The effect of thresholding on temporal avalanche statistics

Abstract

We discuss intermittent time series consisting of discrete bursts or avalanches separated bywaiting or silent times. The short time correlations can be understood to follow from theproperties of individual avalanches, while longer time correlations often present in suchsignals reflect correlations between triggerings of different avalanches. As onepossible source of the latter kinds of correlations in experimental time series, weconsider the effect of a finite detection threshold, due to e.g. experimental noise thatneeds to be removed. To this end, we study a simple toy model of an avalanche,a random walk returning to the origin or a Brownian bridge, in the presenceand absence of superimposed delta-correlated noise. We discuss the propertiesafter thresholding of artificial time series obtained by mixing toy avalanches andwaiting times from a Poisson process. Most of the resulting scalings for individualavalanches and the composite time series can be understood via random walktheory, except for the waiting time distributions when strong additional noise isadded. Then, to compare with a more complicated case we study the Mannasandpile model of self-organized criticality, where some further complicationsappear.