0主震后最大的一次余震, 震源深度约5.0 km, 发震断层为Christchurch南约9 km一条近东西走向逆冲的隐伏断层, 该地震造成Christchurch城内多处建筑物严重损毁. 本文分析了2010年新西兰地震事件M_W7.0主震与M_W6.1余震强地面运动的特征. 新西兰M_W6.1余震近场强地面运动整体高于M_W7.0主震. 将主震和余震的强震观测记录分别与新一代衰减关系(NGA)进行对比, 发现余震强震观测数据整体高于其震级对应的NGA. 分别选取距离主震和余震震中最近且强震观测记录最高的两个台站(GDLC台站和HVSC台站)作为参照台站, 建立动态复合震源模型(DCSM)及有限断层随机振动模型(SFFM)进行强地面运动的模拟计算, 分析两种模型的模拟结果并对比二者的优势及局限, 以便在未来工作中更好地通过模型计算强地面运动特征, 实现区域化特征快速、 实时分析及局部重点、 细致分析相结合的目标.      

基于衰减关系的城镇烈度发生概率快速估计

震后快速产出的震动烈度分布是地震应急救援非常有效的依据, 通常由烈度与地震动参数的经验关系给出. 有台站的场点, 地震动参数可以直接由台站数据给出确定性的结果; 而无台站的场点, 地震动参数只能由衰减关系给出估计值. 目前我国台站覆盖有限, 且难于实时获取, 快速生成的地震动参数主要依赖于地震动衰减关系, 再依据烈度与地震动参数的经验关系, 输出确定性的震动烈度分布. 由于衰减关系本身存在着不确定性, 将其估计值用于生成确定性的震动烈度分布是不准确的. 而且实践证明, 震动烈度与实际调查烈度存在差异. 鉴于此, 从衰减关系模型中的ε出发, 提出了场点(城镇)遭遇不同烈度的概率计算方法: 利用衰减关系的估计值与衰减关系的标准差, 构造峰值加速度(PGA)变化的对数正态分布, 然后以烈度分档对应的PGA范围, 计算震区各城镇遭遇不同烈度的概率及各城镇抗震设防烈度被超越的概率. 具体以1966年3月8日河北邢台M_S6.8地震为例, 说明了此方法的可行性, 认为以概率形式给出城镇可能遭遇的烈度在表述上更为合理, 并建议将场点(城镇)遭遇烈度的概率表达方法用于震害快速评估.      

Mechanism and bounding of earthquake energy input to building structure on surface ground subjected to engineering bedrock motion

The mechanism of earthquake energy input to building structures is clarified by considering the surface ground amplification and soil–structure interaction. The earthquake input energies to superstructures, soil–foundation systems and total swaying–rocking system are obtained by taking the corresponding appropriate free bodies into account and defining the energy transfer functions. It has been made clear that, when the ground surface motion is white, the input energy to the swaying–rocking model is constant regardless of the soil property (input energy constant property). The upper bound of earthquake input energy to the swaying–rocking model is derived for the model including the surface ground amplification by taking full advantage of the above-mentioned input energy constant property and introducing the envelope function for the transfer function of the surface ground amplification. Extension of the theory to a general earthquake ground motion model at the engineering bedrock is also made by taking full advantage of the above-mentioned input energy constant property.