Dynamic structural analysis often requires the selection of input ground motions with a target mean response spectrum. The variance of the target response spectrum is usually ignored or accounted for in an ad hoc manner, which can bias the structural response estimates. This manuscript proposes a computationally efficient and theoretically consistent algorithm to select ground motions that match the target response spectrum mean and variance. The selection algorithm probabilistically generates multiple response spectra from a target distribution, and then selects recorded ground motions whose response spectra individually match the simulated response spectra. A greedy optimization technique further improves the match between the target and the sample means and variances. The proposed algorithm is used to select ground motions for the analysis of sample structures in order to assess the impact of considering ground-motion variance on the structural response estimates. The implications for code-based design and performance-based earthquake engineering are discussed.

Ground motion models (or "attenuation relationships") describe the probability distribution of spectral acceleration at an individual period, given a set of predictor variables such as magnitude and distance, but they do not address the correlations between spectral acceleration values at multiple periods or orientations. Those correlations are needed for several calculations related to seismic hazard analysis and ground motion selection. Four NGA models and the NGA ground motion database are used here to measure these correlations, and predictive equations are fit to the results. The equations are valid for periods from 0.01 seconds to 10 seconds, versus similar previous equations that were valid only between 0.05 and 5 seconds and produced unreasonable results if extrapolated. Use of the new NGA ground motion database also facilitates a first study of correlations from intra-and inter-event residuals. Observed correlations are not sensitive to the choice of accompanying ground motion model, and intra-event, inter-event, and total residuals all exhibit similar correlation structure. A single equation is thus applicable for a variety of correlation predictions. A simple example illustrates the use of the proposed equations for one hazard analysis application.

SUMMARYRisk assessment of spatially distributed building portfolios or infrastructure systems requires quantification of the joint occurrence of ground-motion intensities at several sites, during the same earthquake. The ground-motion models that are used for site-specific hazard analysis do not provide information on the spatial correlation between ground-motion intensities, which is required for the joint prediction of intensities at multiple sites. Moreover, researchers who have previously computed these correlations using observed ground-motion recordings differ in their estimates of spatial correlation. In this paper, ground motions observed during seven past earthquakes are used to estimate correlations between spatially distributed spectral accelerations at various spectral periods. Geostatistical tools are used to quantify and express the observed correlations in a standard format. The estimated correlation model is also compared with previously published results, and apparent discrepancies among the previous results are explained.The analysis shows that the spatial correlation reduces with increasing separation between the sites of interest. The rate of decay of correlation typically decreases with increasing spectral acceleration period. At periods longer than 2 s, the correlations were similar for all the earthquake ground motions considered. At shorter periods, however, the correlations were found to be related to the local-site conditions (as indicated by site V s 30 values) at the ground-motion recording stations. The research work also investigates the assumption of isotropy used in developing the spatial correlation models. It is seen using the Northridge and Chi-Chi earthquake time histories that the isotropy assumption is reasonable at both long and short periods. Based on the factors identified as influencing the spatial correlation, a model is developed that can be used to select appropriate correlation estimates for use in practical risk assessment problems.

Assessment of seismic hazard using conventional probabilistic seismic hazard analysis (PSHA) typically involves the assumption that the logarithmic spectral acceleration values follow a normal distribution marginally. There are, however, a variety of cases in which a vector of ground-motion intensity measures are considered for seismic hazard analysis. In such cases, assumptions regarding the joint distribution of the ground-motion intensity measures are required for analysis. In this article, statistical tests are used to examine the assumption of univariate normality of logarithmic spectral acceleration values and to verify that vectors of logarithmic spectral acceleration values computed at different sites and/or different periods follow a multivariate normal distribution. Multivariate normality of logarithmic spectral accelerations are verified by testing the multivariate normality of interevent and intraevent residuals obtained from ground-motion models. The univariate normality tests indicate that both interevent and intraevent residuals can be well represented by normal distributions marginally. No evidence is found to support truncation of the normal distribution, as is sometimes done in PSHA. The tests for multivariate normality show that interevent and intraevent residuals at a site, computed at different periods, follow multivariate normal distributions. It is also seen that spatially distributed intraevent residuals can be well represented by the multivariate normal distribution. This study provides a sound statistical basis for assumptions regarding the marginal and joint distribution of ground-motion parameters that must be made for a variety of seismic hazard calculations.

SUMMARYProbabilistic seismic risk assessment for spatially distributed lifelines is less straightforward than for individual structures. While procedures such as the 'PEER framework' have been developed for risk assessment of individual structures, these are not easily applicable to distributed lifeline systems, due to difficulties in describing ground-motion intensity (e.g. spectral acceleration) over a region (in contrast to ground-motion intensity at a single site, which is easily quantified using Probabilistic Seismic Hazard Analysis), and since the link between the ground-motion intensities and lifeline performance is usually not available in closed form. As a result, Monte Carlo simulation (MCS) and its variants are well suited for characterizing ground motions and computing resulting losses to lifelines. This paper proposes a simulation-based framework for developing a small but stochastically representative catalog of earthquake ground-motion intensity maps that can be used for lifeline risk assessment. In this framework, Importance Sampling is used to preferentially sample 'important' ground-motion intensity maps, and K -Means Clustering is used to identify and combine redundant maps in order to obtain a small catalog. The effects of sampling and clustering are accounted for through a weighting on each remaining map, so that the resulting catalog is still a probabilistically correct representation. The feasibility of the proposed simulation framework is illustrated by using it to assess the seismic risk of a simplified model of the San Francisco Bay Area transportation network. A catalog of just 150 intensity maps is generated to represent hazard at 1038 sites from 10 regional fault segments causing earthquakes with magnitudes between five and eight. The risk estimates obtained using these maps are consistent with those obtained using conventional MCS utilizing many orders of magnitudes more ground-motion intensity maps. Therefore, the proposed technique can be used to drastically reduce the computational expense of a simulationbased risk assessment, without compromising the accuracy of the risk estimates. This will facilitate computationally intensive risk analysis of systems such as transportation networks. Finally, the study shows that the uncertainties in the ground-motion intensities and the spatial correlations between ground-motion intensities at various sites must be modeled in order to obtain unbiased estimates of lifeline risk.