Amplitude Correlation in First-Passage Problems

A relatively simple technique is presented for estimating the first-passage probability of any stationary Gaussian process. In particular, the technique is shown to give meaningful results for the troublesome case of a process with a bimodal spectral density that cannot be truly classified as either narrowband or broadband. Mathematically, the technique is only a seemingly minor revision of a long-known method for narrowband processes. What is new here is the demonstration of its applicability not only to narrowband processes but also to other spectral densities, including challenging bimodal situations. The fundamental idea is that of a Markovian point process in which the probability of the process never having crossed a given level $u$ is approximated as the probability that all members of the set of amplitude (or envelope) values at a discrete set of time values are below $u$, along with the assumption that the set of amplitude values has Markovian conditioning. The critical parameter that must be determined is a correlation coefficient that appears in the jointly Rayleigh distribution of the amplitude at two time values. A method involving simple numerical integration (or a series solution) is shown to give an appropriate value for this parameter for a given spectral density and a given value of $u$. The model is shown to give generally good agreement with simulation results for processes in which the commonly used Vanmarcke approximation is inadequate.