## Memory efficient simulation of frequency dependent Q

Kyle B. Withers, Kim B. Olsen, & Steven M. DayPublished July 1, 2015, SCEC Contribution #6002

Memory-variable methods have been widely applied to approximate frequency-independent Q in numerical simulation of wave propagation. The frequency-independent model is often appropriate for frequencies up to about 1 Hz, but at higher frequencies is inconsistent with some regional studies of seismic attenuation. We apply the memory-variable approach to frequency-dependent Q models that are constant below, and power-law above, a chosen transition frequency. We present numerical results for the corresponding memory-variable relaxation times and weights, obtained by non-negative least squares ﬁtting of the Q(f) function, for a range of exponent values; these times and weights can be scaled to arbitrary transition frequency and power-law prefactor, respectively. The resulting memory-variable formulation can be used with numerical wave-propagation solvers based on methods such as ﬁnite differences or spectral elements, and may be implemented in either conventional or coarse-grained form. In the coarse-grained approach, we ﬁt ‘effective’ Q for low Q values (< 200) using a nonlinear inversion technique and use an interpolation formula to ﬁnd the corresponding weighting coefficients for arbitrary Q. A 3D staggered-grid ﬁnite difference implementation closely approximates the frequency-wavenumber solution to both a half-space and layered model with a shallow dislocation source for Q as low as 20 over a bandwidth of two decades. We compare the effects of different power-law exponents using a ﬁnite-fault source model of the 2008 Mw 5.4 Chino Hills, CA, earthquake and ﬁnd that Q(f) models generally better ﬁt the strong motion data than constant Q models for frequencies above 1 Hz.

**Citation**

Withers, K. B., Olsen, K. B., & Day, S. M. (2015). Memory efficient simulation of frequency dependent Q. *Bulletin of the Seismological Society of America*, 105, 3129-3142.

**Related Projects & Working Groups**

CME, Ground-Motion Prediction