  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 Benioff strain (BS) 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. A few years ago, on the basis of
a fit of the cumulative BS 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 (DSI). The observable
consequence of DSI 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 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.
