Maximum Likelihood Estimation of Strong‐Motion Attenuation Relationships

Strong‐motion attenuation relations are commonly derived from earthquake ground motion collected on triggered recorders. Parametric attenuation relations are estimated from these data using standard least squares methods. The variance of ground acceleration is relatively large and for any given earthquake, there is a distance range for which only stations with larger than average amplitudes will trigger recording. Consequently observed accelerations at these distances are higher than the mean ground acceleration and a bias may be introduced into an attenuation relation regression by non‐detection data censoring. Introduction of the log‐likelihood function as the objective function is a natural extension of current regression practices. Maximum likelihood estimation (MLE) of parametric forms fitted to data can be applied to peak acceleration data to lessen the bias introduced by the censoring of recordings by non‐triggering. In the absence of censoring information the MLE methods reduce to standard least squares estimation (LSE). An example is shown for peak acceleration data recorded from the Whittier Narrows earthquake of October 1, 1987. Roughly 33 percent of the CDMG stations located between 75 and 110 km from the epicenter did not trigger at the 1 percent g vertical acceleration level. No stations triggered beyond 115 km from the epicenter. A standard parametric form is fit to the vertical peak acceleration data using both LSE and MLE methods. It is found that inclusion of censoring data stabilizes the regression for attenuation parameters and that the MLE solution significantly differs from the LSE solution. For comparison, the LSE fit predicts a mean peak vertical acceleration of 1 percent g at 180 km while the MLE solution predicts a mean peak vertical acceleration of 1 percent g at only 115 km. By assuming a linear relationship between vertical and horizontal acceleration, a MLE fit for horizontal peak accelerations may also be determined.