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A New Method for Estimation of the Attenuation Relationship with Variance Components

Institute of Statistical Science, Academia Sinica
128 Sec. 2 Academia Road
Nankang, Taipei
Taiwan 11529, R.O.C.
yhchen{at}stat.sinica.edu.tw
(Y.-H.C.)
Institute of Earth Sciences, Academia Sinica
128 Sec. 2 Academia Road
Nankang, Taipei
Taiwan 11529, R.O.C.
ctsai{at}earth.sinica.edu.tw
(C.-C.P.T.)

Manuscript received 23 July 2001.

A new approach to estimation of strong-motion attenuation relations with multiple variance components is proposed. In the terminology of Abrahamson and Youngs (1992), the attenuation relationship considered is a "mixed" model, where the regression coefficients are treated as the "fixed effects" and the components of deviations are treated as the "random effects." Following Dempster et al. (1981), an MLR procedure is adopted, which performs the maximum likelihood (ML) estimation of mixed models where the fixed effects are treated as random (R) effects with infinite variance. Both the fixed and random effects are estimated in a unified framework via an expectation-maximization (EM) algorithm. A modification of the MLR procedure is performed to accommodate nonlinear attenuation models. Compared with other methods, our proposal requires no additional regression or searching procedures and hence is neat and simple. Explicit formulae are provided for the variance estimates of the estimated model parameters. Two types of the strong-motion prediction are discussed: the unconditional prediction without accounting for the site-specific deviation, and the conditional prediction that further incorporates that deviation. A simulation study shows that the proposed procedure yields estimates with smallest biases and least computation time, compared with the EM procedure in Brillinger and Preisler (1985) and with the two-stage method in Joyner and Boore (1993). The new method is applied to a dataset of Taiwan's ground motions for illustration. This application reveals that the site-to-site variability in this dataset is remarkable; hence, the ergodic assumption ignoring "the spatial variability of ground motions" (Anderson and Brune, 1999) may not be suitable for probabilistic seismic hazard analyses in Taiwan. Also, in this application the conditional prediction is shown to be much more accurate than the unconditional prediction.

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