Extreme Values of Waves and Ship Responses Subject to the Markov Chain Condition

Abstract

This paper discusses the effect of statistical dependence of the maxima (peak values) of a stationary random process on the magnitude of the extreme values. A theoretical analysis of the extreme values of a stationary normal random process is made, assuming the maxima are subject to the Markov chain condition. For this, the probability distribution function of maxima as well as the joint probability distribution function of two successive maxima of a normal process having an arbitrary spectral bandwidth are applied to Epstein's theorem for evaluating the extreme values in a given sample under the Markov chain condition. A numerical evaluation of the extreme values is then carried out for a total of 14 random processes, including nine ocean wave records, with various spectral bandwidth parameters ranging from 0.11 to 0.78. From the results of the computations, it is concluded that the Markov concept is applicable to the maxima of random processes whose spectral bandwidth parameter, ɛ, is less than 0.5, and that the extreme values with and without the Markov concept are constant irrespective of the e-value, and the former is approximately 10 percent greater than the latter. It is also found that the sample size for which the extreme value reaches a certain level with the Markov concept is much less than that without the Markov concept. For example, the extreme value will reach a level of 4.0 (nondimensional value) in 1100 observations of the maxima with the Markov concept, while the extreme value will reach the same level in 3200 observations of the maxima without the Markov concept.