Joint statistics of combined first- and second-order random processes

Two-term Volterra series are often used to describe non-linear random processes in subject areas as diverse as communication theory and the dynamics of offshore structures. It is well known that the characteristic function, the moments and the cumulants of such processes can be calculated analytically and that the probability density function can be calculated accurately and efficiently. This paper considers the joint statistics between two such random processes by deriving: (i) an exact expression for the joint characteristic function; (ii) an efficient means for calculating the joint moments; (iii) an ‘exact’ numerical means for calculating the joint probability density function (jpdf). For the special case of a combined first- and second-order process and a pure first-order process it is shown that it is possible to derive analytical expressions for the characteristic function and to calculate the jpdf accurately and efficiently using saddle-point integration. In addition to the above, the maximum entropy principle is used to calculate the jpdf.     

Extreme value distributions for nonlinear transformations of vector Gaussian processes

Approximations are developed for the marginal and joint probability distributions for the extreme values, associated with a vector of non-Gaussian random processes. The component non-Gaussian processes are obtained as nonlinear transformations of a vector of stationary, mutually correlated, Gaussian random processes and are thus, mutually dependent. The multivariate counting process, associated with the number of level crossings by the component non-Gaussian processes, is modelled as a multivariate Poisson point process. An analytical formulation is developed for determining the parameters of the multivariate Poisson process. This, in turn, leads to the joint probability distribution of the extreme values of the non-Gaussian processes, over a given time duration. For problems not amenable for analytical solutions, an algorithm is developed to determine these parameters numerically. The proposed extreme value distributions have applications in time-variant reliability analysis of randomly vibrating structural systems. The method is illustrated through three numerical examples and their accuracy is examined with respect to estimates from full scale Monte Carlo simulations of vector non-Gaussian processes.     

The asymptotic behaviour of second-order stochastic Volterra series models of slow drift response

The paper presents a detailed study of the structure and asymptotic behaviour of a second-order stochastic Volterra series model of the slow drift response of large volume compliant offshore structures subjected to random seas. A long standing challenge has been to develop efficient and accurate methods for calculating the response statistics of compliant offshore structures to random seas. Recent work has revealed that the statistical properties of the response process, which consist of a linear, first-order component and a nonlinear, second-order component, is surprisingly complex. The goal of the research work presented here is to complement efforts to develop numerical procedures to calculate the statistics of the response process.