Wave analysis by Slepian models G. Lindgren * Department of Mathematical Statistics, University of Lund, Box 118, S-221 00 Lund, Sweden Abstract Ocean waves exhibit more or less a Gaussian distribution for the instantaneous water surface height, and there is a need to develop simple models for generation of the characteristic non-Gaussian statistics, namely the asymmetric distributions of water surface height and wave slope. We argue that a simple class of non-linear oscillators can reproduce some of the characteristic features of random water wave processes and linear or non-linear response to ocean waves. We describe the Slepian model for the Gaussian case, and explain the use of the regression approximation for level crossing distances and associated variables, such as wave period and amplitude. Finally we speculate about a generalization of the regression technique to the non-linear Markov process case.  2000 Elsevier Science Ltd. All rights reserved. Keywords: Asymmetric waves; Non-Gaussian process; Non-linear models; Random waves; Wave amplitude; Wave period 1. Introduction One of the most exciting challenges in non-linear stochas- tic mechanics is identification and estimation of realistic models, with physical relevance, complex enough to catch important features of the real world, but simple enough to allow numerical computation of important quantities of relevance for structural safety and systems performance. To be useful, the models need to have a balance between the flexibility of a completely non-parametric model, which may be difficult to generalize to other situations, and the generality of a strictly parametric model, which may be easy to transport to other circumstances, but may not suffi- ciently fit the actual data. A physically based wave model is a set of solutions to equations set up to describe the dynamics of traveling water waves. In the simplest of these models waves are built as a sum or integral of non-interacting harmonic traveling waves, and then often a Gaussian assumption is added to make the wave process a Gaussian process. This is often quite realistic and agrees with empirical data for small waves. In this article we shall discuss the use of an even more empirical approach in which the statistical model is used to describe a non-Gaussian character of the wave surface. Although it is not aimed to be a tool for physical understanding we believe that simple non-Gaussian models can be used for example in reliability analysis of marine structures. Thus there is a need for statistical estimation techniques for semi- and non-parametric stochastic mechanical models, as well as a need for numerical techniques which can handle crossings and extreme value problems in such models. In particular, the first passage problem must be given a satis- factory answer for each practical situation. In this article we will argue that the Slepian model and the regression approximation can be generalized to work on the special class of non-Gaussian processes which are the solu- tions of simple non-linear stochastic differential equations driven by Gaussian white noise, and that such models can be successfully used to model common ocean waves as well as important quantities in marine engineering. We first describe the basic principles for the Slepian model used in crossing problems for Gaussian stochastic processes, and the use of the recursive regression technique for numerical computation of crossings and wave distribu- tions, like the distribution of wave period/wave length and the associated wave amplitude. Then we describe some simple types of non-parametric models for non-Gaussian processes appearing in stochastic mechanics, and briefly indicate how such models can be estimated from data. Finally, we indicate how the Slepian model and regression technique could be extended to such non-Gaussian (but Markovian) models. 2. Non-Gaussian data and non-linear models 2.1. Two different types of non-Gaussianity Ocean waves have a complicated stochastic structure, and Probabilistic Engineering Mechanics 15 (2000) 49–57 0266-8920/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S0266-8920(99)00008-9 www.elsevier.com/locate/probengmech * Tel.: +46-222-8547; fax: +46-222-4623. E-mail address: georg@maths.lth.se (G. Lindgren).