Accelerations at a site resulting from earthquakes of a given magnitude and distance are commonly assumed to be lognormally distributed with standard deviation σ (where σ is independent of magnitude and distance. Significantly higher accelerations may be predicted for a given return period when acceleration variability is taken into account than when only median acceleration values are used in seismic hazard calculations. If the magnitude range is not restricted, the acceleration having a given return period increases from ao, obtained using median values, to aoexp(βσ2/2c2), when acceleration variability is included. This result assumes the attenuation function is of the form loge(a) = c1 + c2m + f(R) [where f(R) is a function of distance only], and the magnitude-frequency relationship is loge(N) = α − βm. In this case, the acceleration for a return period is best approximated by using, rather than the median value for each magnitude and distance, the acceleration that is a factor exp(kσ) greater than the median, where k = βσ/2c2.
For a restricted magnitude range, mmin ≦ m ≦ mmax, using only median values limits the calculated accelerations at a site to a range amin ≦ a ≦ amax. If variability is included, the acceleration range is no longer limited; the return period of an acceleration near amin increases, while that of an acceleration near amax decreases. If the distribution of accelerations is truncated at nσ, the maximum acceleration at a site will be amaxexp(nσ). Half or more of the increase in the acceleration level for a return period obtained by including acceleration variability may result from accelerations that are greater than 1.5σ to 2.0σ above the median value. Including variability and using a finite maximum magnitude may give a higher acceleration for a fixed return period than the value calculated using median values and an infinite maximum magnitude.