The effect of directivity on the stress parameter determined from ground motion observations

A key parameter in the stochastic model for predicting ground motions (Hanks and McGuire, 1981; Boore, 1983) is the stress parameter used in relating the corner frequency to the seismic moment. Because the high-frequency spectral level of an w-squared model goes as the stress parameter to the 2/3 power, the ground motions at frequencies above the corner frequency are particularly sensitive to this parameter. (Boore, 1983, shows that log p g a ~ 0.81ogAa, where pga is the peak ground acceleration and Aa is the stress parameter.) In practice, the stress parameter is determined by fitting observed ground motions, including peak acceleration, velocity, and response spectra, to the theoretical predictions. The observations are conveniently summarized by the equations obtained from regression of the ground motion data against magnitude and distance (see Joyner and Boore, 1988, for a recent review). As such, the regressions represent an average of the ground motion data for many take-off angles and azimuths from the earthquake source. To the extent that rupture propagation over the fault plane has influenced the spectra, the derived stress parameter has incorporated this directivity effect in some average way. The stochastic models in current use (e. g., Boore and Atkinson, 1987; Silva and Lee, 1987; Toro and McGuire, 1987) represent the source as a point, with no consideration of rupture over the fault plane. This can lead to different estimates of the stress parameter for the same set of earthquakes, depending on the type of waves used and their take-off angles from the source. To illustrate the problem, consider a set of dynamically similar earthquakes with rupture along the fault strike. The directivity will most strongly affect S waves leaving the source almost horizontally; the effect on teleseismic recordings of P waves will be minor. Thus without explicit consideration of directivity, it is likely that the estimates of the stress parameter from teleseismic P waves would be systematically different from those made using strong-motion recordings of the same set of earthquakes at near distances. This might explain Boore's (1986) finding that strong-motion S-wave data from the western United States led to a stress parameter of 50 bars, while teleseismic P-wave data from a globally distributed set of earthquakes were best fit by a stress parameter closer to 30 bars (although other explanations can be invoked, including regional variations in the stress parameter). The effect of directivity not only can produce systematic variations in estimates of a parameter related to the physics of the faulting process, it can also lead to errors in modeling of motions from an extended fault if the stress parameter obtained from the point-source stochastic model were used in the extended-fault model (such as that of Joyner et al., 1988). The effect of directivity would be included twice: implicitly in the azimuthally averaged ground motion data, and explicitly in the finite-fault calculations. In this note, we illustrate the effects of directivity on the derived stress parameter for two models of directivity. We do not intend the results in this note to be the definitive treatment of directivity; rather, our primary purpose in writing this note will be served if it draws attention to a shortcoming of the standard stochastic model and stimulates discussion and further work on the problem.