Rice's formula and palm probabilities with applications to structural reliability.

摘要

This research concerns the distributions of stochastic processes at times when another random process crosses a level. The theoretical results are related to Rice's formula, which describes the intensity of level crossings by a stationary Gaussian process, and in particular to a general formulation given by Leadbetter. The main theorems provide formulas for the distributions of random processes, called “marks”, at level crossings by another process. In particular the theory applies to a general stationary stochastic process in the case when its derivative is one of the marks considered. This leads to integral formulas for so-called “Palm” distributions which determine the statistical properties of the mark processes at level crossings. In general, Palm distributions describe the probability of an event occurring at random time points. In this research the time points correspond to level crossings by a stochastic process, and the Palm distributions represent the long-term empirical behavior of various marks at the level crossings.; The theory regarding Palm distributions has been used to describe many problems in the scientific literature, including applications, for example, to engineering, climatology and queuing theory. To illustrate potential uses of the theory developed in this research, it is applied to the statistical modeling of structural reliability in naval architecture. Specifically this builds upon a substantial existing literature (e.g. [11] [14] [22] [27] [29] [32] (33]) concerning extreme stress loads which can severely damage a vessel operating in heavy seas. The loads of interest are caused in large part by “slams,” which are sudden impacts between the ship and the ocean surface. By consideration of the ship-sea vertical motion process, slamming is traditionally modeled (see [27]) as instances when this process crosses from above to below a high level. In physical terms this “downcrossing” event represents bow emergence followed by abrupt reentry into the sea. Based on general modeling of the relevant processes, an explicit form for the Palm distribution of total load at slams is then obtained. Within this context a variety of applied methods are developed to facilitate parameter estimation for the mark distributions. The application of fitted mark distributions is then illustrated by examination of lifetime vessel structural reliability.