# Is Earthquake Rupture Deterministic?

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

## Introduction

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.

## and observations

The frequency content of P-wave arrivals is measured through the parameter (Allen and Kanamori, 2003; Olson and Allen, 2005). We calculate in a recursive fashion from a vertical velocity timeseries to generate as a function of time, . Figure 4.1 shows the vertical velocity waveform recorded during a 4.6 earthquake in southern California and the timeseries derived from it. Figure 4.2 shows a similar example but for the 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 is derived. We define the parameter as the maximum data point between 0.05 and 4.0 sec after the P-wave trigger as shown in Figures 4.1 and 4.2.

A total of 71 earthquakes producing 1,842 waveforms recorded within 100 km are used in this study. When is plotted against on a log-linear scale, a scaling relation emerges as shown in Figure 4.3a. The 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 observation for each earthquake using all available data. The best-fit linear relation to the event averages is , 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 observations equivalent to 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.

A second parameter, , is also measured from the timeseries. is the delay of the 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 observations versus magnitude, which shows a general increase in 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 4 the , observation is made before the rupture has ceased. While up to 4 sec of data are used to determine , the average time window of the P-wave required to determine is less than 2 sec for almost all earthquakes in our study.

## Discussion

While there is a ±1 magnitude unit scatter in the 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 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.

## Acknowledgements

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.

## References

Allen, R. M. and H. Kanamori, The potential for earthquake early warning in southern California, Science, 300, 786-789, 2003.

Das, S. and C. H. Scholz. Why large earthquakes do not nucleate at shallow depths, Nature, 305, 621-623, 1983.

Lockman, A. and R. M. Allen. Single station earthquake characterization for early warning, Bull. seism. Soc. Am., 95, 2029-2039, 2005.

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

Nielsen, S. B. and K. B. Olsen. Constraints on stress and friction from dynamic rupture models of the 1994 Northridge, California, earthquake, Pure And Applied Geophysics, 157, 2029-2046, 2000.

Oglesby, D. D. and S. M. Day, Stochastic fault stress: Implications for fault dynamics and ground motion, Bull. seism. Soc. Am., 92, 3006-3021, 2002.

Olson, E. and R. M. Allen, The deterministic nature of earthquake rupture, Nature, 438, 212-215, 2005.

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