Open AccessA statistical evaluation of a ‘stress‐forecast’ earthquake
Author(s) -
Seher T.,
Main I. G.
Publication year - 2004
Publication title -
geophysical journal international
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0956-540X
DOI - 10.1111/j.1365-246x.2004.02186.x
Subject(s) - bayesian information criterion , goodness of fit , residual , mathematics , statistics , residual sum of squares , bayesian probability , data set , poisson distribution , event (particle physics) , least squares function approximation , explained sum of squares , algorithm , physics , non linear least squares , quantum mechanics , estimator
SUMMARY The goodness of fit for competing statistical models with different numbers of degrees of freedom cannot be assessed solely by the residual sum of squares, because more complex models will naturally have lower residuals. A standard approach to hypothesis testing for large data sets is to use the objective Bayesian information criterion (BIC), which penalizes models with larger numbers of free parameters appropriately. We apply this method to the analysis of time delays from data on seismic shear‐wave splitting in SW Iceland. The same data set has previously been used to estimate the time at which stress‐modified micro‐cracking reaches an inferred state of fracture criticality. The method does not forecast the location of the event, however, the time and magnitude were consistent with the actual occurrence of an M = 5 earthquake in the region. The forecast was based on a multi‐line model with 17 degrees of freedom (five straight lines, each with a start and end point, plus the variance, with four endpoints being fixed by the occurrence of a main shock), and a signal‐to‐noise ratio near unity. The BIC is used to assess this forecast in comparison with competing curve fits for Poisson, multiline, sinusoidal, or polynomial (truncated Taylor expansion) hypotheses. The null hypothesis of random occurrence can only be rejected formally for the sinusoidal model, implying cyclical recurrence with a period of 134.6 days. All other models we consider have a lower BIC. We also analyse the selected portion of the data set used to make the forecast of fracture criticality using Gaussian statistics. The 95 per cent confidence intervals on the predicted main shock magnitude range between magnitudes of 3.9 and 6.7. The time range indicated by the same confidence limits starts 42 days before the actual event; a clear end cannot be located. The relation between predicted magnitude and waiting time is not significantly different from that inferred from the background Gutenberg–Richter frequency–magnitude relation within the model uncertainties. Thus, it is not possible, based on the data, to formally reject the hypothesis that the magnitude 5 event was part of the normal background seismicity.