Peak Ground Acceleration Estimation by Linear and Nonlinear Models with Reduced Order Monte Carlo Simulation

Estimation of the peak ground acceleration (PGA) is one of the main tasks in civil and earthquake engineering practice since it is an important factor for the design spectrum. The Boore-Joyner-Fumal (BJF) and the Crouse-McGuire formula are well-known empirical models by estimating the PGA with the magnitude of earthquake, the fault-to-site distance, and the site foundation properties. It is obvious that a predictive model class with more effective free parameters often fit the data better. However, this does not imply that the complicated formula is more realistic since overfitting may happen when the formula has too many free parameters. In this article, 32 linear and 16 nonlinear predictive model classes are constructed and investigated. The Bayesian model class selection approach is utilized to obtain the most suitable predictive model class for the seismic attenuation formula. In this approach, each predictive model class is evaluated by the plausibility conditional on the data and it is proportional to the evidence which involves a high-dimensional integral. This integral has closed-form solution for the linear model classes. Analytic work was done to improve the original asymptotic expansion in this study. For the nonlinear model classes, the evidence integral can be reduced to two-dimensional and then Monte Carlo simulation is utilized to evaluate the double integral. The most plausible model class is robust in the sense that it balances between the data-fitting capability and the sensitivity to noise. A database of 266 strong-motion records, obtained from the China Earthquake Data Center, is utilized for the analysis. The most plausible predictive model class and its updated model parameters are determined. It turns out that the most plausible model class is generally simpler than the full BJF empirical formula. In the case where no single model class has dominant plausibility, one can utilize the multi-model predictive formula that is a plausibility-weighted average of the prediction of different predictive models.