9:15 AM - 9:30 AM
[SSS06-02] Quantifying complexity of observed and synthetic moment rate functions
Keywords:Earthquake source process, Moment rate function, Rupture complexity
Moment Rate Functions (MRFs) estimated for almost every mid-to-large earthquake by seismic inversion analyses are essential to interpret the complexity of their faulting processes and the cause of strong ground motions. In some cases, they show significant fluctuation corresponding to growth, acceleration, deceleration, and re-activation of the coseismic slip, while others are resolved as relatively smooth bell-shaped functions. Moreover, some empirical laws on MRFs have widely been known: a scaling between their moment and duration, ω-2-type spectral shapes, and the GR law. Recently, MRFs have been routinely obtained by some automatic (or at least unified) algorithms (e.g., Vallée et al. 2010; Ye et al. 2018), enabling us to investigate MRFs' statistics, including their averages and variabilities. For example, to quantify the variability of rupture complexity of earthquakes, Kanamori&Rivera (2004) and Ye et al. (2018) defined Moment Rate Function Roughness, γ, as the L2-norm of the moment acceleration function normalized by their theoretical minimum value given their duration, which results in higher values for more fluctuated MRFs. Thanks to the introduction of γ, we can now discuss the statistics of rupture complexity. Thus, we are interested in an empirical frequency distribution of γ and its theoretical modeling.
First, we calculate γ for automatically inverted MRFs in a database (SCARDEC by Vallée et al. 2010) except for too much complicated events showing re-activation after complete stop and fit their distribution with parametric probability density functions. We find that the log-normal distribution with γ=1.73−4.62 as the 25th to 75th percentile well describes the empirical one (note that 1≦γ by definition, and for example, γ=1.68 for the 2016 Kumamoto earthquake, γ=3.46 for the 2004 Mid-Niigata earthquake, and γ=5.17 for the 1999 Chi-Chi earthquake). We also confirm that clipping the initial and final part of MRFs, which might be poorly resolved due to their small amplitude and the coda waves, significantly reduces the γ-value. Furthermore, the MRFs estimated by seismic inversions lack high-frequency content due to the filtering of the observed seismograms, also resulting in some underestimation of the γ-value. Due to both above, the estimated log-normal distribution may give some lower bound of the true distribution.
Next, we reproduce the log-normal distribution based on a synthetic MRF model. Hirano (2022) proposed such a model using a stochastic differential equation to generate random MRFs and showed that the generated functions satisfy multiple empirical laws on MRFs enumerated above (scaling, ω-2-model, and the GR-law). The stochastic model includes a parameter corresponding to the ratio of the two corner frequencies, and their γ-values follow the log-normal distribution independent of the frequency ratio. However, the empirical distribution is biased into the lower value of γ compared to the theoretically obtained distribution; this would be reasonable given that the former is some lower bound of the true distribution.
References
・Vallée et al. (2010). SCARDEC: a new technique for the rapid determination of seismic moment magnitude, focal mechanism and source time functions for large earthquakes using body-wave deconvolution. Geophys. J. Int. doi:10.1111/j.1365-246x.2010.04836.x
・Ye et al. (2018). Global variations of large megathrust earthquake rupture characteristics. Sci. Adv. doi:10.1126/sciadv.aao4915
・Kanamori & Rivera (2004). Static and Dynamic Scaling Relations for Earthquakes and Their Implications for Rupture Speed and Stress Drop. Bull. Seism. Soc. Am. doi:10.1785/0120030159
・Hirano, S. (2022). Source time functions of earthquakes based on a stochastic differential equation. Sci. Rep. doi:10.1038/s41598-022-07873-2
First, we calculate γ for automatically inverted MRFs in a database (SCARDEC by Vallée et al. 2010) except for too much complicated events showing re-activation after complete stop and fit their distribution with parametric probability density functions. We find that the log-normal distribution with γ=1.73−4.62 as the 25th to 75th percentile well describes the empirical one (note that 1≦γ by definition, and for example, γ=1.68 for the 2016 Kumamoto earthquake, γ=3.46 for the 2004 Mid-Niigata earthquake, and γ=5.17 for the 1999 Chi-Chi earthquake). We also confirm that clipping the initial and final part of MRFs, which might be poorly resolved due to their small amplitude and the coda waves, significantly reduces the γ-value. Furthermore, the MRFs estimated by seismic inversions lack high-frequency content due to the filtering of the observed seismograms, also resulting in some underestimation of the γ-value. Due to both above, the estimated log-normal distribution may give some lower bound of the true distribution.
Next, we reproduce the log-normal distribution based on a synthetic MRF model. Hirano (2022) proposed such a model using a stochastic differential equation to generate random MRFs and showed that the generated functions satisfy multiple empirical laws on MRFs enumerated above (scaling, ω-2-model, and the GR-law). The stochastic model includes a parameter corresponding to the ratio of the two corner frequencies, and their γ-values follow the log-normal distribution independent of the frequency ratio. However, the empirical distribution is biased into the lower value of γ compared to the theoretically obtained distribution; this would be reasonable given that the former is some lower bound of the true distribution.
References
・Vallée et al. (2010). SCARDEC: a new technique for the rapid determination of seismic moment magnitude, focal mechanism and source time functions for large earthquakes using body-wave deconvolution. Geophys. J. Int. doi:10.1111/j.1365-246x.2010.04836.x
・Ye et al. (2018). Global variations of large megathrust earthquake rupture characteristics. Sci. Adv. doi:10.1126/sciadv.aao4915
・Kanamori & Rivera (2004). Static and Dynamic Scaling Relations for Earthquakes and Their Implications for Rupture Speed and Stress Drop. Bull. Seism. Soc. Am. doi:10.1785/0120030159
・Hirano, S. (2022). Source time functions of earthquakes based on a stochastic differential equation. Sci. Rep. doi:10.1038/s41598-022-07873-2