5:15 PM - 7:15 PM
[SSS11-P22] A Study on the High-Frequency Shape of the Acceleration Source Spectrum Based on a Revised Haskell Model
Keywords:revised Haskell model, acceleration source spectrum, corner frequency, fmax
The source model proposed by Haskell (1964), named Haskell model, is a simple model in which rupture propagates unilaterally along a rectangular fault, and the slip velocity function at each point on the fault is uniform and given by a boxcar function. This model has served as a fundamental basis for source theory up to the present day. Geller (1976) examined the source spectrum of far-field S-wave predicted by the Haskell model and demonstrated that its shape is characterized by the three corner frequencies fcL, fcW, and fcr, which correspond to the fault length, fault width, and rise time, respectively, and that the overall spectrum follows the omega-cubed model. Since the observed source spectrum generally follows the omega-square model proposed by Aki (1967), modern strong motion researchers regard the Haskell model as an inadequate representation of the physics of earthquake sources.
However, the above argument is based on simplistic assumptions about space-time slip distribution and may be leading to physically incorrect conclusions. Dynamic studies such as those by Day (1982) have shown that (1) the rise time is not uniform across the fault plane, and (2) the slip velocity function does not take the shape of a boxcar function. Regarding (1), as pointed out by Nozu (2004), considering the heterogeneity of the rise time across the fault plane is expected to shift fcr to higher frequencies. Regarding (2), it is expected that not only will fcr shift to higher frequencies, but also that the slope of the acceleration source spectrum in the high-frequency range beyond fcr will be smaller than -1 on a log-log scale. To confirm these hypotheses, this study reformulated the Haskell model with a more realistic space-time slip distribution and examined the shape of the resulting acceleration source spectrum.
As a specific example, we theoretically computed the acceleration source spectrum under the following conditions: the fault length was 5.0 km, the fault width was 3.0 km, the S-wave velocity in the source region was 3.5 km/s, and the rupture propagation velocity was 3.0 km/s. The angle between the normal vector of the fault plane and the vector toward the observation point was set to 45 degrees. The angle between the fault-length direction vector and the vector toward the observation point was set to 90 degrees. For (1), instead of using the theoretical formula of Day (1982), we adopted the equation from Kataoka et al. (2003). For (2), we used the Regularized Yoffe function by Tinti et al. (2005), which reflects dynamic characteristics of slip velocity. The obtained acceleration source spectrum is shown in the figure. From the low-frequency side, fcL and fcW appear, and the spectrum remains flat between fcW and fcr. This indicates that the spectrum follows an omega-square model with two corner frequencies for frequencies lower than fcr. Beyond fcr, the spectrum decays steeply with a slope of -2.5 on a log-log scale, and fcr appears similar to fmax, as pointed out by Hanks (1982).
Regarding the observed fmax, Anderson and Hough (1984) assert the existence of site-controlled fmax, and this view has been widely supported in subsequent studies. On the other hand, although less common, studies such as those by Papageorgiou and Aki (1983) and Gusev (1983) assert the existence of source-controlled fmax. Gusev and Guseva (2016) acknowledged the existence of site-controlled fmax but also argued that the observed acceleration source spectrum exhibits three corner frequencies, with the highest one corresponding to a source-controlled fmax. The acceleration source spectrum obtained in this study is in good agreement with this characteristic.
In this study, we revised the slip velocity function of the Haskell model. The obtained more realistic theoretical acceleration source spectrum is consistent with the shapes observed in previous studies. This suggests that our findings theoretically support the existence of a source-controlled fmax.
However, the above argument is based on simplistic assumptions about space-time slip distribution and may be leading to physically incorrect conclusions. Dynamic studies such as those by Day (1982) have shown that (1) the rise time is not uniform across the fault plane, and (2) the slip velocity function does not take the shape of a boxcar function. Regarding (1), as pointed out by Nozu (2004), considering the heterogeneity of the rise time across the fault plane is expected to shift fcr to higher frequencies. Regarding (2), it is expected that not only will fcr shift to higher frequencies, but also that the slope of the acceleration source spectrum in the high-frequency range beyond fcr will be smaller than -1 on a log-log scale. To confirm these hypotheses, this study reformulated the Haskell model with a more realistic space-time slip distribution and examined the shape of the resulting acceleration source spectrum.
As a specific example, we theoretically computed the acceleration source spectrum under the following conditions: the fault length was 5.0 km, the fault width was 3.0 km, the S-wave velocity in the source region was 3.5 km/s, and the rupture propagation velocity was 3.0 km/s. The angle between the normal vector of the fault plane and the vector toward the observation point was set to 45 degrees. The angle between the fault-length direction vector and the vector toward the observation point was set to 90 degrees. For (1), instead of using the theoretical formula of Day (1982), we adopted the equation from Kataoka et al. (2003). For (2), we used the Regularized Yoffe function by Tinti et al. (2005), which reflects dynamic characteristics of slip velocity. The obtained acceleration source spectrum is shown in the figure. From the low-frequency side, fcL and fcW appear, and the spectrum remains flat between fcW and fcr. This indicates that the spectrum follows an omega-square model with two corner frequencies for frequencies lower than fcr. Beyond fcr, the spectrum decays steeply with a slope of -2.5 on a log-log scale, and fcr appears similar to fmax, as pointed out by Hanks (1982).
Regarding the observed fmax, Anderson and Hough (1984) assert the existence of site-controlled fmax, and this view has been widely supported in subsequent studies. On the other hand, although less common, studies such as those by Papageorgiou and Aki (1983) and Gusev (1983) assert the existence of source-controlled fmax. Gusev and Guseva (2016) acknowledged the existence of site-controlled fmax but also argued that the observed acceleration source spectrum exhibits three corner frequencies, with the highest one corresponding to a source-controlled fmax. The acceleration source spectrum obtained in this study is in good agreement with this characteristic.
In this study, we revised the slip velocity function of the Haskell model. The obtained more realistic theoretical acceleration source spectrum is consistent with the shapes observed in previous studies. This suggests that our findings theoretically support the existence of a source-controlled fmax.