Is Earthquake Rupture Deterministic?

Richard M Allen and Erik L Olson (University of Wisconsin, Madison)


Understanding the earthquake rupture process is key to our understanding of fault systems and earthquake hazards. Over the past 15 years, multiple hypotheses concerning the nature of fault rupture have been proposed but no unifying theory has emerged. The conceptual hypothesis most commonly cited is the cascade model for fault rupture. In the cascade model, slip initiates on a small fault patch and continues to rupture further across a fault plane as long as the conditions are favorable. Two fundamental implications of this domino-like theory are that small earthquakes begin in the same manner as large earthquakes, and that the rupture process is not deterministic, i.e., the size of the earthquake cannot be determined until the cessation of rupture. Here we show that the frequency content of radiated seismic energy within the first few seconds of rupture scales with the final magnitude of the event. Therefore the magnitude of an earthquake can be estimated before the rupture is complete. This finding implies that the rupture process is to some degree deterministic and has far-reaching implications for the physics of the rupture process.

$\tau _p$ and $\tau _d$ observations

The frequency content of P-wave arrivals is measured through the parameter $\tau _p$ (Allen and Kanamori, 2003; Olson and Allen, 2005). We calculate $\tau _p$ in a recursive fashion from a vertical velocity timeseries to generate $\tau _p$ as a function of time, $\tau _{p}(t)$. Figure 4.1 shows the vertical velocity waveform recorded during a $M_{L}$ 4.6 earthquake in southern California and the $\tau _p$ timeseries derived from it. Figure 4.2 shows a similar example but for the $M_{w}$ 8.3 Tokachi-oki earthquake. In this case, only acceleration records are available, which have been recursively integrated in a causal fashion to derive the velocity trace from which $\tau _{p}(t)$ is derived. We define the parameter $\tau _{p}^{max}$ as the maximum $\tau _{p}(t)$ data point between 0.05 and 4.0 sec after the P-wave trigger as shown in Figures 4.1 and 4.2.

Figure 4.1: Example waveform and $\tau _{p}$ calculation for a $M$ 4.6 earthquake in southern California recorded at station GSC 74 km from the epicenter. a) The raw vertical component waveform recorded by a broadband velocity sensor. b) Ten seconds of the velocity waveform after low-pass filtering at 3 Hz. The P-wave trigger time is shown by the vertical line at 13.01 sec. c) $\tau _{p}(t)$ trace calculated in a recursive fashion from the waveform in b showing the change in the frequency content from the pre-trigger noise to the post-trigger P-wave. The $\tau _{p}^{max}$ observation is circled (equal to 0.86 sec in this case), $\tau _{d}$ is the delay of $\tau _{p}^{max}$ with respect to the trigger (0.43 sec in this case).
\epsfig{file=rallen06_1_1.eps, width=8cm}\end{center}\end{figure}

Figure 4.2: Example waveform and $\tau _{p}^{max}$ calculation for the $M_{w}$ 8.3 Tokachi-oki earthquake recorded at station HKD112 71 km from the epicenter. a) The raw vertical component waveform recorded on an accelerometer. b) Ten seconds of the raw acceleration waveform. The P-wave trigger is shown by the vertical line at 35.41 sec. c) Ten seconds of the velocity waveform determined from the acceleration recording using recursive relations only. It has also been low-pass filtered at 3 Hz. d) $\tau _{p}(t)$ trace calculated in a recursive fashion from the waveform in c. The $\tau _{p}^{max}$ observation is circled ( $\tau _{p}^{max}$ = 1.62 sec, $\tau _{d}$ = 1.49 sec), it has a longer period and is observed later than the example in Figure 4.1c due to the larger magnitude of the earthquake.
\epsfig{file=rallen06_1_2.eps, width=8cm}\end{center}\end{figure}

A total of 71 earthquakes producing 1,842 waveforms recorded within 100 km are used in this study. When $\tau _{p}^{max}$ is plotted against $M$ on a log-linear scale, a scaling relation emerges as shown in Figure 4.3a. The $\tau _{p}^{max}$ observations from waveforms at individual stations can exhibit large variability for a single earthquake, which is likely due to measurement error, station and path effects (Lockman and Allen, 2005). Figure 4.3a shows the average $\tau _{p}^{max}$ observation for each earthquake using all available data. The best-fit linear relation to the event averages is $log \tau_{p}^{max} = 0.14M - 0.83$, and the average absolute deviation is 0.54 magnitude units. The dataset has a high linear correlation coefficient of 0.9. Although there is variability in $\tau _{p}^{max}$ observations equivalent to $\pm$ 1 magnitude unit, a scaling relation is clear, implying that information about the final magnitude of an earthquake is available within the first few seconds of its initiation irrespective of the total rupture duration.

Figure 4.3: The relation between $\tau _{p}^{max}$, $\tau _{d}$ and magnitude. a) The scaling relation between event averaged $\tau _{p}^{max}$ and magnitude for earthquakes in southern California (triangles), Japan (circles), Taiwan (squares) and Alaska (star). Observations at individual stations can show large scatter for a given event (small grey shapes). The event averaged values are also shown (large black symbols). The best fit line to the averages is also shown (solid). The average absolute deviation of the observations is 0.54 magnitude units; plus and minus two times the average absolute deviation is shown as dashed lines. b) $\tau _{d}$ plotted against magnitude showing the general increase in the time required to make the $\tau _{p}^{max}$ observation with increasing magnitude. The symbols are the same as in a. Grey shapes show individual station observations, black is the event average. The thick grey bar shows the approximate rupture duration as a function of magnitude indicating that the $\tau _{p}^{max}$ observation is made before rupture ceases for all $M$ $>$ 4 earthquakes in this study. The thin grey line indicates one tenth the rupture duration and is crossed by the trend of the $\tau _{d}$ data.
\epsfig{file=rallen06_1_3.eps, width=16cm}\end{center}\end{figure*}

A second parameter, $\tau _{d}$, is also measured from the $\tau _{p}$ timeseries. $\tau _{d}$ is the delay of the $\tau _{p}^{max}$ observation with respect to the P-wave trigger and is, therefore, in the range of 0.05 to 4 sec (see Figures 4.1 and 4.2). Figure 4.3b plots the event-averaged $\tau _{d}$ observations versus magnitude, which shows a general increase in $\tau _{d}$ with magnitude. Also indicated on Figure 4.3b is the typical rupture duration as a function of magnitude. The relation shown is only approximate as the rupture duration for a given magnitude event can vary by a factor of 2 or 3. Despite the uncertainty in the rupture duration of the specific earthquakes included in this study, it is clear that for earthquakes with $M$ $>$ 4 the $\tau _{p}^{max}$, observation is made before the rupture has ceased. While up to 4 sec of data are used to determine $\tau _{p}^{max}$, the average time window of the P-wave required to determine $\tau _{p}^{max}$ is less than 2 sec for almost all earthquakes in our study.


While there is a 1 magnitude unit scatter in the $\tau _{p}^{max}$ data, the observations show that the rupture process is at least partly deterministic, i.e. the final magnitude of an event is to some degree controlled by processes within the first few seconds (typically $<$ 2 sec) of rupture. The scatter could be due to source processes and/or local site and measurement errors. If the scatter is non-source related, then removal or correction for site and path effects could reduce the scatter in the data points of Figure 4.3a to a single line, implying that the final magnitude of an earthquake is entirely determined within the first few seconds of rupture. Variability in the quality of the $\tau _{p}^{max}$ observations at different stations has already been observed, indicating that site effects do play a role (Lockman and Allen, 2005). Nevertheless, it seems unlikely that all the scatter is due only to site effects. Instead, source related processes including rupture behavior, stress heterogeneity and other on-fault variability probably also play a role.

We propose that the final magnitude of an earthquake is partially controlled by the initiation process within the first few seconds of rupture, and partially by the physical state of the surrounding fault plane. The role played by the initiation process can be understood by considering the energy balance of fault rupture. A rupture can only propagate when the available energy is sufficient to supply the necessary fracture energy (Nielsen and Olsen, 2000; Oglesby and Day, 2002). When a propagating fracture encounters a patch with a lower stress-drop, the total energy in the system will begin to decrease. Depending on the size of the patch, it may cause the rupture to terminate. The total rupture energy available increases with the amount of slip, so a large-slip rupture will propagate further across a heterogeneous fault plane. Therefore, if the rupture pulse initiates with large slip, it is more likely to evolve into a large earthquake. This explanation is consistent with the observation that large earthquakes do not nucleate at shallow depths, but instead at greater depths where the frictional strength and stress drop are greater (Das and Scholz, 1983). A recent study (Mai et al., 2005) also shows that hypocenters are preferentially located within or close to regions of large slip.


We thank Hiroo Kanamori and Stefan Nielsen for helpful discussions, and Yih-Min Wu and Roger Hansen for making waveform data available for the study. Funding for this work was provided by USGS/NEHRP awards 03HQGR0043 and 05HQGR0074.


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Mai, P. M., P. Spudich and J. Boatwright. Hypocenter locations in finite-source rupture models, Bull. seism. Soc. Am., 95, 965-980, 2005.

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