Trends and clustering in the onsets of volcanic eruptions

Abstract

We show how the trend renewal process (TRP) can be used to model the process of eruption onsets. Using combinations of power, exponential and sine functions, trends and/or cyclic activity can be fitted. On 14 example volcanoes, cases of constant activity level, increasing trend, wax and wane of activity, and cyclic plus trend behaviors were all observed. Embedded in the TRP is a renewal function for the detrended interonset times. Using a Weibull distribution, which can measure the degree of clustering/periodicity, we show that the widely observed tendency of eruption onsets to cluster may be best explained by trends in the activity, which is confirmed by a fractal analysis using the Hurst exponent. This means that compensating for extra parameters by means of the Akaike information criterion, the best fitted model for most of the volcanoes appears to be the nonhomogeneous Poisson process special case of the TRP. The TRP lends itself to analytically simple forecasting of the annual eruption probability (AEP) for the next eruption. This AEP can exhibit considerable structure with certain trend functions, but example data from Etna show how the TRP can automatically act to compensate the AEP against incomplete observation by fitting an increasing trend.