Reexamination of log periodicity observed in the seismic precursors of the 1989 Loma Prieta earthquake

Based on several empirical evidence, a series of papers has advocated the concept that seismicity prior to a large earthquake can be understood in terms of the statistical physics of a critical phase transition. In this model, the cumulative seismic Benioff strain release ∈ increases as a power law time-to-failure before the final event. This power law reflects a kind of scale invariance with respect to the distance to the critical point: ∈ is the same up to a simple reseating λz after the time-to-failure has been scaled by a factor λ. A few years ago, on the basis of a fit of the cumulative Benioff strain released prior to the 1989 Loma Prieta earthquake, Sornette and Sammis [1995] proposed that this scale invariance could be partially broken into a discrete scale invariance, defined such that the scale invariance occurs only with respect to specific integer powers of a fundamental scale ratio. The observable consequence of discrete scale invariance takes the form of log-periodic oscillations decorating the accelerating power law. They found that the quality of the fit and the predicted time of the event are significantly improved by the introduction of log periodicity. Here we present a battery of synthetic tests performed to quantify the statistical significance of this claim. We put special attention to the definition of synthetic tests that are as much as possible identical to the real time series except for the property to be tested, namely, log periodicity. Without this precaution, we would conclude that the existence of log periodicity in the Loma Prieta cumulative Benioff strain is highly statistically significant. In contrast, we find that log-periodic oscillations with frequency and regularity similar to those of the Loma Prieta case are very likely to be generated by the interplay of the low-pass filtering step due to the construction of cumulative functions together with the approximate power law acceleration. Thus the single Loma Prieta case alone cannot support the initial claim, and additional cases and further study are needed to increase the signal-to-noise ratio, if any. The present study will be a useful methodological benchmark for future testing of additional events when the methodology and data to construct reliable Benioff strain function become available.