Applicability of time-to-failure analysis to accelerated strain before earthquakes and volcanic eruptions

We examine quantitatively the ranges of applicability of the equation Omega = A + B[1 - t/t(f)](m) for predicting 'system-sized' failure times t(f) in the Earth. In applications Omega is a proxy measure for strain or crack length, and A, B and the index m are model parameters determined by curve fitting. We consider constitutive rules derived from (a) Charles law for subcritical crack growth; (b) Voight's equation; and (c) a simple percolation model, and show in each case that this equation holds only when m < 0. When m > 0, the general solution takes the form Omega = A + B[1 + t/T](m), where T is a positive time constant, and no failure time can be defined. Reported values for Volcanic precursors based on rate data are found to be within the range of applicability of time-to-failure analysis (m < 0). The same applies to seismic moment release before earthquakes, at the expense of poor retrospective predictability of the time of the a posteriori-defined main shock. In contrast, reported values based on increasing cumulative Benioff strain occur in the region where a system-sized failure time cannot be defined (m > 0; commonly m approximate to 0.3). We conclude on physical grounds that cumulative seismic moment is preferred as the most direct measure of seismic strain. If cumulative Benioff strain is to be retained on empirical grounds, then it is important that these data either be re-examined with the independent constraint m < 0, or that for the case 0 < m + 1 < 1, a specific correction for the time-integration of cumulative data be applied, of the form Sigma Omega = At + B'{1 - [1 - t/t(f)](m+1)}.