Outcrossing of Stationary Gaussian Process from 2D Elliptical Region

Abstract

An exact expression for the mean outcrossing rate of a Gaussian vector process from two‐dimensional elliptical regions is derived. The expression is exact under the assumption that the cross correlation between process component 1 and velocity component 2 is zero, and it generally yields a good approximation for nonzero cross correlations. A correction for a nonzero cross correlation is shown to be vanishing to first order in the cross‐correlation coefficient. Closed‐form expressions are derived for the ensemble‐averaged rate for various types of uncertain failure surfaces, and a upper bound for the expected number of outcrossings of an exponentially deteriorating elliptical surface is determined. The analytical formulas are demonstrated to give good results by examples.