First-passage probability estimation by Poisson branching process model

Abstract First-passage failure probability, i.e. the probability that a stochastic process crosses a prescribed threshold level at least once during a finite duration, is an important performance measure of a system under stochastic excitations. However, accurate estimation of the first-passage probability is a challenging task due to the difficulty of incorporating the statistical dependence between crossing events. To overcome this challenge, especially for stationary Gaussian processes, this paper proposes a new method which employs the Poisson Branching Process (PBP) model to effectively describe the statistical dependence between crossings. The PBP-based modeling is motivated from inspection of narrowband responses in which the crossing events generally form clusters. In the proposed method, each cluster of crossings is modelled by a forefront crossing event, namely initiating crossing, and the subsequent crossings in the same cluster. The initiating crossing is modeled as a Poisson point process while each of the subsequent crossings is assumed to branch out from a precedent crossing with a fixed probability, which is termed consecutive crossing probability. This probability can be obtained by numerically integrating the joint probability density of the neighboring extrema values of the process. A new derivation of this joint distribution is proposed based on theories of random vibrations. Finally, the first-passage probability is obtained by deriving the occurrence probability of the initiating crossings in terms of the consecutive crossing probability. The numerical examples demonstrate that the proposed formulation provides more accurate estimations of first-passage probability than existing approaches for stochastic processes having various shapes of power spectral density functions.