Statistical tests of the joint distribution of spectral acceleration values

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.