Re-examination 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 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.Comment: LaTeX, JGR preprint with AGU++ v16.b and AGUTeX 5.0, use packages graphicx and psfrag, 23 eps figures, 17 pages. In press J. Geophys. Re

Oscillatory regimes of the thermomagnetic instability in superconducting films

The stability of superconducting films with respect to oscillatory precursor modes for thermomag- netic avalanches is investigated theoretically. The results for the onset threshold show that previous treatments of non-oscillatory modes have predicted much higher thresholds. Thus, in film supercon- ductors, oscillatory modes are far more likely to cause thermomagnetic breakdown. This explains the experimental fact that flux avalanches in film superconductors can occur even at very small ramping rates of the applied magnetic field. Closed expressions for the threshold magnetic field and temperature, as well oscillation frequency, are derived for different regimes of the oscillatory thermomagnetic instability.Comment: 5 pages, 5 figure

Dendritic flux avalanches in rectangular superconducting films -- numerical simulations

Dendritic flux avalanches is a frequently encountered instability in the vortex matter of type II superconducting films at low temperatures. Previously, linear stability analysis has shown that such avalanches should be nucleated where the flux penetration is deepest. To check this prediction we do numerical simulations on a superconducting rectangle. We find that at low substrate temperature the first avalanches appear exactly in the middle of the long edges, in agreement with the predictions. At higher substrate temperature, where there are no clear predictions from the theory, we find that the location of the first avalanche is decided by fluctuations due to the randomly distributed disorder.Comment: 3 pages, 2 figure

Magnetostrictive behaviour of thin superconducting disks

Flux-pinning-induced stress and strain distributions in a thin disk superconductor in a perpendicular magnetic field is analyzed. We calculate the body forces, solve the magneto-elastic problem and derive formulas for all stress and strain components, including the magnetostriction $\Delta R/R$. The flux and current density profiles in the disk are assumed to follow the Bean model. During a cycle of the applied field the maximum tensile stress is found to occur approximately midway between the maximum field and the remanent state. An effective relationship between this overall maximum stress and the peak field is found.Comment: 8 pages, 6 figures, submitted to Supercond. Sci. Technol., Proceed. of MEM03 in Kyot

Artifactual log-periodicity in finite size data: Relevance for earthquake aftershocks

The recently proposed discrete scale invariance and its associated log-periodicity are an elaboration of the concept of scale invariance in which the system is scale invariant only under powers of specific values of the magnification factor. We report on the discovery of a novel mechanism for such log-periodicity relying solely on the manipulation of data. This ``synthetic'' scenario for log-periodicity relies on two steps: (1) the fact that approximately logarithmic sampling in time corresponds to uniform sampling in the logarithm of time; and (2) a low-pass-filtering step, as occurs in constructing cumulative functions, in maximum likelihood estimations, and in de-trending, reddens the noise and, in a finite sample, creates a maximum in the spectrum leading to a most probable frequency in the logarithm of time. We explore in detail this mechanism and present extensive numerical simulations. We use this insight to analyze the 27 best aftershock sequences studied by Kisslinger and Jones [1991] to search for traces of genuine log-periodic corrections to Omori's law, which states that the earthquake rate decays approximately as the inverse of the time since the last main shock. The observed log-periodicity is shown to almost entirely result from the ``synthetic scenario'' owing to the data analysis. From a statistical point of view, resolving the issue of the possible existence of log-periodicity in aftershocks will be very difficult as Omori's law describes a point process with a uniform sampling in the logarithm of the time. By construction, strong log-periodic fluctuations are thus created by this logarithmic sampling.Comment: LaTeX, JGR preprint with AGU++ v16.b and AGUTeX 5.0, use packages graphicx, psfrag and latexsym, 41 eps figures, 26 pages. In press J. Geophys. Re